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MIME: Mutual Information Minimization and Entropy Maximization for Bayesian Belief Propagation Anand Rangarajan Dept. of Computer and Information Science and Engineering University of Florida Gainesville, FL 32611-6120, US anand@cise.ufl.edu Alan L. Yuille Smith-Kettlewell Eye Research Institute 2318 Fillmore St. San Francisco, CA 94115, US yuille@ski.org Abstract Bayesian belief propagation in graphical models has been recently shown to have very close ties to inference methods based in statistical physics. After Yedidia et al. demonstrated that belief propagation fixed points correspond to extrema of the so-called Bethe free energy, Yuille derived a double loop algorithm that is guaranteed to converge to a local minimum of the Bethe free energy. Yuille’s algorithm is based on a certain decomposition of the Bethe free energy and he mentions that other decompositions are possible and may even be fruitful. In the present work, we begin with the Bethe free energy and show that it has a principled interpretation as pairwise mutual information minimization and marginal entropy maximization (MIME). Next, we construct a family of free energy functions from a spectrum of decompositions of the original Bethe free energy. For each free energy in this family, we develop a new algorithm that is guaranteed to converge to a local minimum. Preliminary computer simulations are in agreement with this theoretical development. 1 Introduction In graphical models, Bayesian belief propagation (BBP) algorithms often (but not always) yield reasonable estimates of the marginal probabilities at each node [6]. Recently, Yedidia et al. [7] demonstrated an intriguing connection between BBP and certain inference methods based in statistical physics. Essentially, they demonstrated that traditional BBP algorithms can be shown to arise from approximations of the extrema of the Bethe and Kikuchi free energies. Next, Yuille [8] derived new double-loop algorithms which are guaranteed to minimize the Bethe and Kikuchi energy functions while continuing to have close ties to the original BBP algorithms. Yuille’s approach relies on a certain decomposition of the Bethe and Kikuchi free energies. In the present work, we begin with a new principle—pairwise mutual information minimization and marginal entropy maximization (MIME)—and derive a new energy function which is shown to be equivalent to the Bethe free energy. After demonstrating this connection, we derive a family of free energies closely related to the MIME principle which also shown to be equivalent, when constraint satisfaction is exact, to the Bethe free energy. For each member in this family of energy functions , we derive a new algorithm that is guaranteed to converge to a local minimum. Moreover, the resulting form of the algorithm is very simple despite the somewhat unwieldy nature of the algebraic development. Preliminary comparisons of the new algorithm with BBP were carried out on spin glass-like problems and indicate that the new algorithm is convergent when BBP is not. However, the effectiveness of the new algorithms remains to be seen. 2 Bethe free energy and the MIME principle In this section, we show that the Bethe free energy can be interpreted as pairwise mutual information minimization and marginal entropy maximization. The Bethe free energy for Bayesian belief propagation is written as FBethe({pij, pi, γij, λij}) = P ij:i>j P xi,xj pij(xi, xj) log pij(xi,xj) φij(xi,xj) −P i(ni −1) P xi pi(xi) log pi(xi) ψi(xi) + P ij:i>j P xj λij(xj)[P xi pij(xi, xj) −pj(xj)] + P ij:i>j P xi λji(xi)[P xj pij(xi, xj) −pi(xi)] + P ij:i>j γij(P xi,xj pij(xi, xj) −1) (1) where φij(xi, xj) def = ψij(xi, xj)ψi(xi)ψj(xj) and ni is the number of neighbors of node i. Link functions ψij > 0 are available relational data between nodes i and j. The singleton function ψi is also available at each node i. The double summation P ij:i>j is carried out only over the nodes that are connected. The Lagrange parameters {λij, γij} are needed in the Bethe free energy (1) to satisfy the following constraints relating the joint probabilities {pij} with the marginals {pi}: X xi pij(xi, xj) = pj(xj), X xj pij(xi, xj) = pi(xi), and X xi,xj pij(xi, xj) = 1. (2) The pairwise mutual information is defined as MIij = X xi,xj pij(xi, xj) log pij(xi, xj) pi(xi)pj(xj) (3) The mutual information is minimized when the joint probability pij(xi, xj) = pi(xi)pj(xj) or equivalently when nodes i and j are independent. When nodes i and j are connected via a non-separable link ψij(xi, xj) they will not be independent. We now state the MIME principle. Statement of the MIME principle: Maximize the marginal entropy and minimize the pairwise mutual information using the available marginal and pairwise link function expectations while satisfying the joint probability constraints. The pairwise MIME principle leads to the following free energy: FMIME({pij, pi, γij, λij}) = P ij:i>j P xi,xj pij(xi, xj) log pij(xi,xj) pi(xi)pj(xj) + P i P xi pi(xi) log pi(xi) −P ij:i>j P xi,xj pij(xi, xj) log ψij(xi, xj) −P i P xi pi(xi) log ψi(xi) + P ij:i>j P xj λij(xj)[P xi pij(xi, xj) −pj(xj)] + P ij:i>j P xi λji(xi)[P xj pij(xi, xj) −pi(xi)] + P ij:i>j γij(P xi,xj pij(xi, xj) −1). (4) In the above free energy, we minimize the pairwise mutual information and maximize the marginal entropies. The singleton and pairwise link functions are additional information which do not allow the system to reach its “natural” equilibrium—a uniform i.i.d. distribution on the nodes. The Lagrange parameters enforce the constraints between the pairwise and marginal probabilities. These constraints are the same as in the Bethe free energy (1). Note that the Lagrange parameter terms vanish if the constraints in (2) are exactly satisfied. This is an important point when considering equivalences between different energy functions. Lemma 1 Provided the constraints in (2) are exactly satisfied, the MIME free energy in (4) is equivalent to the Bethe free energy in (1). Proof: Using the fact that constraint satisfaction is exact and using the identity pij(xi, xj) = pji(xj, xi), we may write − X ij:i>j X xi,xj pij(xi, xj) log pi(xi)pj(xj) = − X ij:i̸=j X xi,xj pij(xi, xj) log pi(xi) = − X i ni X xi pi(xi) log pi(xi), and X ij:i>j X xi,xj pij(xi, xj) log ψi(xi)ψj(xj) = X i ni X xi pi(xi) log ψi(xi). (5) We have shown that a marginal entropy term emerges from the mutual information term in (4) when constraint satisfaction is exact. Collecting the marginal entropy terms together and rearranging the MIME free energy in (4), we get the Bethe free energy in (1). 3 A family of decompositions of the Bethe free energy Recall that the Bethe free energy and the energy function resulting from application of the MIME principle were shown to be equivalent. However, the MIME energy function is merely one particular decomposition of the Bethe free energy. As Yuille mentions [8], many decompositions are possible. The main motivation for considering alternative decompositions is for algorithmic reasons. We believe that certain decompositions may be more effective than others. This belief is based on our previous experience with closely related deterministic annealing algorithms [3, 2]. In this section, we derive a family of free energies that are equivalent to the Bethe free energy provided constraint satisfaction is exact. The family of free energies is inspired by and closely related to the MIME free energy in (4). Lemma 2 The following family of energy functions indexed by the free parameters δ > 0 and {ξi} is equivalent to the original Bethe free energy (1) provided the constraints in (2) are exactly satisfied and the parameters q and r are set to {qi = (1 −δ)ni} and {ri = 1 −niξi} respectively. Fequiv({pij, pi, γij, λij}) = P ij:i>j P xi,xj pij(xi, xj) log pij(xi,xj) [P xj pij(xi,xj)]δ[P xi pij(xi,xj)]δ + P i P xi pi(xi) log pi(xi) −P i qi P xi pi(xi) log pi(xi) −P ij:i>j P xi,xj pij(xi, xj) log ψij(xi, xj)ψξi i (xi)ψξj j (xj) −P i ri P xi pi(xi) log ψi(xi) + P ij:i>j P xj λij(xj)[P xi pij(xi, xj) −pj(xj)] + P ij:i>j P xi λji(xi)[P xj pij(xi, xj) −pi(xi)] + P ij:i>j γij(P xi,xj pij(xi, xj) −1). (6) In (6), the first term is no longer the pairwise mutual information as in (4). And unlike (4), pi(xi) no longer appears in the pairwise mutual information-like term. Proof: We selectively substitute P xi pij(xi, xj) = pj(xj) and P xj pij(xi, xj) = pi(xi) to show the equivalence. First X ij:i>j X xi,xj pij(xi, xj) log[ X xj pij(xi, xj)]δ[ X xi pij(xi, xj)]δ = δ X i ni X xj pi(xi) log pi(xi), X ij:i>j X xi,xj pij(xi, xj) log ψ ξi i (xi)ψ ξj j (xj) = X i niξi X xj pi(xi) log ψi(xi). (7) Substituting the identities in (7) into (6), we see that the free energies are algebraically equivalent. 4 A family of algorithms for belief propagation We now derive descent algorithms for the family of energy functions in (6). All the algorithms are guaranteed to converge to a local minimum of (6) under mild assumptions regarding the number of fixed points. For each member in the family of energy functions, there is a corresponding descent algorithm. Since the form of the free energy in (6) is complex and precludes easy minimization, we use algebraic (Legendre) transformations [1] to simplify the optimization. − X xj pij(xi, xj) log X xj pij(xi, xj) = minσji(xi) −P xj pij(xi, xj) log σji(xi) + σji(xi) −P xj pij(xi, xj) − X xi pij(xi, xj) log X xi pij(xi, xj) = minσij(xj) −P xi pij(xi, xj) log σij(xj) + σij(xj) −P xi pij(xi, xj) −pi(xi) log pi(xi) = min ρi(xi) −pi(xi) log ρi(xi) + ρi(xi) −pi(xi). (8) We now apply the above algebraic transforms. The new free energy is (after some algebraic manipulations) Fequiv({pij, pi, σij, ρi, γij, λij}) = X ij:i>j X xi,xj pij(xi, xj) log pij(xi, xj) σδ ji(xi)σδ ij(xj) +δ X ij:i̸=j X xi σij(xj) + X i X xi pi(xi) log pi(xi) ρqi i (xi) + X i qi X xi ρi(xi) − X ij:i>j X xi,xj pij(xi, xj) log ψij(xi, xj)ψ ξi i (xi)ψ ξj j (xj) − X i ri X xi pi(xi) log ψi(xi) + X ij:i>j X xj λij(xj)[ X xi pij(xi, xj) −pj(xj)] + X ij:i>j X xi λji(xi)[ X xj pij(xi, xj) −pi(xi)] + X ij:i>j γij( X xi,xj pij(xi, xj) −1). (9) We continue to keep the parameters {qi} and {ri} in (9). However, from Lemma 2, we know that the equivalence of (9) to the Bethe free energy is predicated upon appropriate setting of these parameters. In the rest of the paper, we continue to use q and r for the sake of notational simplicity. Despite the introduction of new variables via Legendre transforms, the optimization problem in (9) is still a minimization problem over all the variables. The algebraically transformed energy function in (9) is separately convex w.r.t. {pij, pi} and w.r.t. {σij, ρi} provided δ ∈[0, 1]. Since the overall energy function is not convex w.r.t. all the variables, we pursue an alternating algorithm strategy similar to the double loop algorithm in Yuille [8]. The basic idea is to separately minimize w.r.t. the variables {σij, ρi} and the variables {pij, pi}. The linear constraints in (2) are enforced when minimizing w.r.t the latter and do not affect the convergence properties of the algorithm since the energy function w.r.t. {pij, pi} is convex. We evaluate the fixpoints of {σij, ρi}. Note that (9) is convex w.r.t. {σij, ρi}. σij(xj) = X xi pij(xi, xj), σji(xi) = X xj pij(xi, xj), and ρi(xi) = pi(xi). (10) The fixpoints of {pij, pi} are evaluated next. Note that (9) is convex w.r.t. {pij, pi}. pij(xi, xj) = σδ ji(xi)σδ ij(xj)ψij(xi, xj)ψξi i (xi)ψξj j (xj)e−λij(xj)−λji(xi)−γij−1 pi(xi) = ρqi i (xi)ψri i (xi)e P k λki(xi)−1. (11) The constraint satisfaction equations from (2) can be rewritten as X xj pij(xi, xj) = pi(xi) ⇒ e2λji(xi) = P xj σδ ji(xi)σδ ij(xj)ψij(xi,xj)ψξi i (xi)ψ ξj j (xj)e−λij (xj )−γij −1 ρqi i (xi)ψri i (xi)e P k̸=j λki(xi)−1 (12) Similar relations can be obtained for the other constraints in (2). Consider a Lagrange parameter update sequence where the Lagrange parameter currently being updated is tagged as “new” with the rest designated as “old.” We can then rewrite the Lagrange parameter updates using “old” and “new” values. Please note that each Lagrange parameter update corresponds to one of the constraints in (2). It can be shown that the iterative update of the Lagrange parameters is guaranteed to converge to the unique solution of (2) [8]. While rewriting (12), we multiply the left and right sides with e−2λold ji (xi). e2λnew ji (xi)−2λold ji (xi) = P xj σδ ji(xi)σδ ij(xj)ψij(xi,xj)ψ ξi i (xi)ψ ξj j (xj)e −λold ij (xj )−λold ji (xi)−γold ij −1 ρ qi i (xi)ψ ri i (xi)e P k λold ki (xi)−1 . (13) Using (11), we relate each Lagrange parameter update with an update of pij(xi, xj) and pi(xi). We again invoke the “old” and “new” designations, this time on the probabilities. From (11), (12) and (13), we write the joint probability update pnew ij (xi, xj) pold ij (xi, xj) = e−λnew ji (xi)+λold ji (xi) = s pold i (xi) P xj pold ij (xi, xj) (14) and for the marginal probability update pnew i (xi) pold i (xi) = eλnew ji (xi)−λold ji (xi) = sP xj pold ij (xi, xj) pold i (xi) . (15) From (14) and (15), the update equations for the probabilities are pnew ij (xi, xj) = pold ij (xi, xj) s pold i (xi) P xj pold ij (xi, xj), pnew i (xi) = s pold i (xi) X xj pold ij (xi, xj) (16) With the probability updates in place, we may write down new algorithms minimizing the family of Bethe equivalent free energies using only probability updates. The update equations (16) can be seen to satisfy the first constraint in (2). Similar update equations can be derived for the other constraints in (2). For each Lagrange parameter update, an equivalent, simultaneous probability (joint and marginal) update can be derived similar to (16). The overall family of algorithms can be summarized as shown in the pseudocode. Despite the unwieldy algebraic development preceding it, the algorithm is very simple and straightforward. Set free parameters δ ∈[0, 1] and {ξi}. Initialize {pij, pi}. Set {qi = (1 −δ)ni} and {ri = 1 −niξi}. Begin A: Outer Loop σij(xj) ←P xi pij(xi, xj) σji(xi) ←P xj pij(xi, xj) ρi(xi) ←pi(xi) pij(xi, xj) ←σδ ji(xi)σδ ij(xj)ψij(xi, xj)ψξi i (xi)ψξj j (xj) pi(xi) ←ρqi i (xi)ψri i (xi) Begin B: Inner Loop: Do B until 1 N P ij:i>j[(P xj pij(xi, xj) − pi(xi))2 + (P xi pij(xi, xj) −pj(xj))2] < cthr Simultaneously update pij(xi, xj) and pi(xi) below. pij(xi, xj) ←pij(xi, xj) r pi(xi) P xj pij(xi,xj) pi(xi) ← q pi(xi) P xj pij(xi, xj) Simultaneously update pij(xi, xj) and pj(xj) below. pij(xi, xj) ←pij(xi, xj) r pj(xj) P xi pij(xi,xj) pj(xj) ← q pj(xj) P xi pij(xi, xj) Normalize pij(xi, xj). pij(xi, xj) ← pij(xi,xj) P xi,xj pij(xi,xj) End B End A In the above family of algorithms, the MIME algorithm corresponds to free parameter settings δ = 1 and ξi = 0 which in turn lead to parameter settings qi = 0 and ri = 1. The Yuille [8] double loop algorithm corresponds to the free parameter settings δ = 0 and ξi = 0 which in turn leads to parameter settings qi = ni and ri = 1. A crucial point is that the energy function for every valid parameter setting is equivalent to the Bethe free energy provided constraint satisfaction is exact. The inner loop constraint satisfaction threshold parameter cthr setting is very important in this regard. We are obviously not restricted to the MIME parameter settings. At this early stage of exploration of the inter-relationships between Bayesian belief propagation and inference methods based in statistical physics [7], it is premature to speculate regarding the “best” parameter settings for δ and {ξi}. Most likely, the effectiveness of the algorithms will vary depending on the problem setting which enters into the formulation via the link functions {ψij} and the singleton functions {ψi}. 5 Results We implemented the family of algorithms in C++ and conducted tests on locally connected 50 node graphs and binary state variables. The ψi(xi) and ψij(xi, xj) are of the form e±hi and e±hij where hi and hij are drawn from uniform distributions (in the interval [−1, 1]). Provided the constraint satisfaction theshold parameter cthr was set low enough, the algorithm (for δ = 1 and other parameter settings as described in Figure 1) exhibited monotonic convergence. Figure 2 shows the number of inner loop iterations corresponding to different settings of the constraint satisfaction threshold parameter. We also implemented the BBP algorithm and empirically observed that it often did not converge for these graphs. These results are quite preliminary and far more validation experiments are required. However, they provide a proof of concept for our approach. 6 Conclusion We began with the MIME principle and showed the equivalence of the MIMEbased free energy to the Bethe free energy assuming constraint satisfaction to be exact. Then, we derived new decompositions of the Bethe free energy inspired by the MIME principle, and driven by our belief that certain decompositions may be more effective than others. We then derived a convergent algorithm for each member in the family of MIME-based decompositions. It remains to be seen if the MIME-based algorithms are efficient for a reasonable class of problems. While the MIME-based algorithms derived here use closed-form solutions in the constraint satisfaction inner loop, it may turn out that the inner loop is better handled using preconditioned gradient-based descent algorithms. And it is important to explore the inter-relationships between the convergent MIME-based descent algorithms and other recent related approaches with interesting convergence properties [4, 5]. References [1] E. Mjolsness and C. Garrett. Algebraic transformations of objective functions. Neural Networks, 3:651–669, 1990. [2] A. Rangarajan. Self annealing and self annihilation: unifying deterministic annealing and relaxation labeling. Pattern Recognition, 33:635–649, 2000. [3] A. Rangarajan, S. Gold, and E. Mjolsness. A novel optimizing network architecture with applications. Neural Computation, 8(5):1041–1060, 1996. [4] Y. W. Teh and M. Welling. Passing and bouncing messages for generalized inference. Technical Report GCNU 2001-01, Gatsby Computational Neuroscience Unit, University College, London, 2001. [5] M. Wainwright, T. Jaakola, and A. Willsky. Tree-based reparameterization framework for approximate estimation of stochastic processes on graphs with cycles. Technical Report LIDS P-2510, MIT, Cambridge, MA, 2001. [6] Y. Weiss. Correctness of local probability propagation in graphical models with loops. Neural Computation, 12:1–41, 2000. [7] J. S. Yedidia, W. T. Freeman, and Y. Weiss. Bethe free energy, Kikuchi approximations and belief propagation algorithms. In Advances in Neural Information Processing Systems 13, Cambridge, MA, 2001. MIT Press. [8] A. L. Yuille. A double loop algorithm to minimize the Bethe and Kikuchi free energies. Neural Computation, 2001. (submitted). 0 500 1000 1500 −0.5 −0.4 iteration MIME energy 0 500 1000 1500 −0.5 −0.4 iteration MIME energy 0 500 1000 1500 −0.5 −0.4 iteration MIME energy (a) (b) (c) Figure 1: MIME energy versus outer loop iteration: 50 node, local topology, δ = 1. Constraint satisfaction threshold parameter cthr was set to (a) 10−8 (b) 10−4 (c) 10−2 0 500 1000 1500 0 2 4 6 8 10 12 14 16 18 20 outer loop iteration index total # of inner loop iterations 0 500 1000 1500 1 2 3 4 5 6 7 outer loop iteration index total # of inner loop iterations 0 500 1000 1500 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 outer loop iteration index total # of inner loop iterations (a) (b) (c) Figure 2: Inner loop iterations versus outer loop: 50 node, local topology, δ = 1. Constraint satisfaction threshold parameter cthr was set to (a) 10−8 (b) 10−4 (c) 10−2
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Rates of Convergence of Performance Gradient Estimates Using Function Approximation and Bias in Reinforcement Learning Gregory Z. Grudic University of Colorado, Boulder grudic@cs.colorado.edu Lyle H. Ungar University of Pennsylvania ungar@cis.upenn.edu Abstract We address two open theoretical questions in Policy Gradient Reinforcement Learning. The first concerns the efficacy of using function approximation to represent the state action value function, . Theory is presented showing that linear function approximation representations of can degrade the rate of convergence of performance gradient estimates by a factor of relative to when no function approximation of is used, where is the number of possible actions and is the number of basis functions in the function approximation representation. The second concerns the use of a bias term in estimating the state action value function. Theory is presented showing that a non-zero bias term can improve the rate of convergence of performance gradient estimates by
, where is the number of possible actions. Experimental evidence is presented showing that these theoretical results lead to significant improvement in the convergence properties of Policy Gradient Reinforcement Learning algorithms. 1 Introduction Policy Gradient Reinforcement Learning (PGRL) algorithms have recently received attention because of their potential usefulness in addressing large continuous reinforcement Learning (RL) problems. However, there is still no widespread agreement on how PGRL algorithms should be implemented. In PGRL, the agent’s policy is characterized by a set of parameters which in turn implies a parameterization of the agent’s performance metric. Thus if represents a dimensional parameterization of the agent’s policy and is a performance metric the agent is meant to maximize, then the performance metric must have the form [6]. PGRL algorithms work by first estimating the performance gradient (PG) ! "#! and then using this gradient to update the agent’s policy using: $&%('*)+$-,. / ! !0 (1) where . is a small positive step size. If the estimate of ! "#! is accurate, then the agent can climb the performancegradient in the parameter space, toward locally optimal policies. In practice, ! 12! is estimated using samples of the state action value function . The PGRL formulation is attractive because 1) the parameterization of the policy can directly imply a generalization over the agent’s state space (e.g., can represent the adjustable weights in a neural network approximation), which suggests that PGRL algorithms can work well on very high dimensional problems [3]; 2) the computational cost of estimating !0 "#! is linear in the number of parameters , which contrasts with the computational cost for most RL algorithms which grows exponentially with the dimension of the state space; and 3) PG algorithms exist which are guaranteed to give unbiased estimates of !0 "#! [6, 5, 4, 2, 1]. This paper addresses two open theoretical questions in PGRL formulations. In PGRL formulations performance gradient estimates typically have the following form: / !0 ! ) #'
1' #'
(2) where !"
#! is the estimate of the value of executing action $! in state ! (i.e. the state action value function), # %! the bias subtracted from !& & #! in state ! , ' is the number of steps the agent takes before estimating ! 12!0 , and the form of the function depends on the PGRL algorithm being used (see Section 2, equation (3) for the form being considered here). The effectiveness of PGRL algorithms strongly depends on how !" & #! is obtained and the form of # %! . The aim of this work is to address these questions. The first open theoretical question addressed here is concerned with the use of function approximation (FA) to represent the state action value function , which is in turn used to estimate the performance gradient. The original formulation of PGRL [6], the REINFORCE algorithm, has been largely ignored because of the slow rate of convergence of the PG estimate. The use of FA techniques to represent based on its observations has been suggested as a way of improving convergence properties. It has been proven that specific linear FA formulations can be incorporated into PGRL algorithms, while still guaranteeing convergence to locally optimal solutions [5, 4]. However, whether linear FA representations actually improves the convergence properties of PGRL is an open question. We present theory showing that using linear basis function representations of , rather than direct observations of it, can slow the rate of convergence of PG estimates by a factor of (see Theorem 1 in Section 3.1). This result suggests that PGRL formulations should avoid the use of linear FA techniques to represent . In Section 4, experimental evidence is presented supporting this conjecture. The second open theoretical question addressed here is can a non-zero bias term # in (2) improve the convergence properties of PG estimates? There has been speculation that an appropriate choice of # can improve convergence properties [6, 5], but theoretical support has been lacking. This paper presents theory showing that if # ) #)(* + & 1 , where is the number actions, then the rate of convergence of the PG estimate is improved by # (see Theorem 2 in Section 3.2). This suggests that the convergence properties of PGRL algorithms can be improved by using a bias term that is the average of values in each state. Section 4 gives experimental evidence supporting this conjecture. 2 The RL Formulation and Assumptions The RL problem is modeled as a Markov Decision Process (MDP). The agent’s state at time , . /
0$ 12 is given by $ 43 , 365 87 . At each time step the agent chooses from a finite set of :9 actions $ <; ) ' 1
#= and receives a reward > $ . The dynamics of the environment are characterized by transition probabilities ? * @"@ A )B?C>+- $&%(' )D A&E $ ) + & $ )F G2 and expected rewards H * @ )JIK-> $&%(' E $ )L+ & $ )M G2 , NO+
A 3P & ; . The policy followed by the agent is characterized by a parameter vector , and is defined by the probability distribution Q + & R )S?C>+-% $ )D E $ )S+RT2 , NO U3P & U; . We assume that Q + & R is differentiable with respect to . We use the Policy Gradient Theorem of Sutton et al. [5] and limit our analysis to the start state discount reward formulation. Here the reward function Q and state action value function + & 1 are defined as: Q )I ( $ ' $ > $ / &Q
+
)I ( (' ' > $&% $ ) + & $ ) "Q
where . Then the exact expression for the performance gradient is: ! ! ) @ = !(' !GQ + & ! R ! + & ! (3) where ) ( $ $ $ ) E "Q2 and # . This policy gradient formulation requires that the state-action value function, , under the current policy be estimated. This estimate, , is derived using the observed value ! @ + & ! . We assume that ! @ + & ! has the following form: ! @ /
#! ) + & #! ,#" + & #! where " +
#! has zero mean and finite variance $&% @!' *)( . Therefore, if +
! is an estimate of /
#! obtained by averaging * observations of ! @ + & #! , then the mean and variance are given by: IB /
#! ) + & #! O ,+B + & #! ).-0/ 132 4 ( 5 (4) In addition, we assume that ! @ /
! are independently distributed. This is consistent with the MDP assumption. 3 Rate of Convergence Results Before stating the convergence theorems, we define the following: $6% 798;: ) <>=0? @!@ AB' ! @C ' 'EDEDEDE' =GF $6% @!' * ( H$6% 79I J ) <GKML @N@ A' ! @C ' 'EDEDEDE' =>F $6% @!' * ( (5) where $ % @!' * ( is defined in (4) and O 79I J )QP ( @ % = ( !M(' SR BT @!' *)(VU W!X R W %ZY $ % 79I J O 798;: )QP ( @ % = ( !M(' R BT @!' * ( U W!X R W %ZY $6% 798;: (6) 3.1 Rate of Convergence of PIFA Algorithms Consider the PIFA algorithm [5] which uses a basis function representation for estimated state action value function, , of the following form: /
#! ) *)( )\[ ] '&^ *)( ' ]`_ *)( ' ] (7) where ^ * (3' ] are weights and _ * (3' ] are basis functions defined in + 7 . If the weights ^ * (a' ] are chosen based using the observed ! @ + & ! , and the basis functions, _ *)( ' ] , satisfy the conditions defined in [5, 4], then the performance gradient is given by: ! ! cb ) @ = !(' !Q + & ! R ! * ( (8) The following theorem establishes bounds on the rate of convergencefor this representation of the performance gradient. Theorem 1: Let R R W b be an estimate of (8) obtained using the PIFA algorithm and the basis function representation (7). Then, given the assumptions defined in Section 2 and equations (5) and (6), the rate of convergence of a PIFA algorithm is bounded below and above by: O 79I J * + ! ! b O 798 : * (9) where is the number of basis functions, is the number of possible actions, and * is the number of independent estimates of the performance gradient. Proof: See Appendix. 3.2 Rate of Convergence of Direct Sampling Algorithms In the previous section, the observed ! @ /
#! are used to build a linear basis function representation of the state action value function, + & ! , which is in turn used to estimate the performance gradient. In this section we establish rate of convergence bounds for performance gradient estimates that directly use the observed @ + & #! without the intermediate step of building the FA representation. These bounds are established for the conditions # ) # ( * /
1 and # ) in (3). Theorem 2: Let / R R W be a estimate of (3), be obtained using direct samples of . Then, if #
) , and given the assumptions defined in Section 2 and equations (5) and (6), the rate of convergence of / R R W is bounded by: O 79I J * + / !0 ! O 798;: * (10) where * is the number of independent estimates of the performance gradient. If # ) is defined as: ) = ' + & (11) then the rate of convergence of the performance gradient R R W is bounded by: O 79I J * + ! !0 O 798 : *
(12) where is the number of possible actions. Proof: See Appendix. Thus comparing (12) and (10) to (9) one can see that policy gradient algorithms such as PIFA which build FA representations of converge by a factor of slower than algorithms which directly sample . Furthermore, if the bias term is as defined in (11), the bounds on the variance are further reduced by . In the next section experimental evidence is given showing that these theoretical consideration can be used to improve the convergence properties of PGRL algorithms. 4 Experiments The Simulated Environment: The experiments simulate an agent episodically interacting in a continuous two dimensional environment. The agent starts each episode in the same state ! , and executes a finite number of steps following a policy to a fixed goal state . The stochastic policy is defined by a finite set of Gaussians, each associated with a specific 0 20 40 60 80 100 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Number of Policy Updates ρ(π) Biased Q No Bias Linear FA Q a) Convergence of Algorithms 0 2 4 6 8 10 12 14 10 0 10 1 10 2 10 3 10 4 Number of Possible Actions (M) V[∂ ρ / ∂ θF] / V[∂ ρ / ∂ θ] b) + / R R W b + / R R W 0 2 4 6 8 10 12 14 10 0 10 1 10 2 Number of Possible Actions (M) V[∂ ρ / ∂ θ] / V[∂ ρ / ∂ θb] c) + / R R W + / R R W Figure 1: Simulation Results action. The Gaussian associated with action is defined as: ) Z? 7 ('
% where ) #' 1
7 87 , is the agents state, *'%
7 is the Gaussian center, and *' 1 7 is the variance along each state space dimension. The probability of executing action in state is Q + & R ) = ( (' where ) '' 1 ' ' ' 1 ' 1 1= ' 1 1= 7 = ' = 7 defines the policy parameters that dictate the agent’s actions. Action ' directs the agent toward the goals state , while the remaining actions (for ) 0T ) direct the agent towards the corresponding Gaussian center *'%
7 . Noise is modeled using a uniform random distribution between $ denoted by $ , such that the noise in dimension is given by: ! @ ) , `) #$ 2 where 9 is the magnitude of the noise, @ is the state the agent observes and uses to choose actions, and is the actual state of the agent. The agent receives a reward of +1 when it reaches the goal state, otherwise it receives a reward of: > ) ) Z? 7 (' % Thus the agent gets negative rewards the closer it gets to the origin of the state space, and a positive reward whenever it reaches the goal state. Implementation of the PGRL algorithms: All the PGRL formulations studied here require observations (i.e. samples) of the state action value function. @ + & ! is sampled by executing action T! in state and thereafter following the policy. In the episodic formulation, where the agent executes a maximum of ' steps during each episode, at the end of each episode, @ $ & $ for step , can be evaluated as follows: @ $ & $ ) (' ' > $&% E $ )+
$ ) &Q Thus, given that the agent executes a complete episode '
1'
#( following the policy Q , at the completion of the episode we can calculate @ ' & ' 1 @ & . This gives samples of ' state action value pairs. Equation (3) tells us that we require a total of ' state action value function observations to estimate a performance gradient (assuming the agent can execute actions). Therefore, we can obtain the remaining + ' observations of @ by sending the agent out on ' epsisodes, each time allowing it to follow the policy Q for all ' steps, with the exception that action $ )U is executed when ! @ $ & is being observed. This sampling procedure requires a total of ' episodes and gives a complete set of @ state action pairs for any path ' & ' 1
. For the direct sampling algorithms in Section 3.2, these observations are directly used to estimate the performance gradient. For the linear basis function based PGRL algorithm in Section 3.1, these observations are first used to calculate the ^ * ( ' ] as defined in [5, 4], and then the performance gradient is calculated using (8). Experimental Results: Figure 1b shows a plot of average + / ! "#! b + / ! 12!0 values over 10,000 estimates of the performance gradient. For each estimate, the goal state, start state, and Gaussian centers are all chosen using a uniform random distribution + ; the Gaussian variances are sampled from a uniform distribution $ / . As predicted by Theorem 1 in Section 3.1 and Theorem 2 in Section 3.2, as the number of actions increases, this ratio also increases. Note that Figure 1b plots average variance ratios, not the bounds in variance given in Theorem 1 and Theorem 2 (which have not been experimentally sampled), so the ratio predicted by the theorems is supported by the increase in the ratio as increases. Figure 1c shows a plot of average + / ! 12!0 + / ! "#! values over 10,000 estimates of the performance gradient. As above, for each estimate, the goal state, start state, and Gaussian centers are all chosen using a uniform random distribution + ; the Gaussian variances are sampled from a uniform distribution `) + . This also follows the predicted trends of Theorem 1 and Theorem 2. Finally, Figure 1a shows the average reward over 100 runs as the three algorithms converge on a two action problem. Each algorithm is given the same number of ! @ samples to estimate the gradient before each update. Because / !0 "#! has the least variance, it allows the policy Q to converge to the highest reward value Q . Similarly, because / ! 12!0 b has the highest variance, its policy updates converge to the worst Q . Note that because all three algorithms will converge to the same locally optimal policy given enough samples of ! @ , Figure 1a simply demonstrates that / ! 12! b requires more samples than / ! 1#! , which in turn requires more samples than / ! "#! . 5 Conclusion The theoretical and experimental results presented here indicate that how PGRL algorithms are implemented can substantially affect the number of observations of the state action value function ( ) needed to obtain good estimates of the performance gradient. Furthermore, they suggest that an appropriately chosen bias term, specifically the average value of over all actions, and the direct use of observed values can improve the convergence of PGRL algorithms. In practice linear basis function representations of can significantly degrade the convergence properties of policy gradient algorithms. This leaves open the question of whether any (i.e. nonlinear) function approximation representation of value functions can be used to improve convergence of such algorithms. References [1] Jonathan Baxter and Peter L. Bartlett, Reinforcement learning in pomdp’s via direct gradient ascent, Proceedings of the Seventeenth International Conference on Machine Learning (ICML’2000) (Stanford University, CA), June 2000, pp. 41–48. [2] G. Z. Grudic and L. H. Ungar, Localizing policy gradient estimates to action transitions, Proceedings of the Seventeenth International Conference on Machine Learning, vol. 17, Morgan Kaufmann, June 29 - July 2 2000, pp. 343–350. [3] , Localizing search in reinforcement learning, Proceedings of the Seventeenth National Conference on Artificial Intelligence, vol. 17, Menlo Park, CA: AAAI Press / Cambridge, MA: MIT Press, July 30 - August 3 2000, pp. 590–595. [4] V. R. Konda and J. N. Tsitsiklis, Actor-critic algorithms, Advances in Neural Information Processing Systems (Cambridge, MA) (S. A. Solla, T. K. Leen, and K.-R. Mller, eds.), vol. 12, MIT Press, 2000. [5] R. S. Sutton, D. McAllester, S. Singh, and Y. Mansour, Policy gradient methods for reinforcement learning with function approximation, Advances in Neural Information Processing Systems (Cambridge, MA) (S. A. Solla, T. K. Leen, and K.-R. Mller, eds.), vol. 12, MIT Press, 2000. [6] R. J. Williams, Simple statistical gradient-following algorithms for connectionist reinforcement learning, Machine Learning 8 (1992), no. 3, 229–256. Appendix: Proofs of Theorems 1 and 2 Proof of Theorem 1: Consider the definition of * ( given in (7). In [5] it is shown that there exist ^ *Z( ' ] and _ *)( ' ] such that: I @ = !(' !Q + & ! R ! * ( @ = !M(' !Q + & ! R ! + & ! ) (13) Let R R W ! @ be the observation of R R W (3) after a single episode. Using (13), we get the following: R R W ! @ ) ( @ = ( !(' R BT @N' * ( U W!X R W ! @ /
#! R R W , " ) P ( @ = ( !M ' R BT @!' *Z( U W;X R W + & ! Y , " ) P ( @ = ( !M ' R BT @!' * ( U W;X R W *)( Y ,#" ) P ( @ = ( !M ' R BT @!' * ( U W;X R W [ ( ] ' ^ *)( ' ]`_ *)( ' ] Y ,#" ) P = ( !M(' [ ( ] (' ^ *Z( ' ] ( @ R BT @!' * ( U W X R W _ *Z( ' ] Y , " = ( !M ' [ ( ] (' ^ *)( ' ] ! ] ,#" where the basis functions ! ] have the form ! ] ) @ !GQ /
! R ! _ * ( ' ] and I " ) , with variance + " )+ P ! ! @ Y ) @ % = !(' !Q + & ! R ! % $ % @!' * ( Denoting R R W b as the least squares (LS) estimate of (3), its form is given by: ! !0 b ) = [ (' (14) where are LS estimates of the weights ^ *)( ' ] and correspond to the basis functions ! ] . Then, it can be shown that any linear system of the type given in (14) has a rate of convergence given by: + ! ! b ) * + " ) * @ % = !(' !GQ + & #!R ! % $ % @!' * ( Substituting (5) and (6) into the above equation completes the proof. Proof of Theorem 2: We prove equation (10) first. For * estimates of the performance gradient, we get * independent samples of each ! @ /
#! . These examples are averaged and therefore: I / ! ! ) @ = !(' !GQ /
! R ! /
! Because each ! @ + & ! is independently distributed, the variance of the estimate is given by + / ! ! ) * @ % = !(' !Q + & #!R ! % $ % @N' *)( (15) Given (5) the worst rate of convergence is bounded by: + / ! ! @ % = !(' !Q + & ! R ! % $ % 798 : * ) O 798 : * A similarly argument applies to the lower bound on convergence completing the proof for (10). Following the same argument for (12), we have + ! ! ) * @ % = !(' !Q + & ! R ! % + +
! = (' +
Where + + & ! ' = = ( (' + & )+ = ' = + & ! ' = = ( (' ! /
) = ' =
% $6% @!' *)( , = ( (' ! ' =
% $6% @N' * (16) Given (5) the variance + on the far left of (16) is bounded by: = ' =
% $6% @!' *)( , = ( (' ! ' =
% $6% @!' * 798;: ) = ' =
% $6% 798;: , = ( (' ! ' =
% $6% 798 : ) = ' =
% , ' =
% $6% 798;: ) ' =
$6% 798 : Plugging the above into (16) and inserting O * from (6) completes the proof for the upper bound. The proof for the lower bound in the variance follows similar reasoning.
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A Generalization of Principal Component Analysis to the Exponential Family Michael Collins Sanjoy Dasgupta Robert E. Schapire AT&T Labs Research 180 Park Avenue, Florham Park, NJ 07932 mcollins, dasgupta, schapire @research.att.com Abstract Principal component analysis (PCA) is a commonly applied technique for dimensionality reduction. PCA implicitly minimizes a squared loss function, which may be inappropriate for data that is not real-valued, such as binary-valued data. This paper draws on ideas from the Exponential family, Generalized linear models, and Bregman distances, to give a generalization of PCA to loss functions that we argue are better suited to other data types. We describe algorithms for minimizing the loss functions, and give examples on simulated data. 1 Introduction Principal component analysis (PCA) is a hugely popular dimensionality reduction technique that attempts to find a low-dimensional subspace passing close to a given set of points
. More specifically, in PCA, we find a lower dimensional subspace that minimizes the sum of the squared distances from the data points to their projections in the subspace, i.e.,
(1) This turns out to be equivalent to choosing a subspace that maximizes the sum of the squared lengths of the projections , which is the same as the (empirical) variance of these projections if the data happens to be centered at the origin (so that ). PCA also has another convenient interpretation that is perhaps less well known. In this probabilistic interpretation, each point is thought of as a random draw from some unknown distribution "! , where denotes a unit Gaussian with mean #$ . The purpose then of PCA is to find the set of parameters % that maximizes the likelihood of the data, subject to the condition that these parameters all lie in a low-dimensional subspace. In other words, are considered to be noise-corrupted versions of some true points & which lie in a subspace; the goal is to find these true points, and the main assumption is that the noise is Gaussian. The equivalence of this interpretation to the ones given above follows simply from the fact that negative log likelihood under this Guassian model is equal (ignoring constants) to Eq. (1). This Gaussian assumption may be inappropriate, for instance if data is binary-valued, or integer-valued, or is nonnegative. In fact, the Gaussian is only one of the canonical distributions that make up the exponential family, and it is a distribution tailored to real-valued data. The Poisson is better suited to integer data, and the Bernoulli to binary data. It seems natural to consider variants of PCA which are founded upon these other distributions in place of the Gaussian. We extend PCA to the rest of the exponential family. Let be any parameterized set of distributions from the exponential family, where is the natural parameter of a distribution. For instance, a one-dimensional Poisson distribution can be parameterized by , corresponding to mean and distribution
Given data & , the goal is now to find parameters which lie in a low-dimensional subspace and for which the log-likelihood !
is maximized. Our unified approach effortlessly permits hybrid dimensionality reduction schemes in which different types of distributions can be used for different attributes of the data. If the data have a few binary attributes and a few integer-valued attributes, then some coordinates of the corresponding can be parameters of binomial distributions while others are parameters of Poisson distributions. (However, for simplicity of presentation, in this abstract we assume all distributions are of the same type.) The dimensionality reduction schemes for non-Gaussian distributions are substantially different from PCA. For instance, in PCA the parameters , which are means of Gaussians, lie in a space which coincides with that of the data . This is not the case in general, and therefore, although the parameters lie in a linear subspace, they typically correspond to a nonlinear surface in the space of the data. The discrepancy and interaction between the space of parameters and the space of the data is a central preoccupation in the study of exponential families, generalized linear models (GLM’s), and Bregman distances. Our exposition is inevitably woven around these three intimately related subjects. In particular, we show that the way in which we generalize PCA is exactly analogous to the manner in which regression is generalized by GLM’s. In this respect, and in others which will be elucidated later, it differs from other variants of PCA recently proposed by Lee and Seung [7], and by Hofmann [4]. We show that the optimization problem we derive can be solved quite naturally by an algorithm that alternately minimizes over the components of the analysis and their coefficients; thus, the algorithm is reminiscent of Csisz´ar and Tusn´ady’s alternating minization procedures [2]. In our case, each side of the minimization is a simple convex program that can be interpreted as a projection with respect to a suitable Bregman distance; however, the overall program is not generally convex. In the case of Gaussian distributions, our algorithm coincides exactly with the power method for computing eigenvectors; in this sense it is a generalization of one of the oldest algorithms for PCA. Although omitted for lack of space, we can show that our procedure converges in that any limit point of the computed coefficients is a stationary point of the loss function. Moreover, a slight modification of the optimization criterion guarantees the existence of at least one limit point. Some comments on notation: All vectors in this paper are row vectors. If is a matrix, we denote its ’th row by ! and its #" ’th element by $ &% . 2 The Exponential Family, GLM’s, and Bregman Distances 2.1 The Exponential Family and Generalized Linear Models In the exponential family of distributions the conditional probability of a value ' given parameter value takes the following form: ' ( ) +* ' -, '. 0/ (2) Here, is the “natural parameter” of the distribution, and can usually take any value in the reals. / is a function that ensures that the sum (integral) of ' over the domain of ' is 1. From this it follows that / ( ) * ' . We use to denote the domain of ' . The sum is replaced by an integral in the continuous case, where defines a density over . * is a term that depends only on ' , and can usually be ignored as a constant during estimation. The main difference between different members of the family is the form of / . We will see that almost all of the concepts of the PCA algorithms in this paper stem directly from the definition of / . A first example is a normal distribution, with mean and unit variance, which has a density that is usually written as ' ( ) ' . It can be verified that this is a member of the exponential family with * ' ' , , and / . Another common case is a Bernoulli distribution for the case of binary outcomes. In this case . The probability of ' is usually written ' + -
where is a parameter in . This is a member of the exponential family with * ' , ( ) , and / , . A critical function is the derivative / , which we will denote as throughout this paper. By differentiating / * ' , it is easily verified that ' , the expectation of ' under ' . In the normal distribution, ' , and in the Bernoulli case ' . In the general case, ' is referred to as the “expectation parameter”, and defines a function from the natural parameter values to the expectation parameter values. Our generalization of PCA is analogous to the manner in which generalized linear models (GLM’s) [8] provide a unified treatment of regression for the exponential family by generalizing least-squares regression to loss functions that are more appropriate for other members of this family. The regression set-up assumes a training sample of pairs, where is a vector of attributes, and is some response variable. The parameters of the model are a vector . The dot product is taken to be an approximation of . In least squares regression the optimal parameters are set to be ! " $#&%(' *) + & . In GLM’s, , - is taken to approximate the expectation parameter of the exponential model, where , is the inverse of the “link function” [8]. A natural choice is to use the “canonical link”, where , , being the derivative /. . In this case the natural parameters are directly approximated by / , and the log-likelihood ' is simply ( ) * , 01
/ 2
. In the case of a normal distribution with fixed variance, / and it follows easily that the maximum-likelihood criterion is equivalent to the least squares criterion. Another interesting case is logistic regression where / , , and the negative log-likelihood for parameters is 3 , ) 546 ! 7 8 !:9 where if , if . 2.2 Bregman Distances and the Exponential Family Let ;2<>=@? be a differentiable and strictly convex function defined on a closed, convex set =BA . The Bregman distance associated with ; is defined for
C D= to be EGF H JI C ; H + ; C LK C M C where K ' ; ' . It can be shown that, in general, every Bregman distance is nonnegative and is equal to zero if and only if its two arguments are equal. For the exponential family the log-likelihood ( ) ' is directly related to a Bregman normal Bernoulli Poisson
"!# !# $ % &('*) % + ! # , .-/ 0 .-/ 21 -3 0 41 -/ 0 .-/ 1 56.-/87 $39 ' .-3 ;: ' 9 : <>= ?A@"B0 C? 1 B0 ? 0 ED FG H1 ?/ 0 ' 9 D ' 9 F ? 0 GD FE B 1 ? < = I-A@ $ 0J .1 0
"! 9 : 6 # where -LKH7 1M ! # 1 -N 0 1 Table 1: Various functions of interest for three members of the exponential family distance. Specifically, [1, 3] define a “dual” function ; through / and : ; , / (3) It can be shown under fairly general conditions that K ' ' . Application of these identities implies that the negative log-likelihood of a point can be expressed through a Bregman distance [1, 3]: ( ) ' ( ) +* ' ; ' , E F ' I (4) In other words, negative log-likelihood can always be written as a Bregman distance plus a term that is constant with respect to and which therefore can be ignored. Table 1 summarizes various functions of interest for examples of the exponential family. We will find it useful to extend the idea of Bregman distances to divergences between vectors and matrices. If , O are vectors, and P , Q are matrices, then we overload the notation as E F I O E F ' I and E F P I Q % E F SR % IUT % . (The notion of Bregman distance as well as our generalization of PCA can be extended to vectors in a more general manner; here, for simplicity, we restrict our attention to Bregman distances and PCA problems of this particular form.) 3 PCA for the Exponential Family We now generalize PCA to other members of the exponential family. We wish to find ’s that are “close” to the ’s and which belong to a lower dimensional subspace of parameter space. Thus, our approach is to find a basis V % VXW in and to represent each as the linear combination of these elements ZY R Y V Y that is “closest” to
. Let [ be the ]\_^ matrix whose ’th row is . Let ` be the a \b^ matrix whose c ’th row is V Y , and let P be the d\ a matrix with elements R Y . Then e PG` is an d\f^ matrix whose ’th row is as above. This is a matrix of natural parameter values which define the probability of each point in [ . Following the discussion in Section 2, we consider the loss function taking the form g ` P [ P ` % ' &% &% Zh , % ' &% % , / % where h is a constant term which will be dropped from here on. The loss function varies depending on which member of the exponential family is taken, which simply changes the form of / . For example, if [ is a matrix of real values, and the normal distribution is appropriate for the data, then / and the loss criterion is the usual squared loss for PCA. For the Bernoulli distribution, / , ) . If we define ' &% ' &% , then g ` P %- , 6 ! i ! i . From the relationship between log-likelihood and Bregman distances (see Eq. (4)), the loss can also be written as g ` P % E F ' &% I % E F I (where we allow to be applied to vectors and matrices in a pointwise manner). Once ` and P have been found for the data points, the ’th data point can be represented as the vector in the lower dimensional space W . Then are the coefficients which define a Bregman projection of the vector : " #&% ' E F I 3` (5) The generalized form of PCA can also be considered to be search for a low dimensional basis (matrix ` ) which defines a surface that is close to all the data points . We define the set of points ` to be ` 3` W . The optimal value for ` then minimizes the sum of projection distances: ` " # %(' #&%(' E F I
. Note that for the normal distribution and the Bregman distance is Euclidean distance so that the projection operation in Eq. (5) is a simple linear projection ( "
` ). ` is also simplified in the normal case, simply being the hyperplane whose basis is ` . To summarize, once a member of the exponential family — and by implication a convex function / — is chosen, regular PCA is generalized in the following way: The loss function is negative log-likelihood, ( ) ' ' , / -, constant. The matrix e PG` is taken to be a matrix of natural parameter values. The derivative of / defines a matrix of expectation parameters, PG` . A function ; is derived from / and . A Bregman distance E F is derived from ; . The loss is a sum of Bregman distances from the elements ' % to values % % . PCA can also be thought of as search for a matrix ` that defines a surface ` which is “close” to all the data points. The normal distribution is a simple case because , and the divergence is Euclidean distance. The projection operation is a linear operation, and ` is the hyperplane which has ` as its basis. 4 Generic Algorithms for Minimizing the Loss Function We now describe a generic algorithm for minimization of the loss function. First, we concentrate on the simplest case where there is just a single component so that a . (We drop the c subscript from R Y and % Y .) The method is iterative, with an initial random choice for the value of ` . Let ` , P , etc. denote the values at the ’th iteration, and let ` * be the initial random choice. We propose the iterative updates P " # %(' g ` P and ` " # %(' g ` P . Thus g is alternately minimized with respective to its two arguments, each time optimizing one argument while keeping the other one fixed, reminiscent of Csisz´ar and Tusn´ady’s alternating minization procedures [2]. It is useful to write these minimization problems as follows: For , R ! " $#&%(' % E F 3 ' &% I *R % 9 For " ^ , % ! " $#&%(' E F 3 ' % I SR 9 . We can then see that there are , ^ optimization problems, and that each one is essentially identical to a GLM regression problem (a very simple one, where there is a single parameter being optimized over). These sub-problems are easily solved, as the functions are convex in the argument being optimized over, and the large literature on maximumlikelihood estimation in GLM’s can be directly applied to the problem. These updates take a simple form for the normal distribution: P [ ` ` , and ` P [ P . It follows that ` ` [ [ h , where h is a scalar value. The method is then equivalent to the power method (see Jolliffe [5]) for finding the eigenvector of [ [ with the largest eigenvalue, which is the best single component solution for ` . Thus the generic algorithm generalizes one of the oldest algorithms for solving the regular PCA problem. The loss is convex in either of its arguments with the other fixed, but in general is not convex in the two arguments together. This makes it very difficult to prove convergence to the global minimum. The normal distribution is an interesting special case in this respect — the power method is known to converge to the optimal solution, in spite of the non-convex nature of the loss surface. A simple proof of this comes from properties of eigenvectors (Jolliffe [5]). It can also be explained by analysis of the Hessian : for any stationary point which is not the global minimum, is not positive semi-definite. Thus these stationary points are saddle points rather than local minima. The Hessian for the generalized loss function is more complex; it remains an open problem whether it is also not positive semidefinite at stationary points other than the global minimum. It is also open to determine under which conditions this generic algorithm will converge to a global minimum. In preliminary numerical studies, the algorithm seems to be well behaved in this respect. Moreover, any limit point of the sequence e P ` will be a stationary point. However, it is possible for this sequence to diverge since the optimum may be at infinity. To avoid such degenerate choices of e , we can use a modified loss % E F ' &% I % , E F * I % where is a small positive constant, and +* is any value in the range of (and therefore for which * is finite). This is roughly equivalent to adding a conjugate prior and finding the maximum a posteriori solution. It can be proved, for this modified loss, that the sequence e remains in a bounded region and hence always has at least one limit point which must be a stationary point. (All proofs omitted for lack of space.) There are various ways to optimize the loss function when there is more than one component. We give one algorithm which cycles through the a components, optimizing each in turn while the others are held fixed: //Initialization Set 7 , 7 //Cycle through components times For 7 0 , 7 0
: //Now optimize the ’th component with other components fixed Initialize & + randomly, and set 7 For 7 0 convergence For ! 7
, &#" + 7$ % '&)(#*,+ < = 3 @ $ &#" 9 ' + . 9 For / 7 8 10 , &#" + 7$ % '&)(#*32 < = 3 @ $ &" + . 9 The modified Bregman projections now include a term 4 &% representing the contribution of the a fixed components. These sub-problems are again a standard optimization problem regarding Bregman distances, where the terms 4 % form a “reference prior”. 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 X Y data PCA exp 0 20 40 60 80 100 0 50 100 150 200 250 300 350 400 450 500 X Y data PCA exp Figure 1: Regular PCA vs. PCA for the exponential distribution. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 A C’ B B’ C 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 C C’ D B B’ A D’ Figure 2: Projecting from 3- to 1-dimensional space, via Bernoulli PCA. Left: the three points E h are projected onto a one-dimensional curve. Right: point is added. 5 Illustrative examples Exponential distribution. Our generalization of PCA behaves rather differently for different members of the exponential family. One interesting example is that of the exponential distributions on nonnegative reals. For one-dimensional data, these densities are usually written as , where is the mean. In the uniform system of notation we have been using, we would instead index each distribution by a single natural parameter (basically, ), and write the density as ' , where / ' . The link function in this case is , the mean of the distribution. Suppose we are given data [ and want to find the best one-dimensional approximation: a vector V and coefficients such that the approximation *R V & has minimum loss. The alternating minimization procedure of the previous section has a simple closed form in this case, consisting of the iterative update rule V
^ [ [bV Here the shorthand denotes a componentwise reciprocal, i.e., ) . Notice the similarity to the update rule of the power method for PCA: V
[ [bV . Once V is found, we can recover the coefficients ^ The points R V lie on a line through the origin. Normally, we would not expect the points to also lie on a straight line; however, in this case they do, because any point of the form SR V R , can be written as and so must lie in the direction . Therefore, we can reasonably ask how the lines found under this exponential assumption differ from those found under a Gaussian assumption (that is, those found by regular PCA), provided all data is nonnegative. As a very simple illustration, we conducted two toy experiments with twenty data points in (Figure 1). In the first, the points all lay very close to a line, and the two versions of PCA produced similar results. In the second experiment, a few of the points were moved farther afield, and these outliers had a larger effect upon regular PCA than upon its exponential variant. Bernoulli distribution. For the Bernoulli distribution, a linear subspace of the space of parameters is typically a nonlinear surface in the space of the data. In Figure 2 (left), three points in the three-dimensional hypercube are mapped via our PCA to a onedimensional curve. The curve passes through one of the points ( ); the projections of the two other ( E ? E and h ? h ) are indicated. Notice that the curve is symmetric about the center of the hypercube, . In Figure 2 (right), another point (D) is added, and causes the approximating one-dimensional curve to swerve closer to it. 6 Relationship to Previous Work Lee and Seung [6, 7] and Hofmann [4] also describe probabilistic alternatives to PCA, tailored to data types that are not gaussian. In contrast to our method, [4, 6, 7] approximate mean parameters underlying the generation of the data points, with constraints on the matrices P and ` ensuring that the elements of PG` are in the correct domain. By instead choosing to approximate the natural parameters, in our method the matrices P and ` do not usually need to be constrained—instead, we rely on the link function to give a transformed matrix PG` which lies in the domain of the data points. More specifically, Lee and Seung [6] use the loss function % ' &% ( ) % , &% (ignoring constant factors, and again defining &%$ Y R Y Y % ). This is optimized with the constraint that P and ` should be positive. This method has a probabilistic interpretation, where each data point ' &% is generated from a Poisson distribution with mean parameter % . For the Poisson distribution, our method uses the loss function % ' % &% , ! i , but without any constraints on the matrices P and ` . The algorithm in Hofmann [4] uses a loss function % ' &% ( ) % , where the matrices P and ` are constrained such that all the % ’s are positive, and also such that % % . Bishop and Tipping [9] describe probabilistic variants of the gaussian case. Tipping [10] discusses a model that is very similar to our case for the Bernoulli family. Acknowledgements. This work builds upon intuitions about exponential families and Bregman distances obtained largely from interactions with Manfred Warmuth, and from his papers. Thanks also to Andreas Buja for several helpful comments. References [1] Katy S. Azoury and M. K. Warmuth. Relative loss bounds for on-line density estimation with the exponential family of distributions. Machine Learning, 43:211–246, 2001. [2] I. Csisz´ar and G. Tusn´ady. Information geometry and alternating minimization procedures. Statistics and Decisions, Supplement Issue, 1:205–237, 1984. [3] J¨urgen Forster and Manfred Warmuth. Relative expected instantaneous loss bounds. Journal of Computer and System Sciences, to appear. [4] Thomas Hofmann. Probabilistic latent semantic indexing. In Proceedings of the 22nd Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, 1999. [5] I. T. Jolliffe. Principal Component Analysis. Springer-Verlag, 1986. [6] D. D. Lee and H. S. Seung. Learning the parts of objects with nonnegative matrix factorization. Nature, 401:788, 1999. [7] Daniel D. Lee and H. Sebastian Seung. Algorithms for non-negative matrix factorization. In Advances in Neural Information Processing Systems 13, 2001. [8] P. McCullagh and J. A. Nelder. Generalized Linear Models. CRC Press, 2nd edition, 1990. [9] M. E. Tipping and C. M. Bishop. Probabilistic principal component analysis. Journal of the Royal Statistical Society, Series B, 61(3):611–622, 1999. [10] Michael E. Tipping. Probabilistic visualisation of high-dimensional binary data. In Advances in Neural Information Processing Systems 11, pages 592–598, 1999.
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Minimax Probability Machine Gert R.G. Lanckriet* Department of EECS University of California, Berkeley Berkeley, CA 94720-1770 gert@eecs.berkeley.edu Chiranjib Bhattacharyya Department of EECS University of California, Berkeley Berkeley, CA 94720-1776 chiru@eecs.berkeley.edu Laurent EI Ghaoui Department of EECS University of California, Berkeley Berkeley, CA 94720-1770 elghaoui@eecs.berkeley.edu Michael I. Jordan Computer Science and Statistics University of California, Berkeley Berkeley, CA 94720-1776 jordan@cs.berkeley.edu Abstract When constructing a classifier, the probability of correct classification of future data points should be maximized. In the current paper this desideratum is translated in a very direct way into an optimization problem, which is solved using methods from convex optimization. We also show how to exploit Mercer kernels in this setting to obtain nonlinear decision boundaries. A worst-case bound on the probability of misclassification of future data is obtained explicitly. 1 Introduction Consider the problem of choosing a linear discriminant by minimizing the probabilities that data vectors fall on the wrong side of the boundary. One way to attempt to achieve this is via a generative approach in which one makes distributional assumptions about the class-conditional densities and thereby estimates and controls the relevant probabilities. The need to make distributional assumptions, however, casts doubt on the generality and validity of such an approach, and in discriminative solutions to classification problems it is common to attempt to dispense with class-conditional densities entirely. Rather than avoiding any reference to class-conditional densities, it might be useful to attempt to control misclassification probabilities in a worst-case setting; that is, under all possible choices of class-conditional densities. Such a minimax approach could be viewed as providing an alternative justification for discriminative approaches. In this paper we show how such a minimax programme can be carried out in the setting of binary classification. Our approach involves exploiting the following powerful theorem due to Isii [6], as extended in recent work by Bertsimas • http://robotics.eecs.berkeley.edur gert/ and Sethuraman [2]: where y is a random vector, where a and b are constants, and where the supremum is taken over all distributions having mean y and covariance matrix ~y. This theorem provides us with the ability to bound the probability of misclassifying a point, without making Gaussian or other specific distributional assumptions. We will show how to exploit this ability in the design of linear classifiers. One of the appealing features of this formulation is that one obtains an explicit upper bound on the probability of misclassification of future data: 1/(1 + rP). A second appealing feature of this approach is that, as in linear discriminant analysis [7], it is possible to generalize the basic methodology, utilizing Mercer kernels and thereby forming nonlinear decision boundaries. We show how to do this in Section 3. The paper is organized as follows: in Section 2 we present the minimax formulation for linear classifiers, while in Section 3 we deal with kernelizing the method. We present empirical results in Section 4. 2 Maximum probabilistic decision hyperplane In this section we present our minimax formulation for linear decision boundaries. Let x and y denote random vectors in a binary classification problem, with mean vectors and covariance matrices given by x '" (x, ~x) and y '" (y, ~y) , respectively, where ""," means that the random variable has the specified mean and covariance matrix but that the distribution is otherwise unconstrained. Note that x, x, y, Y E JRn and ~x, ~y E JRnxn. We want to determine the hyperplane aT z = b (a, z E JRn and b E JR) that separates the two classes of points with maximal probability with respect to all distributions having these means and covariance matrices. This boils down to: or, max a s.t. inf Pr{ aT x 2: b} 2: a (2) a ,a ,b max a s.t. a,a,b 1 - a 2: sup Pr{ aT x :s b} 1- a 2: sup Pr{aT y 2: b}. (3) Consider the second constraint in (3). Recall the result of Bertsimas and Sethuraman [2]: 1 supPr{aTY2:b}=-d2' with d2 = inf (Y_Yf~y-1(y_y) (4) 1 + aTy?b We can write this as d2 = infcTw>d wT w, where w = ~y -1/2 (y_y), cT = aT~y 1/2 and d = b - aTy. To solve this,-first notice that we can assume that aTy :S b (i.e. y is classified correctly by the decision hyperplane aT z = b): indeed, otherwise we would find d2 = 0 and thus a = 0 for that particular a and b, which can never be an optimal value. So, d> o. We then form the Lagrangian: £(w, >.) = wT w + >.(d - cT w), (5) which is to be maximized with respect to A 2: 0 and minimized with respect to w . At the optimum, 2w = AC and d = cT W , so A = -!#c and w = c%c. This yields: (6) Using (4), the second constraint in (3) becomes 1-0: 2: 1/(I+d2 ) or ~ 2: 0:/(1-0:). Taking (6) into account, this boils down to: b-aTY2:,,(o:)/aT~ya where ,,(0:)=) 0: (7) V 1-0: We can handle the first constraint in (3) in a similar way (just write aT x ::::: b as _aT x 2: -b and apply the result (7) for the second constraint). The optimization problem (3) then becomes: max 0: s.t. a ,a,b -b + aTx 2: ,,(o:)JaT~xa b - aTy 2: "(o:h/aT~ya. Because "(0:) is a monotone increasing function of 0:, we can write this as: max" s.t. ""a,b b - aTy 2: "JaT~ya. From both constraints in (9), we get aTy + "JaT~ya::::: b::::: aTx - "JaT~xa, which allows us to eliminate b from (9): aTy + "JaT~ya::::: aTx - "JaT~xa. I<,a max" s.t. (8) (9) (10) (11) Because we want to maximize ", it is obvious that the inequalities in (10) will become equalities at the optimum. The optimal value of b will thus be given by where a* and "* are the optimal values of a and " respectively. constraint in (11), we get: aT(x - y) 2:" (JaT~xa+ JaT~ya). (12) Rearranging the (13) The above is positively homogeneous in a: if a satisfies (13), sa with s E 114 also does. Furthermore, (13) implies aT(x - y) 2: O. Thus, we can restrict a to be such that aT(x - y) = 1. The optimization problem (11) then becomes max" s.t. I<,a which allows us to eliminate ,,: ~ 2: JaT~xa + JaT~ya aT (x-Y)=I, m~n JaT~xa + JaT~ya s.t. aT(x - y) = 1, (14) (15) or, equivalently (16) This is a convex optimization problem, more precisely a second order cone program (SOCP) [8,5]. Furthermore, notice that we can write a = ao +Fu, where U E Il~n-l, ao = (x - y)/llx - y112, and F E IRnx (n-l) is an orthogonal matrix whose columns span the subspace of vectors orthogonal to x - y. Using this we can write (16) as an unconstrained SOCP: (17) We can solve this problem in various ways, for example using interior-point methods for SOCP [8], which yield a worst-case complexity of O(n3 ). Of course, the first and second moments of x, y must be estimated from data, using for example plug-in estimates X, y, :Ex, :Ey for respectively x, y, ~x, ~y. This brings the total complexity to O(ln3 ), where l is the number of data points. This is the same complexity as the quadratic programs one has to solve in support vector machines. In our implementations, we took an iterative least-squares approach, which is based on the following form, equivalent to (17): (18) At iteration k, we first minimize with respect to 15 and E by setting 15k = II~x 1/2(ao + Fuk- d112 and Ek = II~y 1/2(ao + Fuk - 1)112. Then we minimize with respect to U by solving a least squares problem in u for 15 = 15k and E = Ek, which gives us Uk. Because in both update steps the objective of this COP will not increase, the iteration will converge to the global minimum II~xl/2(ao + Fu*)112 + II~yl /2(ao + Fu*)lb with u* an optimal value of u. We then obtain a* as ao + Fu* and b* from (12) with "'* = l/h/ar~xa* + Jar~ya*). Classification of a new data point Znew is done by evaluating sign( a;; Znew - b*): if this is + 1, Znew is classified as from class x, otherwise Znew is classified as from class y. It is interesting to see what happens if we make distributional assumptions; in particular, let us assume that x "" N(x, ~x) and y "" N(y, ~y). This leads to the following optimization problem: max a S.t. -b + aTx ::::: <I>-l(a)JaT~xa o:,a ,b (19) where <I>(z) is the cumulative distribution function for a standard normal Gaussian distribution. This has the same form as (8), but now with ",(a) = <I>-l(a) instead of ",(a) = V l~a (d. a result by Chernoff [4]). We thus solve the same optimization problem (a disappears from the optimization problem because ",(a) is monotone increasing) and find the same decision hyperplane aT z = b. The difference lies in the value of a associated with "'*: a will be higher in this case, so the hyperplane will have a higher predicted probability of classifying future data correctly. 3 Kernelization In this section we describe the "kernelization" of the minimax approach described in the previous section. We seek to map the problem to a higher dimensional feature space ]Rf via a mapping cP : ]Rn 1-+ ]Rf, such that a linear discriminant in the feature space corresponds to a nonlinear discriminant in the original space. To carry out this programme, we need to try to reformulate the minimax problem in terms of a kernel function K(Z1' Z2) = cp(Z1)T CP(Z2) satisfying Mercer's condition. Let the data be mapped as x 1-+ cp(x) ""' (cp(X) , ~cp(x)) and Y 1-+ cp(y) ""' (cp(y) , ~cp(y)) where {Xi}~1 and {Yi}~1 are training data points in the classes corresponding to x and Y respectively. The decision hyperplane in ]Rf is then given by aT cp(Z) = b with a, cp(z) E ]Rf and b E ]R. In ]Rf, we need to solve the following optimization problem: mln Jr-aT-~-cp-(-x)-a + J aT~cp(y)a s.t. aT (cp(X) - cp(y)) = 1, (20) where, as in (12), the optimal value of b will be given by b* = a; cp(x) "'*Jar~cp(x)a* = a; cp(y) + "'*Jar~cp(y)a*, (21) where a* and "'* are the optimal values of a and '" respectively. However, we do not wish to solve the COP in this form, because we want to avoid using f or cp explicitly. If a has a component in ]Rf which is orthogonal to the subspace spanned by CP(Xi), i = 1,2, ... , N x and CP(Yi), i = 1,2, ... , Ny, then that component won't affect the objective or the constraint in (20) . This implies that we can write a as N. Ny a = LaiCP(Xi) + L;)jCP(Yj). (22) i=1 j=1 Substituting expression (22) for a and estimates ;Pw = J. 2:~1 CP(Xi) , ;p(Y) = 1 Ny A _ 1 N. .....--.. .....--.. T A _ Ny 2:i=l cp(Yi), ~cp(x) N. 2:i=1 (cp(Xi) - cp(X)) (cp(Xi) - cp(x)) and ~cp(y) N .....--.. .....--.. J 2:i~1(CP(Yi) - cp(y))(cp(Yi) - cp(y))T for the means and the covariance matriy ces in the objective and the constraint of the optimization problem (20), we see that both the objective and the constraints can be written in terms of the kernel function K(Zl' Z2) = CP(Z1)T cp(Z2) . We obtain: T "f (kx - ky) = 1, (23) T N N . where "f = [a1 a2 ... aN. ;)1 ;)2 ... ;)Nyl , kx E ]R .+ y WIth [kxli = J. 2:f;1 K(xj, Zi), ky E ]RN. +Ny with [kyli = Jy 2:f~l K(Yj, Zi), Zi = Xi for i = 1,2, ... ,Nx and Zi = Yi- N. for i = Nx + 1, Nx + 2, ... ,Nx + Ny . K is defined as: K = (Kx -IN.~~) = (*x) Ky -lNy ky Ky (24) where 1m is a column vector with ones of dimension m. Kx and Ky contain respectively the first N x rows and the last Ny rows of the Gram matrix K (defined as Kij = cp(zdTcp(zj) = K(Zi,Zj)). We can also write (23) as Kx I I Ky I T m~n II ~"f12 + I .jlV;"f 12 s.t. "f (kx - ky) = 1, (25) which is a second order cone program (SOCP) [5] that has the same form as the SOCP in (16) and can thus be solved in a similar way. Notice that, in this case, the optimizing variable is "f E ~Nz +Ny instead of a E ~n. Thus the dimension of the optimization problem increases, but the solution is more powerful because the kernelization corresponds to a more complex decision boundary in ~n . Similarly, the optimal value b* of b in (21) will then become (26) where "f* and "'* are the optimal values of "f and", respectively. Once "f* is known, we get "'* = 1/ ( J ~z "f;K~Kx"f* + J ~y "f;K~Ky "f* ) and then b* from (26). Classification of a new data point Znew is then done by evaluating sign(a; <p(znew) -b*) = sign ( (L~l+Ny b*]iK(Zi, Znew) ) - b*) (again only in terms of the kernel function): if this is + 1, Znew is classified as from class x, otherwise Znew is classified as from class y. 4 Experiments In this section we report the results of experiments that we carried out to test our algorithmic approach. The validity of 1 - a as the worst case bound on the probability of misclassification of future data is checked, and we also assess the usefulness of the kernel trick in this setting. We compare linear kernels and Gaussian kernels. Experimental results on standard benchmark problems are summarized in Table 1. The Wisconsin breast cancer dataset contained 16 missing examples which were not used. The breast cancer, pima, diabetes, ionosphere and sonar data were obtained from the VCI repository. Data for the twonorm problem data were generated as specified in [3]. Each dataset was randomly partitioned into 90% training and 10% test sets. The kernel parameter (u) for the Gaussian kernel (e-llx-yI12/,,) was tuned using cross-validation over 20 random partitions. The reported results are the averages over 50 random partitions for both the linear kernel and the Gaussian kernel with u chosen as above. The results are comparable with those in the existing literature [3] and with those obtained with Support Vector Machines. Also, we notice that a is indeed smaller Table 1: a and test-set accuracy (TSA) compared to BPB (best performance in [3]) and to the performance of an SVM with linear kernel (SVML) and an SVM with Gaussian kernel (SVMG) Dataset Linear kernel Gaussian kernel BPB SVML SVMG a TSA: a TSA: Twonorm 80.2 % 96.0 % 83.6 % 97.2 % 96.3 % 95.6 % 97.4 % Breast cancer 84.4 % 97.2 % 92.7 % 97.3 % 96.8 % 92.6 % 98.5 % Ionosphere 63.3 % 85.4 % 89.9 % 93.0 % 93.7 % 87.8 % 91.5 % Pima diabetes 31.2 % 73.8 % 33.0 % 74.6 % 76.1 % 70.1 % 75.3 % Sonar 62.4 % 75.1 % 87.1 % 89.8 % 75.9 % 86.7 % than the test-set accuracy in all cases. Furthermore, a is smaller for a linear decision boundary then for the nonlinear decision boundary obtained via the Gaussian kernel. This clearly shows that kernelizing the method leads to more powerful decision boundaries. 5 Conclusions The problem of linear discrimination has a long and distinguished history. Many results on misclassification rates have been obtained by making distributional assumptions (e.g., Anderson and Bahadur [1]). Our results, on the other hand, make use of recent work on moment problems and semidefinite optimization to obtain distribution-free results for linear discriminants. We have also shown how to exploit Mercer kernels to generalize our algorithm to nonlinear classification. The computational complexity of our method is comparable to the quadratic program that one has to solve for the support vector machine (SVM). While we have used a simple iterative least-squares approach, we believe that there is much to gain from exploiting analogies to the SVM and developing specialized, more efficient optimization procedures for our algorithm, in particular tools that break the data into subsets. The extension towards large scale applications is a current focus of our research, as is the problem of developing a variant of our algorithm for multiway classification and function regression. Also the statistical consequences of using plug-in estimates for the mean vectors and covariance matrices needs to be investigated. Acknowledgements We would like to acknowledge support from ONR MURI N00014-00-1-0637, from NSF grants IIS-9988642 and ECS-9983874 and from the Belgian American Educational Foundation. References [1] Anderson, T . W . and Bahadur, R. R. (1962) Classification into two multivariate Normal distributions with different covariance matrices. Annals of Mathematical Statistics 33(2): 420-431. [2] Bertsimas, D. and Sethuraman, J. (2000) Moment problems and semidefinite optimization. Handbook of Semidefinite Optimization 469-509, Kluwer Academic Publishers. [3] Breiman L. (1996) Arcing classifiers. Technical Report 460, Statistics Department, University of California, December 1997. [4] Chernoff H. (1972) The selection of effective attributes for deciding between hypothesis using linear discriminant functions. In Frontiers of Pattern Recognition, (S. Watanabe, ed.), 55-60. New York: Academic Press. [5] Boyd, S. and Vandenberghe, L. (2001) Convex Optimization. Course notes for EE364, Stanford University. Available at http://www . stanford. edu/ class/ee364. [6] Isii, K. (1963) On the sharpness of Chebyshev-type inequalities. Ann. Inst. Stat. Math. 14: 185-197. [7] Mika, M. Ratsch, G., Weston, J., SchOikopf, B., and Mii11er, K.-R. (1999) Fisher discriminant analysis with kernels. In Neural Networks for Signal Processing IX, 41- 48, New York: IEEE Press. [8] Nesterov, Y. and Nemirovsky, A. (1994) Interior Point Polynomial Methods in Convex Programming: Theory and Applications. Philadelphia, PA: SIAM.
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Agglomerative Multivariate Information Bottleneck Noam Sionim Nir Friedman Naftali Tishby School of Computer Science & Engineering, Hebrew University, Jerusalem 91904, Israel {noamm, nir, tishby } @cs.huji.ac.il Abstract The information bottleneck method is an unsupervised model independent data organization technique. Given a joint distribution peA, B), this method constructs a new variable T that extracts partitions, or clusters, over the values of A that are informative about B. In a recent paper, we introduced a general principled framework for multivariate extensions of the information bottleneck method that allows us to consider multiple systems of data partitions that are inter-related. In this paper, we present a new family of simple agglomerative algorithms to construct such systems of inter-related clusters. We analyze the behavior of these algorithms and apply them to several real-life datasets. 1 Introduction The information bottleneck (IB) method of Tishby et al [14] is an unsupervised nonparametric data organization technique. Given a joint distribution P(A, B), this method constructs a new variable T that represents partitions of A which are (locally) maximizing the mutual information about B. In other words, the variable T induces a sufficient partition, or informative features of the variable A with respect to B. The construction of T finds a tradeoff between the information about A that we try to minimize, J(T; A), and the information about B which we try to maximize, J(T; B). This approach is particularly useful for co-occurrence data, such as words and documents [12], where we want to capture what information one variable (e.g., use of a word) contains about the other (e.g., the document). In a recent paper, Friedman et al. [4] introduce multivariate extension of the IB principle. This extension allows us to consider cases where the data partition is relevant with respect to several variables, or where we construct several systems of clusters simultaneously. In this framework, we specify the desired interactions by a pair of Bayesian networks. One network, Gin, represents which variables are compressed versions of the observed variables - each new variable compresses its parents in the network. The second network, Gout> defines the statistical relationship between these new variables and the observed variables that should be maintained. Similar to the original IB, in Friedman et al. we formulated the general principle as a tradeoff between the (multi) information each network carries. On the one hand, we want to minimize the information maintained by G in and on the other to maximize the information maintained by Gout. We also provide a characterization of stationary points in this tradeoff as a set of self-consistent equations. Moreover, we prove that iterations of these equations converges to a (local) optimum. Then, we describe a deterministic annealing procedure that constructs a solution by tracking the bifurcation of clusters as it traverses the tradeoff curve, similar to the original IB method. In this paper, we consider an alternative approach to solving multivariate IB problems which is motivated by the success of the agglomerative IB of Slonim and Tishby [11]. As shown there, a bottom-up greedy agglomeration is a simple heuristic procedure that can yield good solutions to the original IB problem. Here we extend this idea in the context of multivariate IB problems. We start by analyzing the cost of agglomeration steps within this framework. This both elucidates the criteria that guides greedy agglomeration and provides for efficient local evaluation rules for agglomeration steps. This construction results with a novel family of information theoretic agglomerative clustering algorithms, that can be specified using the graphs Gin and G out. We demonstrate the performance of some of these algorithms for document and word clustering and gene expression analysis. 2 Multivariate Information Bottleneck A Bayesian network structure G is a DAG that specifies interactions among variables [8]. A distribution P is consistent with G (denoted P F G), if P(Xl , ... , X n) = I1 P(Xi I Pa<fJ, where Pa<fi are the parents of X i in G. Our main interest is in the information that the variables Xl " '" X n contain about each other. A quantity that captures this is the multi-information given by where V(Pllq) is the familiar Kullback-Liebler divergence [2]. Proposition 2.1 [4] Let G be a DAG over {Xl , ... , X n}, and let P F G be a distribution. Then, I G = I(Xl' ... , X n) = L i I(Xi ; Pa<fi ). That is, the multi-information is the sum of local mutual information terms between each variable and its parents (denoted I G). Friedman et al. define the multivariate IE problem as follows. Suppose we are given a set of observed variables, X = {Xl , ... , X n} and their joint distribution P (X l , ... , X n). We want to "construct" new variables T , where the relations between the observed variables and these new compression variables are specified using a DAG Gin over X U T where the variables in T are leafs. Thus, each Tj is a stochastic function of a set of variables U j = Pa~;in ~ X. Once these are set, we have a joint distribution over the combined set of variables: P(X, T) = P(X) ITj P(Tj I U j ). The "relevant" information that we want to preserve is specified by another DAG, Gout . This graph specifies, for each Tj which variables it predicts. These are simply its children in G out . More precisely, we want to predict each Xi (or Tj ) by V X i = Pa~;"t (resp. V T; = Pa~;out ), its parents in G out. Thus, we think ofIGout as a measure of how much information the variables in T maintain about their target variables. The Lagrangian can then be defined as (1) with a tradeoff parameter (Lagrange multiplier) (3. 1 The variation is done subject to the normalization constraints on the partition distributions. Thus, we balance between the information T loses about X in G in and the information it preserves in G out. Friedman et al. [4] show that stationary points of this Lagrangian satisfy a set of selfconsistent equations. Moreover, they show that iterating these equations converges to a INotice that under this formulation we would like to maximize £. An equivalent definition [4] would be to minimize £ = 'LGin - (J . 'LGO"t . stationary point of the tradeoff. Then, extending the procedure of Tishby et al [14], they propose a procedure that searches for a solution of the IB equations using a 'deterministic annealing' approach [9]. This is a top-down hierarchical algorithm that starts from a single cluster for each Tj at j3 -+ 0, and then undergoes a cascade of cluster splits as j3 is being "cooled". These determines "soft" trees of clusters (one for each Tj ) that describe solutions at different tradeoff values of j3. 3 The Agglomerative Procedure For the original IB problem, Slonim and Tishby [11] introduced a simpler procedure that performs greedy bottom-up merging of values. Several successful applications of this algorithm are already presented for a variety of real-world problems [10, 12, 13, 15]. The main focus of the current work is in extending this approach for the multivariate IB problem. As we will show, this will lead to further insights about the method, and also provide a rather simple and intuitive clustering procedures. We consider procedures that start with a set of clusters for each Tj (usually the most fine-grained solution we can consider where Tj = U j ) and then iteratively reduce the cardinality of one of the Tj 's by merging two values t~ and tj of Tj into a single value lj. To formalize this notion we must define the membership probability of a new cluster lj, resulting from merging {t~, tj} '* lj in Tj . This is done rather naturally by (2) In other words, we view the event l j as the union of the events t~ and tj. Given the membership probabilities, at each step we can also draw the connection between Tj and the other variables. This is done using the following proposition which is based on the conditional independence assumptions given in Gin. Proposition 3.1 Let Y , Z C X U T \ {Tj} then, (3) h II { } { p( t ~ I Z) p(t j IZ) }. h d· ·b· d·· d were Z = 1f1,Z, 1fr ,z = p(t; IZ)' p(t; IZ) IS t e merger 1Str1 uttOn can ltzone on z. In particular, this proposition allows us to evaluate all the predictions defined in G out and all the informations terms in £ that involve Tj . The crucial question in an agglomerative process is of course which pair to merge at each step. We know that the merger "cost" in our terms is exactly the difference in the values of £ , before and after the merger. Let T}ef and T/ft denote the random variables that correspond to Tj , before and after the merger, respectively. Thus, the values of £ before and after the merger are calculated based on Trf and Ttt. The merger cost is then simply given by, (4) The greedy procedure evaluates all the potential mergers (for all Tj ) and then applies the best one (i.e., the one that minimizes 6.£( t~ , tj). This is repeated until all the variables in T degenerate into trivial clusters. The resulting set of trees describes a range of solutions at different resolutions. This agglomerative approach is different in several important aspects from the deterministic annealing approach described above. In that approach, by "cooling" (i.e., increasing) j3, we move along a tradeoff curve from the trivial - single cluster - solution toward solutions with higher resolutions that preserve more information in G out. In contrast, in the agglomerative approach we progress in the opposite direction. We start with a high resolution clustering and as the merging process continues we move toward more and more compact solutions. During this process (3 is kept constant and the driving force is the reduction in the cardinality of the T/s. Therefore, we are able to look for good solutions in different resolutions for ajixed tradeoff parameter (3. Since the merging does not attempt directly to maintain the (stationary) self-consistent "soft" membership probabilities, we do not expect the self-consistent equations to hold at solutions found by the agglomerative procedure. On the other hand, the agglomerative process is much simpler to implement and fully deterministic. As we will show, it provides sufficiently good solutions for the IB problem in many situations. 4 Local Merging Criteria In the procedure we outline above, at every step there are O(ITj 12) possible mergers of values of Tj (for every j). A direct calculation of the costs of all these potential mergers is typically infeasible. However, it turns out that one may calculate t:...c (t; , tj) while examining only the probability distributions that involve t; and tj directly. Generalizing the results of [11] for the original IB, we now develop a closed-form formula for t:...c(t;, tj). To describe this result we need the following definition. The Jensen-Shannon ( J S) divergence [7, 3] between two probabilities PI , P2 is given by where II = {7rl' 7r2} is a normalized probability and p = 7rlPl + 7r2P 2 . The J S divergence is equal zero if and only if both its arguments are identical. It is upper bounded and symmetric, though it is not a metric. One interpretation of the J S -divergence relates it to the (logarithmic) measure of the likelihood that the two sample distributions originate by the most likely common source, denoted by p. In addition, we need the notation V-X~ = V X i - {Tj} (similarly for VT!). Theorem 4.1 Let t; , tj E Tj be two clusters. Then, t:...c( t; , tj) = p(tj) . d(t;, tj) where d(t; , tj) L Ep(' lt;) [JSrrv _; (P(Xi 1 t;, V-X~ ),p(Xi 1 tj, V -X{))] i:T; EVXi Xi + L Ep(' lt; ) [JSrrv _; (p(Te 1 t;, V Tj) ,p(Te 1 tj , VT! ))] e:T;EvT£ T£ + JSrr(p(VT; 1 t;) ,p(VT; 1 tj)) - (3-1. JSrr(p(Uj 1 t;) ,p(Uj 1 tj)) A detailed proof of this theorem will be given elsewhere. Thus, the merger cost is a multiplication of the weight of the merger components (P(tj)) with their "distance" given by d(t; , tj). Notice that due to the properties of the JS-divergence, this distance is symmetric. In addition, the last term in this distance has the opposite sign to the first three terms. Thus, the distance between two clusters is a tradeoff between these two factors. Roughly speaking, we may say that the distance is minimized for pairs that give similar predictions about the variables connected with Tj in Gout and have different predictions (minimum overlap) about the variables connected with Tj in Gin. We notice also the analogy between this result and the main theorem in [4]. In [4] the optimization is governed by the KL divergences between data and cluster's centroids, or by the likelihood that the data was generated by the centroid distribution. Here the optimization is controlled through the J S divergences, i.e. the likelihood that the two clusters have a common source. Next, we notice that after applying a merger, only a small portion of the other mergers costs change. The following proposition characterizes these costs. r 0~ n T} Tz ~ o 8 n T,( TIJ n A 8 Gin G out Gin G out Gin G out I(T; B) - (3-1 I(T; A) I(T1, T2; B) I(TA;TB ) _(3-" (I(T1 ; A) + I(T2; A)) _((3-" - l)(I(TA; A) + I(TB; B)) (a) Original Bottleneck (b) Parallel Bottleneck (c) Symmetric Bottleneck Figure 1: The source and target networks and the corresponding Lagrangian for the three examples we consider. Proposition4.2 The merger {t; ,tj} :::} tj in Tj can change the cost 6..c(t~ , tc ) only if p(tj , te) > 0 and Tj , Te co-appear in some information term in r Gout • This proposition is particularly useful, when we consider "hard" clustering where Tj is a (deterministic) function ofUj . In this case, p(tj,te) is often zero (especially when Tj and Te compressing similar variables, i.e., U j n U e =I- 0). In particular, after the merger {t;, tj} :::} tj, we do not have to reevaluate merger costs of other values of Tj, except for mergers of tj with each of these values. In the case of hard clustering we also find thatI(Tj; U j ) = H(Tj) (where H(P) is Shannon's entropy). Roughly speaking, we may say that H(P) is decreasing for less balanced probability distributions p. Therefore, increasing (3-1 will result with a tendency to look for less balanced "hard" partitions and vice verse. This is reflected by the fact that the last term in d( t; , tj) is then simplified through J Sn (p(U j I t;), p(U j I tj)) = H (II). 5 Examples We now briefly consider three examples of the general methodology. For brevity we focus on the simpler case of hard clustering. We first consider the example shown in figure I(a). This choice of graphs results in the original IB problem. The merger cost in this case is given by, 6..c(tl, n = p(t) . (JSn(p(B I tl),p(B I n) - (3-1 H(II)) . (5) Note that for (3-1 -+ 0 we get exactly the algorithm presented in [11]. One simple extension of the original IB is the parallel bottleneck [4]. In this case we introduce two variables T1 and T2 as in Figure I(b), both of them are functions of A. Similarly to the original IB, Gout specifies that T1 and T2 should predict B. We can think of this requirement as an attempt to decompose the information A contains about B into two "orthogonal" components. In this case, the merger cost for T1 is given by, 6..c(ti, tD = p(t1) . (Ep(.lld[JSnT2 (P(B I ti,T2),p(B I tLT2))]- (3-1 H(II)) . (6) Finally, we consider the symmetric bottleneck [4, 12]. In this case, we want to compress A into T A and B into T B so that T A extracts the information A contains about B, and at the same time TB extracts the information B contains about A. The DAG Gin of figure I(c) captures the form of the compression. The choice of G out is less obvious and several alternatives are described in [4]. Here, we concentrate only in one option, shown in figure I(c). In this case we attempt to make each ofTA and TB sufficient to separate A from B. Thus, on one hand we attempt to compress, and on the other hand we attempt to make T A and T B as informative about each other as possible. The merger cost in T A is given by 6..c(t~, tA) = P(tA) . JSn(p(TB I t~) , p(TB ItA)) - ((3-1 - l)H(II)), (7) while for merging in TB we will get an analogous expression. 6 Applications We examine a few applications of the examples presented above. As one data set we used a subset ofthe 20 newsgroups corpus [6] where we randomly choose 2000 documents evenly distributed among the 4 science discussion groups (sci. crypt, sci. electronics, sci.med and sci.space).2 Our pre-processing included ignoring file headers (and the subject lines), lowering upper case and ignoring words that contained non 'a .. z' characters. Given this document set we can evaluate the joint probability p(W, D), which is the probability that a random word position is equal to w E Wand at the same time the document is dE D . We sort all words by their contribution to I(W; D) and used only the 2000 'most informative' ones, ending up with a joint probability with I W I = ID I = 2000. We first used the original IB to cluster W , while trying to preserve the information about D. This was already done in [12] with (3-1 = 0, but in this new experiment we took (3-1 = 0.15. Recall that increasing (3-1 results in a tendency for finding less balanced clusters. Indeed, while for (3- 1 = 0 we got relatively balanced word clusters (high H(Tw )), for (3-1 = 0.15 the probability p(Tw) is much less smooth. For 50 word clusters, one cluster contained almost half of the words, while the other clusters were typically much smaller. Since the algorithm also tries to maximize I(Tw; D), the words merged into the big cluster are usually the less informative words about D. Thus, a word must be highly informative to stay out of this cluster. In this sense, increasing (3-1 is equivalent for inducing a "noise filter", that leave only the most informative features in specific clusters. In figure 2 we present p( D I tw) for several clusters tw E Tw. Clearly, words that passed the "filter" form much more informative clusters about the real structure of D. A more formal demonstration of this effect is given in the right panel of figure 2. For a given compression level (i.e. a given I(Tw; W)), we see that taking (3-1 = 0.15 preserve much more information aboutD. While an exact implementation of the symmetric IB will require alternating mergers in Tw and TD, an approximated approach require only two steps. First we find Tw. Second, we project each d E D into the low dimensional space defined by Tw , and use this more robust representation to extract document clusters TD. Approximately, we are trying to find Tw and TD that will maximize I(Tw; TD)' This two-phase IB algorithm was shown in [12] to be significantly superior to six other document clustering methods, when the performance are measured by the correlation of the obtained document clusters with the real newsgroup categories. Here we use the same procedure, but for finding Tw we take (3-1 = 0.15 (instead of zero). Using the above intuition we predict this will induce a cleaner representation for the document set. Indeed, the averaged correlation of TD (for lTD I = 4) with the original categories was 0.65, while for (3-1 = 0 it was 0.58 (the average is taken over different number of word clusters, ITw I = 10, 11...50). Similar results were obtained for all the 9 other subsets of the 20 newsgroups corpus described in [12]. As a second data set we used the gene expression measurements of rv 6800 genes in 72 samples of Leukemia [5]. The sample annotations included type of leukemia (ALL vs. AML), type of cells, source of sample, gender and donating hospital. We removed genes that were not expressed in the data and normalized the measurements of each sample to get a joint probability P(G, A) over genes and samples (with uniform prior on samples). We sorted all genes by their contribution to I(G; A) and chose the 500 most informative ones, which capture 47% of the original information, ending up with a joint probability with IAI = 72 and IGI = 500. We first used an exact implementation of the symmetric IB with alternating mergers be2We used the same subset already used in [12]. aotthetoand 0·04'---~----r~_~CC= , ,~ 905;O=w~ ord~ S 0.03 0.02 aCldvltammGaIClumlntakekldDey ... 0.04 c4,35words 0.03 0.02 algonthm secure secunty enayptlon ClaSSlIJed ... analog mOde signaimput output ... 0.04 c2, 20 words 0.04,---~-~~ c3~, 19;O=w~ ord~ S 0.03 0.02 ames planetary nasa spaceanane ... 0·04'---~-~~ C5o=" 35;O=w~ ord~ s 0.03 0.02 0.03 0.02 0.01 00 500 1000 1500 2000 sCience dataset, lnl0rmallon curves 1 .5~~~------, I ~-:=O I ~ 1 o ~ 0.5 IfTw·W\ Figure 2: P(D I tw) for 5 word clusters, tw E Tw. Documents 1 - 500 belong to sci. crypt category, 501 - 1000 to sci. electronics, 1001 - 1500 to sci.med and 1501 - 2000 to sci. space. In the title of each panel we see the 5 most frequent words in the cluster. The 'big' cluster (upper left panel) is clearly less informative about the structure of D. In the lower right panel we see the two information curves. Given some compression level, for (3- 1 = 0.15 we preserve much more information about D than for (3-1 = O. tween both clustering hierarchies (and /3-1 = 1). For ITA I = 2 we found an almost perfect correlation with the ALL vs. AML annotations (with only 4 exceptions). For ITA I = 8 and ITGI = 10 we found again high correlation between our sample clusters and the different sample annotations. For example, one cluster contained 10 samples that were all annotated as ALL type, taken from male patients in the same hospital. Almost all of these 10 were also annotated as T-cells, taken from bone marrow. Looking at p(TA I TG) we see that given the third genes cluster (which contained 17 genes) the probability of the above specific samples cluster is especially high. Further such analysis might yield additional insights about the structure of this data and will be presented elsewhere. Finally, to demonstrate the performance of the parallel IB we apply it to the same data. Using the parallel IB algorithm (with /3-1 = 0) we clustered the arrays A into two clustering hierarchies, T1 and T2 , that try together to capture the information about G. For ITj I = 4 we find that each I(Tj; G) preserve about 15% of the original information. However, taking ITj I = 2 (i.e. again, just 4 clusters) we see that the combination of the hierarchies, I(T1, T2 ; G), preserve 21 % of the original information. We then compared the two partitions we found against sample annotations. We found that the first hierarchy with IT11 = 2 almost perfectly match the split between B-cells and T-cells (among the 47 samples for which we had this annotation). The second hierarchy, with IT21 = 2 separates a cluster of 18 samples, almost all of which are ALL samples taken from the bone marrow of patients from the same hospital. These results demonstrate the ability of the algorithm to extract in parallel different meaningful independent partitions of the data. 7 Discussion The analysis presented by this work enables to implement a family of novel agglomerative clustering algorithms. All of these algorithms are motivated by one variational framework given by the multivariate IB method. Unlike most other clustering techniques, this is a principled model independent approach, which aims directly at the extraction of informative structures about given observed variables. It is thus very different from maximumlikelihood estimation of some mixture model and relies on fundamental information theoretic notions, similar to rate distortion theory and channel coding. In fact the multivariate IB can be considered as a multivariate coding result. The fundamental tradeoff between the compressed multi-information rGin and the preserved multi-information r G ou, provides a generalized coding limiting function, similar to the information curve in the original IB and to the rate distortion function in lossy compression. Despite the only local-optimality of the resulting solutions this information theoretic quantity - the fraction of the multiinformation that is extracted by the clusters - provides an objective figure of merit for the obtained clustering schemes. The suggested approach of this paper has several practical advantages over the 'deterministic annealing' algorithms suggested in [4], as it is simpler, fully deterministic and non-parametric. There is no need to identify cluster splits which is usually rather tricky. Though agglomeration procedures do not scale linearly with the sample size as top down methods do, there exist several heuristics to improve the complexity of these algorithms (e.g. [1]). While a typical initialization of an agglomerative procedure induces "hard" clustering solutions, all of the above analysis holds for "soft" clustering as well. Moreover, as already noted in [11], the obtained "hard" partitions can be used as a platform to find also "soft" solutions through a process of "reverse annealing". This raises the possibility for using an agglomerative procedure over "soft" clustering solutions, which we leave for future work. We could describe here only a few relatively simple examples. These examples show promising results on non trivial real life data. Moreover, other choices of Gin and Gout can yield additional novel algorithms with applications over a variety of data types. Acknowledgements This work was supported in part by the Israel Science Foundation (ISF), the Israeli Ministry of Science, and by the US-Israel Bi-national Science Foundation (BSF). N. Slonim was also supported by an Eshkol fellowship. N. Friedman was also supported by an Alon fellowship and the Harry & Abe Sherman Senior Lectureship in Computer Science. References [I] L. D. Baker and A. K. McCallum. Distributional clustering of words for text classification. In ACM SIGIR 98. [2] T. M. Cover and J. A. Thomas. Elements of Information Theory. 1991. [3] R. EI-Yaniv, S. Fine, and N. Tishby. Agnostic classification of Markovian sequences. In NIPS'97. [4] N. Friedman, O. Mosenzon, N. Sionim and N. Tishby Multivariate Infonnation Bottleneck UAI,2001. [5] T. Golub, D. Slonim, P. Tamayo, C.M. Huard, J.M. Caasenbeek, H. Coller, M. Loh, J. Downing, M. Caligiuri, C. Bloomfield, and E. Lander. Molecular classification of cancer: class discovery and class prediction by gene expression monitoring Science 286, 531- 537,1999. [6] K. Lang. Learning to filter netnews. In ICML'95. [7] J. Lin. Divergence Measures Based on the Shannon Entropy. IEEE Trans. Info. Theory, 37(1):145-151 , 1991. [8] J. Pearl. Probabilistic Reasoning in Intelligent Systems. 1988. [9] K. Rose. Detenninistic annealing for clustering, compression, classification, regression, and related optimization problems. Proc. IEEE, 86:2210--2239,1998. [10] N. Sionim, R. Somerville, N. Tishby, and O. Lahav. Objective spectral classification of galaxies using the infonnation bottleneck method. in "Monthly Notices of the Royal Astronomical Society", MNRAS, 323, 270, 2001. [II] N. Slonim and N. Tishby. Agglomerative Infonnation Bottleneck. In NIPS'99. [12] N. Sionim and N. Tishby. Document clustering using word clusters via the infonnation bottleneck method. InACM SIGIR 2000. [13] N. Slonim and N. Tishby. The power of word clusters for text classification. In ECIR, 2001. [14] N. Tishby, F. Pereira, and W. Bialek. The Infonnation Bottleneck method. In Proc. 37th Allerton Conference on Communication and Computation. 1999. [15] N. Tishby and N. Slonim. Data clustering by markovian relaxation and the infonnation bottleneck method. In NIPS'OO.
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Duality, Geometry, and Support Vector Regression Jinbo Bi and Kristin P. Bennett Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy, NY 12180 bij2@rpi.edu, bennek@rpi.edu Abstract We develop an intuitive geometric framework for support vector regression (SVR). By examining when ϵ-tubes exist, we show that SVR can be regarded as a classification problem in the dual space. Hard and soft ϵ-tubes are constructed by separating the convex or reduced convex hulls respectively of the training data with the response variable shifted up and down by ϵ. A novel SVR model is proposed based on choosing the max-margin plane between the two shifted datasets. Maximizing the margin corresponds to shrinking the effective ϵ-tube. In the proposed approach the effects of the choices of all parameters become clear geometrically. 1 Introduction Support Vector Machines (SVMs) [6] are a very robust methodology for inference with minimal parameter choices. Intuitive geometric formulations exist for the classification case addressing both the error metric and capacity control [1, 2]. For linearly separable classification, the primal SVM finds the separating plane with maximum hard margin between two sets. The equivalent dual SVM computes the closest points in the convex hulls of the data from each class. For the inseparable case, the primal SVM optimizes the soft margin of separation between the two classes. The corresponding dual SVM finds the closest points in the reduced convex hulls. In this paper, we derive analogous arguments for SVM regression (SVR). We provide a geometric explanation for SVR with the ϵ-insensitive loss function. From the primal perspective, a linear function with no residuals greater than ϵ corresponds to an ϵ-tube constructed about the data in the space of the data attributes and the response variable [6] (see e.g. Figure 1(a)). The primary contribution of this work is a novel geometric interpretation of SVR from the dual perspective along with a mathematically rigorous derivation of the geometric concepts. In Section 2, for a fixed ϵ > 0 we examine the question “When does a “perfect” or “hard” ϵ-tube exist?”. With duality analysis, the existence of a hard ϵ-tube depends on the separability of two sets. The two sets consist of the training data augmented with the response variable shifted up and down by ϵ. In the dual space, regression becomes the classification problem of distinguishing between these two sets. The geometric formulations developed for the classification case [1] become applicable to the regression case. We call the resulting formulation convex SVR (C-SVR) since it is based on convex hulls of the augmented training data. Much like in SVM classification, to compute a hard ϵ-tube, C-SVR computes the nearest points in the convex hulls of the augmented classes. The corresponding maximum margin (max-margin) planes define the effective ϵ-tube. The size of margin determines how much the effective ϵ-tube shrinks. Similarly, to compute a soft ϵ-tube, reduced-convex SVR (RC-SVR) finds the closest points in the reduced convex hulls of the two augmented sets. This paper introduces the geometrically intuitive RC-SVR formulation which is a variation of the classic ϵ-SVR [6] and ν-SVR models [5]. If parameters are properly tuned, the methods perform similarly although not necessarily identically. RCSVR eliminates the pesky parameter C used in ϵ-SVR and ν-SVR. The geometric role or interpretation of C is not known for these formulations. The geometric roles of the two parameters of RC-SVR, ν and ϵ, are very clear, facilitating model selection, especially for nonexperts. Like ν-SVR, RC-SVR shrinks the ϵ-tube and has a parameter ν controlling the robustness of the solution. The parameter ϵ acts as an upper bound on the size of the allowable ϵ-insensitive error function. In addition, RC-SVR can be solved by fast and scalable nearest-point algorithms such as those used in [3] for SVM classification. 2 When does a hard ϵ-tube exist? y x ε ε (a) x y D D + (b) x y (c) D D + x y D D + (d) Figure 1: The (a) primal hard ϵ0-tube, and dual cases: (b) dual strictly separable ϵ > ϵ0, (c) dual separable ϵ = ϵ0, and (d) dual inseparable ϵ < ϵ0. SVR constructs a regression model that minimizes some empirical risk measure regularized to control capacity. Let x be the n predictor variables and y the dependent response variable. In [6], Vapnik proposed using the ϵ-insensitive loss function Lϵ(x, y, f) = |y −f(x)|ϵ = max(0, |y −f(x)| −ϵ), in which an example is in error if its residual |y −f(x)| is greater than ϵ. Plotting the points in (x, y) space as in Figure 1(a), we see that for a “perfect” regression model the data fall in a hard ϵ-tube about the regression line. Let (Xi, yi) be an example where i = 1, 2, · · · , m, Xi is the ith predictor vector, and yi is its response. The training data are then (X, y) where Xi is a row of the matrix X ∈Rm×n and y ∈Rm is the response. A hard ϵ-tube for a fixed ϵ > 0 is defined as a plane y = w′x+b satisfying −ϵe ≤y −Xw −be ≤ϵe where e is an m-dimensional vector of ones. When does a hard ϵ-tube exist? Clearly, for ϵ large enough such a tube always exists for finite data. The smallest tube, the ϵ0-tube, can be found by optimizing: min w,b,ϵϵ s.t. −ϵe ≤y −Xw −be ≤ϵe (1) Note that the smallest tube is typically not the ϵ-SVR solution. Let D+ and D−be formed by augmenting the data with the response variable respectively increased and decreased by ϵ, i.e. D+ = {(Xi, yi + ϵ), i = 1, · · · , m} and D−= {(Xi, yi − ϵ), i = 1, · · · , m}. Consider the simple problem in Figure 1(a). For any fixed ϵ > 0, there are three possible cases: ϵ > ϵ0 in which strict hard ϵ-tubes exist, ϵ = ϵ0 in which only ϵ0-tubes exist, and ϵ < ϵ0 in which no hard ϵ-tubes exist. A strict hard ϵ-tube with no points on the edges of the tube only exists for ϵ > ϵ0. Figure 1(b-d) illustrates what happens in the dual space for each case. The convex hulls of D+ and D−are drawn along with the max-margin plane in (b) and the supporting plane in (c) for separating the convex hulls. Clearly, the existence of the tube is directly related to the separability of D+ and D−. If ϵ > ϵ0 then a strict tube exists and the convex hulls of D+ and D−are strictly separable1. There are infinitely many possible ϵ-tubes when ϵ > ϵ0. One can see that the max-margin plane separating D+ and D−corresponds to one such ϵ. In fact this plane forms an ˆϵ tube where ϵ > ˆϵ ≥ϵ0. If ϵ = ϵ0, then the convex hulls of D+ and D−are separable but not strictly separable. The plane that separates the two convex hulls forms the ϵ0 tube. In the last case, where ϵ < ϵ0, the two sets D+ and D−intersect. No ϵ-tubes or max-margin planes exist. It is easy to show by construction that if a hard ϵ-tube exists for a given ϵ > 0 then the convex hulls of D+ and D−will be separable. If a hard ϵ-tube exists, then there exists (w, b) such that (y + ϵe) −Xw −be ≥0, (y −ϵe) −Xw −be ≤0. (2) For any convex combination of D+, X′ (y+ϵe)′ u where e′u = 1, u ≥0 of points (Xi, yi + ϵ), i = 1, 2, · · · , m, we have (y + ϵe)′u −w′(X′u) −b ≥0. Similarly for D−, X′ (y−ϵe)′ v where e′v = 1, v ≥0 of points (Xi, yi −ϵ), i = 1, 2, · · · , m, we have (y−ϵe)′v−w′(X′v)−b ≤0. Then the plane y = w′x+b in the ϵ-tube separates the two convex hulls. Note the separating plane and the ϵ-tube plane are the same. If no separating plane exists, then there is no tube. Gale’s Theorem2 of the alternative can be used to precisely characterize the ϵ-tube. Theorem 2.1 (Conditions for existence of hard ϵ-tube) A hard ϵ-tube exists for a given ϵ > 0 if and only if the following system in (u, v) has no solution: X′u = X′v, e′u = e′v = 1, (y + ϵe)′u −(y −ϵe)′v < 0, u ≥0, v ≥0. (3) Proof A hard ϵ-tube exists if and only if System (2) has a solution. By Gale’s Theorem of the alternative [4], system (2) has a solution if and only if the following alternative system has no solution: X′u = X′v, e′u = e′v, (y+ϵe)′u−(y −ϵe)′v = −1, u ≥0, v ≥0. Rescaling by 1 σ where σ = e′u = e′v > 0 yields the result. 1We use the following definitions of separation of convex sets. Let D+ and D−be nonempty convex sets. A plane H = {x : w′x = α} is said to separate D+ and D−if w′x ≥α, ∀x ∈D+ and w′x ≤α, ∀x ∈D−. H is said to strictly separate D+ and D−if w′x ≥α + ∆for x ∈D+, and w′x ≤α −∆for each x ∈D−where ∆is a positive scalar. 2The system Ax ≤c has a (or has no) solution if and only if the alternative system A′y = 0, c′y = −1, y ≥0 has no (or has a) solution. Note that if ϵ ≥ϵ0 then (y + ϵe)′u −(y −ϵe)′v ≥0. for any (u, v) such that X′u = X′v, e′u = e′v = 1, u, v ≥0. So as a consequence of this theorem, if D+ and D−are separable, then a hard ϵ-tube exists. 3 Constructing the ϵ-tube For any ϵ > ϵ0 infinitely many possible ϵ-tubes exist. Which ϵ-tube should be used? The linear program (1) can be solved to find the smallest ϵ0-tube. But this corresponds to just doing empirical risk minimization and may result in poor generalization due to overfitting. We know capacity control or structural risk minimization is fundamental to the success of SVM classification and regression. We take our inspiration from SVM classification. In hard-margin SVM classification, the dual SVM formulation constructs the max-margin plane by finding the two nearest points in the convex hulls of the two classes. The max-margin plane is the plane bisecting these two points. We know that the existence of the tube is linked to the separability of the shifted sets, D+ and D−. The key insight is that the regression problem can be regarded as a classification problem between D+ and D−. The two sets D+ and D−defined as in Section 2 both contain the same number of data points. The only significant difference occurs along the y dimension as the response variable y is shifted up by ϵ in D+ and down by ϵ in D−. For ϵ > ϵ0, the max-margin separating plane corresponds to a hard ˆϵ-tube where ϵ > ˆϵ ≥ϵ0. The resulting tube is smaller than ϵ but not necessarily the smallest tube. Figure 1(b) shows the max-margin plane found for ϵ > ϵ0. Figure 1(a) shows that the corresponding linear regression function for this simple example turns out to be the ϵ0 tube. As in classification, we will have a hard and soft ϵ-tube case. The soft ϵ-tube with ϵ ≤ϵ0 is used to obtain good generalization when there are outliers. 3.1 The hard ϵ-tube case We now apply the dual convex hull method to constructing the max-margin plane for our augmented sets D+ and D−assuming they are strictly separable, i.e. ϵ > ϵ0. The problem is illustrated in detail in Figure 2. The closest points of D+ and D−can be found by solving the following dual C-SVR quadratic program: min u,v 1 2
X′ (y+ϵe)′ u − X′ (y−ϵe)′ v
2 s.t. e′u = 1, e′v = 1, u ≥0, v ≥0. (4) Let the closest points in the convex hulls of D+ and D−be c = X′ (y+ϵe)′ ˆu and d = X′ (y−ϵe)′ ˆv respectively. The max-margin separating plane bisects these two points. The normal (ˆw, ˆδ) of the plane is the difference between them, i.e., ˆw = X′ˆu −X′ˆv, ˆδ = (y + ϵe)′ˆu −(y −ϵe)′ˆv. The threshold, ˆb, is the distance from the origin to the point halfway between the two closest points along the normal: ˆb = ˆw′ X′ˆu+X′ˆv 2 +ˆδ y′ˆu+y′ˆv 2 . The separating plane has the equation ˆw′x+ˆδy−ˆb = 0. Rescaling this plane yields the regression function. Dual C-SVR (4) is in the dual space. The corresponding Primal C-SVR is:
Figure 2: The solution ˆϵ-tube found by C-SVR can have ˆϵ < ϵ. Squares are original data. Dots are in D+. Triangles are in D−. Support Vectors are circled. min w,δ,α,β 1 2 ∥w∥2 + 1 2δ2 −(α −β) s.t. Xw + δ(y + ϵe) −αe ≥0 Xw + δ(y −ϵe) −βe ≤0. (5) Dual C-SVR (4) can be derived by taking the Wolfe or Lagrangian dual [4] of primal C-SVR (5) and simplifying. We prove that the optimal plane from C-SVR bisects the ˆϵ tube. The supporting planes for class D+ and class D−determines the lower and upper edges of the ˆϵ-tube respectively. The support vectors from D+ and D−correspond to the points along the lower and upper edges of the ˆϵ-tube. See Figure 2. Theorem 3.1 (C-SVR constructs ˆϵ-tube) Let the max-margin plane obtained by C-SVR (4) be ˆw′x+ˆδy−ˆb = 0 where ˆw = X′ˆu−X′ˆv, ˆδ = (y+ϵe)′ˆu−(y−ϵe)′ˆv, and ˆb = ˆw′ X′ˆu+X′ˆv 2 + ˆδ y′ˆu+y′ˆv 2 . If ϵ > ϵ0, then the plane y = w′x + b corresponds to an ˆϵ-tube of training data (Xi, yi), i = 1, 2, · · · , m where w = −ˆw ˆδ , b = ˆb ˆδ and ˆϵ = ϵ −ˆα−ˆβ 2ˆδ < ϵ. Proof First, we show ˆδ > 0. By the Wolfe duality theorem [4], ˆα −ˆβ > 0, since the objective values of (5) and the negative objective value of (4) are equal at optimality. By complementarity, the closest points are right on the margin planes ˆw′x + ˆδy −ˆα = 0 and ˆw′x + ˆδy −ˆβ = 0 respectively, so ˆα = ˆw′X′ˆu + ˆδ(y + ϵe)′ˆu and ˆβ = ˆw′X′ˆv+ˆδ(y−ϵe)′ˆv. Hence ˆb = ˆα+ ˆβ 2 , and ˆw, ˆδ, ˆα, and ˆβ satisfy the constraints of problem (5), i.e., Xˆw+ˆδ(y+ϵe)−ˆαe ≥0, Xˆw+ˆδ(y−ϵe)−ˆβe ≤0. Then subtract the second inequality from the first inequality: 2ˆδϵ −ˆα + ˆβ ≥0, that is, ˆδ ≥ˆα−ˆβ 2ϵ > 0 because ϵ > ϵ0 ≥0. Rescale constraints by −ˆδ < 0, and reverse the signs. Let w = −ˆw ˆδ , then the inequalities become Xw −y ≤ϵe −ˆα ˆδ e, Xw −y ≥−ϵe − ˆβ ˆδ e. Let b = ˆb ˆδ, then ˆα ˆδ = b + ˆα−ˆβ 2ˆδ and ˆβ ˆδ = b −ˆα−ˆβ 2ˆδ . Substituting into the previous inequalities yields Xw−y ≤ ϵ −ˆα−ˆβ 2ˆδ e−be, Xw−y ≥− ϵ −ˆα−ˆβ 2ˆδ e−be. Denote ˆϵ = ϵ−ˆα−ˆβ 2ˆδ < ϵ. These inequalities become Xw+be−y ≤ˆϵe, Xw+be−y ≥−ˆϵe. Hence the plane y = w′x + b is in the middle of the ˆϵ < ϵ tube. 3.2 The soft ϵ-tube case For ϵ < ϵ0, a hard ϵ-tube does not exist. Making ϵ large to fit outliers may result in poor overall accuracy. In soft-margin classification, outliers were handled in the
2ε^ y x Figure 3: Soft ˆϵ-tube found by RC-SVR: left: dual, right: primal space. dual space by using reduced convex hulls. The same strategy works for soft ϵ-tubes, see Figure 3. Instead of taking the full convex hulls of D+ and D−, we reduce the convex hulls away from the difficult boundary cases. RC-SVR computes the closest points in the reduced convex hulls min u,v 1 2
X′ (y+ϵe)′ u − X′ (y−ϵe)′ v
2 s.t. e′u = 1, e′v = 1, 0 ≤u ≤De, 0 ≤v ≤De. (6) Parameter D determines the robustness of the solution by reducing the convex hull. D limits the influence of any single point. As in ν-SVM, we can parameterize D by ν. Let D = 1 νm where m is the number of points. Figure 3 illustrates the case for m = 6 points, ν = 2/6, and D = 1/2. In this example, every point in the reduced convex hull must depend on at least two data points since Pm i=1 ui = 1 and 0 ≤ui ≤1/2. In general, every point in the reduced convex hull can be written as the convex combination of at least ⌈1/D⌉= ⌈ν ∗m⌉. Since these points are exactly the support vectors and there are two reduced convex hulls, 2 ∗⌈νm⌉is a lower bound on the number of support vectors in RC-SVR. By choosing ν sufficiently large, the inseparable case with ϵ ≤ϵ0 is transformed into a separable case where once again our nearest-points-in-the-convex-hull-problem is well defined. As in classification, the dual reduced convex hull problem corresponds to computing a soft ϵ-tube in the primal space. Consider the following soft tube version of the primal C-SVR (7) which has its Wolfe Dual RC-SVR (6): min w,δ,α,β,ξ,η 1 2 ∥w∥2 + 1 2δ2 −(α −β) + C(e′ξ + e′η) s.t. Xw + δ(y + ϵe) −αe + ξ ≥0, ξ ≥0 Xw + δ(y −ϵe) −βe −η ≤0, η ≥0 (7) The results of Theorem 3.1 can be easily extended to soft ϵ-tubes. Theorem 3.2 (RC-SVR constructs soft ˆϵ-tube) Let the soft max-margin plane obtained by RC-SVR (6) be ˆw′x + ˆδy −ˆb = 0 where ˆw = X′ˆu −X′ˆv, ˆδ = (y+ϵe)′ˆu−(y −ϵe)′ˆv, and ˆb = X′ˆu+X′ˆv 2 ′ ˆw+ y′ˆu+y′ ˆv 2 ˆδ. If 0 < ϵ ≤ϵ0, then the plane y = w′x + b corresponds to a soft ˆϵ = ϵ −˜α−˜β 2ˆδ < ϵ-tube of training data (Xi, yi), i = 1, 2, · · · , m, i.e., a ˆϵ-tube of reduced convex hull of training data where w = −ˆw ˆδ , b = ˆb ˆδ and ˜α = ˆw′X′ˆu + ˆδ(y + ϵe)′ˆu, ˜β = ˆw′X′ˆv + ˆδ(y −ϵe)′ˆv. Notice that the ˜α and ˜β determine the planes parallel to the regression plane and through the closest points in each reduced convex hull of shifted data. In the inseparable case, these planes are parallel but not necessarily identical to the planes obtained by the primal RC-SVR (7). Nonlinear C-SVR and RC-SVR can be achieved by using the usual kernel trick. Let Φ by a nonlinear mapping of x such that k(Xi, Xj) = Φ(Xi) · Φ(Xj). The objective function of C-SVR (4) and RC-SVR (6) applied to the mapped data becomes 1 2 Pm i=1 Pm j=1 ((ui −vi)(uj −vj)(Φ(Xi) · Φ(Xj) + yiyj)) + 2ϵ Pm i=1 (yi(ui −vi)) = 1 2 Pm i=1 Pm j=1 ((ui −vi)(uj −vj)(k(Xi, Xj) + yiyj)) + 2ϵ Pm i=1 (yi(ui −vi)) (8) The final regression model after optimizing C-SVR or RC-SVR with kernels takes the form of f(x) = Pm i=1 (¯ui −¯vi) k(Xi, x) + ¯b, where ¯ui = ˆui ˆδ , ¯vi = ˆvi ˆδ , ˆδ = (ˆu − ˆv)′y+2ϵ, and the intercept term ¯b = (ˆu+ˆv)′K(ˆu−ˆv) 2ˆδ + (ˆu+ˆv)′y 2 where Kij = k(Xi, Xj). 4 Computational Results We illustrate the difference between RC-SVR and ϵ-SVR on a toy linear problem3. Figure 4 depicts the functions constructed by RC-SVR and ϵ-SVR for different values of ϵ. For large ϵ, ϵ-SVR produces undesirable results. RC-SVR constructs the same function for ϵ sufficiently large. Too small ϵ can result in poor generalization. −1 0 1 2 3 4 5 6 −0.5 0 0.5 1 1.5 2 2.5 ε = 0.75 ε = 0.45 ε = 0.25 ε = 0.15 (a) −1 0 1 2 3 4 5 6 −0.5 0 0.5 1 1.5 2 2.5 ε = 0.75, 0.45, 0.25 (b) Figure 4: Regression lines from (a) ϵ-SVR and (b) RC-SVR with distinct ϵ. In Table 1, we compare RC-SVR, ϵ-SVR and ν-SVR on the Boston Housing problem. Following the experimental design in [5] we used RBF kernel with 2σ2 = 3.9, C = 500·m for ϵ-SVR and ν-SVR, and ϵ = 3.0 for RC-SVR. RC-SVR, ϵ-SVR, and ν-SVR are computationally similar for good parameter choices. In ϵ-SVR, ϵ is fixed. In RC-SVR, ϵ is the maximum allowable tube width. Choosing ϵ is critical for ϵ-SVR but less so for RC-SVR. Both RC-SVR and ν-SVR can shrink or grow the tube according to desired robustness. But ν-SVR has no upper ϵ bound. 5 Conclusion and Discussion By examining when ϵ-tubes exist, we showed that in the dual space SVR can be regarded as a classification problem. Hard and soft ϵ-tubes are constructed by separating the convex or reduced convex hulls respectively of the training data with the response variable shifted up and down by ϵ. We proposed RC-SVR based on choosing the soft max-margin plane between the two shifted datasets. Like ν-SVM, RC-SVR shrinks the ϵ-tube. The max-margin determines how much the tube can shrink. Domain knowledge can be incorporated into the RC-SVR parameters ϵ 3The data consist of (x,y): (0 0), (1 0.1), (2 0.7), (2.5 0.9), (3 1.1) and (5 2). The CPLEX 6.6 optimization package was used. Table 1: Testing Results for Boston Housing, MSE= average of mean squared errors of 25 testing points over 100 trials, STD: standard deviation 2ν 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 RC-SVR MSE 37.3 11.2 10.7 9.6 8.9 10.6 11.5 12.5 STD 72.3 7.6 7.3 7.4 8.4 9.1 9.3 9.8 ϵ 0 1 2 3 4 5 6 7 ϵ-SVR MSE 11.2 10.8 9.5 10.3 11.6 13.6 15.6 17.2 STD 8.3 8.2 8.2 7.3 5.8 5.8 5.9 5.8 ν 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ν-SVR MSE 9.6 8.9 9.5 10.8 10.9 11.0 11.2 11.1 STD 5.8 7.9 8.3 8.2 8.3 8.4 8.5 8.4 and ν. The parameter C in ν-SVM and ϵ-SVR has been eliminated. Computationally, no one method is superior for good parameter choices. RC-SVR alone has a geometrically intuitive framework that allows users to easily grasp the model and its parameters. Also, RC-SVR can be solved by fast nearest point algorithms. Considering regression as a classification problem suggests other interesting SVR formulations. We can show ϵ-SVR is equivalent to finding closest points in a reduced convex hull problem for certain C, but the equivalent problem utilizes a different metric in the objective function than RC-SVR. Perhaps other variations would yield even better formulations. Acknowledgments Thanks to referees and Bernhard Sch¨olkopf for suggestions to improve this work. This work was supported by NSF IRI-9702306, NSF IIS-9979860. References [1] K. Bennett and E. Bredensteiner. Duality and Geometry in SVM Classifiers. In P. Langley, eds., Proc. of Seventeenth Intl. Conf. on Machine Learning, p 57–64, Morgan Kaufmann, San Francisco, 2000. [2] D. Crisp and C. Burges. A Geometric Interpretation of ν-SVM Classifiers. In S. Solla, T. Leen, and K. Muller, eds., Advances in Neural Info. Proc. Sys., Vol 12. p 244–251, MIT Press, Cambridge, MA, 1999. [3] S.S. Keerthi, S.K. Shevade, C. Bhattacharyya and K.R.K. Murthy, A Fast Iterative Nearest Point Algorithm for Support Vector Machine Classifier Design, IEEE Transactions on Neural Networks, Vol. 11, pp.124-136, 2000. [4] O. Mangasarian. Nonlinear Programming. SIAM, Philadelphia, 1994. [5] B. Sch¨olkopf, P. Bartlett, A. Smola and R. Williamson. Shrinking the Tube: A New Support Vector Regression Algorithm. In M. Kearns, S. Solla, and D. Cohn eds., Advances in Neural Info. Proc. Sys., Vol 12, MIT Press, Cambridge, MA, 1999. [6] V. Vapnik. The Nature of Statistical Learning Theory. Wiley, New York, 1995.
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BLIND SOURCE SEPARATION VIA MULTINODE SPARSE REPRESENTATION Michael Zibulevsky Department of Electrical Engineering Technion, Haifa 32000, Israel mzib@ee.technion.ac. if Yehoshua Y. Zeevi Department of Electrical Engineering Technion, Haifa 32000, Israel zeevi@ee.technion.ac. if Pavel Kisilev Department of Electrical Engineering Technion, Haifa 32000, Israel paufk@tx.technion.ac. if Barak Pearlmutter Department of Computer Science University of New Mexico Albuquerque, NM 87131 USA bap@cs. unm. edu Abstract We consider a problem of blind source separation from a set of instantaneous linear mixtures, where the mixing matrix is unknown. It was discovered recently, that exploiting the sparsity of sources in an appropriate representation according to some signal dictionary, dramatically improves the quality of separation. In this work we use the property of multi scale transforms, such as wavelet or wavelet packets, to decompose signals into sets of local features with various degrees of sparsity. We use this intrinsic property for selecting the best (most sparse) subsets of features for further separation. The performance of the algorithm is verified on noise-free and noisy data. Experiments with simulated signals, musical sounds and images demonstrate significant improvement of separation quality over previously reported results. 1 Introduction In the blind source separation problem an N-channel sensor signal x(~ ) is generated by M unknown scalar source signals s rn(~) , linearly mixed together by an unknown N x M mixing, or crosstalk, matrix A, and possibly corrupted by additive noise n(~): x(~) = As(~) + n(~ ). (1) The independent variable ~ is either time or spatial coordinates in the case of images. We wish to estimate the mixing matrix A and the M-dimensional source signal s(~). The assumption of statistical independence of the source components Srn(~) , m = 1, ... , M leads to the Independent Component Analysis (lCA) [1], [2]. A stronger assumption is the °Supported in part by the Ollendorff Minerva Center, by the Israeli Ministry of Science, by NSF CAREER award 97-02-311 and by the National Foundation for Functional Brain Imaging sparsity of decomposition coefficients, when the sources are properly represented [3]. In particular, let each 8 m (~ ) have a sparse representation obtained by means of its decomposition coefficients Cmk according to a signal dictionary offunctions Y k (~ ): 8m (~ ) = L Cmk Yk(~ )' (2) k The functions Yk (~ ) are called atoms or elements of the dictionary. These elements do not have to be linearly independent, and instead may form an overcomplete dictionary, e.g. wavelet-related dictionaries (wavelet packets, stationary wavelets, etc., see for example [9]). Sparsity means that only a small number of coefficients Cmk differ significantly from zero. Then, unmixing of the sources is performed in the transform domain, i.e. in the domain of these coefficients Cmk. The property of sparsity often yields much better source separation than standard ICA, and can work well even with more sources than mixtures. In many cases there are distinct groups of coefficients, wherein sources have different sparsity properties. The key idea in this study is to select only a subset of features (coefficients) which is best suited for separation, with respect to the following criteria: (1) sparsity of coefficients (2) separability of sources' features. After this subset is formed, one uses it in the separation process, which can be accomplished by standard ICA algorithms or by clustering. The performance of our approach is verified on noise-free and noisy data. Our experiments with ID signals and images demonstrate that the proposed method further improves separation quality, as compared with result obtained by using sparsity of all decomposition coefficients. 2 Two approaches to sparse source separation: InfoMax and Clustering Sparse sources can be separated by each one of several techniques, e.g. the Bell-Sejnowski Information Maximization (BS InfoMax) approach [1], or by approaches based on geometric considerations (see for example [8]). In the former case, the algorithm estimates the unmixing matrix W = A - I, while in the later case the output is the estimated mixing matrix. In both cases, these matrices can be estimated only up to a column permutation and a scaling factor [4]. InfoMax. Under the assumption of a noiseless system and a square mixing matrix in (1), the BS InfoMax is equivalent to the maximum likelihood (ML) formulation of the problem [4], which is used in this section. For the sake of simplicity of the presentation, let us consider the case where the dictionary of functions used in a source decomposition (2) is an orthonormal basis. (In this case, the corresponding coefficients Cmk =< 8m, 'Pk >, where < ',' > denotes the inner product). From (1) and (2) the decomposition coefficients of the noiseless mixtures, according to the same signal dictionary of functions Y k (~ ) ' are: Ak= ACk, (3) where M -dimensional vector Ck forms the k-th column of the matrix C = { Cmk}. Let Y be thefeatures, or (new) data, matrix of dimension M x K , where K is the number of features. Its rows are either the samples of sensor signals (mixtures), or their decomposition coefficients. In the later case, the coefficients Ak'S form the columns ofY. (In the following discussion we assume this setting for Y , if not stated other). We are interested in the maximum likelihood estimate of A given the data Y. Let the corresponding coefficients Cmk be independent random variables with a probability density function (pdf) of an exponential type (4) where the scalar function v(·) is a smooth approximation of an absolute value function. Such kind of distribution is widely used for modeling sparsity [5]. In view of the independence of Cmk, and (4), the prior pdf of C is p(C) ex II exp{ - V(Cmk)}. (5) m,k Taking into account that Y = AC, the parametric model for the pdf of Y with respect to parameters A is (6) Let W = A -I be the unmixing matrix, to be estimated. Then, substituting C = WY, combining (6) with (5) and taking the logarithm we arrive at the log-likelihood function: M K Lw(Y) = Klog ldetWI- L LV((WY)mk). (7) m=l k = l Maximization of Lw(Y) with respect to W is equivalent to the BS InfoMax, and can be solved efficiently by the Natural Gradient algorithm [6]. We used this algorithm as implemented in the ICAlEEG Matlab toolbox [7]. Clustering. In the case of geometry based methods, separation of sparse sources can be achieved by clustering along orientations of data concentration in the N-dimensional space wherein each column Yk of the matrix Y represents a data point (N is the number of mixtures). Let us consider a two-dimensional noiseless case, wherein two source signals, Sl(t) and S2(t), are mixed by a 2x2 matrix A, arriving at two mixtures Xl(t) and X2(t). (Here, the data matrix is constructed from these mixtures Xl (t) and xd t)). Typically, a scatter plot of two sparse mixtures X1(t) versus X2(t), looks like the rightmost plot in Figure 2. If only one source, say Sl (t), was present, the sensor signals would be Xl (t) = all Sl (t) and X2(t) = a21s1 (t) and the data points at the scatter diagram of Xl (t) versus X2(t) would belong to the straight line placed along the vector [ana21]T. The same thing happens, when two sparse sources are present. In this sparse case, at each particular index where a sample of the first source is large, there is a high probability, that the corresponding sample of the second source is small, and the point at the scatter diagram still lies close to the mentioned straight line. The same arguments are valid for the second source. As a result, data points are concentrated around two dominant orientations, which are directly related to the columns of A. Source signals are rarely sparse in their original domain. In contrast, their decomposition coefficients (2) usually show high sparsity. Therefore, we construct the data matrix Y from the decomposition coefficients of mixtures (3), rather than from the mixtures themselves. In order to determine orientations of scattered data, we project the data points onto the surface of a unit sphere by normalizing corresponding vectors, and then apply a standard clustering algorithm. This clustering approach works efficiently even if the number of sources is greater than the number of sensors. Our clustering procedure can be summarized as follows: 1. Form the feature matrix Y , by putting samples of the sensor signals or (subset of) their decomposition coefficients into the corresponding rows ofthe matrix; 2. Normalize feature vectors (columns ofY): Yk = Yk/II Yk I12' in order to project data points onto the surface of a unit sphere, where 11 · 11 2 denotes the l2 norm. Before nonnalization, it is reasonable to remove data points with a very small norm, since these very likely to be crosstalk-corrupted by small coefficients from others' sources. 3. Move data points to a half-sphere, e.g. by forcing the sign of the first coordinate yk to be positive: IF yk < 0 THEN Yk = - Yk. Without this operation each set oflineariy (i.e., along a line) clustered data points would yield two clusters on opposite sides of the sphere. · : -s tOO 200 300 ~oo »:I eoo 700 ~ 900 tOC>:l ,: 5 100 200 300 ~OO 500 600 700 1\00 900 1000 , : -s tOO 200 300 ~OO 500 eoo 700 1\00 900 tOC>:l , : -5 100 200 300 ~OO 500 600 700 1\00 900 1():Xl Figure 1: Random block signals (two upper) and their mixtures (two lower) 4. Estimate cluster centers by using a clustering algorithm. The coordinates of these centers will form the columns of the estimated mixing matrix A. We used Fuzzy C-means (FCM) clustering algorithm as implemented in Matlab Fuzzy Logic Toolbox. Sources recovery. The estimated unmixing matrix A-I is obtained by either the BS InfoMax or the above clustering procedure, applied to either complete data set, or to some subsets of data (to be explained in the next section). Then, the sources are recovered in their original domain by s(t) = A - lX(t). We should stress here that if the clustering approach is used, the estimation of sources is not restricted to the case of square mixing matrices, although the sources recovery is more complicated in the rectangular cases (this topic is out of scope of this paper). 3 Multinode based source separation Motivating example: sparsity of random blocks in the Haar basis. To provide intuitive insight into the practical implications of our main idea, we first use ID block functions, that are piecewise constant, with random amplitude and duration of each constant piece (Figure 1). It is known, that the Haar wavelet basis provides compact representation of such functions. Let us take a close look at the Haar wavelet coefficients at different resolution levels j =O, 1, ... ,1. Wavelet basis functions at the finest resolution level j =J are obtained by translation of the Haar mother wavelet: <p(t) = {I, ift E [0, 1); - I , ift E [1, 2); 0 otherwise}. Taking the scalar product ofa function s(t) with the wavelet <PJ(t - T), we produce a finite differentiation of the function s(t) at the point t = T. This means that the number of non-zero coefficients at the finest resolution for a block function will correspond roughly to the number of jumps ofthis function. Proceeding to the next, coarser resolution level, we have <P J - l (t) = {I , ift E [0, 2); - 1, if t E [2,4); ° otherwise}. At this level, the number of non-zero coefficients still corresponds to the number of jumps, but the total number of coefficients at this level is halved, and so is the sparsity. If we further proceed to coarser resolutions, we will encounter levels where the support of a wavelet <Pj(t) is comparable to the typical distance between jumps in the function s(t). In this case, most of the coefficients are expected to be nonzero, and, therefore, sparsity will fade away. To demonstrate how this influences accuracy of a blind source separation, we randomly generated two block-signal sources (Figure 1, two upper plots.), and mixed them by the crosstalk matrix A with colwnns [0.83 -0.55] and [0.62 0.78]. Resulting sensor signals, or mixtures, X l (t) and X2 (t) are shown in the two lower plots of Figure l. The scatter plot of X l (t) versus X2( t) does not exhibit any visible distinct orientations (Figure 2, left). Similarly, in the scatter plot of the wavelet coefficients at the lowest resolution distinct orientations are hardly detectable (Figure 2, middle). In contrast, the scatter plot of the wavelet coefficients at the highest resolution (Figure 2, right) depicts two distinct orientations, which correspond to the columns of the mixing matrix. InfoMax FCM Raw signals : :"~~:·;K·;\. " , " "1 .::;; Of :~~.: • •••• l.93 1.78 All wavelet coefficients 0.183 0.058 High resolution WT coefficients 1·. ~ ."'~'>!~ . : >/_~: .~ -I ; /" , .,/', " "" 0.005 0.002 Figure 2: Separation of block signals: scatter plots of sensor signals (left), and of their wavelet coefficients (middle and right). Lower colwnns present the normalized meansquared separation error (%) corresponding to the Bell-Sejnowski InfoMax, and to the Fuzzy C-Means clustering, respectively. Since a crosstalk matrix A is estimated only up to a column permutation and a scaling factor, in order to measure the separation accuracy, we normalize the original sources sm(t) and their corresponding estimated sources sm(t). The averaged (over sources) normalized squared error (NSE) is then computed as: NSE = it 2:~= 1 (ilsm - sm ll§/llsmll§)· Resulting separation errors for block sources are presented in the lower part of Figure 2. The largest error (l.93%) is obtained on the raw data, and the smallest «0.005%) - on the wavelet coefficients at the highest resolution, which have the best sparsity. Using all wavelet coefficients yields intermediate sparsity and performance. Multinode representation. Our choice of a particular wavelet basis and of the sparsest subset of coefficients was obvious in the above example: it was based on knowledge of the structure of piecewise constant signals. For sources having oscillatory components (like sounds or images with textures), other systems of basis functions, such as wavelet packets and trigonometric function libraries [9], might be more appropriate. The wavelet packet library consists of the triple-indexed family of functions: i.f!j ,i,q(t) = 2j / 2i.f!q(2j t - i), j , i E Z , q E N,where j , i are the scale and shift parameters, respectively, and q is the frequency parameter. [Roughly speaking, q is proportional to the nwnber of oscillations of a mother wavelet i.f!q(t)]. These functions form a binary tree whose nodes are indexed by the depth of the level j and the node number q = 0, 1, 2, 3, ... , 2j - l at the specified level j. This same indexing is used for corresponding subsets of wavelet packet coefficients (as well as in scatter diagrams in the section on experimental results). Adaptive selection of sparse subsets. When signals have a complex nature, it is difficult to decide in advance which nodes contain the sparsest sets of coefficients. That is why we use the following simple adaptive approach. First, for every node of the tree, we apply our clustering algorithm, and compute a measure of clusters' distortion. In our experiments we used a standard global distortion, the mean squared distance of data points to the centers of their own (closest) clusters (here again, the weights of the data points can be incorporated): d=2:f=l min II U m - Yk II ,where K is the nwnber of data points, U m is the m-th centroid m coordinates, Yk is the k-th data point coordinates, and 11 . 11 is the sum-of-squares distance. Second, we choose a few best nodes with the minimal distortion, combine their coefficients into one data set, and apply a separation algorithm (clustering or Infomax) to these data. 4 Experimental results The proposed blind separation method based on the wavelet-packet representation, was evaluated by using several types of signals. We have already discussed the relatively simple example of a random block signal. The second type of signal is a frequency modulated (FM) sinusoidal signal. The carrier frequency is modulated by either a sinusoidal function (FM signal) or by random blocks (BFM signal). The third type is a musical recording of flute sounds. Finally, we apply our algorithm to images. An example of such images is presented in the left part of Figure 3. 111 '10 ' 11 '~ • • t: , ' ' 12 ' 13 'JJ SI •• .. '~' 00 0° • . '. , 8 . , foo 0 , . , • 8 ' 22 11 S. Ss ~ :. , :Y6~ , ",t, " ' ' 26 \; '21 'lI '11 t , "*, ' , :, '8 ' " Figure 3: Left: two source images (upper pair), their mixtures (middle pair) and estimated images (lower pair). Right: scatter plots ofthe wavelet packet (WP) coefficients of mixtures of images; subsets are indexed on the WP tree. In order to compare accuracy of our adaptive best nodes method with that attainable by standard methods, we form the following feature sets: (1) raw data, (2) Short Time Fourier Transform (STFT) coefficients (in the case of ID signals), (3) Wavelet Transform coefficients (4) Wavelet packet coefficients at the best nodes found by our method, while using various wavelet families with different smoothness (haar, db-4, db-S). In the case of image separation, we used the Discrete Cosine Transform (DCT) instead of the STFT, and the sym4 and symS mother wavelet instead of db-4 and db-S, when using wavelet transform and wavelet packets. The right part of Figure 3 presents an example of scatter plots of the wavelet packet coefficients obtained at various nodes of the wavelet packet tree. The upper left scatter plot, marked with 'C', corresponds to the complete set of coefficients at all nodes. The rest are the scatter plots of sets of coefficients indexed on a wavelet packet tree. Generally speaking, the more distinct the two dominant orientations appear on these plots, the more precise is the estimation of the mixing matrix, and, therefore, the better is the quality of separation. Note, that only two nodes, C22 and C23, show clear orientations. These nodes will most likely be selected by the algorithm for further estimation process. Signals raw STFT WT WT WP WP data db8 haar db8 haar Blocks 10.16 2.669 0.174 0.037 0.073 0.002 BFM sine 24.51 0.667 0.665 2.34 0.2 0.442 FM sine 25.57 0.32 1.032 6.105 0.176 0.284 Flutes 1.48 0.287 0.355 0.852 0.154 0.648 raw OCT WT WT WP WP Images data sym8 haar sym8 haar 4.88 3.651 l.l64 l.l14 0.365 0.687 Table 1: Experimental results: normalized mean-squared separation error (%) for noisefree signals and images, applying the FCM separation to raw data and decomposition coefficients in various domains. In the case of wavelet packets (WP) the best nodes selected by our algorithm were used. Table 1 summarizes results of experiments in which we applied our approach of the best features selection along with the FCM separation to each noise-free feature set. In these experiments, we compared the quality of separation of deterministic signals by calculating N SE's (i.e., residual crosstalk errors). In the case of random block and BFM signals, we performed 100 Monte-Carlo simulations and calculated the normalized mean-squared errors (N M SE) for the above feature sets. From Table 1 it is clear that using our adaptive best nodes method outperforms all other feature sets (including complete set of wavelet coefficients), for each type of signals. Similar improvement was achieved by using our method along with the BS InfoMax separation, which provided even better results for images. In the case of the random block signals, using the Haar wavelet function for the wavelet packet representation yields a better separation than using some smooth wavelet, e.g. db-S. The reason is that these block signals, that are not natural signals, have a sparser representation in the case of the Haar wavelets. In contrast, as expected, natural signals such as the Flute's signals are better represented by smooth wavelets, that in turn provide a better separation. This is another advantage of using sets of features at multiple nodes along with various families of 'mother' functions: one can choose best nodes from several decomposition trees simultaneously. In order to verify the performance of our method in presence of noise, we added various types of noise (white gaussian and salt&pepper) to three mixtures of three images at various signal-to-noise energy ratios (SNR). Table 2 summarizes these experiments in which we applied our approach along with the BS InfoMax separation. It turns out that the ideas used in wavelet based signal denoising (see for example [10] and references therein), are applied to signal separation from noisy mixtures. In particular, in case of white gaussian noise, the noise energy is uniformly distributed over all wavelet coefficients at various scales. Therefore, at sufficiently high SNR's, the large coefficients of the signals are only slightly distorted by the noise coefficients, and the estimation of the unmixing matrix is almost not affected by the presence of noise. (In contrast, the BS InfoMax applied to three noisy mixtures themselves, failed completely, arriving at N S E of 19% even in the case of SNR=12dB). We should stress here that, although our adaptive best nodes method performs reasonably well in the presence of noise, it is not supposed to further denoise the reconstructed images (this can be achieved by some denoising method, after source signals are separated). More experimental results, as well as parameters of simulations, can be found in [11]. SNR [dB] Mixtures w. white gaussian noise Mixtures w. salt&pepper noise Table 2: Perfonnance of the algorithm in presence of various sources of noise in mixtures of images: nonnalized mean-squared separation error (%), applying our adaptive approach along with the BS InfoMax separation. 5 Conclusions Experiments with both one- and two-dimensional simulated and natural signals demonstrate that multinode sparse representations improve the efficiency of blind source separation. The proposed method improves the separation quality by utilizing the structure of signals, wherein several subsets of the wavelet packet coefficients have significantly better sparsity and separability than others. In this case, scatter plots of these coefficients show distinct orientations each of which specifies a column of the mixing matrix. We choose the 'good subsets' according to the global distortion adopted as a measure of cluster quality. Finally, we combine together coefficients from the best chosen subsets and restore the mixing matrix using only this new subset of coefficients by the Infomax algorithm or clustering. This yields significantly better results than those obtained by applying standard Infomax and clustering approaches directly to the raw data. The advantage of our method is in particular noticeable in the case of noisy mixtures. References [1] A. 1. Bell and T. 1. Sejnowski, "An information-maximization approach to blind separation and blind deconvolution," Neural Computation, vol. 7, no. 6, pp. 1129- 1159, 1995. [2] A. Hyvarinen, "Survey on independent component analysis," Neural Computing Surveys, no. 2, pp. 94- 128, 1999. [3] M. Zibulevsky and B. A. Pearlmutter, "Blind separation of sources with sparse representations in a given signal dictionary," Neural Computation, vol. l3, no. 4, pp. 863882,2001. [4] 1.-F. Cardoso. "Infomax and maximum likelihood for blind separation," IEEE Signal Processing Letters 4 112-114, 1997. [5] M. S. Lewicki and T. 1. Sejnowski, "Learning overcomplete representations," Neural Computation, 12(2): 337-365, 2000. [6] S. Amari, A. Cichocki, and H. H. Yang, "A new learning algorithm for blind signal separation," In Advances in Neural Information Processing Systems 8. MIT Press. 1996. [7] S. Makeig, ICAlEEG toolbox. Computational Neurobiology Laboratory, the Salk Institute. http://www.cnl.salk.edurtewonlica _ cnl.html, 1999. [8] A. Prieto, C. G. Puntonet, and B. Prieto, "A neural algorithm for blind separation of sources based on geometric prperties.," Signal Processing, vol. 64, no. 3, pp. 315- 331, 1998. [9] S. Mallat, A Wavelet Tour of Signal Processing. Academic Press, 1998. [10] D. L. Donoho, "De-Noising by Soft Thresholding," IEEE Trans. Inf. Theory, vol. 41, 3, 1995, pp.613-627. [11] P. Kisilev, M. Zibulevsky, Y. Y. Zeevi, and B. A. Pearlmutter, Multiresolution frameworkfor sparse blind source separation, CCIT Report no.317, June 2000
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Estimating the Reliability of leA Projections F. Meinecke l ,2, A. Ziehel , M. Kawanabel and K.-R. Miillerl ,2* 1 Fraunhofer FIRST.IDA, Kekuh~str. 7, 12489 Berlin, Germany 2University of Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany {meinecke,ziehe,nabe,klaus}©first.fhg.de Abstract When applying unsupervised learning techniques like ICA or temporal decorrelation, a key question is whether the discovered projections are reliable. In other words: can we give error bars or can we assess the quality of our separation? We use resampling methods to tackle these questions and show experimentally that our proposed variance estimations are strongly correlated to the separation error. We demonstrate that this reliability estimation can be used to choose the appropriate ICA-model, to enhance significantly the separation performance, and, most important, to mark the components that have a actual physical meaning. Application to 49-channel-data from an magneto encephalography (MEG) experiment underlines the usefulness of our approach. 1 Introduction Blind source separation (BSS) techniques have found wide-spread use in various application domains, e.g. acoustics, telecommunication or biomedical signal processing. (see e.g. [9, 5, 6, 1, 2, 4, 14, 8]). BSS is a statistical technique to reveal unknown source signals when only mixtures of them can be observed. In the following we will only consider linear mixtures; the goal is then to estimate those projection directions, that recover the source signals. Many different BSS algorithms have been proposed, but to our knowledge, so far, no principled attempts have been made to assess the reliability of BSS algorithms, such that error bars are given along with the resulting projection estimates. This lack of error bars or means for selecting between competing models is of course a basic dilemma for most unsupervised learning algorithms. The sources of potential unreliability of unsupervised algorithms are ubiquous, i.e. noise, non-stationarities, small sample size or inadequate modeling (e.g. sources are simply dependent instead of independent). Unsupervised projection techniques like PCA or BSS will always give an answer that is found within their model class, e.g. PCA will supply an orthogonal basis even if the correct modeling might be non-orthogonal. But how can we assess such a miss-specification or a large statistical error? Our approach to this problem is inspired by the large body of statistics literature on • To whom correspondence should be addressed. resampling methods (see [12] or [7] for references), where algorithms for assessing the stability of the solution have been analyzed e.g. for peA [3]. We propose reliability estimates based on bootstrap resampling. This will enable us to select a good BSS model, in order to improve the separation performance and to find potentially meaningful projection directions. In the following we will give an algorithmic description of the resampling methods, accompanied by some theoretical remarks (section 2) and show excellent experimental results (sections 3 and 4). We conclude with a brief discussion. 2 Resampling Techniques for BSS 2.1 The leA Model In blind source separation we assume that at time instant t each component Xi(t) of the observed n-dimensional data vector, x(t) is a linear superposition of m ::::: n statistically independent signals: m Xi(t) = LAijSj(t) j=l (e.g. [8]). The source signals Sj(t) are unknown, as are the coefficients Aij of the mixing matrix A. The goal is therefore to estimate both unknowns from a sample of the x(t), i.e. y(t) = s(t) = Wx(t), where W is called the separating matrix. Since both A and s(t) are unknown, it is impossible to recover the scaling or the order of the columns of the mixing matrix A. All that one can get are the projection directions. The mixing/ demixing process can be described as a change of coordinates. From this point of view the data vector stays the same, but is expressed in different coordinate systems (passive transformation). Let {ed be the canonical basis of the true sources s = 'E eiSi. Analogous, let {fj} be the basis of the estimated leA channels: y = 'E fjYj. Using this, we can define a component-wise separation error Ei as the angle difference between the true direction of the source and the direction of the respective leA channel: Ei = arccos ("e~i: ~ifill) . To calculate this angle difference, remember that component-wise we have Yj 'E WjkAkisi. With Y = s, this leads to: fj = 'E ei(WA)ij1, i.e. fj is the j-th column of (WA)- l. In the following, we will illustrate our approach for two different source separation algorithms (JADE, TDSEP). JADE [4] using higher order statistics is based on the joint diagonalization of matrices obtained from 'parallel slices' of the fourth order cumulant tensor. TDSEP [14] relies on second order statistics only, enforcing temporal decorrelation between channels. 2.2 About Resampling The objective of resampling techniques is to produce surrogate data sets that eventually allow to approximate the 'separation error' by a repeated estimation of the parameters of interest. The underlying mixing should of course be independent of the generation process of the surrogate data and therefore remain invariant under resampling. Bootstrap Resampling The most popular res amp ling methods are the Jackknife and the Bootstrap (see e.g. [12, 7]) The Jackknife produces surrogate data sets by just deleting one datum each time from the original data. There are generalizations of this approach like k-fold cross-validation which delete more than one datum at a time. A more general approach is the Bootstrap. Consider a block of, say, N data points. For obtaining one bootstrap sample, we draw randomly N elements from the original data, i.e. some data points might occur several times, others don't occur at all in the bootstrap sample. This defines a series {at} with each at telling how often the data point x(t) has been drawn. Then, the separating matrix is computed on the full block and repeatedly on each of the N -element bootstrap samples. The variance is computed as the squared average difference between the estimate on the full block and the respective bootstrap unmixings. (These resampling methods have some desirable properties, which make them very attractive; for example, it can be shown that for iid data the bootstrap estimators of the distributions of many commonly used statistics are consistent.) It is straight forward to apply this procedure to BSS algorithms that do not use time structure; however, only a small modification is needed to take time structure into account. For example, the time lagged correlation matrices needed for TDSEP, can be obtained from {ad by 1 N Cij(T) = N 2: at 'Xi(t)Xj(t+T) t= l with L at = N and at E {O, 1, 2, ... }. Other resampling methods Besides the Bootstrap, there are other res amp ling methods like the Jackknife or cross-validation which can be understood as special cases of Bootstrap. We have tried k-fold cross-validation, which yielded very similar results to the ones reported here. 2.3 The Resampling Algorithm After performing BSS, the estimated ICA-projections are used to generate surrogate data by resampling. On the whitenedl surrogate data, the source separation algorithm is used again to estimate a rotation that separates this surrogate data. In order to compare different rotation matrices, we use the fact that the matrix representation of the rotation group SO(N) can be parameterized by with (Mab)ij r5~r5t r5~r5b , where the matrices Mij are generators of the group and the aij are the rotation parameters (angles) of the rotation matrix R. Using this parameterization we can easily compare different N-dimensional rotations by comparing the rotation parameters aij. Since the sources are already separated, the estimated rotation matrices will be in the vicinity of the identity matrix.2 . IThe whitening transformation is defined as x' = Vx with V = E[xxTtl/2. 21t is important to perform the resampling when the sources are already separated, so that the aij are distributed around zero, because SO(N) is a non-Abelian group; that means that in general R(a)R«(3) of- R«(3)R(a). Var(aij) measures the instability of the separation with respect to a rotation in the (i, j)-plane. Since the reliability of a projection is bounded by the maximum angle variance of all rotations that affect this direction, we define the uncertainty of the i-th ICA-Projection as Ui := maxj Var(aij). Let us summarize the resampling algorithm: 1. Estimate the separating matrix W with some ICA algorithm. Calculate the ICA-Projections y = Wx 2. Produce k surrogate data sets from y and whiten these data sets 3. For each surrogate data set: do BSS, producing a set of rotation matrices 4. Calculate variances of rotation parameters (angles) aij 5. For each ICA component calculate the uncertainty Ui = maxVar(aij). J 2.4 Asymptotic Considerations for Resampling Properties of res amp ling methods are typically studied in the limit when the number of bootstrap samples B -+ 00 and the length of signal T -+ 00 [12]. In our case, as B -+ 00, the bootstrap variance estimator Ut(B) computed from the aiJ's converge to Ut(oo) := maxj Varp[aij] where aij denotes the res amp led deviation and F denotes the distribution generating it. Furthermore, if F -+ F, Ut (00) converges to the true variance Ui = maxj VarF[aij ] as T -+ 00. This is the case, for example, if the original signal is i.i.d. in time. When the data has time structure, F does not necessarily converge to the generating distribution F of the original signal anymore. Although we cannot neglect this difference completely, it is small enough to use our scheme for the purposes considered in this paper, e.g. in TDSEP, where the aij depend on the variation of the time-lagged covariances Cij(T) of the signals, we can show that their estimators Ctj (T) are unbiased: Furthermore, we can bound the difference t:.ijkl(T,V) = COVF [Cij(T),Ckl(V)] COV p [Ctj ( T), Ckl (v)] between the covariance of the real matrices and their bootstrap estimators as if :3a < 1, M ;::: 1, Vi: ICii (T) I :S M aJLICii(O) I. In our experiments, however, the bias is usually found to be much smaller than this upper bound. 3 Experiments 3.1 Comparing the separation error with the uncertainty estimate To show the practical applicability of the resampling idea to ICA, the separation error Ei was compared with the uncertainty Ui. The separation was performed on different artificial 2D mixtures of speech and music signals and different iid data sets of the same variance. To achieve different separation qualities, white gaussian noise of different intensity has been added to the mixtures. Uj = 0.015 U. = 0.177 ' - - -' j o L---~~~~~~~~--~ o 0.2 0.4 0.6 0.8 separation error Ej 0.7,-----------------------_ 0.6 ur ~ 0.5 § 0.4 ~ ~0.3 c . ~ 0.2 0.1 o L-----~----~----~--~ 0.05 0.15 0.25 0.35 0.45 Figure 1: (a) The probability distribution for the separation error for a small uncertainty is close to zero, for higher uncertainty it spreads over a larger range. (b) The expected error increases with the uncertainty. Figure 1 relates the uncertainty to the separation error for JADE (TDSEP results look qualitatively the same) . In Fig.1 (left) we see the separation error distribution which has a strong peak for small values of our uncertainty measure, whereas for large uncertainties it tends to become flat, i.e. - as also seen from Fig.1 (right) the uncertainty reflects very well the true separation error. 3.2 Selecting the appropriate BSS algorithm As our variance estimation gives a high correlation to the (true) separation error, the next logical step is to use it as a model selection criterion for: (a) selecting some hyperparameter of the BSS algorithm, e.g. choosing the lag values for TDSEP or (b) choosing between a set of different algorithms that rely on different assumptions about the data, i.e. higher order statistics (e.g. JADE, INFO MAX, FastICA, ... ) or second order statistics (e.g. TDSEP). It could, in principle, be much better to extract the first component with one and the next with another assumption/ algorithm. To illustrate the usefulness of our reliability measure, we study a five-channel mixture of two channels of pure white gaussian noise, two audio signals and one channel of uniformly distributed noise. The reliability analysis for higher order statistics (JADE) temporal decorrelation (TDSEP) 0.3 0.3 0.25 0.25 TDSEP 3 9.17.10-5 ~- 0.2 ~- 0.2 TDSEP 4 E E 1.29.10-5 :rg 0.15 :rg 0.15 ,---,---g g ,---::J 0.1 ::J 0.1 0.05 0.05 3 3 ICA Channel i ICA Channel i Figure 2: Uncertainty of leA projections of an artificial mixture using JADE and TDSEP. Resampling displays the strengths and weaknesses of the different models JADE gives the advice to rely only on channels 3,4,5 (d. Fig.2 left). In fact, these are the channels that contain the audio signals and the uniformly distributed noise. The same analysis applied to the TDSEP-projections (time lag = 0, ... ,20) shows, that TDSEP can give reliable estimates only for the two audio sources (which is to be expected; d. Fig.2 right). According to our measure, the estimation for the audio sources is more reliable in the TDSEP-case. Calculation of the separation error verifies this: TDSEP separates better by about 3 orders of magnitude (JADE: E3 = 1.5 . 10- 1 , E4 = 1.4 . 10- 1 , TDSEP: E3 = 1.2 . 10- 4 , E4 = 8.7· 10- 5). Finally, in our example, estimating the audio sources with TDSEP and after this applying JADE to the orthogonal subspace, gives the optimal solution since it combines the small separation errors E3, E4 for TDSEP with the ability of JADE to separate the uniformly distributed noise. 3.3 Blockwise uncertainty estimates For a longer time series it is not only important to know which ICA channels are reliable, but also to know whether different parts of a given time series are more (or less) reliable to separate than others. To demonstrate these effects, we mixed two audio sources (8kHz, lOs - 80000 data points), where the mixtures are partly corrupted by white gaussian noise. Reliability analysis is performed on windows of length 1000, shifted in steps of 250; the resulting variance estimates are smoothed. Fig.3 shows again that the uncertainty measure is nicely correlated with the true separation error, furthermore the variance goes systematically up within the noisy part but also in other parts of the time series that do not seem to match the assumptions underlying the algorithm.3 So our reliability estimates can eventually Figure 3: Upper panel: mixtures, partly corrupted by noise. Lower panel: the blockwise variance estimate (solid line) vs the true separation error on this block (dotted line). be used to improve separation performance by removing all but the 'reliable' parts of the time series. For our example this reduces the overall separation error by 2 orders of magnitude from 2.4.10- 2 to 1.7.10-4 . This moving-window resampling can detect instabilities of the projections in two different ways: Besides the resampling variance that can be calculated for each window, one can also calculate the change of the projection directions between two windows. The later has already been used successfully by Makeig et. al. [10]. 4 Assigning Meaning: Application to Biomedical Data We now apply our reliability analysis to biomedical data that has been produced by an MEG experiment with acoustic stimulation. The stimulation was achieved by presenting alternating periods of music and silence, each of 30s length, to the subjects right ear during 30 min. of total recording time (for details see [13]). The measured DC magnetic field values, sampled at a frequency of 0.4 Hz, gave a total number of 720 sample points for each of the 49 channels. While previously 3For example, the peak in the last third of the time series can be traced back to the fact that the original time series are correlated in this region. [13] analysing the data, we found that many of the ICA components are seemingly meaningless and it took some medical knowledge to find potential meaningful projections for a later close inspection. However, our reliability assessment can also be seen as indication for meaningful projections, i.e. meaningful components should have low variance. In the experiment, BSS was performed on the 23 most powerful principal components using (a) higher order statistics (JADE) and (b) temporal decorrelation (TDSEP, time lag 0 .. 50). The results in Fig.4 show that none of higher order statistics (JADE) temporal decorrelation (TDSEP) 0.35 0.35 0.3 0.3 0.25 0.25 ::J ::J ~ 0.2 ~ 0.2 i ~ g 0.1 5 g 0.15 ::J ::J 0.1 0.1 0.05 0.05 ,~ 10 15 20 10 15 20 leA-Channel i leA-Channel i Figure 4: Resampling on the biomedical data from MEG experiment shows: (a) no JADE projection is reliable (has low uncertainty) (b) TDSEP is able to identify three sources with low uncertainty. the JADE-projections (left) have small variance whereas TDSEP (right) identifies three sources with a good reliability. In fact, these three components have physical meaning: while component 23 is an internal very low frequency signal (drift) that is always present in DC-measurements, component 22 turns out to be an artifact of the measurement; interestingly component 6 shows a (noisy) rectangular waveform that clearly displays the 1/308 on/off characteristics of the stimulus (correlation to stimulus 0.7; see Fig.5) . The clear dipole-structure of the spatial field pattern in 0.5 ~ ~O In -0.5 ~ stimulUS 1 234 5 6 7 t[min) Figure 5: Spatial field pattern, frequency content and time course of TDSEP channel 6. Fig.5 underlines the relevance of this projection. The components found by JADE do not show such a clear structure and the strongest correlation of any component to the stimulus is about 0.3, which is of the same order of magnitude as the strongest correlated PCA-component before applying JADE. 5 Discussion We proposed a simple method to estimate the reliability of ICA projections based on res amp ling techniques. After showing that our technique approximates the separation error, several directions are open(ed) for applications. First, we may like to use it for model selection purposes to distinguish between algorithms or to chose appropriate hyperparameter values (possibly even component-wise). Second, variances can be estimated on blocks of data and separation performance can be enhanced by using only low variance blocks where the model matches the data nicely. Finally reliability estimates can be used to find meaningful components. Here our assumption is that the more meaningful a component is, the more stably we should be able to estimate it. In this sense artifacts appear of course also as meaningful, whereas noisy directions are discarded easily, due to their high uncertainty. Future research will focus on applying res amp ling techniques to other unsupervised learning scenarios. We will also consider Bayesian modelings where often a variance estimate comes for free, along with the trained model. Acknowledgments K-R.M thanks Guido Nolte and the members of the Oberwolfach Seminar September 2000 in particular Lutz Dumbgen and Enno Mammen for helpful discussions and suggestions. K -R. M and A. Z. acknowledge partial funding by the EU project (IST-1999-14190 - BLISS). We thank the Biomagnetism Group of the PhysikalischTechnische Bundesanstalt (PTB) for providing the MEG-DC data. References [1] S. Amari, A. Cichocki, and H. H. Yang. A new learning algorithm for blind signal separation. In D.S. Touretzky, M.C. Mozer, and M.E. Hasselmo, editors, Advances in Neural Information Processing Systems (NIPS 95), volume 8, pages 882-893. The MIT Press, 1996. [2] A. J. Bell and T. J. Sejnowski. An information maximisation approach to blind separation and blind deconvolution. Neural Computation, 7:1129- 1159, 1995. [3] R. Beran and M.S. Srivastava. Bootstrap tests and confidence regions for functions of a covariance matrix. Annals of Statistics, 13:95- 115, 1985. [4] J.-F. Cardoso and A. Souloumiac. Blind beamforming for non Gaussian signals. IEEE Proceedings-F, 140(6):362- 370, December 1994. [5] P. Comon. Independent component analysis, a new concept? Signal Processing, 36(3):287-314, 1994. [6] G. Deco and D. Obradovic. An information-theoretic approach to neural computing. Springer, New York, 1996. [7] B. Efron and R.J. Tibshirani. An Introduction to the Bootstrap. Chapman & Hall, first edition, 1993. [8] A. Hyviirinen, J. Karhunen, and E. Oja. Independent Component Analysis. Wiley, 200l. [9] Ch. Jutten and J. Herault. Blind separation of sources, part I: An adaptive algorithm based on neuromimetic architecture. Signal Processing, 24:1- 10, 1991. [10] S. Makeig, S. Enghoff, T.-P. Jung, and T. Sejnowski. Moving-window ICA decomposition of EEG data reveals event-related changes in oscillatory brain activity. In Proc. 2nd Int. Workshop on Independent Component Analysis and Blind Source Separation (ICA '2000), pages 627- 632, Helsinki, Finland, 2000. [11] F. Meinecke, A. Ziehe, M. Kawanabe, and K-R. Muller. Assessing reliability of ica projections - a resampling approach. In ICA '01. T.-W. Lee, Ed., 200l. [12] J. Shao and D. Th. The Jackknife and Bootstrap. Springer, New York, 1995. [13] G. Wubbeler, A. Ziehe, B.-M. Mackert, K-R. Muller, L. Trahms, and G. Curio. Independent component analysis of non-invasively recorded cortical magnetic dc-fields in humans. IEEE Transactions on Biomedical Engineering, 47(5):594-599, 2000. [14] A. Ziehe and K-R. Muller. TDSEP - an efficient algorithm for blind separation using time structure. In L. Niklasson, M. Boden, and T. Ziemke, editors, Proc. Int. Conf. on Artificial Neural Networks (ICANN'9S), pages 675 - 680, Skiivde, Sweden, 1998. Springer Verlag.
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Learning from Infinite Data in Finite Time Pedro Domingos Geoff H ulten Department of Computer Science and Engineering University of Washington Seattle, WA 98185-2350, U.S.A. {pedrod, ghulten} @cs.washington.edu Abstract We propose the following general method for scaling learning algorithms to arbitrarily large data sets. Consider the model Mii learned by the algorithm using ni examples in step i (ii = (nl , ... ,nm)), and the model Moo that would be learned using infinite examples. Upper-bound the loss L(Mii' Moo ) between them as a function of ii, and then minimize the algorithm's time complexity f(ii) subject to the constraint that L(Moo , Mii ) be at most f with probability at most 8. We apply this method to the EM algorithm for mixtures of Gaussians. Preliminary experiments on a series of large data sets provide evidence of the potential of this approach. 1 An Approach to Large-Scale Learning Large data sets make it possible to reliably learn complex models. On the other hand, they require large computational resources to learn from. While in the past the factor limiting the quality of learnable models was typically the quantity of data available, in many domains today data is super-abundant, and the bottleneck is the time required to process it. Many algorithms for learning on large data sets have been proposed, but in order to achieve scalability they generally compromise the quality of the results to an unspecified degree. We believe this unsatisfactory state of affairs is avoidable, and in this paper we propose a general method for scaling learning algorithms to arbitrarily large databases without compromising the quality of the results. Our method makes it possible to learn in finite time a model that is essentially indistinguishable from the one that would be obtained using infinite data. Consider the simplest possible learning problem: estimating the mean of a random variable x. If we have a very large number of samples, most of them are probably superfluous. If we are willing to accept an error of at most f with probability at most 8, Hoeffding bounds [4] (for example) tell us that, irrespective of the distribution of x, only n = ~(R/f)2 1n (2/8) samples are needed, where R is x's range. We propose to extend this type of reasoning beyond learning single parameters, to learning complex models. The approach we propose consists of three steps: 1. Derive an upper bound on the relative loss between the finite-data and infinite-data models, as a function of the number of samples used in each step of the finite-data algorithm. 2. Derive an upper bound on the time complexity of the learning algorithm, as a function of the number of samples used in each step. 3. Minimize the time bound (via the number of samples used in each step) subject to target limits on the loss. In this paper we exemplify this approach using the EM algorithm for mixtures of Gaussians. In earlier papers we applied it (or an earlier version of it) to decision tree induction [2J and k-means clustering [3J. Despite its wide use, EM has long been criticized for its inefficiency (see discussion following Dempster et al. [1]), and has been considered unsuitable for large data sets [8J. Many approaches to speeding it up have been proposed (see Thiesson et al. [6J for a survey). Our method can be seen as an extension of progressive sampling approaches like Meek et al. [5J: rather than minimize the total number of samples needed by the algorithm, we minimize the number needed by each step, leading to potentially much greater savings; and we obtain guarantees that do not depend on unverifiable extrapolations of learning curves. 2 A Loss Bound for EM In a mixture of Gaussians model, each D-dimensional data point Xj is assumed to have been independently generated by the following process: 1) randomly choose a mixture component k; 2) randomly generate a point from it according to a Gaussian distribution with mean f-Lk and covariance matrix ~k. In this paper we will restrict ourselves to the case where the number K of mixture components and the probability of selection P(f-Lk) and covariance matrix for each component are known. Given a training set S = {Xl, ... , X N }, the learning goal is then to find the maximumlikelihood estimates of the means f-Lk. The EM algorithm [IJ accomplishes this by, starting from some set of initial means, alternating until convergence between estimating the probability p(f-Lk IXj) that each point was generated by each Gaussian (the Estep), and computing the ML estimates of the means ilk = 2::;':1 WjkXj / 2::f=l Wjk (the M step), where Wjk = p(f-Lklxj) from the previous E step. In the basic EM algorithm, all N examples in the training set are used in each iteration. The goal in this paper is to speed up EM by using only ni < N examples in the ith iteration, while guaranteeing that the means produced by the algorithm do not differ significantly from those that would be obtained with arbitrarily large N. Let Mii = (ill , . . . ,ilK) be the vector of mean estimates obtained by the finite-data EM algorithm (i.e., using ni examples in iteration i), and let Moo = (f-L1, ... ,f-LK) be the vector obtained using infinite examples at each iteration. In order to proceed, we need to quantify the difference between Mii and Moo . A natural choice is the sum of the squared errors between corresponding means, which is proportional to the negative log-likelihood of the finite-data means given the infinite-data ones: K K D L(Mii' Moo) = L Ililk - f-Lkl12 = L L lilkd f-Lkdl 2 (1) k=l k=ld=l where ilkd is the dth coordinate of il, and similarly for f-Lkd. After any given iteration of EM, lilkd - f-Lkdl has two components. One, which we call the sampling error, derives from the fact that ilkd is estimated from a finite sample, while J-Lkd is estimated from an infinite one. The other component, which we call the weighting error, derives from the fact that, due to sampling errors in previous iterations, the weights Wjk used to compute the two estimates may differ. Let J-Lkdi be the infinite-data estimate of the dth coordinate of the kth mean produced in iteration i, flkdi be the corresponding finite-data estimate, and flkdi be the estimate that would be obtained if there were no weighting errors in that iteration. Then the sampling error at iteration i is Iflkdi J-Lkdi I, the weighting error is Iflkdi flkdi I, and the total error is Iflkdi J-Lkdi 1 :::; Iflkdi flkdi 1 + Iflkdi J-Lkdi I· Given bounds on the total error of each coordinate of each mean after iteration i-I, we can derive a bound on the weighting error after iteration i as follows. Bounds on J-Lkd ,i-l for each d imply bounds on p(XjlJ-Lki ) for each example Xj, obtained by substituting the maximum and minimum allowed distances between Xjd and J-Lkd ,i-l into the expression of the Gaussian distribution. Let P}ki be the upper bound on P(XjlJ-Lki) , and Pjki be the lower bound. Then the weight of example Xj in mean J-Lki can be bounded from below by wjki = PjkiP(J-Lk)/ ~~= l P}k'iP(J-LU, and from above by W}ki = min{p}kiP(J-Lk)/ ~~=l Pjk'iP(J-LU, I}. Let w;t: = W}ki if Xj ::::: 0 and (+) h' d 1 t (- ) -'f > 0 d (- ) + th . W jki W jki ot erWlse, an e W jki W jki 1 Xj _ an W jki W jki 0 erWlse. Then Iflkdi flkdi 1 I , ~7~1 Wjki Xj I J-Lkdi ",ni uj=l Wjki {I ", ni (+) II ",ni ( - ) I} , uj=l W jki Xj , uj=l W jki Xj max J-Lkdi ",ni _ ,J-Lkdi ",ni + uj=l w jki uj=l w jki (2) < A corollary of Hoeffding's [4] Theorem 2 is that, with probability at least 1 - 8, the sampling error is bounded by Iflkdi J-Lkdi 1 :::; (3) where Rd is the range of the dth coordinate of the data (assumed known 1). This bound is independent of the distribution of the data, which will ensure that our results are valid even if the data was not truly generated by a mixture of Gaussians, as is often the case in practice. On the other hand, the bound is more conservative than distribution-dependent ones, requiring more samples to reach the same guarantees. The initialization step is error-free, assuming the finite- and infinite-data algorithms are initialized with the same means. Therefore the weighting error in the first iteration is zero, and Equation 3 bounds the total error. From this we can bound the weighting error in the second iteration according to Equation 2, and therefore bound the total error by the sum of Equations 2 and 3, and so on for each iteration until the algorithms converge. If the finite- and infinite-data EM converge in the same number of iterations m, the loss due to finite data is L(Mii" Moo ) = ~f= l ~~= llflkdm J-Lkdml 2 (see Equation 1). Assume that the convergence criterion is ~f= l IIJ-Lki J-Lk,i-111 2 :::; f. In general 1 Although a normally distributed variable has infinite range, our experiments show that assuming a sufficiently wide finite range does not significantly affect the results. (with probability specified below), infinite-data EM converges at one of the iterations for which the minimum possible change in mean positions is below ,,/, and is guaranteed to converge at the first iteration for which the maximum possible change is below "(. More precisely, it converges at one of the iterations for which ~~= l ~~= l (max{ IPkd,i- l - Pkdil-IPkd,i- l - ftkd,i - ll-IPkdi - ftkdil, O})2 ::; ,,/, and is guaranteed to converge at the first iteration for which ~~=l ~~=l (IPkd,i-l Pkdil + IPkd,i-l - ftkd,i-ll + IPkdi - ftkdil)2 ::; "/. To obtain a bound for L(Mn, Moo), finite-data EM must be run until the latter condition holds. Let I be the set of iterations at which infinite-data EM could have converged. Then we finally obtain where m is the total number of iterations carried out. This bound holds if all of the Hoeffding bounds (Equation 3) hold. Since each of these bounds fails with probability at most 8, the bound above fails with probability at most 8* = K Dm8 (by the union bound). As a result, the growth with K, D and m of the number of examples required to reach a given loss bound with a given probability is only O(v'lnKDm). The bound we have just derived utilizes run-time information, namely the distance of each example to each mean along each coordinate in each iteration. This allows it to be tighter than a priori bounds. Notice also that it would be trivial to modify the treatment for any other loss criterion that depends only on the terms IPkdm - ftkdm I (e.g., absolute loss) . 3 A Fast EM Algorithm We now apply the previous section's result to reduce the number of examples used by EM at each iteration while keeping the loss bounded. We call the resulting algorithm VFEM. The goal is to learn in minimum time a model whose loss relative to EM applied to infinite data is at most f* with probability at least 1 - 8*. (The reason to use f* instead of f will become apparent below.) Using the notation of the previous section, if ni examples are used at each iteration then the running time of EM is O(KD ~::l ni) , and can be minimized by minimizing ~::l ni. Assume for the moment that the number of iterations m is known. Then, using Equation 1, we can state the goal more precisely as follows. Goal: Minimize ~ ::l ni, subject to the constraint that ~~=l IIPkm - ftkml12 ::; f* with probability at least 1 - 8* . A sufficient condition for ~~=l IIPkm - ftkml12 ::; f* is that Vk IIPkm - ftkmll ::; Jf*/K. We thus proceed by first minimizing ~::l ni subject to IIPkm - ftkmll ::; J f* / K separately for each mean.2 In order to do this, we need to express IIPkm ftkm II as a function of the ni 'so By the triangle inequality, IIPki - ftki II ::; IIPki - ftki II + Ilftki - ftk& By Equation 3, Ilftki - ftki II::; ~R2ln(2/8) ~;~ l w;kd(~;~ l Wjki)2, where R2 = ~~=l RJ and 8 = 8* / K Dm per the discussion following Equation 4. The (~;~ l Wjki)2 / ~;~ l W;ki term is a measure of the diversity of the weights, 2This will generally lead to a suboptimal solution; improving it is a matter for future work. being equal to 1 l/Gini(W~i)' where W~i is the vector of normalized weights wjki = wjkd 2:j,i=l Wjl ki. It attains a minimum of! when all the weights but one are zero, and a maximum of ni when all the weights are equal and non-zero. However, we would like to have a measure whose maximum is independent of ni, so that it remains approximately constant whatever the value of ni chosen (for sufficiently large ni). The measure will then depend only on the underlying distribution of the data. Thus we define f3ki = (2:7~1 Wjki)2 /(ni 2:7~1 W]ki)' obtaining IliLki - ILkill :::; JR2ln(2/8)/(2f3kini). Also, IIP-ki-iLkill = J2:~= llP-kdi - iLkdil2, with lP-kdi-iLkdil bounded by Equation 2. To keep the analysis tractable, we upper-bound this term by a function proportional to IIP-kd,i-1 - ILkd,i-111. This captures the notion than the weighting error in one iteration should increase with the total error in the previous one. Combining this with the bound for IliLki - ILkill, we obtain R2 ln(2/8) 2f3kini (5) where CXki is the proportionality constant. Given this equation and IIP-kO - ILkO II = 0, it can be shown by induction that m IIP-km - ILkmll :::; ~ ~ (6) where (7) The target bound will thus be satisfied by minimizing 2::1 ni subject to 2::1 (rkd,;niJ = J E* / K. 3 Finding the n/s by the method of Lagrange multipliers yields ni = ~ (f ~rkir%j) 2 )=1 (8) This equation will produce a required value of ni for each mean. To guarantee the desired E*, it is sufficient to make ni equal to the maximum of these values. The VFEM algorithm consists of a sequence of runs of EM, with each run using more examples than the last, until the bound L(Mii' Moo) :::; E* is satisfied, with L(Mii' Moo) bounded according to Equation 4. In the first run, VFEM postulates a maximum number of iterations m, and uses it to set 8 = 8* / K Dm. If m is exceeded, for the next run it is set to 50% more than the number needed in the current run. (A new run will be carried out if either the 8* or E* target is not met.) The number of examples used in the first run of EM is the same for all iterations, and is set to 1.1(K/2)(R/E*)2ln(2/8). This is 10% more than the number of examples that would theoretically be required in the best possible case (no weighting errors in the last 3This may lead to a suboptimal solution for the ni's, in the unlikely case that Ilflkm Jtkm II increases with them. iteration, leading to a pure Hoeffding bound, and a uniform distribution of examples among mixture components). The numbers of examples for subsequent runs are set according to Equation 8. For iterations beyond the last one in the previous run, the number of examples is set as for the first run. A run of EM is terminated when L~= l L~= l (Iflkd,i- l - flkdi 1 + Iflkd,i-l - ILkd,i-l l + Iflkdi - ILkdi 1)2 :s: "( (see discussion preceding Equation 4), or two iterations after L~=l IIILk i - ILk,i-1 112 :s: "( 13, whichever comes first. The latter condition avoids overly long unproductive runs. If the user target bound is E, E* is set to min{ E, "( 13}, to facilitate meeting the first criterion above. When the convergence threshold for infinite-data EM was not reached even when using the whole training set, VFEM reports that it was unable to find a bound; otherwise the bound obtained is reported. VFEM ensures that the total number of examples used in one run is always at least twice the number n used in the previous run. This is done by, if L ni < 2n, setting the ni's instead to n~ = 2n(nil L ni). If at any point L ni > mN, where m is the number of iterations carried out and N is the size of the full training set, Vi ni = N is used. Thus, assuming that the number of iterations does not decrease with the number of examples, VFEM's total running time is always less than three times the time taken by the last run of EM. (The worst case occurs when the one-but-last run is carried out on almost the full training set.) The run-time information gathered in one run is used to set the n/s for the next run. We compute each Ctki as Ilflki - Pkill/llflk,i-l - ILk,i-lll. The approximations made in the derivation will be good, and the resulting ni's accurate, if the means' paths in the current run are similar to those in the previous run. This may not be true in the earlier runs, but their running time will be negligible compared to that of later runs, where the assumption of path similarity from one run to the next should hold. 4 Experiments We conducted a series of experiments on large synthetic data sets to compare VFEM with EM. All data sets were generated by mixtures of spherical Gaussians with means ILk in the unit hypercube. Each data set was generated according to three parameters: the dimensionality D, the number of mixture components K , and the standard deviation (Y of each coordinate in each component. The means were generated one at a time by sampling each dimension uniformly from the range (2(Y,1 - 2(Y). This ensured that most of the data points generated were within the unit hypercube. The range of each dimension in VFEM was set to one. Rather than discard points outside the unit hypercube, we left them in to test VFEM's robustness to outliers. Any ILk that was less than (vD 1 K)(Y away from a previously generated mean was rejected and regenerated, since problems with very close means are unlikely to be solvable by either EM or VFEM. Examples were generated by choosing one of the means ILk with uniform probability, and setting the value of each dimension of the example by randomly sampling from a Gaussian distribution with mean ILkd and standard deviation (Y. We compared VFEM to EM on 64 data sets of 10 million examples each, generated by using every possible combination of the following parameters: D E {4, 8,12, 16}; K E {3, 4, 5, 6}; (Y E {.01, .03, .05, .07}. In each run the two algorithms were initialized with the same means, randomly selected with the constraint that no two be less than vD 1 (2K) apart. VFEM was allowed to converge before EM's guaranteed convergence criterion was met (see discussion preceding Equation 4). All experiments were run on a 1 GHz Pentium III machine under Linux, with "( = O.OOOlDK, 8* = 0.05, and E* = min{O.Ol, "(}. Table 1: Experimental results. Values are averages over the number of runs shown. Times are in seconds, and #EA is the total number of example accesses made by the algorithm, in millions. Runs Algorithm #Runs Time #EA Loss D K rr Bound VFEM 40 217 1.21 2.51 10.5 4.2 0.029 EM 40 3457 19.75 2.51 10.5 4.2 0.029 No bound VFEM 24 7820 43.19 1.20 9.1 4.9 0.058 EM 24 4502 27.91 1.20 9.1 4.9 0.058 All VFEM 64 3068 16.95 2.02 10 4.5 0.04 EM 64 3849 22.81 2.02 10 4.5 0.04 The results are shown in Table 1. Losses were computed relative to the true means, with the best match between true means and empirical ones found by greedy search. Results for runs in which VFEM achieved and did not achieve the required E* and 8* bounds are reported separately. VFEM achieved the required bounds and was able to stop early on 62.5% of its runs. When it found a bound, it was on average 16 times faster than EM. When it did not, it was on average 73% slower. The losses of the two algorithms were virtually identical in both situations. VFEM was more likely to converge rapidly for higher D's and lower K's and rr's. When achieved, the average loss bound for VFEM was 0.006554, and for EM it was 0.000081. In other words, the means produced by both algorithms were virtually identical to those that would be obtained with infinite data.4 We also compared VFEM and EM on a large real-world data set, obtained by recording a week of Web page requests from the entire University of Washington campus. The data is described in detail in Wolman et al. [7], and the preprocessing carried out for these experiments is described in Domingos & Hulten [3]. The goal was to cluster patterns of Web access in order to support distributed caching. On a dataset with D = 10 and 20 million examples, with 8* = 0.05, I = 0.001, E* = 1/3, K = 3, and rr = 0.01, VFEM achieved a loss bound of 0.00581 and was two orders of magnitude faster than EM (62 seconds vs. 5928), while learning essentially the same means. VFEM's speedup relative to EM will generally approach infinity as the data set size approaches infinity. The key question is thus: what are the data set sizes at which VFEM becomes worthwhile? The tentative evidence from these experiments is that they will be in the millions. Databases of this size are now common, and their growth continues unabated, auguring well for the use of VFEM. 5 Conclusion Learning algorithms can be sped up by minimizing the number of examples used in each step, under the constraint that the loss between the resulting model and the one that would be obtained with infinite data remain bounded. In this paper we applied this method to the EM algorithm for mixtures of Gaussians, and observed the resulting speedups on a series of large data sets. 4The much higher loss values relative to the true means, however, indicate that infinitedata EM would often find only local optima (unless the greedy search itself only found a suboptimal match). Acknowledgments This research was partly supported by NSF CAREER and IBM Faculty awards to the first author, and by a gift from the Ford Motor Company. References [1] 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. [2] P. Domingos and G. Hulten. Mining high-speed data streams. In Proceedings of the Sixth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 71- 80, Boston, MA, 2000. ACM Press. [3] P. Domingos and G. Hulten. A general method for scaling up machine learning algorithms and its application to clustering. In Proceedings of the Eighteenth International Conference on Machine Learning, pp. 106-113, Williamstown, MA, 2001. Morgan Kaufmann. [4] W. Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58:13- 30, 1963. [5] C. Meek, B. Thiesson, and D. Heckerman. The learning-curve method applied to clustering. Technical Report MSR-TR-01-34, Microsoft Research, Redmond, WA,2000. [6] B. Thiesson, C. Meek, and D. Heckerman. Accelerating EM for large databases. Technical Report MSR-TR-99-31, Microsoft Research, Redmond, WA, 2001. [7] A. Wolman, G. Voelker, N. Sharma, N. Cardwell, M. Brown, T. Landray, D. Pinnel, A. Karlin, and H. Levy. Organization-based analysis of Web-object sharing and caching. In Proceedings of the Second USENIX Conference on Internet Technologies and Systems, pp. 25- 36, Boulder, CO, 1999. [8] T. Zhang, R. Ramakrishnan, and M. Livny. BIRCH: An efficient data clustering method for very large databases. In Proceedings of the 1996 A CM SIGMOD International Conference on Management of Data, pp. 103- 114, Montreal, Canada, 1996. ACM Press.
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The 9 Factor: Relating Distributions on Features to Distributions on Images James M. Coughlan and A. L. Yuille Smith-Kettlewell Eye Research Institute, 2318 Fillmore Street, San Francisco, CA 94115, USA. Tel. (415) 345-2146/2144. Fax. (415) 345-8455. Email: coughlan@ski.org.yuille@ski.org Abstract We describe the g-factor, which relates probability distributions on image features to distributions on the images themselves. The g-factor depends only on our choice of features and lattice quantization and is independent of the training image data. We illustrate the importance of the g-factor by analyzing how the parameters of Markov Random Field (i.e. Gibbs or log-linear) probability models of images are learned from data by maximum likelihood estimation. In particular, we study homogeneous MRF models which learn image distributions in terms of clique potentials corresponding to feature histogram statistics (d. Minimax Entropy Learning (MEL) by Zhu, Wu and Mumford 1997 [11]) . We first use our analysis of the g-factor to determine when the clique potentials decouple for different features. Second, we show that clique potentials can be computed analytically by approximating the g-factor. Third, we demonstrate a connection between this approximation and the Generalized Iterative Scaling algorithm (GIS), due to Darroch and Ratcliff 1972 [2], for calculating potentials. This connection enables us to use GIS to improve our multinomial approximation, using Bethe-Kikuchi[8] approximations to simplify the GIS procedure. We support our analysis by computer simulations. 1 Introduction There has recently been a lot of interest in learning probability models for vision. The most common approach is to learn histograms of filter responses or, equivalently, to learn probability distributions on features (see right panel of figure (1)). See, for example, [6], [5], [4]. (In this paper the features we are considering will be extracted from the image by filters - hence we use the terms "features" and "filters" synonymously. ) An alternative approach, however, is to learn probability distributions on the images themselves. The Minimax Entropy Learning (MEL) theory [11] uses the maximum entropy principle to learn MRF distributions in terms of clique potentials determined by the feature statistics (i.e. histograms of filter responses). (We note that the maximum entropy principle is equivalent to performing maximum likelihood estimation on an MRF whose form is determined by the choice of feature statistics.) When applied to texture modeling it gives a way to unify the filter based approaches (which are often very effective) with the MRF distribution approaches (which are theoretically attractive). ) \ Figure 1: Distributions on images vs. distributions on features. Left and center panels show a natural image and its image gradient magnitude map, respectively. Right panel shows the empirical histogram (i.e. a distribution on a feature) of the image gradient across a dataset of natural images. This feature distribution can be used to create a MRF distribution over images[10]. This paper introduces the g-factor to examine connections between the distribution over images and the distribution over features. As we describe in this paper (see figure (1)), distributions on images and on features can be related by a g-factor (such factors arise in statistical physics, see [3]) . Understanding the g-factor allows us to approximate it in a form that helps explain why the clique potentials learned by MEL take the form that they do as functions of the feature statistics. Moreover, the MEL clique potentials for different features often seem to be decoupled and the g-factor can explain why, and when, this occurs. (I.e. the two clique potentials corresponding to two features A and B are identical whether we learn them jointly or independently). The g-factor is determined only by the form of the features chosen and the spatial lattice and quantization of the image gray-levels. It is completely independent of the training image data. It should be stressed that the choice of image lattice, gray-level quantization and histogram quantization can make a big difference to the g-factor and hence to the probability distributions which are the output of MEL. In Section (2), we briefly review Minimax Entropy Learning. Section (3) introduces the g-factor and determines conditions for when clique potentials are decoupled. In Section (4) we describe a simple approximation which enables us to learn the clique potentials analytically, and in Section (5) we discuss connections between this approximation and the Generalized Iterative Scaling (GIS) algorithm. 2 Minimax Entropy Learning Suppose we have training image data which we assume has been generated by an (unknown) probability distribution PT(X) where x represents an image. Minimax Entropy Learning (MEL) [11] approximates PT(X) by selecting the distribution with maximum entropy constrained by observed feature statistics i(X) = ;fobs. This gives >:. ¢(£) P(xIA) = e Z [>:] ,where A is a parameter chosen such that Lx P(xIA)¢>(X) = 'l/Jobs· Or equivalently, so that <910;{[>:] = ;fobs. We will treat the special case where the statistics i are the histogram of a shiftinvariant filter {fi(X) : i = 1, ... , N} , where N is the total number of pixels in the image. So 'l/Ja = ¢>a(x) = -tv L~l ba,' i(X) where a = 1, ... , Q indicates the (quantized) ~ ~ Q N filter response values. The potentials become A·¢>(X) = -tv La=l Li=l A(a)ba,fi(X) = -tv L~l A(fi(X)). Hence P(xl,X) becomes a MRF distribution with clique potentials given by A(fi (x)). This determines a Markov random field with the clique structure given by the filters {fd. MEL also has a feature selection stage based on Minimum Entropy to determine which features to use in the Maximum Entropy Principle. The features are evaluated by computing the entropy - Lx P(xl,X) log P(xl,X) for each choice of features (with small entropies being preferred). A filter pursuit procedure was described to determine which filters/features should be considered (our approximations work for this also). 3 The g-Factor This section defines the g-factor and starts investigating its properties in subsection (3.1). In particular, when, and why, do clique potentials decouple? More precisely, when do the potentials for filters A and B learned simultaneously differ from the potentials for the two filters when they are learned independently? We address these issues by introducing the g-factor g(;f) and the associated distribution Po (;f): x space -----+ iii space GG g(ijiJ = number of images x with histogram iii (1) Figure 2: The g-factor g(;f) counts the number of images x that have statistics ;f. Note that the g-factor depends only on the choice of filters and is independent of the training image data. Here L is the number of grayscale levels of each pixel, so that LN is the total number of possible images. The g-factor is essentially a combinational factor which counts the number of ways that one can obtain statistics ;f, see figure (2). Equivalently, Po is the default distribution on ;f if the images are generated by white noise (i.e. completely random images). We can use the g-factor to compute the induced distribution P(~I'x) on the statistics determined by MEL: A ~ ~ L ~ ~ g( ~)eX.,j; ~ L ~ X,j; P(1/1 I'\) = 6;;: 2(-)P(xl'\) = ~, Z[,\] = g(1/1)e· . (2) X'j','j' x Z[,\] ,j; Observe that both P(~I'x) and log Z[,X] are sufficient for computing the parameters X. The ,X can be found by solving either of the following two (equivalent) equations: A ~ ~ ~ ~ 8 10 zrXl ~ L:,j; P(1/1I,\) 1/1 = 1/1obs, or ;X = 1/1obs, which shows that knowledge of the g-factor and eX. ,j; are all that is required to do MEL. Observe from equation (2) that we have P(~I'x = 0) = Po(~) . In other words, setting ,X = 0 corresponds to a uniform distribution on the images x. 3.1 Decoupling Filters We now derive an important property of the minimax entropy approach. As mentioned earlier, it often seems that the potentials for filters A and B decouple. In other words, if one applies MEL to two filters A, B simultaneously bv letting ...... ....A ...... B...... ....A -B ...... "'""'A ...... B . :..tA ...... B 1/1 = (1/1 ,1/1 ), '\ = (,\ ,'\ ), and 1/1obs = (1/1 obs' 1/1 obs)' then the solutIOns'\ , '\ to the equations: LP(xl,XA, ,XB)(iA(x),iB(x)) = (~:bs'~!s)' (3) x are the same (approximately) as the solutions to the equations L:x p(xl,XA )iA(x) = ~!s and L:x P(xl,XB)iB(x) = ~!s, see figure (3) for an example. Figure 3: Evidence for decoupling of features. The left and right panels show the clique potentials learned for the features a I ax and a I ay respectively. The solid lines give the potentials when they are learned individually. The dashed lines show the potentials when they are learned simultaneously. Figure courtesy of Prof. Xiuwen Liu, Florida State University. We now show how this decoupling property arises naturally if the g-factor for the two filters factorizes. This factorization, of course, is a property only of the form of the statistics and is completely independent of whether the statistics of the two filters are dependent for the training data. Property I: Suppose we have two sufficient statistics iA(x), iB (x) which are independent on the lattice in the sense that g(~A,~B ) = gA (~A )gB(~B) , then logZ[,XA,,XB] = logZA[,XA] + logZB[,XB] and p(~A,~B ) = pA(~A)pB(~B ). This implies that the parameters XA, XB can be solved from the independent . 81ogZA[XA] _ -A 8 1ogZB[XB ] equatwns 8XA 'ljJobs' 8XB _ -B A A -A -A -A 'ljJobs or L.,j;A P ('ljJ)'ljJ = 'ljJobs' L.,j;B pB(;fB );fB = ;f~s ' Moreover, the resulting distribution PUC) can be obtained by multiplying the distributions (l/ZA)eXA .,j;A(x) and (l/ZB) eXB.,j;B(x) together. The point here is that the potential terms for the two statistics ;fA,;fB decouple if the phase factor g(;fA,;fB) can be factorized. We conjecture that this is effectively the case for many linear filters used in vision processing. For example, it is plausible that the g-factor for features 0/ ox and 0/ oy factorizes - and figure (3) shows that their clique potentials do decouple (approximately). Clearly, if factorization between filters occurs then it gives great simplification to the system. 4 Approximating the g-factor for a Single Histogram We now consider the case where the statistic is a single histogram. Our aim is to understand why features whose histograms are of stereotypical shape give rise to potentials of the form given by figure (3). Our results, of course, can be directly extended to multiple histograms if the filters decouple, see subsection (3.1). We first describe the approximation and then discuss its relevance for filter pursuit. We rescale the X variables by N so that we have: eNX.¢(x) A _ _ eNX.,j; P(X'I-\) = Z[X] , P('ljJ I-\) = g('ljJ) Z[X] , (4) We now consider the approximation that the filter responses {Ii} are independent of each other when the images are uniformly distributed. This is the multinomial approximation. (We attempted a related approximation [1] which was less successful.) It implies that we can express the phase factor as being proportional to a multinomial distribution: (nt:) LN N! N1/Jl N1/JQ n (nt:) _ N! N1/Jl N1/JQ 9 <P = (N'ljJd!. .. (N'ljJQ)!o ... 0Q ' TO <p (N'ljJd!. .. (N'ljJQ)!Ol "'OQ (5) where L.~= 1 'ljJa = 1 (by definition) and the {oa} are the means of the components Na } with respect to the distribution Po (;f). As we will describe later, the {oa} will be determined by the filters {fi}. See Coughlan and Yuille, in preparation, for details of how to compute the {oa}. This approximation enables us to calculate MEL analytically. Theorem With the multinomial approximation the log partition function is: Q log Z[X] = N log L + N log{~= e " a+1og aa } , (6) a=l and the "potentials" P a} can be solved in terms of the observed data {'ljJobs,a} to be: \ - I 'ljJobs,a Aa - og--, Oa a = 1, ... ,Q. (7) Figure 4: Top row: the multinomial approximation. Bottom row: full implementation of MEL (see text). (Left panels) the potentials, (center panels) synthesized images, and (right panels) the difference between the observed histogram (dashed line) and the histogram of the synthesized images (bold line). Filters were d/dx and d/dy. We note that there is an ambiguity Aa r-+ Aa + K where K is an arbitrary number (recall that L~=l 'IjJ(a) = 1). We fix this ambiguity by setting X = 0 if a. = "Jobs. Proof. Direct calculation. Our simulation results show that this simple approximation gives the typical potential forms generated by Markov Chain Monte Carlo (MCMC) algorithms for Minimax Entropy Learning. Compare the multinomial approximation results with those obtained from a full implementation of MEL by the algorithm used in [11], see figure (4). Filter pursuit is required to determine which filters carry most information. MEL [11] prefers filters (statistics) which give rise to low entropy distributions (this is the "Min" part of Minimax). The entropy is given by H(P) = - Lx P(xIX) log P(xIX) = log Z[X] L~=l Aa'IjJa · For the multinomial approximation this can be computed to be N log L - N L~= l 'ljJa log ~. This gives an intuitive interpretation of feature pursuit: we should prefer filters whose statistical response to the image training data is as large as possible from their responses to uniformly distributed images. This is measured by the Kullback-Leibler divergence L~= l 'ljJa log ~. Recall that if the multinomial approximation is used for multiple filters then we should simply add together the entropies of different filters. 5 Connections to Generalized Iterative Scaling In this section we demonstrate a connection between the multinomial approximation and Generalized Iterative Scaling (GIS)[2]. GIS is an iterative procedure for calculating clique potentials that is guaranteed to converge to the maximum likelihood values of the potentials given the desired empirical filter marginals (e.g. filter histograms). We show that estimating the potentials by the multinomial approximation is equivalent to the estimate obtained after performing the first iteration of GIS. We also outline an efficient procedure that allows us to continue additional GIS iterations to improve upon the multinomial approximation. The GIS procedure calculates a sequence of distributions on the entire image (and is guaranteed to converge to the correct maximum likelihood distribution), with an update rule given by p(t+1)(x) ex P(O)(x)Il~=l{ :F; } <pa(x), where 'lfJit ) =< <Pa(X) >P(t)(x) is the expected histogram for the distribution at time t. This implies that the corresponding clique potential update equation is given by: >.it +1) = >.it ) + log 'lfJ~bs - log 'lfJit ). If we initialize GIS so that the initial distribution is the uniform distribution, i.e. p(O) (x) = L -N, then the distribution after one iteration is p(1) (x) ex e2::a <Pa(X) log (1j;~bs /aa) . In other words, the distribution after one iteration is the MEL distribution with clique potential given by the multinomial approximation. (The result can be adapted to the case of multiple filters, as explained in Coughlan and Yuille, in preparation.) We can iterate GIS to improve the estimate of the clique potentials beyond the accuracy of the multinomial approximation. The main difficulty lies in estimating 'lfJit ) for t > 0 (at t = 0 this expectation is just the mean histogram with respect to the uniform distribution, <l:a, which may be calculated efficiently as described in Coughlan and Yuille, in preparation). One way to approximate these expectations is to apply a Bethe-Kikuchi approximation technique [8], used for estimating marginals on Markov Random Fields, to our MEL distribution. Our technique, which was inspired by the Unified Propagation and Scaling Algorithm [7], consists of writing the Bethe free energy [8] for our 2-d image lattice, simplifying it using the shift invariance of the lattice (which enables the algorithm to run swiftly), and using the Convex-Concave Procedure (CCCP) [9] procedure to obtain an iterative update equation to estimate the histogram expectations. The GIS algorithm is then run using these histogram expectations (the results were accurate and did not improve appreciably by using the higher-order Kikuchi free energy approximation). See Coughlan and Yuille, in preparation, for details of this procedure. 6 Discussion This paper describes the g-factor, which depends on the lattice and quantization and is independent of the training image data. Alternatively it can be thought of as being proportional to the distribution of feature responses when the input images are uniformly distributed. We showed that the g-factor can be used to relate probability distributions on features to distributions on images. In particular, we described approximations which, when valid, enable MEL to be computed analytically. In addition, we can determine when the clique potentials for features decouple, and evaluate how informative each feature is. Finally, we establish a connection between the multinomial approximation and GIS, and outline an efficient procedure based on Bethe-Kikuchi approximations that allows us to continue additional GIS iterations to improve upon the multinomial approximation. Acknowledgements We would like to thank Michael Jordan and Yair Weiss for introducing us to Generalized Iterative Scaling and related algorithms. We also thank Anand Rangarajan, Xiuwen Liu, and Song Chun Zhu for helpful conversations. Sabino Ferreira gave useful feedback on the manuscript. This work was supported by the National Institute of Health (NEI) with grant number R01-EY 12691-01. References [1] J.M. Coughlan and A.L. Yuille. "A Phase Space Approach to Minimax Entropy Learning and The Minutemax approximation". In Proceedings NIPS'98. 1998. [2] J. N. Darroch and D. Ratcliff. "Generalized Iterative Scaling for Log-Linear Models". The Annals of Mathematical Statistics. 1972. Vol. 43, No.5, 14701480. [3] C. Domb and M.S. Green (Eds). Phase Transitions and Critical Phenomena. Vol. 2. Academic Press. London. 1972. [4] S. M. Konishi, A.L. Yuille, J.M. Coughlan and Song Chun Zhu. "Fundamental Bounds on Edge Detection: An Information Theoretic Evaluation of Different Edge Cues." In Proceedings Computer Vision and Pattern Recognition CVPR'99. Fort Collins, Colorado. June 1999. [5] A.B. Lee, D.B. Mumford, and J. Huang. "Occlusion Models of Natural Images: A Statistical Study of a Scale-Invariant Dead Leaf Model". International Journal of Computer Vision. Vol. 41, No.'s 1/2. January/February 2001. [6] J. Portilla and E. P. Simoncelli. "Parametric Texture Model based on Joint Statistics of Complex Wavelet Coefficients". International Journal of Computer Vision. October 2000. [7] Y. W. Teh and M. Welling. "The Unified Propagation and Scaling Algorithm." In Proceedings NIPS'01. 2001. [8] J.S. Yedidia, W.T. Freeman, Y. Weiss, "Generalized Belief Propagation." In Proceedings NIPS'OO. 2000. [9] A.L. Yuille. "CCCP Algorithms to Minimize the Bethe and Kikuchi Free Energies," Neural Computation. In press. 2002. [10] S.C. Zhu and D. Mumford. "Prior Learning and Gibbs Reaction-Diffusion." PAMI vo1.19, no.11, pp1236-1250, Nov. 1997. [11] S.C. Zhu, Y. Wu, and D. Mumford. "Minimax Entropy Principle and Its Application to Texture Modeling". Neural Computation. Vol. 9. no. 8. Nov. 1997.
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Distribution of Mutual Information Marcus Hutter IDSIA, Galleria 2, CH-6928 Manno-Lugano, Switzerland marcus@idsia.ch http://www.idsia.ch/- marcus Abstract The mutual information of two random variables z and J with joint probabilities {7rij} is commonly used in learning Bayesian nets as well as in many other fields. The chances 7rij are usually estimated by the empirical sampling frequency nij In leading to a point estimate J(nij In) for the mutual information. To answer questions like "is J (nij In) consistent with zero?" or "what is the probability that the true mutual information is much larger than the point estimate?" one has to go beyond the point estimate. In the Bayesian framework one can answer these questions by utilizing a (second order) prior distribution p( 7r) comprising prior information about 7r. From the prior p(7r) one can compute the posterior p(7rln), from which the distribution p(Iln) of the mutual information can be calculated. We derive reliable and quickly computable approximations for p(Iln). We concentrate on the mean, variance, skewness, and kurtosis, and non-informative priors. For the mean we also give an exact expression. Numerical issues and the range of validity are discussed. 1 Introduction The mutual information J (also called cross entropy) is a widely used information theoretic measure for the stochastic dependency of random variables [CT91, SooOO]. It is used, for instance, in learning Bayesian nets [Bun96, Hec98] , where stochastically dependent nodes shall be connected. The mutual information defined in (1) can be computed if the joint probabilities {7rij} of the two random variables z and J are known. The standard procedure in the common case of unknown chances 7rij is to use the sample frequency estimates n~; instead, as if they were precisely known probabilities; but this is not always appropriate. Furthermore, the point estimate J (n~; ) gives no clue about the reliability of the value if the sample size n is finite. For instance, for independent z and J, J(7r) =0 but J(n~;) = O(n- 1/ 2 ) due to noise in the data. The criterion for judging dependency is how many standard deviations J(":,;) is away from zero. In [KJ96, Kle99] the probability that the true J(7r) is greater than a given threshold has been used to construct Bayesian nets. In the Bayesian framework one can answer these questions by utilizing a (second order) prior distribution p(7r),which takes account of any impreciseness about 7r. From the prior p(7r) one can compute the posterior p(7rln), from which the distribution p(Iln) of the mutual information can be obtained. The objective of this work is to derive reliable and quickly computable analytical expressions for p(1ln). Section 2 introduces the mutual information distribution, Section 3 discusses some results in advance before delving into the derivation. Since the central limit theorem ensures that p(1ln) converges to a Gaussian distribution a good starting point is to compute the mean and variance of p(1ln). In section 4 we relate the mean and variance to the covariance structure of p(7rln). Most non-informative priors lead to a Dirichlet posterior. An exact expression for the mean (Section 6) and approximate expressions for the variance (Sections 5) are given for the Dirichlet distribution. More accurate estimates of the variance and higher central moments are derived in Section 7, which lead to good approximations of p(1ln) even for small sample sizes. We show that the expressions obtained in [KJ96, Kle99] by heuristic numerical methods are incorrect. Numerical issues and the range of validity are briefly discussed in section 8. 2 Mutual Information Distribution We consider discrete random variables Z E {l, ... ,r} and J E {l, ... ,s} and an i.i.d. random process with samples (i,j) E {l , ... ,r} x {l, ... ,s} drawn with joint probability 7rij. An important measure of the stochastic dependence of z and J is the mutual information T S 7rij 1( 7r) = L L 7rij log ~ = L 7rij log7rij - L 7ri+ log7ri+ - L 7r +j log7r +j' (1) i=1 j = 1 H +J ij i j log denotes the natural logarithm and 7ri+ = Lj7rij and 7r +j = L i7rij are marginal probabilities. Often one does not know the probabilities 7rij exactly, but one has a sample set with nij outcomes of pair (i,j). The frequency irij := n~j may be used as a first estimate of the unknown probabilities. n:= L ijnij is the total sample size. This leads to a point (frequency) estimate 1(ir) = Lij n~j logn:~:j for the mutual information (per sample). Unfortunately the point estimation 1(ir) gives no information about its accuracy. In the Bayesian approach to this problem one assumes a prior (second order) probability density p( 7r) for the unknown probabilities 7rij on the probability simplex. From this one can compute the posterior distribution p( 7rln) cxp( 7r) rr ij7r~;j (the nij are multinomially distributed). This allows to compute the posterior probability density of the mutual information.1 p(Iln) = f 8(1(7r) - I)p(7rln)dTS7r (2) 2The 80 distribution restricts the integral to 7r for which 1(7r) =1. For large sam1 I(7r) denotes the mutual information for the specific chances 7r, whereas I in the context above is just some non-negative real number. I will also denote the mutual information random variable in the expectation E [I] and variance Var[I]. Expectaions are always w.r.t. to the posterior distribution p(7rln). 2Since O~I(7r) ~Imax with sharp upper bound Imax :=min{logr,logs}, the integral may be restricted to J:mam, which shows that the domain of p(Iln) is [O,Imax] . pIe size n ---+ 00, p(7rln) is strongly peaked around 7r = it and p(Iln) gets strongly peaked around the frequency estimate I = I(it). The mean E[I] = fooo Ip(Iln) dI = f I(7r)p(7rln)dTs7r and the variance Var[I] =E[(I - E[I])2] = E[I2]- E[Ij2 are of central interest. 3 Results for I under the Dirichlet P (oste )rior Most3 non-informative priors for p(7r) lead to a Dirichlet posterior distribution ( I) IT nij -1 ·th· t t t· -,,, h ' th b p 7r n ex: ij 7rij WI III erpre a IOn nij - nij + nij , were nij are e num er of samples (i,j), and n~j comprises prior information (1 for the uniform prior, ~ for Jeffreys' prior, 0 for Haldane's prior, -?:s for Perks' prior [GCSR95]). In principle this allows to compute the posterior density p(Iln) of the mutual information. In sections 4 and 5 we expand the mean and variance in terms of n- 1 : E[I] ~ nij I nijn (r - 1)(8 - 1) O( -2) L...J og --- + + n , .. n ni+n+j 2n 'J (3) Var[I] The first term for the mean is just the point estimate I(it). The second term is a small correction if n » r· 8. Kleiter [KJ96, Kle99] determined the correction by Monte Carlo studies as min {T2~1 , 8;;;,1 }. This is wrong unless 8 or rare 2. The expression 2E[I]/n they determined for the variance has a completely different structure than ours. Note that the mean is lower bounded by co~st. +O(n- 2 ), which is strictly positive for large, but finite sample sizes, even if z and J are statistically independent and independence is perfectly represented in the data (I (it) = 0). On the other hand, in this case, the standard deviation u= y'Var(I) '" ~ ",E[I] correctly indicates that the mean is still consistent with zero. Our approximations (3) for the mean and variance are good if T~8 is small. The central limit theorem ensures that p(Iln) converges to a Gaussian distribution with mean E[I] and variance Var[I]. Since I is non-negative it is more appropriate to approximate p(II7r) as a Gamma (= scaled X2 ) or log-normal distribution with mean E[I] and variance Var[I], which is of course also asymptotically correct. A systematic expansion in n -1 of the mean, variance, and higher moments is possible but gets arbitrarily cumbersome. The O(n- 2) terms for the variance and leading order terms for the skewness and kurtosis are given in Section 7. For the mean it is possible to give an exact expression 1 E[I] = - L nij[1jJ(nij + 1) -1jJ(ni+ + 1) -1jJ(n+j + 1) + 1jJ(n + 1)] (4) n .. 'J with 1jJ(n+1)=-,),+L~= lt=logn+O(~) for integer n. See Section 6 for details and more general expressions for 1jJ for non-integer arguments. There may be other prior information available which cannot be comprised in a Dirichlet distribution. In this general case, the mean and variance of I can still be 3But not all priors which one can argue to be non-informative lead to Dirichlet posteriors. Brand [Bragg] (and others), for instance, advocate the entropic prior p( 7r) ex e-H(rr). related to the covariance structure of p(7fln), which will be done in the following Section. 4 Approximation of Expectation and Variance of I In the following let frij := E[7fij]. Since p( 7fln) is strongly peaked around 7f = fr for large n we may expand J(7f) around fr in the integrals for the mean and the variance. With I:::..ij :=7fij -frij and using L:ij7fij = 1 = L:ijfrij we get for the expansion of (1) ( fr .. ) 1:::..2 . 1:::..2 1:::..2 . J(7f) = J(fr) + 2)og ~ I:::..ij + L ----}J--L ~-L ~+O(1:::..3). (5) .. 7fi+7f+j .. 27fij . 27fi+ . 27f+j 2J 2J 2 J Taking the expectation, the linear term E[ I:::..ij ] = a drops out. The quadratic terms E[ I:::..ij I:::..kd = Cov( 7fij ,7fkl) are the covariance of 7f under distribution p( 7fln) and are proportional to n- 1 . It can be shown that E[1:::..3] ,,-,n-2 (see Section 7). [ ] ( A) 1", (bikbjl bik bjl) ( ) (-2) EJ = J7f +-~ -A- -A- -ACOV7fij,7fkl +On . 2 ijkl 7fij 7fi+ 7f +j (6) The Kronecker delta bij is 1 for i = j and a otherwise. The variance of J in leading order in n - 1 is (7) where :t means = up to terms of order n -2. So the leading order variance and the leading and next to leading order mean of the mutual information J(7f) can be expressed in terms of the covariance of 7f under the posterior distribution p(7fln). 5 The Second Order Dirichlet Distribution Noninformative priors for p(7f) are commonly used if no additional prior information is available. Many non-informative choices (uniform, Jeffreys' , Haldane's, Perks', prior) lead to a Dirichlet posterior distribution: 1 II n;j - 1 ( ) N(n) .. 7fij b 7f++ - 1 with normalization 2J N(n) (8) where r is the Gamma function, and nij = n~j + n~j, where n~j are the number of samples (i,j), and n~j comprises prior information (1 for the uniform prior, ~ for Jeffreys' prior, a for Haldane's prior, -!s for Perks' prior). Mean and covariance of p(7fln) are A E[] nij 7fij:= 7fij =-, n (9) Inserting this into (6) and (7) we get after some algebra for the mean and variance of the mutual information I(7r) up to terms of order n- 2 : E[I] J (r - 1)(8 - 1) O( -2) + 2(n + 1) + n , (10) Var[I] ~1 (K - J2) + 0(n-2), n+ (11) (12) (13) J and K (and L, M, P, Q defined later) depend on 7rij = ":,j only, i.e. are 0(1) in n. Strictly speaking we should expand n~l = ~+0(n-2), i.e. drop the +1, but the exact expression (9) for the covariance suggests to keep the +1. We compared both versions with the exact values (from Monte-Carlo simulations) for various parameters 7r. In most cases the expansion in n~l was more accurate, so we suggest to use this variant. 6 Exact Value for E[I] It is possible to get an exact expression for the mean mutual information E[I] under the Dirichlet distribution. By noting that xlogx= d~x,6I,6= l' (x = {7rij,7ri+ ,7r+j}), one can replace the logarithms in the last expression of (1) by powers. From (8) we see that E[ (7rij ),6] = ~i~:~ t~~~;l. Taking the derivative and setting ,8 = 1 we get d 1 E[7rij log 7rij] = d,8E[(7rij) ,6],6=l = ;;: 2:::: nij[1j!(nij + 1) -1j!(n + 1)]. "J The 1j! function has the following properties (see [AS74] for details) dlogf(z) f'(z) 1 1 1 1j!(z) = dz = f(z)' 1j!(z + 1) = log z + 2z 12z2 + O( Z4)' n - l 1 n 1 1j!(n) = -"( + L k' 1j!(n +~) = -"( + 2log2 + 2 L 2k _ l' (14) k=l k=l The value of the Euler constant "( is irrelevant here, since it cancels out. Since the marginal distributions of 7ri+ and 7r+j are also Dirichlet (with parameters ni+ and n+j) we get similarly 1 - L n+j[1j!(n+j + 1) -1j!(n + 1)]. n . J Inserting this into (1) and rearranging terms we get the exact expression4 1 E[I] = - L nij[1j!(nij + 1) -1j!(ni+ + 1) -1j!(n+j + 1) + 1j!(n + 1)] (15) n .. 4This expression has independently been derived in [WW93]. For large sample sizes, 'Ij;(z+ 1) ~ logz and (15) approaches the frequency estimate I(7r) as it should be. Inserting the expansion 'Ij;(z+ 1) = logz + 2\ + ... into (15) we also get the correction term (r - 11~s - 1) of (3). The presented method (with some refinements) may also be used to determine an exact expression for the variance of I(7f). All but one term can be expressed in terms of Gamma functions. The final result after differentiating w.r.t. (31 and (32 can be represented in terms of 'Ij; and its derivative 'Ij;' . The mixed term E[( 7fi+ )131 (7f +j )132] is more complicated and involves confluent hypergeometric functions, which limits its practical use [WW93] . 7 Generalizations A systematic expansion of all moments of p(Iln) to arbitrary order in n -1 is possible, but gets soon quite cumbersome. For the mean we already gave an exact expression (15), so we concentrate here on the variance, skewness and the kurtosis of p(Iln). The 3rd and 4th central moments of 7f under the Dirichlet distribution are ( )2( ) [27ra7rb7rc - 7ra7rbc5bc - 7rb7rcc5ca - 7rc7rac5ab + 7rac5abc5bc] n+l n+2 (16) ~2 [37ra7rb7rc7rd - jrc!!d!!a c5ab - A7rbjrdA7rac5ac - A7rbA7rcA7rac5ad (17) -7fa7fd7fbc5bc - 7fa7fc7fbc5bd - 7fa7fb7fcc5cd +7ra7rcc5abc5cd + 7ra7rbc5acc5bd + 7ra7rbc5adc5bc] + O(n-3) with a=ij, b= kl, ... E {1, ... ,r} x {1, ... ,8} being double indices, c5ab =c5ik c5jl , ... 7rij = n~j • Expanding D..k = (7f_7r)k in E[D..aD..b ... ] leads to expressions containing E[7fa7fb ... ], which can be computed by a case analysis of all combinations of equal/unequal indices a,b,c, ... using (8). Many terms cancel leading to the above expressions. They allow to compute the order n- 2 term of the variance of I(7f). Again, inspection of (16) suggests to expand in [(n+l)(n+2)]-1, rather than in n-2 . The variance in leading and next to leading order is Var[I] K - J2 + M + (r - 1)(8 1)(~ - J) - Q + O(n- 3) n + 1 (n + l)(n + 2) (18) M L (~- _1 _ _ _ 1_ +~) nij log nijn , ij nij ni+ n+j n ni+n+j (19) Q 2 l-L~· ij ni+n+j (20) J and K are defined in (12) and (13). Note that the first term ~+f also contains second order terms when expanded in n -1. The leading order terms for the 3rd and 4th central moments of p(Iln) are L .'""" nij I nij n ~- og--j n ni+n+j 32 [K - J 2F + O(n- 3 ), n from which the skewness and kurtosis can be obtained by dividing by Var[Ij3/2 and Var[IF respectively. One can see that the skewness is of order n- 1/ 2 and the kurtosis is 3 + 0 (n - 1). Significant deviation of the skewness from a or the kurtosis from 3 would indicate a non-Gaussian I. They can be used to get an improved approximation for p(Iln) by making, for instance, an ansatz and fitting the parameters b, c, jJ" and (j-2 to the mean, variance, skewness, and kurtosis expressions above. Po is the Normal or Gamma distribution (or any other distribution with Gaussian limit). From this, quantiles p(I>I*ln):= fI:'p(Iln) dI, needed in [KJ96, Kle99], can be computed. A systematic expansion of arbitrarily high moments to arbitrarily high order in n- 1 leads, in principle, to arbitrarily accurate estimates. 8 Numerics There are short and fast implementations of'if;. The code of the Gamma function in [PFTV92], for instance, can be modified to compute the 'if; function. For integer and half-integer values one may create a lookup table from (14). The needed quantities J, K, L, M, and Q (depending on n) involve a double sum, P only a single sum, and the r+s quantities Ji+ and J+j also only a single sum. Hence, the computation time for the (central) moments is of the same order O(r·s) as for the point estimate (1). "Exact" values have been obtained for representative choices of 7rij, r, s, and n by Monte Carlo simulation. The 7rij := Xij / x++ are Dirichlet distributed, if each Xij follows a Gamma distribution. See [PFTV92] how to sample from a Gamma distribution. The variance has been expanded in T~S, so the relative error Var [I]app"o.-Var[I] .. act of the approximation (11) and (18) are of the order of T'S and Var[Il e• act n (T~S)2 respectively, if z and J are dependent. If they are independent the leading term (11) drops itself down to order n -2 resulting in a reduced relative accuracy O( T~S) of (18). Comparison with the Monte Carlo values confirmed an accurracy in the range (T~S)1...2. The mean (4) is exact. Together with the skewness and kurtosis we have a good description for the distribution of the mutual information p(Iln) for not too small sample bin sizes nij' We want to conclude with some notes on useful accuracy. The hypothetical prior sample sizes n~j = {a, -!S' ~,1} can all be argued to be non-informative [GCSR95]. Since the central moments are expansions in n- 1 , the next to leading order term can be freely adjusted by adjusting n~j E [0 ... 1]. So one may argue that anything beyond leading order is free to will, and the leading order terms may be regarded as accurate as we can specify our prior knowledge. On the other hand, exact expressions have the advantage of being safe against cancellations. For instance, leading order of E[I] and E[I2] does not suffice to compute the leading order of Var[I]. Acknowledgements I want to thank Ivo Kwee for valuable discussions and Marco Zaffalon for encouraging me to investigate this topic. This work was supported by SNF grant 200061847.00 to Jiirgen Schmidhuber. References [AS74] [Bra99] [Bun96] [CT91] M. Abramowitz and 1. A. Stegun, editors. Handbook of mathematical functions. Dover publications, inc., 1974. M. Brand. Structure learning in conditional probability models via an entropic prior and parameter extinction. Neural Computation, 11(5):1155- 1182, 1999. W. Buntine. A guide to the literature on learning probabilistic networks from data. IEEE Transactions on Knowledge and Data Engineering, 8:195- 210, 1996. T. M. Cover and J. A. Thomas. Elements of Information Theory. Wiley Series in Telecommunications. John Wiley & Sons, New York, NY, USA, 1991. [GCSR95] A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin. Bayesian Data Analysis. Chapman, 1995. [Hec98] D. Heckerman. A tutorial on learning with Bayesian networks. Learnig in Graphical Models, pages 301-354, 1998. [KJ96] G. D. Kleiter and R. Jirousek. Learning Bayesian networks under the control of mutual information. Proceedings of the 6th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU-1996), pages 985- 990, 1996. [Kle99] G. D. Kleiter. The posterior probability of Bayes nets with strong dependences. Soft Computing, 3:162- 173, 1999. [PFTV92] 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, second edition, 1992. [SooOO] E. S. Soofi. Principal information theoretic approaches. Journal of the American Statistical Association, 95:1349- 1353, 2000. [WW93] D. R. Wolf and D. H. Wolpert. Estimating functions of distributions from A finite set of samples, part 2: Bayes estimators for mutual information, chisquared, covariance and other statistics. Technical Report LANL-LA-UR-93833, Los Alamos National Laboratory, 1993. Also Santa Fe Insitute report SFI-TR-93-07 -047.
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A Maximum-Likelihood Approach to Modeling Multisensory Enhancement Hans Colonius* Institut fUr Kognitionsforschung Carl von Ossietzky Universitat Oldenburg, D-26111 hans. colonius@uni-oldenburg.de Adele Diederich School of Social Sciences International University Bremen Bremen, D-28725 a. diederich@iu-bremen.de Abstract Multisensory response enhancement (MRE) is the augmentation of the response of a neuron to sensory input of one modality by simultaneous input from another modality. The maximum likelihood (ML) model presented here modifies the Bayesian model for MRE (Anastasio et al.) by incorporating a decision strategy to maximize the number of correct decisions. Thus the ML model can also deal with the important tasks of stimulus discrimination and identification in the presence of incongruent visual and auditory cues. It accounts for the inverse effectiveness observed in neurophysiological recording data, and it predicts a functional relation between uni- and bimodal levels of discriminability that is testable both in neurophysiological and behavioral experiments. 1 Introduction In a typical environment stimuli occur at various positions in space and time. In order to produce a coherent assessment of the external world an individual must constantly discriminate between signals relevant for action planning (targets) and signals that need no immediate response (distractors). Separate sensory channels process stimuli by modality, but an individual must determine which stimuli are related to one another, i.e., it is must construct a perceptual event by integrating information from several modalities. For example, stimuli that occur at the same time and space are likely to be interrelated by a common cause. However, if the visual and auditory cues are incongruent, e.g., when dubbing one syllable onto a movie showing a person mouthing a different syllable, listeners typically report hearing a third syllable that represents a combination of what was seen and heard (McGurk effect, cf. [1]). This indicates that cross-modal synthesis is particularly important for stimulus identification and discrimination, not only for detection. Evidence for multisensory integration at the neural level has been well documented in a series of studies in the mammalian midbrain by Stein, Meredith and Wallace (e.g., [2]; for a review, see [3]). The deep layers of the superior colliculus (DSC) • www.uni-oldenburg.de/psychologie /hans.colonius /index.html integrate multisensory input and trigger orienting responses toward salient targets. Individual DSC neurons can receive inputs from multiple sensory modalities (visual, auditory, and somatosensory), there is considerable overlap between the receptive fields of these individual multisensory neurons, and the number of neural impulses evoked depends on the spatial and temporal relationships of the multisensory stimuli. Multisensory response enhancement refers to the augmentation of the response of a DSC neuron to a multisensory stimulus compared to the response elicited by the most effective single modality stimulus. A quantitative measure of the percent enhancement is MRE = CM - SMmax x 100, SMmax (1) where CM is the mean number of impulses evoked by the combined-modality stimulus in a given time interval, and S Mmax refers to the response of the most effective single-modality stimulus (cf. [4]). Response enhancement in the DSC neurons can be quite impressive, with values of M RE sometimes reaching values above 1000. Typically, this enhancement is most dramatic when the unimodal stimuli are weak and/or ambiguous, a principle referred to in [4] as "inverse effectiveness" . Since DSC neurons play an important role in orienting responses (like eye and head movements) to exogenous target stimuli, it is not surprising that multisensory enhancement is also observed at the behavioral level in terms of, for example, a lowering of detection thresholds or a speed-up of (saccadic) reaction time (e.g., [5], [6], [7]; see [8] for a review). Inverse effectiveness makes intuitive sense in the behavioral situation: the detection probability for a weak or ambiguous stimulus gains more from response enhancement by multisensory integration than a highintensity stimulus that is easily detected by a single modality alone. A model of the functional significance of multisensory enhancement has recently been proposed by Anastasio, Patton, and Belkacem-Boussaid [9]. They suggested that the responses of individual DSC neurons are proportional to the Bayesian probability that a target is present given their sensory inputs. Here, this Bayesian model is extended to yield a more complete account of the decision situation an organism is faced with. As noted above, in a natural environment an individual is confronted with the task of discriminating between stimuli important for survival (" targets") and stimuli that are irrelevant (" distractors"). Thus, an organism must not only keep up a high rate of detecting targets but, at the same time, must strive to minimize "false alarms" to irrelevant stimuli. An optimally adapted system will be one that maximizes the number of correct decisions. It will be shown here that this can be achieved already at the level of individual DSC neurons by appealing to a maximum-likelihood principle, without requiring any more information than is assumed in the Bayesian model. The next section sketches the Bayesian model by Anastasio, Patton, and BelkacemBoussaid (Bayesian model, for short), after which a maximum-likelihood model of multisensory response enhancement will be introduced. 2 The Bayesian Model of Multisensory Enhancement DSC neurons receive input from the visual and auditory systems elicited by stimuli occurring within their receptive fields! According to the Bayesian model, these vii An extension to the trimodal situation, including somatosensory input, could be easily attained in the models discussed here. sual and auditory inputs are represented by random variables V and A, respectively. A binary random variable T indicates whether a signal is present (T = 1) or not (T = 0). The central assumption of the model is that a DSC neuron computes the Bayesian (posterior) probability that a target is present in its receptive field given its sensory input: P(T = I V = A = ) = P(V = v, A = a I T = I)P(T = 1) 1 v, a P(V = v, A = a) , (2) where v and a denote specific values of the sensory input variables. Analogous expressions hold for the two unimodal situations. The response of the DSC neuron (number of spikes in a unit time interval) is postulated to be proportional to these probabilities. In order to arrive at quantitative predictions two more specific assumptions are made: (1) the distributions of V and A, given T = 1 or T = 0, are conditionally independent, i.e., P(V = v, A = a I T) = P(V = v IT) P(A = a I T) for any v, a; (2) the distribution of V , given T = 1 or T = 0, is Poisson with Al or Ao, resp., and the distribution of A, given T = 1 or T = 0, is Poisson with {-tl or {-to, resp. The conditional independence assumption means that the visibility of a target indicates nothing about its audibility, and vice-versa. The choice of the Poisson distribution is seen as a reasonable first approximation that requires only one single parameter per distribution. Finally, the computation of the posterior probability that a target is present requires specification of the a-priori probability of a target, P(T = 1). The parameters Ao and {-to denote the mean intensity of the visual and auditory input, resp., when no target is present (spontaneous input), while Al and {-tl are the corresponding mean intensities when a target is present (driven input). By an appropriate choice of parameter values, Anastasio et al. [9] show that the Bayesian model reproduces values of multisensory response enhancement in the order of magnitude observed in neurophysiological experiments [10]. In particular, the property of inverse effectiveness, by which the enhancement is largest for combined stimuli that evoke only small unimodal responses, is reflected by the model. 3 The Maximum Likelihood Model of Multisensory Enhancement 3.1 The decision rule The maximum likelihood model (ML model, for short) incorporates the basic decision problem an organism is faced with in a typical environment: to discriminate between relevant stimuli (targets), i.e., signals that require immediate reaction, and irrelevant stimuli (distractors), i.e., signals that can be ignored in a given situation. In the signal-detection theory framework (cf. [11]), P(Yes I T = 1) denotes the probability that the organism (correctly) decides that a target is present (hit), while P(Yes I T = 0) denotes the probability of deciding that a target is present when in fact only a distractor is present (false alarm). In order to maximize the probability of a correct response, P(C) = P(Yes I T = 1) P(T = 1) + [1- P(Yes I T = O)]P(T = 0), (3) the following maximum likelihood decision rule must be adopted (cf. [12]) for, e.g., the unimodal visual case: If P(T = 11 V = v) > P(T = 0 I V = v), then decide "Yes", otherwise decide "No" . The above inequality is equivalent to P(T=IIV=v) P(T=I)P(v=vIT=I) P(T = 0 I V = v) P(T = 0) P(V = v IT = 0) > 1, where the right-most ratio is a function of V , L(V), the likelihood ratio. Thus, the above rule is equivalent to: If L(v) > 1 - P then decide "Yes" otherwise decide "No" , , , P with p = P(T = 1). Since L(V) is a random variable, the probability to decide "Yes" , given a target is present, is P (Yes I T = 1) = P (L(V) > 1; PIT = 1) . Assuming Poisson distributions, this equals with P (exP(Ao - Ad U~) v > ~ I T = 1) = P(V > ciT = 1), In (l;P) + Al - AO c=---'--------'-----;-----;--In U~) In analogy to the Bayesian model, the ML model postulates that the response of a DSC neuron (number of spikes in a unit time interval) to a given target is proportional to the probability to decide that a target is present computed under the optimal (maximum likelihood) strategy defined above. 3.2 Predictions for Hit Probabilities In order to compare the predictions of the ML model for unimodal vs. bimodal inputs, consider the likelihood ratio for bimodal Poisson input under conditional independence: L(V, A) P(V = v, A = a I T = 1) P(V = v, A = a I T = 0) exp(Ao _ Ad (~~) v exp(po _ pd (~~) A The probability to decide "Yes" given bimodal input amounts to, after taking logarithms, P (In (~~) V + In (~~) A > In (1; p) + Al - AO + PI -Po IT = 1) Table 1: Hit probabilities and MRE for different bimodal inputs Mean Driven Input Prob (Hit) Al J.Ll V Driven A Driven V A Driven MRE Low 6 7 .000 .027 .046 704 7 7 .027 .027 .117 335 8 8 .112 .112 .341 204 8 9 .112 .294 .528 79 8 10 .112 .430 .562 31 Medium 12 12 .652 .652 .872 33 12 13 .652 .748 .895 20 High 16 16 .873 .873 .984 13 16 20 .873 .961 .990 3 Note: A-priori target probability is set at p = O.l. Visual and auditory inputs have spontaneous means of 5 impulses per unit time. V Driven (A Driven, V A Driven) columns refer to the hit probabilities given a unimodal visual (resp. auditory, bimodal) target. Multisensory response enhancement (last column) is computed using Eq. (1) For Ad Ao = J.Ld J.Lo this probability is computed directly from the Poisson distribution with mean (AI + J.Ld. Otherwise, hit probabilities follow the distribution of a linear combination of two Poisson distributed variables. Table 1 presents2 hit probabilities and multisensory response enhancement values for different levels of mean driven input. Obviously, the ML model imitates the inverse effectiveness relation: combining weak intensity unimodal stimuli leads to a much larger response enhancement than medium or high intensity stimuli. 3.3 Predictions for discriminability measures The ML model allows to assess the sensitivity of an individual DSC neuron to discriminate between target and distract or signals. Intuitively, this sensitivity should be a (decreasing) function of the amount of overlap between the driven and the spontaneous likelihood (e.g., P(V = v IT = 1) and P(V = v I T = 0)). One possible appropriate measure of sensitivity for the Poisson observer is (cf. [12]) Al - Ao J.Ll J.Lo Dy = (AI AO)I/4 and DA = (J.LIJ.LO)l /4 (4) for the visual and auditory unimodal inputs, resp. A natural choice for the bimodal measure of sensitivity then is D (AI + J.Ll) (J.Lo + Ao) y A = [(AI + J.Ld(Ao + J.Lo)Jl/4 . (5) Note that, unlike the hit probabilities, the relative increase in discriminability by combining two unimodal inputs does not decrease with the intensity of the driven input (see Table 2). Rather, the relation between bimodal and unimodal discriminability measures for the input values in Table 2 is approximately of Euclidean 2For input combinations with >'1 =I- J.t1 hit probabilities are estimated from samples of 1,000 pseudo-random numbers. Table 2: Discriminability measure values and % increase for different bimodal inputs Mean Driven Input Discriminability Value Al J.Ll Dv DA DVA % Increase 7 7 .82 .82 1.16 41 8 8 1.19 1.19 1.69 41 8 10 1.19 1.88 2.18 16 12 12 2.52 2.52 3.57 41 16 16 3.68 3.68 5.20 41 16 20 3.68 4.74 5.97 26 Note: Visual and auditory inputs have spontaneous means of 5 impulses per unit time. % Increase of Dv A over Dv and DA (last column) is computed in analogy to Eq. (1) distance form: (6) For Al = J.Ll this amounts to Dv A = V2Dv yielding the 41 % increase in discriminability. The fact that the discriminability measures do not follow the inverse effectiveness rule should not be not surprising: whether two stimuli are easy or hard to discriminate depends on their signal-to-noise ratio, but not on the level of intensity. 4 Discussion and Conclusion The maximum likelihood model of multisensory enhancement developed here assumes that the response of a DSC neuron to a target stimulus is proportional to the hit probability under a maximum likelihood decision strategy. Obviously, no claim is made here that the neuron actually performs these computations, only that its behavior can be described approximately in this way. Similar to the Bayesian model suggested by Anastasio et al. [9], the neuron's behavior is solely based on the a-priori probability of a target and the likelihood function for the different sensory inputs. The ML model predicts the inverse effectiveness observed in neurophysiological experiments. Moreover, the model allows to derive a measure of the neuron's ability to discriminate between targets and non-targets. It makes specific predictions how un i- and bimodal discriminability measures are related and, thereby, opens up further avenues for testing the model assumptions. The ML model, like the Bayesian model, operates at the level of a single DSC neuron. However, an extension of the model to describe multisensory population responses is desirable: First, this would allow to relate the model predictions to numerous behavioral studies about multisensory effects (e.g., [13], [14]), and, second, as a recent study by Kadunce et al. [15) suggests, the effects of multisensory spatial coincidence observed in behavioral experiments may only be reconcilable with the degree of spatial resolution achievable by a population of DSC neurons with overlapping receptive fields. Moreover, this extension might also be useful to relate behavioral and single-unit recording results to recent findings on multisensory brain areas using functional imaging techniques (e.g., King and Calvert [16]). Acknowledgments This research was partially supported by a grant from Deutsche Forschungsgemeinschaft-SFB 517 Neurokognition to the first author. References [1] McGurk, H. & MacDonald, J. (1976). Hearing lips and seeing voices. Nature, 264, 746-748. [2] Wallace, M. T ., Meredith, M. A., & Stein, B. E. (1993) . Converging influences from visual, auditory, and somatosensory cortices onto output neurons of the superior colliculus. Journal of Neurophysiology, 69, 1797-1809. [3] Stein, B. E., & Meredith, M. A. (1996). The merging of the senses. Cambridge, MA: MIT Press. [4] Meredith, M. A. & Stein, B. E. (1986a). Spatial factors determine the activity of multisensory neurons in cat superior colliculus. Brain Research, 365(2), 350-354. [5] Frens, van Opstal, & van der Willigen (1995) . Spatial and temporal factors determine auditory-visual interactions in human saccadic eye movements. Perception fj Psychophysics, 57, 802-816. [6] Colonius, H. & Arndt, P. A. (2001). A two stage-model for visual-auditory interaction in saccadic latencies. Perception fj Psychophysics, 63, 126-147. [7] Stein, B. E., Meredith, M. A., Huneycutt, W. S., & McDade, L. (1989). Behavioral indices of multisensory integration: Orientation to visual cues is affected by auditory stimuli. Journal of Cognitive Neurosciences, 1, 12-24. [8] Welch, R. B., & Warren, D. H. (1986). Intersensory interactions. In K R. Boff, L. Kaufman, & J. P. Thomas (eds.), Handbook of perception and human performance, Volume I: Sensory process and perception (pp. 25-1-25-36) New York: Wiley [9] Anastasio" T. J., Patton, P. E., & Belkacem-Boussaid, K (2000). Using Bayes' rule to model multisensory enhancement in the superior colliculus. Neural Computation, 12, 1165-1187. [10] Meredith, M. A. & Stein, B. E. (1986b). Visual, auditory, and somatosensory convergence on cells in superior colliculus results in multisensory integration. Journal of Neurophysiology, 56(3), 640-662. [11] Green, D. M., & Swets, J. A. (1974). Signal detection theory and psychophysics. New York: Krieger Pub!. Co. [12] Egan, J. P. (1975) . Signal detection theory and ROC analysis. New York: Academic Press. [13] Craig, A., & Colquhoun, W. P. (1976). Combining evidence presented simultaneously to the eye and the ear: A comparison of some predictive models. Perception fj Psychophysics, 19, 473-484. [14] Stein, B. E., London, N., Wilkinson, L. K , & Price, D. D. (1996). Enhancement of perceived visual intensity by auditory stimuli: A psychophysical analysis. Journal of Cognitive Neuroscience, 8, 497-506. [15] Kadunce, D. C., Vaughan, J. W ., Wallace, M. T ., & Stein, B. E. (2001) . The influence of visual and auditory receptive field organization on multisensory integration in the superior colliculus. Experimental Brain Research, 139, 303-310. [16] King, A. J., & Calvert, G. A. (2001). Multisensory integration: Perceptual grouping by eye and ear. Current Biology, 11, 322-325.
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Model-Free Least Squares Policy Iteration Michail G. Lagoudakis Department of Computer Science Duke University Durham, NC 27708 mgl@cs.duke.edu Ronald Parr Department of Computer Science Duke University Durham, NC 27708 parr@cs.duke.edu Abstract We propose a new approach to reinforcement learning which combines least squares function approximation with policy iteration. Our method is model-free and completely off policy. We are motivated by the least squares temporal difference learning algorithm (LSTD), which is known for its efficient use of sample experiences compared to pure temporal difference algorithms. LSTD is ideal for prediction problems, however it heretofore has not had a straightforward application to control problems. Moreover, approximations learned by LSTD are strongly influenced by the visitation distribution over states. Our new algorithm, Least Squares Policy Iteration (LSPI) addresses these issues. The result is an off-policy method which can use (or reuse) data collected from any source. We have tested LSPI on several problems, including a bicycle simulator in which it learns to guide the bicycle to a goal efficiently by merely observing a relatively small number of completely random trials. 1 Introduction Linear least squares function approximators offer many advantages in the context of reinforcement learning. While their ability to generalize is less powerful than black box methods such as neural networks, they have their virtues: They are easy to implement and use, and their behavior is fairly transparent, both from an analysis standpoint and from a debugging and feature engineering standpoint. When linear methods fail, it is usually relatively easy to get some insight into why the failure has occurred. Our enthusiasm for this approach is inspired by the least squares temporal difference learning algorithm (LSTD) [4]. LSTD makes efficient use of data and converges faster than other conventional temporal difference learning methods. Although it is initially appealing to attempt to use LSTD in the evaluation step of a policy iteration algorithm, this combination can be problematic. Koller and Parr [5] present an example where the combination of LSTD style function approximation and policy iteration oscillates between two bad policies in an MDP with just 4 states. This behavior is explained by the fact that linear approximation methods such as LSTD compute an approximation that is weighted by the state visitation frequencies of the policy under evaluation. Further, even if this problem is overcome, a more serious difficulty is that the state value function that LSTD learns is of no use for policy improvement when a model of the process is not available. This paper introduces the Least Squares Policy Iteration (LSPI) algorithm, which extends the benefits of LSTD to control problems. First, we introduce LSQ, an algorithm that learns least squares approximations of the state-action ( ) value function, thus permitting action selection and policy improvement without a model. Next we introduce LSPI which uses the results of LSQ to form an approximate policy iteration algorithm. This algorithm combines the policy search efficiency of policy iteration with the data efficiency of LSTD. It is completely off policy and can, in principle, use data collected from any reasonable sampling distribution. We have evaluated this method on several problems, including a simulated bicycle control problem in which LSPI learns to guide the bicycle to the goal by observing a relatively small number of completely random trials. 2 Markov Decision Processes We assume that the underlying control problem is a Markov Decision Process (MDP). An MDP is defined as a 4-tuple
where: is a finite set of states; is a finite set of actions; is a Markovian transition model where
represents the probability of going from state to state with action ; and is a reward function IR, such that !
represents the reward obtained when taking action in state and ending up in state " . We will be assuming that the MDP has an infinite horizon and that future rewards are discounted exponentially with a discount factor #%$'& ()+*"
. (If we assume that all policies are proper, our results generalize to the undiscounted case.) A stationary policy , for an MDP is a mapping ,-./ 10 , where ,2!"
is the action the agent takes at state . The state-action value function 43 5
is defined over all possible combinations of states and actions and indicates the expected, discounted total reward when taking action in state and following policy , thereafter. The exact -values for all state-action pairs can be found by solving the linear system of the Bellman equations : 3 !6 7
28'9:!6 7
<;=#>@?BA !
3 ! ,2
C where 9:7
D8FE ? A GH
I J!6 KL
. In matrix format, the system becomes M3 8 9N;O#QP 3R 3 , where S3 and 9 are vectors of size T DTUT 0MT and P 3 is a stochastic matrix of size T DTLT 0 T7OT DTLT 0MT
. P 3 describes the transitions from pairs 7
to pairs !@,2!H
@
. For every MDP, there exists an optimal policy, ,V , which maximizes the expected, discounted return of every state. Policy iteration is a method of discovering this policy by iterating through a sequence of monotonically improving policies. Each iteration consists of two phases. Value determination computes the state-action values for a policy ,.WYX[Z by solving the above system. Policy improvement defines the next policy as ,.WYX[\^]@ZC"
J8`_a bc_d)e 3fhgLi 7
. These steps are repeated until convergence to an optimal policy, often in a surprisingly small number of steps. 3 Least Squares Approximation of Q Functions Policy iteration relies upon the solution of a system of linear equations to find the Q values for the current policy. This is impractical for large state and action spaces. In such cases we may wish to approximate 43 with a parametric function approximator and do some form of approximate policy iteration. We now address the problem of finding a set of parameters that maximizes the accuracy of our approximator. A common class of approximators is the so called linear architectures, where the value function is approximated as a linear weighted combination of basis functions (features): 3 !
8 > ] !6 7
8 !7
S where is a set of weights (parameters). In general,
-T DTLT 0 T and so, the linear system above now becomes an overconstrained system over the parameters : 9F;O#QP 3 # P 3
9 where is a T DTLT 0MT! )
matrix. We are interested in a set of weights 3 that yields a fixed point in value function space, that is a value function 3 8 3 that is invariant under one step of value determination followed by orthogonal projection to the space spanned by the basis functions. Assuming that the columns of are linearly independent this is
] 9N;O#QP 3 3
8 3 8 3 8 # P 3
] 9 We note that this is the standard fixed point approximationmethod for linear value functions with the exception that the problem is formulated in terms of Q values instead of state values. For any P 3 , the solution is guaranteed to exist for all but finitely many # [5]. 4 LSQ: Learning the State-Action Value Function In the previous section we assumed that a model M P 3
of the underlying MDP is available. In many practical applications, such a model is not available and the value function or, more precisely, its parameters have to be learned from sampled data. These sampled data are tuples of the form: L
, meaning that in state , action was taken, a reward was received, and the resulting state was . These data can be collected from actual (sequential) episodes or from random queries to a generative model of the MDP. In the extreme case, they can be experiences of other agents on the same MDP. We know that the desired set of weights can be found as the solution of the system, 3 8! , where 8 # P 3
and 8 9 . The matrix P 3 and the vector 9 are unknown and so, and cannot be determined a priori. However, and can be approximated using samples. Recall that , P 3 , and 9 are of the form: "$# %& & & ' (*),+.-0/213- 4 5 60606 (*),+./7184 5 60606 (*),+:9 ;<9=/ 139 >?9=4 5 @ A A A B CED "$# %& & & ' FHG A:I ),+.-0/ 1J-0/ +0KL42(?),+0KM/ N),+K4 4 5 6O606 FPG A I ),+./21J/+ K 42(?),+ K / NQ),+ K 4 4 5 6O606 FPG A I ),+ 9 ;<9 /21 9 >*9 /+ K 42(?),+ K / NQ),+ K 4 4 5 @ A A A B RS# % & & & ' F G ATI ),+ /21 /+ K 42UV),+ /21 / + K 4 606O6 FHG A I ),+:/71J/+ K 42UV),+./ 1J/ + K 4 606O6 F G A I ),+T9 ;J9=/ 139 >?9W/+0KL42UV),+:9 ;<9X/ 139 >?9Y/+0K4 @ A A A B Given a set of samples, Z 8\["^]0_GJ]0_ K ]0_ :]0_@
+T<`M8 *6OaRbcb dfe , where the :]0_G J]0_I
are sampled from according to distribution g and the 6 ] _ are sampled according to K ]0_ T :]0_GJ]0_I
, we can construct approximate versions of , P 3 , and 9 as follows : 8 hi i i j ]lk ]lk
bc ] _C ] _I
bc !:]m <]m
nbo o o p q P 3 8 hi i i i j !K ]lk @,sr K ]lk
t cb K ] _ @, r K ] _
t cb !H ]m @, r K ]m
t nbo o o o p 9 8 hi i i j ]lk bb ] _ bb :]m nbo o o p These approximations can be thought of as first sampling rows from according to g and then, conditioned on these samples, as sampling terms from the summations in the corresponding rows of P 3 and 9 . The sampling distribution from the summations is governed by the underlying dynamics ( U
) of the process as the samples in Z are taken directly from the MDP. Given , q P 3 , and 9 , and can be approximated as 8 # q P 3
and 8 9 With d uniformly distributed samples over pairs of states and actions !7
, the approximations and are consistent approximations of the true and :
8 d T TUT T and
28 d T TUT T The Markov property ensures that the solution 3 will converge to the true solution 3 for sufficiently large d whenever 3 exists: 3
8 ] C
8 d T DTUT T
] d T DTUT T
28 ] 8$ 3 In the more general case, where g is not uniform, we will compute a weighted projection, which minimizes the g weighted distance in the projection step. Thus, state is implicitly assigned weight g!K
and the projection minimizes the weighted sum of squared errors with respect to g . In LSTD, for example, g is the stationary distribution of P 3 , giving high weight to frequently visited states, and low weight to infrequently visited states. As with LSTD, it is easy to see that approximations ( ] ] 6 ) derived from different sets of samples ( Z ] Z ) can be combined additively to yield a better approximation that corresponds to the combined set of samples: 8 ] ; and 8 ] ; This observation leads to an incremental update rule for and . Assume that initially 8 ( and 8 ( . For a fixed policy, a new sample !6
contributes to the approximation according to the following update equation : ; 5
+ 5
# ! ,2
@
and .; 7
We call this new algorithm LSQ due to its similarity to LSTD. However, unlike LSTD, it computes Q functions and does not expect the data to come from any particular Markov chain. It is a feature of this algorithm that it can use the same set of samples to compute Q values for any policy representation that offers an action choice for each in the set. The policy merely determines which !,2K!@,2!H
@
is added to for each sample. Thus, LSQ can use every single sample available to it no matter what policy is under evaluation. We note that if a particular set of projection weights are desired, it is straightforward to reweight the samples as they are added to . Notice that apart from storing the samples, LSQ requires only J7
space independently of the size of the state and the action space. For each sample in Z , LSQ incurs a cost of J
to update the matrices and and a one time cost of J7 "
to solve the system and find the weights. Singular value decomposition (SVD) can be used for robust inversion of as it is not always a full rank matrix. LSQ includes LSTD as a special case where there is only one action available. It is also possible to extend LSQ to LSQ( ) in a way that closely resembles LSTD( ) [3], but in that case the sample set must consist of complete episodes generated using the policy under evaluation, which again raises the question of bias due to sampling distribution, and prevents the reusability of samples. LSQ is also applicable in the case of infinite and continuous state and/or action spaces with no modification. States and actions are reflected only through the basis functions of the linear approximation and the resulting value function can cover the entire state-action space with the appropriate set of continuous basis functions. 5 LSPI: Least Squares Policy Iteration The LSQ algorithm provides a means of learning an approximate state-action value function, S3 !6 7
, for any fixed policy , . We now integrate LSQ into an approximate policy iteration algorithm. Clearly, LSQ is a candidate for the value determination step. The key insight is that we can achieve the policy improvement step without ever explicitly representing our policy and without any sort of model. Recall that in policy improvement, , WYX[\^]@Z will pick the action that maximizes 43 !7
. Since LSQ computes Q functions directly, we do not need a model to determine our improved policy; all the information we need is contained implicitly in the weights parameterizing our Q functions1: , WUX[\^]IZ !
8 _a bc_d e 7
8 _a b2c_d e !7
We close the loop simply by requiring that LSQ performs this maximization for each when constructing the matrix for a policy. For very large or continuous action spaces, explicit maximization over may be impractical. In such cases, some sort of global nonlinear optimization may be required to determine the optimal action. Since LSPI uses LSQ to compute approximate Q functions, it can use any data source for samples. A single set of samples may be used for the entire optimization, or additional samples may be acquired, either through trajectories or some other scheme, for each iteration of policy iteration. We summarize the LSPI algorithm in Figure 1. As with any approximate policy iteration algorithm, the convergence of LSPI is not guaranteed. Approximate policy iteration variants are typically analyzed in terms of a value function approximation error and an action selection error [2]. LSPI does not require an approximate policy representation, e.g., a policy function or “actor” architecture, removing one source of error. Moreover, the direct computation of linear Q functions from any data source, including stored data, allows the use of all available data to evaluate every policy, making the problem of minimizing value function approximation error more manageable. 6 Results We initially tested LSPI on variants of the problematic MDP from Koller and Parr [5], essentially simple chains of varying length. LSPI easily found the optimal policy within a few iterations using actual trajectories. We also tested LSPI on the inverted pendulum problem, where the task is to balance a pendulum in the upright position by moving the cart to which it is attached. Using a simple set of basis functions and samples collected from random episodes (starting in the upright position and following a purely random policy), LSPI was able to find excellent policies using a few hundred such episodes [7]. Finally, we tried a bicycle balancing problem [12] in which the goal is to learn to balance and ride a bicycle to a target position located 1 km away from the starting location. Initially, the bicycle’s orientation is at an angle of 90 to the goal. The state description is a sixdimensional vector D
, where is the angle of the handlebar, is the vertical 1This is the same principle that allows action selection without a model in Q-learning. To our knowledge, this is the first application of this principle in an approximate policy iteration algorithm. LSPI ( /( / //*N./ ) // : Number of basis functions // ( : Basis functions // : Discount factor // : Stopping criterion // N : Initial policy, given as , N # N),+./ 4 (default: # ) // : Initial set of samples, possibly empty # N K # N // In essence, K # repeat Update (optional) // Add/remove samples, or leave unchanged N # N K // # K N K = LSQ ( / /( / /*N ) // K = LSQ ( / /( /
/ ) until ( NPN K ) // that is, ( K ) return N // return Figure 1: The LSPI algorithm. angle of the bicycle, and is the angle of the bicycle to the goal. The actions are the torque applied to the handlebar (discretized to [ a7 (RC; a<e ) and the displacement of the rider (discretized to [ (3 ( aR()G;(3 ( aJe ). In our experiments, actions are restricted to be either or (or nothing) giving a total of 5 actions2. The noise in the system is a uniformly distributed term in & ( (8a7G;( (8a added to the displacement component of the action. The dynamics of the bicycle are based on the model described by Randløv and Alstrøm [12] and the time step of the simulation is set to (3 (R* seconds. The state-action value function !7
for a fixed action is approximated by a linear combination of 20 basis functions: <* D D
C where 8 , for ( and 8 , for ( . Note that the state variable is completely ignored. This block of basis functions is repeated for each of the 5 actions, giving a total of 100 basis functions and weights. Training data were collected by initializing the bicycle to a random state around the equilibrium position and running small episodes of 20 steps each using a purely random policy. LSPI was applied on training sets of different sizes and the average performance is shown in Figure 2(a). We used the same data set for each run of policy iteration and usually obtained convergence in 6 or 7 iterations. Successful policies usually reached the goal in approximately 1 km total, near optimal performance. We also show an annotated set of trajectories to demonstrate the performance improvement over multiple steps of policy iteration in Figure 2(b). The following design decisions influenced the performance of LSPI on this problem: As is typical with this problem, we used a shaping reward [10] for the distance to the goal. In this case, we used (3 (R* of the net change (in meters) in the distance to the goal. We found that when using full random trajectories, most of our sample points were not very useful; they occurred after the bicycle had already entered into a “death spiral” from which recovery was impossible. This complicated our learning efforts by biasing the samples towards hopeless parts of the space, so we decided to cut off trajectories after 20 steps. This created an additional problem because there was no terminating reward signal to indicate failure. We approximated this with an additional shaping reward, which was proportional to the 2Results are similar for the full 9-action case, but required more training data. 0 500 1000 1500 2000 2500 3000 0 10 20 30 40 50 60 70 80 90 100 Number of training episodes Percentage of trials reaching the goal −200 0 200 400 600 800 1000 1200 −800 −600 −400 −200 0 200 1st iteration 2nd iteration (crash) 4th and 8th iteration 5th and 7th iteration 3rd iteration 6th iteration (crash) Starting Position Goal (a) (b) Figure 2: The bicycle problem: (a) Percentage of final policies that reach the goal, averaged over 200 runs of LSPI for each training set size; (b) A sample run of LSPI based on 2500 training trials. This run converged in 8 iterations. Note that iterations 5 and 7 had different Q-values but very similar policies. This was true of iterations 4 and 8 as well. The weights of the ninth differed from the eighth by less than *H( ] in , indicate convergence. The curves at the end of the trajectories indicating where the bicycle has looped back for a second pass through the goal. net change in the square of the vertical angle. This roughly approximated the likeliness of falling at the end of a truncated trajectory. Finally, we used a discount of ( ( , which seemed to yield more robust performance. We admit to some slight unease about the amount of shaping and adjusting of parameters that was required to obtain good results on this problem. To verify that we had not eliminated the learning problem entirely through shaping, we reran some experiments using a discount of ( . In this case LSQ simply projects the immediate reward function into the column space of the basis functions. If the problem were tweaked too much, acting to maximize the projected immediate reward would be sufficient to obtain good performance. On the contrary, these runs always produced immediate crashes in trials. 7 Discussion and Conclusions We have presented a new, model-free approximate policy iteration algorithm called LSPI, which is inspired by the LSTD algorithm. This algorithm is able to use either a stored repository of samples or samples generated dynamically from trajectories. It performs action selection and approximate policy iteration entirely in value function space, without any need for model. In contrast to other approaches to approximate policy iteration, it does not require any sort of approximate policy function. In comparison to the memory based approach of Ormoneit and Sen [11], our method makes stronger use of function approximation. Rather than using our samples to implicitly construct an approximate model using kernels, we operate entirely in value function space and use our samples directly in the value function projection step. As noted by Boyan [3] the matrix used by LSTD and LSPI can be viewed as an approximate, compressed model. This is most compelling if the columns of are orthonormal. While this provides some intuitions, a proper transition function cannot be reconstructed directly from , making a possible interpretation of LSPI as a model based method an area for future research. In comparison to direct policy search methods [9, 8, 1, 13, 6], we offer the strength of policy iteration. Policy search methods typically make a large number of relatively small steps of gradient-based policy updates to a parameterized policy function. Our use of policy iteration generally results in a small number of very large steps directly in policy space. Our experimental results demonstrate the potential of our method. We achieved good performance on the bicycle task using a very small number of randomly generated samples that were reused across multiple steps of policy iteration. Achieving this level of performance with just a linear value function architecture did require some tweaking, but the transparency of the linear architecture made the relevant issues much more salient than would be the case with any “black box” approach. We believe that the direct approach to function approximation and data reuse taken by LSPI will make the algorithm an intuitive and easy to use first choice for many reinforcement learning tasks. In future work, we plan to investigate the application of our method to multi-agent systems and the use of density estimation to control the projection weights in our function approximator. Acknowledgments We would like to thank J. Randløv and P. Alstrøm for making their bicycle simulator available. We also thank C. Guestrin, D. Koller, U. Lerner and M. Littman for helpful discussions. The first author would like to thank the Lilian-Boudouri Foundation in Greece for partial financial support. References [1] J. Baxter and P.Bartlett. Reinforcement learning in POMDP’s via direct gradient ascent. In Proc. 17th International Conf. on Machine Learning, pages 41–48. Morgan Kaufmann, San Francisco, CA, 2000. [2] D. Bertsekas and J. Tsitsiklis. Neuro-Dynamic Programming. Athena Scientific, Belmont, Massachusetts, 1996. [3] Justin A. Boyan. Least-squares temporal difference learning. In I. Bratko and S. Dzeroski, editors, Machine Learning: Proceedings of the Sixteenth International Conference, pages 49– 56. Morgan Kaufmann, San Francisco, CA, 1999. [4] S. Bradtke and A. Barto. Linear least-squares algorithms for temporal difference learning. Machine Learning, 22(1/2/3):33–57, 1996. [5] D. Koller and R. Parr. Policy iteration for factored mdps. In Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence (UAI-00). Morgan Kaufmann, 2000. [6] V. Konda and J. Tsitsiklis. Actor-critic algorithms. In NIPS 2000 editors, editor, Advances in Neural Information Processing Systems 12: Proceedings of the 1999 Conference. MIT Press, 2000. [7] M. G. Lagoudakis and R. Parr. Model-Free Least-Squares policy iteration. Technical Report CS-2001-05, Department of Computer Science, Duke University, December 2001. [8] A. Ng and M. Jordan. PEGASUS: A policy search method for large MDPs and POMDPs. In Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence (UAI-00). Morgan Kaufmann, 2000. [9] A. Ng, R. Parr, and D. Koller. Policy search via density estimation. In Advances in Neural Information Processing Systems 12: Proceedings of the 1999 Conference. MIT Press, 2000. [10] Andrew Y. Ng, Daishi Harada, and Stuart Russell. Policy invariance under reward transformations: theory and application to reward shaping. In Proc. 16th International Conf. on Machine Learning, pages 278–287. Morgan Kaufmann, San Francisco, CA, 1999. [11] D. Ormoneit and S. Sen. Kernel-based reinforcement learning. To appear, Machine Learning, 2001. [12] J. Randløv and P. Alstrøm. Learning to drive a bicycle using reinforcement learning and shaping. In The Fifteenth International Conference on Machine Learning, 1998. Morgan Kaufmann. [13] R. Sutton, D. McAllester, S. Singh, and Y. Mansour. Policy gradient methods for reinforcement learning with function approximation. In Advances in Neural Information Processing Systems 12: Proceedings of the 1999 Conference, 2000. MIT Press.
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Fragment completion in humans and machines David Jacobs NEC Research Institute 4 Independence Way, Princeton, NJ 08540 dwj@research.nj.nec.com Bas Rokers Psychology Department at UCLA PO Box 951563, Los Angeles, CA 90095 rokers@psych.ucla.edu Archisman Rudra CS Department at NYU 251 Mercer St., New York, NY 10012 archi@cs.nyu.edu Zili Liu Psychology Department at UCLA PO Box 951563, Los Angeles CA 90095 zili@psych.ucla.edu Abstract Partial information can trigger a complete memory. At the same time, human memory is not perfect. A cue can contain enough information to specify an item in memory, but fail to trigger that item. In the context of word memory, we present experiments that demonstrate some basic patterns in human memory errors. We use cues that consist of word fragments. We show that short and long cues are completed more accurately than medium length ones and study some of the factors that lead to this behavior. We then present a novel computational model that shows some of the flexibility and patterns of errors that occur in human memory. This model iterates between bottom-up and top-down computations. These are tied together using a Markov model of words that allows memory to be accessed with a simple feature set, and enables a bottom-up process to compute a probability distribution of possible completions of word fragments, in a manner similar to models of visual perceptual completion. 1 Introduction This paper addresses the problem of retrieving items in memory from partial information. Human memory is remarkable for its flexibility in handling a wide range of possible retrieval cues. It is also very accurate, but not perfect; some cues are more easily used than others. We hypothesize that memory errors occur in part because a trade-off exists between memory accuracy and the complexity of neural hardware needed to perform complicated memory tasks. If this is true, we can gain insight into mechanisms of human memory by studying the patterns of errors humans make, and we can model human memory with systems that produce similar patterns as a result of constraints on computational resources. We experiment with word memory questions of the sort that arise in a game called superghost. Subjects are presented with questions of a form: ‘*p*l*c*’. They must find a valid English word that matches this query, by replacing each ‘*’ with zero or more letters. So for this example, ‘place’, ’application’, and ‘palace’ would all be valid answers. In effect, the subject is given a set of letters and must think of a word that contains all of those letters, in that order, with other letters added as needed. Most of the psychological literature on word completion involves the effects of priming certain responses with recent experience (Shacter and Tulving[18]). However, priming is only able to account for about five percent of the variance in a typical fragment completion task (Olofsson and Nyberg[13], Hintzman and Hartry[6]). We describe experiments that show that the difficulty of a query depends on what we call its redundancy. This measures the extent to which all the letters in the query are needed to find a valid answer. We show that when we control for the redundancy of queries, we find that the difficulty of answering questions increases with their length; queries with many letters tend to be easy only because they tend to be highly redundant. We then describe a model that mimics these and other properties of human memory. Our model is based on the idea that a large memory system can gain efficiency by keeping the comparison between input and items in memory as simple as possible. All comparisons use a small, fixed set of features. To flexibly handle a range of queries, we add a bottomup process that computes the probability that each feature is present in the answer, given the input and a generic, Markov model of words. So the complexity of the bottom-up computation does not grow with the number of items in memory. Finally, the system is allowed to iterate between this bottom up and a top down process, so that a new generic model of words is constructed based on a current probability distribution over all words in memory, and this new model is combined with the input to update the probability that each feature is present in the answer. Previous psychological research has compared performance of word-stem and wordfragment completion. In the former a number of letters (i.e. a fragment) is given beginning with the first letter(s) of the word. In the latter, the string of letters given may begin at any point in the word, and adjacent letters in the fragment do not need, but may, be adjacent in the completed word. For example, for stem completion the fragment “str” may be completed into “string”, but for fragment completion also into “satire”. Performance for wordfragment completion is lower than word-stem completion (Olofsson and Nyberg[12]). In addition words, for which the ending fragment is given, show performance closer to wordstem completion than to word-fragment completion (Olofsson and Nyberg[13]). Seidenberg[17] proposed a model based on tri-grams. Srinivas et al.[21] indicate that assuming orthographic encoding is in most cases sufficient to describe word completion performance in humans. Orthographic Markov models of words have often been used computationally, as, for example, in Shannon’s[19] famous work. Following this work, our model is also orthographic. We find that a bigram rather than a trigram representation is sufficient, and leads to a simpler model. Contradicting evidence exists for the influence of fragment length on word completion. Oloffsson and Nyberg [12] failed to find a difference between two and three letter fragments on words of length of five to eight letters. However this might have been due to the fact that in their task, each fragment has a unique completion. Many recurrent neural networks have been proposed as models of associative memory (Anderson[1] contains a review). Perhaps most relevant to our work are models that use an input query to activate items from a complete dictionary in memory, and then use these items to alter the activations of the input. For example, in the Interactive Activation model of Rumelhart and McClelland[16], the presence of letters activates words, which boost the activity of the letters they contain. In Adaptive Resonance models (Carpenter and Grossberg[3]) activated memory items are compared to the input query and de-activated if they do not match. Also similar in spirit to our approach is the bidirectional model of Kosko[10] (for more recent work see, eg., Sommer and Palm[20]). Other models iteratively combine top-down and bottom-up information (eg., Hinton et al.[5], Rao and Ballard[14]), although these are not used as part of a memory system with complete items stored in memory. Our model differs from all of these in using a Markov model as an intermediate layer between the input and the dictionary. This allows the model to answer superghost queries, and leads to different computational mechanisms that we will detail. We find that superghost queries seem more natural to people than associative memory word problems (compare the superghost query “think of a word with an a” to the associative memory query “think of a word whose seventh letter is an a”). However, it is not clear how to extend most models of associative memory to handle superghost problems. Our use of features is more related to feedforward neural nets, and especially the “information bottleneck” approach of Tishby, Pereira and Bialek[22] (see also Baum, et al.[2]). Our work differs from feedforward methods in that our method is iterative, and uses features symmetrically to relate the memory to input in both directions. Our approach is also related to work on visual object recognition that combines perceptual organization and top-down knowledge (see Ullman[23]). Our model is inspired by Mumford’s[11] and Williams and Jacobs’[24] use of Markov models of contours for bottom-up perceptual completion. Especially relevant to our work is that of Grimes and Mozer[4]. Simultaneous with our work ([8]) they use a bigram model to solve anagram problems, in which letters are unscrambled to match words in a dictionary. They also use a Markov model to find letter orderings that conform with the statistics of English spelling. Their model is quite different in how this is done, due to the different nature of the anagram problem. They view anagram solving as a mix of low-level processing and higher level cognitive processes, while it is our goal to focus just on lower level memory. 2 Experiments with Human Subjects In our experiments, fragments and matching words were drawn from a large standard corpus of English text. The frequency of a word is the number of times it appears in this corpus. The frequency of a fragment is the sum of the frequency of all words that the fragment matches. We used fragments of length two to eight, discarding any fragments with frequency lower than one thousand. Fragments selected for an experiment were presented in random order. In our first experiment we systematically varied the length of the fragments, but otherwise selected them from a uniform, random distribution. Consequently, shorter fragments tended to match more words, with greater total frequency. In the second experiment, fragments were selected so that a uniform distribution of frequencies was ensured over all fragment lengths. For example, we used length two fragments that matched unusually few words. As a result the average frequency in experiment two is also much lower than in experiment one. A fragment was presented on a computer screen with spaces interspersed, indicating the possibility of letter insertion. The subject was required to enter a word that would fit the fragment. A subject was given 10 seconds to produce a completion, with the possibility to give up. For each session 50 fragments were presented, with a similar number of fragments of each length. Reaction times were recorded by measuring the time elapsed between the fragment first appearing on screen and the subject typing the first character of a matching word. Words that did not match the fragment or did not exist in the corpus were marked as not completed. Each experiment was completed by thirty-one subjects. The subjects were undergraduate students at Rutgers University, participating in the experiment for partial credit. Total time 1 2 3 4 5 6 7 8 9 0 0.2 0.4 0.6 0.8 1 Fragment Length Fraction Completed 1 2 3 4 5 6 7 8 9 0 0.2 0.4 0.6 0.8 1 Fragment Length Fraction Completed 1 2 3 4 5 6 7 8 9 0 0.2 0.4 0.6 0.8 1 Fragment Length Fraction Completed R0 R1 R2 R3 R4 Figure 1: Fragment completion as a function of fragment length for randomly chosen cues (top-left) and cues of equal frequency (top-right). On the bottom, the equal frequency cues are divided into five groups, from least redundancy (R0) to most (R5) . spent on the task varied from 15 minutes to close to one hour. Results For each graph we plot the number of fragments completed divided by the number of fragments presented (Figure 1). Error bars are calculated as
, where is the percent correct in the sample, and
is the number of trials. This assumes that all decisions are independent and correct with probability ; more precise results can be obtained by accounting for between-subject variance, but roughly the same results hold. For random, uniformly chosen fragments, there is a U-shaped dependence of performance on length. Controlling for frequency reduces performance because on average lower frequency fragments are selected. The U-shaped curve is flattened, but persists; hence Ushaped performance is not just due to frequency Finally, we divide the fragments from the two experiments into five groups, according to their redundancy. This is a rough measure of how important each letter is in finding a correct answer to the overall question. It is the probability that if we randomly delete a letter from the fragment and find a matching word, that this word will match the full fragment. Specifically, let denote the frequency of a query fragment of length (total frequency of words that match it). Let denote the frequency of the fragment that results when we delete the ’th letter from the query (note, ). Then redundancy is: ! "# . In all cases where there is a significant difference, greater redundancy leads to better performance. In almost all cases, when we control for redundancy performance decreases with length. We will discuss the implications of these experiments after describing corresponding experiments with our model. 3 Using Markov Models for Word Retrieval We now describe a model of word memory in which matching between the query and memory is mediated by a simple set of features. Specifically, we use bigrams (adjacent pairs of letters) as our feature set. We denote the beginning and end of a word using the symbols ‘0’ and ‘1’, respectively, so that bigram probabilities also indicate how often individual letters begin or end a word. Bottom up processing of a cue is done using this as a Markov model of words. Then bigram probabilities are used to trigger words in memory that might match the query. Our algorithm consists of three steps. First, we compute a prior distribution on how likely each word in memory is to match our query. In our simulations, we just use a uniform distribution. However, this distribution could reflect the frequency with which each word occurs in English. It could also be used to capture priming phenomena; for example, if a word has been recently seen, its prior probability could increase, making it more likely that the model would retrieve this word. Then, using these we compute a probability that each bigram will appear if we randomly select a bigram from a word selected according to our prior distribution. Second, we use these bigram probabilities as a Markov model, and compute the expected number of times each bigram will occur in the answer, conditioned on the query. That is, as a generic model of words we assume that each letter in the word depends on the adjacent letters, but is conditionally independent of all others. This conditional independence allows us to decompose our problem into a set of small, independent problems. For example, consider the query ‘*p*l*c*’. Implicitly, each query begins with ‘0’ and ends with ‘1’, so the expected number of times any bigram will appear in the completed word is the sum of the number of times it appears in the completions of the fragments: ‘0*p’, ‘p*l’, ‘l*c’, and ‘c*1’. To compute this, we assume a prior distribution on the number of letters that will replace a ‘*’ in the completed word. We use an exponential model, setting the probability of
letters to be (in practice we truncate
at 5 and normalize the probabilities). A similar model is used in the perceptual completion of contours ([11, 24]). Using these priors, it becomes straightforward to compute a probability distribution on the bigrams that will appear in the completed cue. For a fixed
, we structure this problem as a belief net with
bigrams, and each bigram depending on only its neighbors. The conditional probability of each bigram given its neighbor comes from the Markov model, and we can solve the problem with belief propagation. Beginning the third step of the algorithm, we know the expected number of times that each bigram appears in the completed cue. Each bigram then votes for all words containing that bigram. The weight of this vote is the expected number of times each bigram appears in the completed cue, divided by the prior probability of each bigram, computed in step 1. We combine these votes multiplicatively. We update the prior for each word as the product of these votes with the previous probability. We can view this an approximate computation of the probability of each word being the correct answer, based on the likelihood that a bigram appears in the completed cue, and our prior on each word being correct. After the third step, we once again have a probability that each word is correct, and can iterate, using this probability to initialize step one. After a small number of iterations, we terminate the algorithm and select the most probable word as our answer. Empirically, we find that the answer the algorithm produces often changes in the first one or two iterations, and then generally remains the same. The answer may or may not actually match the input cue, and by this we judge whether it is correct or incorrect. We can view this algorithm as an approximate computation of the probability that each 1 2 3 4 5 6 7 8 9 0.4 0.5 0.6 0.7 0.8 0.9 1 Fragment Length Fraction Completed 1 2 3 4 5 6 7 8 9 0.4 0.5 0.6 0.7 0.8 0.9 1 Fragment Length Fraction Completed 1 2 3 4 5 6 7 8 9 0.4 0.5 0.6 0.7 0.8 0.9 1 Fragment Length Fraction Completed R0 R1 R2 R3 R4 Figure 2: Performance as a function of cue length, for cues of frequency between 4 and 22 (top-left) and between 1 and 3 (top-right). On the bottom, we divide the first set of cues into five groups ranging from the least redundant (R0) to the most (R4). word matches the cue, where the main approximation comes from using a small set of features to bring the cue into contact with items in memory. Denote the number of features by (with a bigram representation, ), the number of features in each word by (ie., the word length plus one), the number of words by , and the maximum number of blanks replacing a ‘*’ by
. Then steps one and three require O(mw) computation, and step two requires O(Fn) computation. In a neural network, the primary requirement would be bidirectional connections between each feature (bigram) and each item in memory. Therefore, computational simplicity is gained by using a small feature set, at the cost of some approximation in the computation. Experiments We have run experiments to compare the performance of this model to that of human subjects. For simplicity, we used a memory of 6,040 words, each with eight characters. First, we simulated the conditions described in Olofsson and Nyberg[12] comparing word stem and word fragment completion. To match their experiments, we used a modified algorithm that handled cues in which the number of missing letters can be specified. We used cues that specified the first three letters of a word, the last three letters, or three letters scattered throughout the word. The algorithm achieved accuracy of 95% in the first case, 87% in the second, and 80% in the third. This qualitatively matches the results for human subjects. Note that our algorithm treats the beginning and end of words symmetrically. Therefore, the fact that it performs better when the first letters of the word are given than when the last are given is due to regularities in English spelling, and is not built into the algorithm. Next we simulated conditions comparable to our own experiments on human subjects, using superghost cues. First we selected cues of varying length that match between four and twenty-two words in the dictionary. Figure 2-top-left shows the percentage of queries the algorithm correctly answered, for cues of lengths two to seven. This figure shows a U-shaped performance curve qualitatively similar to that displayed by human subjects. We also ran these experiments using cues that matched one to three words (Figure 2-topright). These very low frequency cues did not display this U-shaped behavior. The algorithm performs differently on fragments with very low frequency because in our corpus the shorter of these cues had especially low redundancy and the longer fragments had especially high redundancy, in comparison to fragments with frequencies between 4 and 22. Next (Figure 2-bottom) we divided the cues into five groups of equal size, according to their redundancy. We can see that performance increases with redundancy and decreases with cue length. Discussion Our experiments indicate two main effects in human word memory that our model also shares. First, performance improves with the redundancy of cues. Second, when we control for this, performance drops with cue length. Since redundancy tends to increase with cue length, this creates two conflicting tendencies that result in a U-shaped memory curve. We conjecture that these factors may be present in many memory tasks, leading to U-shaped memory curves in a number of domains. In our model, the fact that performance drops with cue length is a result of our use of a simple feature set to mediate matching the cue to words in memory. This means that not all the information present in the cue is conveyed to items in memory. When the length of a cue increases, but its redundancy remains low, all the information in the cue remains important in getting a correct answer, but the amount of information in the cue increases, making it harder to capture it all with a limited feature set. This can account for the performance of our model; similar mechanisms may account for human performance as well. On the other hand, the extent to which redundancy grows with cue length is really a product of the specific words in memory and the cues chosen. Therefore, the exact shape of the performance curve will also depend on these factors. This may partly explain some of the quantitative differences between our model and human performance. Finally, we also point out that our measure of redundancy is rather crude. In particular, it tends to saturate at very high or very low levels. So, for example, if we add a letter to a cue that is already highly redundant, the new letter may not be needed to find a correct answer, but that is not reflected by much of an increase in the cue’s redundancy. 4 Conclusions We have proposed superghost queries as a domain for experimenting with word memory, because it seems a natural task to people, and requires models that can flexibly handle somewhat complicated questions. We have shown that in human subjects, performance on superghost improves with the redundancy of a query, and otherwise tends to decrease with word length. Together, these effects results in a U-shaped performance curve. We have proposed a computational model that uses a simple, generic model of words to map a superghost query onto a simple feature set of bigrams. This means that somewhat complicated questions can be answered while keeping comparisons between the fragments and words in memory very simple. Our model displays the two main trends we have found in human memory. It also does better at word stem completion than word fragment completion, which agrees with previous work on human memory. Future work will investigate the modification of our model to account for priming effects in memory. References [1] J. Anderson. An Introduction to Neural Networks, MIT Press, Cambridge MA. 1995. [2] E. Baum, J. Moody and F. Wilczek. “Internal Representations for Associative Memory,” Biological Cybernetics, 59:217-228, 1988. [3] G. Carpenter, and S. Grossberg. “ART 2: Self-Organization of Stable Category Recognition Codes for Analog Input Patterns,”Applied Optics, 26:4919-4930, 1987. [4] D. Grimes and M. Mozer. “The interplay of symbolic and subsymbolic processes in anagram problem solving,”NIPS, 2001. [5] G. Hinton, P. Dayan, B. Frey, and R. Neal. “The ‘Wake-Sleep’ Algorithm for Unsupervised Neural Networks,”Science, 268:1158-1161, 1995. [6] D.L. Hintzman and A.L. Hartry. Item effects in recognition and fragment completion: Contingency relations vary for different sets of words. JEP: Learning, Memory and Cognition, 17: 341-345, 1990. [7] J. Hopfield. “Neural networks and Physical Systems with Emergent Collective Computational Abilities.”Proc. of the Nat. Acad. of Science, 79:2554-2558, 1982. [8] D. Jacobs and A. Rudra. “An Iterative Projection Model of Memory,” NEC Research Institute Technical Report, 2000. [9] G.V. Jones. Fragment and schema models for recall. Memory and Cognition, 12(3):250-63, 1984. [10] B. Kosko. “Adaptive Bidirectional Associative Memory”, Applied Optics, 26(23):4947-60, 1987. [11] D. Mumford. “Elastica and Computer Vision.”C. Bajaj (Ed), Algebraic Geometry and its Applications New York: Springer-Verlag. 1994. [12] U. Olofsson and L. Nyberg. Swedish norms for completion of word stems and unique word fragments. Scandinavian Journal of Psychology, 33(2):108-16, 1992. [13] U. Olofsson and L. Nyberg. Determinants of word fragment completion. Scandinavian Journal of Psychology, 36(1):59-64, 1995. [14] R. Rao and D. Ballard. “Dynamic Model of Visual Recognition Predicts Neural Response Properties in the Visual Cortex,”Neural Computation, 9(4):721-763, 1997. [15] R.H. Ross and G.H. Bower. Comparisons of models of associative recall. Memory and Cognition, 9(1):1-16, 1981. [16] D. Rumelhart and J. McClelland. “An interactive activation model of context effects in letter perception: part 2. The contextual enhancement effect and some tests and extensions of the model”, Psychological Review, 89:60-94, 1982. [17] M.S. Seidenberg. Sublexical structures in visual word recognition: Access units or orthographic redundancy? In M. Coltheart (Ed.), Attention and performance XII, 245-263. Hillsdale, NJ: Erlbaum. 1987. [18] D.L. Shacter and E. Tulving. Memory systems. Cambridge, MA: MIT Press. 1994. [19] C. Shannon. “Prediction and Entropy of Printed English,” Bell Systems Technical Journal, 30:50-64, 1951. [20] Sommer, F., and Palm, G., 1997, NIPS:676-681. [21] K. Srinivas, H.L. Roediger 3d and S. Rajaram. The role of syllabic and orthographic properties of letter cues in solving word fragments. Memory and Cognition, 20(3):219-30, 1992. [22] N. Tishby, F. Pereira and W. Bialek. “The Information Bottleneck Method,”37th Allerton Conference on Communication, Control, and Computing. 1999. [23] S. Ullman. High-level Vision, MIT Press, Cambridge, MA. 1996. [24] L. Williams & D. Jacobs. “Stochastic Completion Fields: A Neural Model of Illusory Contour Shape and Salience”. Neural Computation, 9:837–858, 1997. Acknowledgements The authors would like to thank Nancy Johal for her assistance in conducting the psychological experiments presented in this paper.
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K-Local Hyperplane and Convex Distance Nearest Neighbor Algorithms Pascal Vincent and Yoshua Bengio Dept. IRO, Universit´e de Montr´eal C.P. 6128, Montreal, Qc, H3C 3J7, Canada vincentp,bengioy @iro.umontreal.ca http://www.iro.umontreal.ca/ vincentp Abstract Guided by an initial idea of building a complex (non linear) decision surface with maximal local margin in input space, we give a possible geometrical intuition as to why K-Nearest Neighbor (KNN) algorithms often perform more poorly than SVMs on classification tasks. We then propose modified K-Nearest Neighbor algorithms to overcome the perceived problem. The approach is similar in spirit to Tangent Distance, but with invariances inferred from the local neighborhood rather than prior knowledge. Experimental results on real world classification tasks suggest that the modified KNN algorithms often give a dramatic improvement over standard KNN and perform as well or better than SVMs. 1 Motivation The notion of margin for classification tasks has been largely popularized by the success of the Support Vector Machine (SVM) [2, 15] approach. The margin of SVMs has a nice geometric interpretation1: it can be defined informally as (twice) the smallest Euclidean distance between the decision surface and the closest training point. The decision surface produced by the original SVM algorithm is the hyperplane that maximizes this distance while still correctly separating the two classes. While the notion of keeping the largest possible safety margin between the decision surface and the data points seems very reasonable and intuitively appealing, questions arise when extending the approach to building more complex, non-linear decision surfaces. Non-linear SVMs usually use the “kernel trick” to achieve their non-linearity. This conceptually corresponds to first mapping the input into a higher-dimensional feature space with some non-linear transformation and building a maximum-margin hyperplane (a linear decision surface) there. The “trick” is that this mapping is never computed directly, but implicitly induced by a kernel. In this setting, the margin being maximized is still the smallest Euclidean distance between the decision surface and the training points, but this time measured in some strange, sometimes infinite dimensional, kernel-induced feature space rather than the original input space. It is less clear whether maximizing the margin in this new space, is meaningful in general (see [16]). 1for the purpose of this discussion, we consider the original hard-margin SVM algorithm for two linearly separable classes. A different approach is to try and build a non-linear decision surface with maximal distance to the closest data point as measured directly in input space (as proposed in [14]). We could for instance restrict ourselves to a certain class of decision functions and try to find the function with maximal margin among this class. But let us take this even further. Extending the idea of building a correctly separating non-linear decision surface as far away as possible from the data points, we define the notion of local margin as the Euclidean distance, in input space, between a given point on the decision surface and the closest training point. Now would it be possible to find an algorithm that could produce a decision surface which correctly separates the classes and such that the local margin is everywhere maximal along its surface? Surprisingly, the plain old Nearest Neighbor algorithm (1NN) [5] does precisely this! So why does 1NN in practice often perform worse than SVMs? One typical explanation, is that it has too much capacity, compared to SVM, that the class of function it can produce is too rich. But, considering it has infinite capacity (VC-dimension), 1NN is still performing quite well... This study is an attempt to better understand what is happening, based on geometrical intuition, and to derive an improved Nearest Neighbor algorithm from this understanding. 2 Fixing a broken Nearest Neighbor algorithm 2.1 Setting and definitions The setting is that of a classical classification problem in (the input space). We are given a training set of points
and their corresponding class label
! #"$ %" '&(
)!* where )!* is the number of different classes. The + , pairs are assumed to be samples drawn from an unknown distribution ./ 0 . Barring duplicate inputs, the class labels associated to each 1 define a partition of : let * 1 32 4 576 . The problem is to find a decision function 8 97: <; " that will generalize well on new points drawn from .= >0 . 8 9 should ideally minimize the expected classification error, i.e. minimize ?A@CB DFE G'HJILKM NFO$P where ?A@ denotes the expectation with respect to .= 0 and DFE G(HJQKM N R denotes the indicator function, whose value is & if 8 9 TS and U otherwise. In the previous and following discussion, we often refer to the concept of decision surface, also known as decision boundary. The function 8 9 corresponding to a given algorithm defines for any class 6 #" two regions of the input space: the region V * W 2 8 9 5X6 and its complement Y V * . The decision surface for class 6 is the “boundary” between those two regions, i.e. the contour of V * , and can be seen as a Z Y & dimensional manifold (a “surface” in ) possibly made of several disconnected components. For simplicity, when we mention the decision surface in our discussion we consider only the case of two class discrimination, in which there is a single decision surface. When we mention a test point, we mean a point 1 that does not belong to the training set and for which the algorithm is to decide on a class 8 9 . By distance, we mean the usual Euclidean distance in input-space . The distance between two points [ and \ will be written ] [ \ or alternatively ^ [ Y \(^ . The distance between a single point and a set of points _ is the distance to the closest point of the set: ] + _ `ba<cJdfe g'h ] + i . The K-neighborhood jLk of a test point is the set of the l points of whose distance to is smallest. The K-c-neighborhood jmk * of a test point is the set of l points of * whose distance to is smallest. By Nearest Neighbor algorithm (1NN) we mean the following algorithm: the class of a test point is decided to be the same as the class of its closest neighbor in _ . By K-Nearest Neighbor algorithm (KNN) we mean the following algorithm: the class of a test point is decided to be the same as the class appearing most frequently among the K-neighborhood of . 2.2 The intuition Figure 1: A local view of the decision surface produced by the Nearest Neighbor (left) and SVM (center) algorithms, and how the Nearest Neighbor solution gets closer to the SVM solution in the limit, if the support for the density of each class is a manifold which can be considered locally linear (right). Figure 1 illustrates a possible intuition about why SVMs outperforms 1NNs when we have a finite number of samples. For classification tasks where the classes are considered to be mostly separable,2 we often like to think of each class as residing close to a lowerdimensional manifold (in the high dimensional input space) which can reasonably be considered locally linear3. In the case of a finite number of samples, “missing” samples would appear as “holes” introducing artifacts in the decision surface produced by classical Nearest Neighbor algorithms. Thus the decision surface, while having the largest possible local margin with regard to the training points, is likely to have a poor small local margin with respect to yet unseen samples falling close to the locally linear manifold, and will thus result in poor generalization performance. This problem fundamentally remains with the K Nearest Neighbor (KNN) variant of the algorithm, but, as can be seen on the figure, it does not seem to affect the decision surface produced by SVMs (as the surface is constrained to a particular smooth form, a straight line or hyperplane in the case of linear SVMs). It is interesting to notice, if the assumption of locally linear class manifolds holds, how the 1NN solution approaches the SVM solution in the limit as we increase the number of samples. To fix this problem, the idea is to somehow fantasize the missing points, based on a local linear approximation of the manifold of each class. This leads to modified Nearest Neighbor algorithms described in the next sections.4 2By “mostly separable” we mean that the Bayes error is almost zero, and the optimal decision surface has not too many disconnected components. 3i.e. each class has a probability density with a “support” that is a lower-dimensional manifold, and with the probability quickly fading, away from this support. 4Note that although we never generate the “fantasy”points explicitly, the proposed algorithms are really equivalent to classical 1NN with fantasized points. 2.3 The basic algorithm Given a test point , we are really interested in finding the closest neighbor, not among the training set , but among an abstract, virtually enriched training set that would contain all the fantasized “missing” points of the manifold of each class, locally approximated by an affine subspace. We shall thus consider, for each class 6 , the local affine subspace that passes through the l points of the K-c neighborhood of . This affine subspace is typically l Y & dimensional or less, and we will somewhat abusively call it the “local hyperplane”.5 Formally, the local hyperplane can be defined as k * 5 i i k N ) N
(1) where )
>) j k * . Another way to define this hyperplane, that gets rid of the constraint & , is to take a reference point within the hyperplane as an origin, for instance the centroid6 ) k k N ) . This same hyperplane can then be expressed as k * 5 i i ) k N Y ; (2) where Y ; ) Y ) . Our modified nearest neighbor algorithm then associates a test point to the class 6 whose hyperplane k * is closest to . Formally 8 9 A $a<c d * g ] + k * , where ] F k * , is logically called K-local Hyperplane Distance, hence the name K-local Hyperplane Distance Nearest Neighbor algorithm (HKNN in short). Computing, for each class 6 ] F k * , a!c d e g !#" $ H QK ^ Y i ^ a<cJd % g'& (*),+ + + + + Y ) Y k N Y ; + + + + + (3) amounts to solving a linear system in , that can be easily expressed in matrix form as: .- /0 / 1- / Y ) (4) where and ) are Z dimensional column vectors,
k - , and is a Z2 l matrix whose columns are the Y ; vectors defined earlier.7 5Strictly speaking a hyperplane in an 3 dimensional input space is an 35476 affine subspace, while our “local hyperplanes”can have fewer dimensions. 6We could be using one of the 8 neighbors as the reference point, but this formulation with the centroid will prove useful later. 7Actually there is an infinite number of solutions to this system since the 4 9 :7; are linearly dependent: remember that the initial formulation had an equality constraint and thus only 8<4=6 effective degrees of freedom. But we are interested in >@?BADCFEHGJI K ?BA LFL not in M so any solution will do. Alternatively, we can remove one of the 4 9 :7; from the system so that it has a unique solution. 2.4 Links with other paradigms The proposed HKNN algorithm is very similar in spirit to the Tangent Distance Algorithm [13]. k * can be seen as a tangent hyperplane representing a set of local directions of transformation (any linear combination of the Y ; vectors) that do not affect the class identity. These are invariances. The main difference is that in HKNN these invariances are inferred directly from the local neighborhood in the training set, whereas in Tangent Distance, they are based on prior knowledge. It should be interesting (and relatively easy) to combine both approaches for improved performance when prior knowledge is available. Previous work on nearest-neighbor variations based on other locally-defined metrics can be found in [12, 9, 6, 7], and is very much related to the more general paradigm of Local Learning Algorithms [3, 1, 10]. We should also mention close similarities between our approach and the recently proposed Local Linear Embedding [11] method for dimensionality reduction. The idea of fantasizing points around the training points in order to define the decision surface is also very close to methods based on estimating the class-conditional input density [14, 4]. Besides, it is interesting to look at HKNN from a different, less geometrical angle: it can be understood as choosing the class that achieves the best reconstruction (the smallest reconstruction error) of the test pattern through a linear combination of l particular prototypes of that class (the l neighbors). From this point of view, the algorithm is very similar to the Nearest Feature Line (NFL) [8] method. They differ in the fact that NFL considers all pairs of points for its search rather than the local l neighbors, thus looking at many ( ) lines (i.e. 2 dimensional affine subspaces), rather than at a single l Y & dimensional one. 3 Fixing the basic HKNN algorithm 3.1 Problem arising for large K One problem with the basic HKNN algorithm, as previously described, arises as we increase the value of l , i.e. the number of points considered in the neighborhood of the test point. In a typical high dimensional setting, exact colinearities between input patterns are rare, which means that as soon as l Z , any pattern of (including nonsensical ones) can be produced by a linear combination of the l neighbors. The “actual” dimensionality of the manifold may be much less than l . This is due to “near-colinearities” producing directions associated to small eigenvalues of the covariance matrix /0 that are but noise, that can lead the algorithm to mistake those noise directions for “invariances”, and may hurt its performance even for smaller values of l . Another related issue is that the linear approximation of the class manifold by a hyperplane is valid only locally, so we might want to restrict the “fantasizing” of class members to a smaller region of the hyperplane. We considered two ways of dealing with these problems.8 3.2 The convex hull solution One way to avoid the above mentioned problems is to restrict ourselves to considering the convex hull of the neighbors, rather than the whole hyperplane they support (of which the convex hull is a subset). This corresponds to adding a constraint of U to equation (1). Unlike the problem of computing the distance to the hyperplane, the distance to the convex hull cannot be found by solving a simple linear system, but typically requires solving a quadratic programming problem (very similar to the one of SVMs). While this 8A third interesting avenue, which we did not have time to explore, would be to keep only the most relevant principal components of : , ignoring those corresponding to small eigenvalues. is more complex to implement, it should be mentioned that the problems to be solved are of a relatively small dimension of order l , and that the time of the whole algorithm will very likely still be dominated by the search of the l nearest neighbors within each class. This algorithm will be referred to as K-local Convex Distance Nearest Neighbor Algorithm (CKNN in short). 3.3 The “weight decay” penalty solution This consists in incorporating a penalty term to equation (3) to penalize large values of (i.e. it penalizes moving away from the centroid, especially in non essential directions): ] + k * a<cJd % g'& (0)+ + + + + Y ) Y k N Y ; + + + + + k N (5) The solution for is given by solving the linear system / D / / Y ) where D is the Z72TZ identity matrix. This is equation (4) with an additional diagonal term. The resulting algorithm is a generalization of HKNN (basic HKNN corresponds to U ). 4 Experimental results We performed a number of experiments, to highlight different properties of the algorithms: A first 2D toy example (see Figure 2) graphically illustrates the qualitative differences in the decision surfaces produced by KNN, linear SVM and CKNN. Table 1 gives quantitative results on two real-world digit OCR tasks, allowing to compare the performance of the different old and new algorithms. Figure 3 illustrates the problem arising with large l , mentioned in Section 3, and shows that the two proposed solutions: CKNN and HKNN with an added weight decay , allow to overcome it. In our final experiment, we wanted to see if the good performance of the new algorithms absolutely depended on having all the training points at hand, as this has a direct impact on speed. So we checked what performance we could get out of HKNN and CKNN when using only a small but representative subset of the training points, namely the set of support vectors found by a Gaussian Kernel SVM. The results obtained for MNIST are given in Table 2, and look very encouraging. HKNN appears to be able to perform as well or better than SVMs without requiring more data points than SVMs. Figure 2: 2D illustration of the decision surfaces produced by KNN (left, K=1), linear SVM (middle), and CKNN (right, K=2). The “holes”are again visible in KNN. CKNN doesn’t suffer from this, but keeps the objective of maximizing the margin locally. 5 Conclusion From a few geometrical intuitions, we have derived two modified versions of the KNN algorithm that look very promising. HKNN is especially attractive: it is very simple to implement on top of a KNN system, as it only requires the additional step of solving a small and simple linear system, and appears to greatly boost the performance of standard KNN even above the level of SVMs. The proposed algorithms share the advantages of KNN (no training required, ideal for fast adaptation, natural handling of the multi-class case) and its drawbacks (requires large memory, slow testing). However our latest result also indicate the possibility of substantially reducing the reference set in memory without loosing on accuracy. This suggests that the algorithm indeed captures essential information in the data, and that our initial intuition on the nature of the flaw of KNN may well be at least partially correct. References [1] C. G. Atkeson, A. W. Moore, and S. Schaal. Locally weighted learning. Artificial Intelligence Review, 1996. [2] B. Boser, I. Guyon, and V. Vapnik. An algorithm for optimal margin classifiers. In Fifth Annual Workshop on Computational Learning Theory, pages 144–152, Pittsburgh, 1992. [3] L. Bottou and V. Vapnik. Local learning algorithms. Neural Computation, 4(6):888–900, 1992. [4] Olivier Chapelle, Jason Weston, L´eon Bottou, and Vladimir Vapnik. Vicinal risk minimization. In T.K. Leen, T.G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems, volume 13, pages 416–422, 2001. [5] T.M. Cover and P.E. Hart. Nearest neighbor pattern classification. IEEE Transactions on Information Theory, 13(1):21–27, 1967. [6] J. Friedman. Flexible metric nearest neighbor classification. Technical Report 113, Stanford University Statistics Department, 1994. [7] Trevor Hastie and Robert Tibshirani. Discriminant adaptive nearest neighbor classification and regression. In David S. Touretzky, Michael C. Mozer, and Michael E. Hasselmo, editors, Advances in Neural Information Processing Systems, volume 8, pages 409–415. The MIT Press, 1996. [8] S.Z. Li and J.W. Lu. Face recognition using the nearest feature line method. IEEE Transactions on Neural Networks, 10(2):439–443, 1999. [9] J. Myles and D. Hand. The multi-class measure problem in nearest neighbour discrimination rules. Pattern Recognition, 23:1291–1297, 1990. [10] D. Ormoneit and T. Hastie. Optimal kernel shapes for local linear regression. In S. A. Solla, T. K. Leen, and K-R. Mller, editors, Advances in Neural Information Processing Systems, volume 12. MIT Press, 2000. [11] Sam Roweis and Lawrence Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500):2323–2326, Dec. 2000. [12] R. D. Short and K. Fukunaga. The optimal distance measure for nearest neighbor classification. IEEE Transactions on Information Theory, 27:622–627, 1981. [13] P. Y. Simard, Y. A. LeCun, J. S. Denker, and B. Victorri. Transformation invariance in pattern recognition — tangent distance and tangent propagation. Lecture Notes in Computer Science, 1524, 1998. [14] S. Tong and D. Koller. Restricted bayes optimal classifiers. In Proceedings of the 17th National Conference on Artificial Intelligence (AAAI), pages 658–664, Austin, Texas, 2000. [15] V.N. Vapnik. The Nature of Statistical Learning Theory. Springer, New York, 1995. [16] Bin Zhang. Is the maximal margin hyperplane special in a feature space? Technical Report HPL-2001-89, Hewlett-Packards Labs, 2001. Table 1: Test-error obtained on the USPS and MNIST digit classification tasks by KNN, SVM (using a Gaussian Kernel), HKNN and CKNN. Hyper parameters were tuned on a separate validation set. Both HKNN and CKNN appear to perform much better than original KNN, and even compare favorably to SVMs. Data Set Algorithm Test Error Parameters used USPS KNN 4.98% l & (6291 train, SVM 4.33% & U(U 1000 valid., HKNN 3.93% l &f U 2007 test points) CKNN 3.98% l U MNIST KNN 2.95% l (50000 train, SVM 1.30%
& U'U 10000 valid., HKNN 1.26% l & U 10000 test points) CKNN 1.46% l
U 0.012 0.014 0.016 0.018 0.02 0.022 0.024 0.026 0.028 0.03 0.032 0 20 40 60 80 100 120 error rate K CKNN basic HKNN HKNN, lambda=1 HKNN, lambda=10 Figure 3: Error rate on MNIST as a function of l for CKNN, and HKNN with different values of . As can be seen the basic HKNN algorithm performs poorly for large values of l . As expected, CKNN is relatively unaffected by this problem, and HKNN can be made robust through the added “weight decay”penalty controlled by . Table 2: Test-error obtained on MNIST with HKNN and CKNN when using a reduced training set made of the 16712 support vectors retained by the best Gaussian Kernel SVM. This corresponds to 28% of the initial 60000 training patterns. Performance is even better than when using the whole dataset. But here, hyper parameters l and were chosen with the test set, as we did not have a separate validation set in this setting. It is nevertheless remarkable that comparable performances can be achieved with far fewer points. Data Set Algorithm Test Error Parameters used MNIST (16712 train s.v., HKNN 1.23% l U & U 10000 test points) CKNN 1.36% l
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Thomas L .. Griffiths & Joshua B. Tenenbaum Department of Psychology Stanford University, Stanford, CA 94305 {gruffydd,jbt}©psych. stanford. edu Abstract Estimating the parameters of sparse multinomial distributions is an important component of many statistical learning tasks. Recent approaches have used uncertainty over the vocabulary of symbols in a multinomial distribution as a means of accounting for sparsity. We present a Bayesian approach that allows weak prior knowledge, in the form of a small set of approximate candidate vocabularies, to be used to dramatically improve the resulting estimates. We demonstrate these improvements in applications to text compression and estimating distributions over words in newsgroup data. 1 Introduction Sparse multinomial distributions arise in many statistical domains, including natural language processing and graphical models. Consequently, a number of approaches to parameter estimation for sparse multinomial distributions have been suggested [3]. These approaches tend to be domain-independent: they make little use of prior knowledge about a specific domain. In many domains where multinomial distributions are estimated there is often at least weak prior knowledge about' the potential structure of distributions, such as a set of hypotheses about restricted vocabularies from which the symbols might be generated. Such knowledge can be solicited from experts or obtained from unlabeled data. We present a method for Bayesian_parameter estimation in sparse discrete domains that exploits this weak form of prior knowledge to improve estimates over knowledge-free approaches. 1.1 Bayesian parameter estimation for multinomial distributions Following the presentation in [4], we consider a language ~ containing L distinct symbols. A multinomial distribution is specified by a parameter vector f) == (Ol, ... ,f)L), where f)i is the probability of an observation being symbol i. Consequently, we have the constraints that Ef==l f)i == 1 and (h ~ 0, i == 1, ... ,L. The task of multinomial estimation is to take a data set D and produce a'vector f) that results in a good approximation to the distribution that produced D. In this case, D consists of N independent observations Xl, ... xN drawn from the distribution to be estimated, which can be summarized by the statistics Ni specifying the number of times the ith symbol occurs in the data. D also determines the set ~o of symbols that occur in the data. Stated in this way, multinomial estimation involves predicting the next observation based on the data. Specifically, we wish to calculate P(XN+1ID). The Bayesian estimate for this probability is given by PL(xN+lID) =I p(XN+1IB)P(BID)dB where P(XN +1 10) follows from the multinomial distribution corresponding to O. The posterior probability P(OID) can be obtained via Bayes rule L P(OID) oc P(DIO)P(O) == P(8}II ONi i==l where P(O) is the prior probability of a given O. Laplace used this method with a uniform prior over 0 to give the famous "law of succession" [6J. A more general approach is to assume a Dirichlet prior over (), which is conjugate to the multinomial distribution and gives P(XN+l = ilD) = Ni +LCY.i (1) N + l:j==l O!.j where the ai are the hyperparameters of the Dirichlet distribution. Different estimates are obtained for different choices of the ai, with most approaches making the simplifying assumption that ai == O!. for all i. Laplace's law results from a == 1. The case with a == 0.5 is the Jeffreys-Perks law or Expected Likelihood Estimation [2] [5J [9J, while using arbitrary O!. is Lidstone's law [7]. 1.2 EstiIllating sparse Illultinomial distributions Several authors have extended the Bayesian approach to sparse multinomial distributions, in which only a restricted vocabulary of symbols are used, by maintaining uncertainty over these vocabularies. In [10], Ristad uses assumptions about the probability of strings based upon different vocabularies to give the estimate { (Ni + l)/(N +L) if kO == L PR (X N +1 == ilD) == (Ni + l)(N + 1 - kO)/(N2 + N + 2kO) if kO < L 1\ Ni > 0 kO(kO + l)/(L - kO)(N2 + N + 2kO) otherwise where kO == I~oIis the size of the smallest vocabulary consistent with the data. A different approach is taken by Friedman and Singer in [4], who point out that Ristad's method is a special case of their framework. Friedman and Singer consider the vocabulary V ~ :E to be a random variable, allowing them to write p.(XN +1 == ilD) == L p(XN +1 == ilV, D)P(VID) (2) v where P{XN +1 == ilV, D) results from a Dirichlet prior over the symbols in V, p(XN +1 == ilV, D) == {~it'ja if i E v: (3) o otherWIse and by Bayes' rule and the properties of Dirichlet priors P(VID) oc P(DIV)P(V) { ~fJ~~(a) niE~O r(~·t~a) P(V) EO ~ V ( ) otherwise 4 Friedman and Singer assume a hierarchical prior over V, such that all vocabularies of cardinality k are given equal probability, namely P(S == k)/(t), where P(S == k) is the probability that the size of the vocabulary (IVI) is k. It follows that if i E ~o, p(XN +I == ilD) == Lk :+1~P(S == kiD). If i ¢ ~o, it is necessary to estimate the proportion of V that contain i for a given k. The simplified result is PF(XN +1 == ilD) == { %tt~aC(D,L) L-kO(1- C(D,L)) where . h P(S k) - k! r(ka:) WIt mk ==. == (k-kO)!· r(N+ka:) . 2 Ivlaking use of weak prior knowiedge if i E ~o otherwise (5) Friedman and Singer assume a prior that gives equal probability to all vocabularies of a given cardinality. However, many real-world tasks provide limited knowledge about the structure of distributions that we can build into our methods for parameter estimation. In the context of sparse multinomial estimation, one instance of such knowledge the importance of specific vocabularies. For example, in predicting the next character in a file, our predictions could be facilitated by considering the fact that most files either use a vocabulary consisting of ASCII printing characters (such as text files), or all possible characters (suc~ as object files). This kind of structural knowledge about a domain is typically easier to solicit from experts than accurate distributional information, and forms a valuable informational resource. If we have this kind of prior knowledge, we can restrict our attention to a subset of the 2L possible vocabularies. fu particular, we can specify a set of vocabularies V which we consider as hypotheses for the vocabulary used in producing D, where the elements of V are specified by our knowledge of the domain. This stands as a compromise between Friedman and Singer's approach, in which V consists of all vocabularies, and traditional Bayesian parameter estimation as represented by Equation 1, in which V consists of only the vocabulary containing all words. To do this, we explicitly evaluate the sum given in Equation 2, where the sum over V includes all V E V. This sum remains tractable when V is a small subset of the possible vocabularies, and the efficiency is aided by the fact that P(DIV) shares common terms across all V which can cancel in normalization. The intuition behind this approach is that it attempts to classify thetarget distribution as using one of a known set of vocabularies, where the vocabularies are obtained either from experts or from unlabeled data. Applying standard Bayesian multinomial estimation within this vocabulary gives enough flexibility for the method to capture a range of distributions, while making use of our weak prior knowledge. 2.1 An illustration: Text compression Text compression is an effective test of methods for multinomial estimation. Adaptive coding can be performed by specifying a method for calculating a distribution over the probability of the next byte in a file based upon the preceding bytes [1]. The extent to which the file is compressed depends upon the quality of these predictions. To illustrate the utility of including prior knowledge, we follow Ristad in using the Calgary text compression corpus [1]. This corpus consists of 19 files of Table 1: Text compression lengths (in bytes) on the Calgary corpus file size kO NH(Ni/N ) Pv PF PR PL PJ bib 111261 81 72330 18 89 92 269 174 book1 768771 82 435043 219 105 116 352 219 book2 610856 96 365952 94 115 124 329 212 geo 102400 256 72274 161 162 165 165 161 nellS 377109 98 244633 89 113 116 304 201 obj1 21504 256 15989 126 127 129 129 126 obj2 246814 256 193144 182 184 190 189 182 paper1 53161 95 33113 71 94 100 236 156 paper2 82199 91 47280 75 94 105 259 167 paper3 46526 84 27132 70 85 92 238 154 paper4 13286 80 7806 58 72 79 190 126 paper5 11954 91 7376 57 79 83 181 122 paper6 38105 93 23861 68 90 95 223 149 pic 513216 159 77636 205 16~ 216 323 205 progc 39611 92 25743 68 - 89 91 222 150 progl 71646 87 42720 74 91 97 253 164 progp 49379 89 30052 71 89 94 236 155 trans 93695 99 64800 169 101 105 252 169 several different types, each using some subset of 256 possible characters (L == 256). The files include Bib'IEXsource (bib), formatted English text (book*, paper*), geological data (geo), newsgroup articles (news), object files (obj*), a bit-mapped picture (pic), programs in three different languages (prog*) and a terminal transcript (trans). The task was to estimate the distribution from which characters in the file were drawn based upon the first N characters and thus predict the N + 1st character. Performance was measured in terms of the length of the resulting file, where the contribution of the N + 1st character to the length is log2 P(XN+lID). The results are expressed as the number of bytes required to encode the file relative to the empirical entropy NH(Ni/N) as assessed by Ristad [10]. Results are shown in Table 1. Pv is the restricted vocabulary model outlined above, with V consisting of just two hypotheses: one corresponding to binary files, containing all 256 characters, and one consisting of a 107 character vocabulary representing formatted English. The latter vocabulary was estimated from 5MB of English text, C code, Bib'IEXsource, and newsgroup data from outside the Calgary corpus. PF is Friedman and Singer's method. For both of these approaches, a was set to 0.5, to allow direct comparison to the Jeffreys-Perks law, PJo PR and PL are Ristad's and Laplace's laws respectively. Py outperformed the other methods on all files based upon English text, bar bookl, and all files using all 256 symbolsl . The high performance followed from rapid classification of these files as using the appropriate vocabulary in V. When the vocabulary included all symbols Py performed as PJ, which gave the best predictions for these files. 1A number of excellent techniques for· text compression exist that outperform all of those presented here. We have not included these techniques for comparison because our interest is in using text compression as a means of assessing estimation procedures, rather than as an end in itself. We thus consider only methods for multinomial estimation as our comparison. group. 2.2 Maintaining uncertainty in vocabularies The results for book1 illustrate a weakness of the approach outlined above. The file length for Py is higher than those for PF and PR , despite the fact that the file uses a text-based vocabulary. This file contains two characters that were not encountered in the data used to construct V. These characters caused Py to default to the unrestricted vocabulary of all 256 characters. From that point Py corresponded to PJ, which gave poor results on this file. This behavior results from the assumption that the candidate vocabularies in V are completely accurate. Since in many cases the knowledge that informs the vocabularies in V may be imperfect, it is desirable to allow for uncertainty in vocabularies. This uncertainty will be reflected in the fact that symbols outside V are expected to occur with a vocabulary-specific probability ty, p(XN+1 == ilV, D) == { (1 - (L -IVI)ty) N~~t~la if i E V ty otherwise where Ny == I:iEY N i · It follows that r(IVla) r(Ni + a) P(DIV) = (1 - (L -IVJ)€V)NV€t"-Nv r(N + 1V1a:) r(a:) y iE:EonY which replaces Equations 3-4 in specifying Py . When V is determined by the judgments of domain experts, ty is the probability that an unmentioned word actually belongs to a particular vocabulary. While it may not be the most efficient use of such data, the V E V can also be estimated from some form of unlabeled data. In this case, Friedman and Singer's approach provides a means of setting ty. Friedman and Singer explicitly calculate the probability that an unseen word is in V based upon a dataset: from the second condition of Equation 5, we find that we should set ty == L_1IYI (1- C(D, L)). We use this method below. 3 Bayesian parameter estimation in natural language Statistical natural language processing often uses sparse multinomial distributions over large vocabularies of words. In different contexts, different vocabularies will be used. By specifying a basis set ofvocabularies, we can perform parameter estimation by classifying distributions according to their vocabulary. This idea was examined using data from 20 different Usenet newsgroups. This dataset is commonly used in testing text classification algorithms (eg. [8]). Ten newsgroups were used to estimate a set of vocabularies V with corresponding ty. These vocabularies were used in estimating multinomial distributions on these newsgroups and ten others. The dataset was 20news-18827, which consists of the 20newsgroups data with headers and duplicates removed, and was preprocessed to remove all punctuation, capitalization, and distinct numbers. The articl~s in each of the 20 newsgroups were then divided into three sets. The first 500 articles from ten newsgroups were used to estimate the candidate vocabularies V and uncertainty parameters ty. Articles 501700 for all 20 newsgroups were used as training data for multinomial estimation. Articles 701-900 for all 20 newgroups were used as testing data. Following [8], a dictionary was built up by running over the 13,000 articles resulting from this division, and all words that occurred only once were mapped to an "unknown" word. The resulting dictionary contained L == 54309 words. As before, the restricted vocabulary method (Py), Friedman and Singer's method (PF), and Ristad's (PR ), Laplace's (PL ) and the Jeffreys-Perks (PJ ) laws were ap~~~ soc.religion.christian talk.politics.guns comp.sys.ibm.pc.hardware ~~~ rec.sport.hockey scLelectronics comp.windows.x ~~~ rec.autos rec.sport.baseball scLcrypt ~~~ scLmed comp.os.ms-windows.misc misc.forsale ~::=:= ~-;""", r.?';'~;:;" ~~~ comp.sys.mac.hardware talk.religion.misc comp.graphics ~~~ rec.motorcycles scLspace alt.atheism .... " talk.politics. mise talk.politics.mideast 17 '. 18 . 16 18 11 100 10000 50000 100 10000 Number of words F~gure 1: Cross-entropy of predictions on newsgroup data as a function of the logarithm of the number of words. The abscissa is at the empirical entropy of the test distribution. The top ten panels (talk.polities.mideast and those to its right) are for the newsgroups with unknown vocabularies. The bottom ten are for those that contributed vocabularies to V, trained and tested on novel data. PL and PJ are both indicated with dotted lines, but PJ always performs better than PL. The box on talk.polities.mideast indicates the point at which Pv defaults to the full vocabulary, as the number of unseen words makes this vocabulary more likely. At this point, the line for Pv joins the line for PJ , since both methods give the same estimates of the distribution. plied to the task. Both Pv and PF used a == 0.5 to facilitate comparison with PJ . 'V featured one vocabulary that contained all words in the dictionary, and ten vocabularies each corresponding to the words used in the first 500 articles of one of the newsgroups designated for this purpose. €y was estimated as outlined above. Testing for each newsgroup consisted of taking words from the 200 articles assigned for training purposes, estimating a. distribution using each method, and then computing the cross-entropy between that distribution and an empirical estimate of the true distribution. The cross-entropy is H{Q; P) == Ei Qi log2 Pi, where Q is the true distribution and P is the distribution produced by the estimation method. Q was given by the maximum likelihood estimate formed from the word frequencies in all 200 articles assigned for testing purposes. The testing procedure was conducted with just 100 words, and then in increments of 450 up to a total of 10000 words. Long-run performance was examined on talk.polities.mideast and talk.polities.mise, each trained with 50000 words. The results are shown in Figure 1. As expected, Py consistently outperformed the other methods on the newsgroups that contributed to V. However, performance on novel newsgroups was also greatly improved. As can be seen in Figure 2, the novel newsgroups were classified to appropriate vocabularies - for example all words rec.autos I-----------------rec.motorcycles r--+-ta1k.politics.guns ,-------T---+----------- talk.politics.mideast rl\--_f-------T----'r----------- alt.atheism frt-'\.;:::::::::::;f:=.='=f--t---------- soc.religion.christian l-+-if-Hf-------t---------- scLspace ,-~~?.;~~~~~~g~ey scLmed rec.sport.baseball scLcrypt misc.forsale comp.graphics comp.sys.mac.hardware talk.religion. misc talk.politics.misc comp.os.ms-windows.m~c'-----------------com~sy&ibm.p~hardware o 10000 Number of words Figure 2: Classification of newsgroup vocabularies. The lines illustrate the vocabulary which had maximum posterior probability for each of the ten test newsgroups after exposure to differing numbers of words. The vocabularies in V are listed along the left hand side of the axis, and the lines are identified with newsgroups by the labels on the right hand side. Lines are offset to facilitate identification. talk.religion.misc had the highest posterior probability for alt.atheism and soc. religion. christian, while rec. autos had highest posterior probability for rec .motorcycles. The proportion of word types occurring in the test data but not the vocabulary to which the novel newsgroups were classified ranged between 30.5% and 66.2%, with a mean of 42.2%. This illustrates that even approximate knowledge can facilitate predictions: the basis set of vocabularies allowed the high frequency words in the data to be modelled effectively, without excess mass being attributed to the low frequency novel word tokens. Long-run performance on talk.politics .mideast illustrates the same defaulting behavior that was shown for text classification: when the data become more probable under the vocabulary containing all words than under a restricted vocabulary the method defaults to the Jeffreys-Perks law. This guarantees that the method will tend to perform no worse than PJ when unseen words are encountered in sufficient proportions. This is desirable, since PJ gives good estimates once N becomes large. 4 Discussion Bayesian approaches to parameter estimation for sparse multinomial distributions have employed the notion of a restricted vocabulary from which symbols are produced. In many domains where such distributions are estimated; there is often at least limited knowledge about the structure of these vocabularies. By considering just the vocabularies suggested by such knowledge, together with some uncertainty concerning those vocabularies, we can achieve very good estimates of distributions in these domains. We have presented a Bayesian approach that employs limited prior knowledge, and shown that it outperforms a range of approaches to multinomial estimation for both text compression and a task involving natural language. While our applications in this paper estimated approximate vocabularies from data, the real promise of this approach lies with domain knowledge solicited from experts. Experts are typically better at providing qualitative structural information than quantitative distributional information, and our approach provides a way of using this information in parameter estimation. For example, in the context of parameter estimation for graphical models to be used in medical diagnosis, distinguishing classes of symptoms might be informative in determining the parameters governing their relationship to diseases. This form of knowledge is naturally translated into a set of vocabularies to be considered for each such distribution. More complex applications to natural language lllay also be possible, such as using syntactic information in estimating probabilities for n-gram models. The approach we have presented in this paper provides a simple way to allow this kind of limited domain knowledge to be useful in Bayesian parameter estimation. References [1] T. C. Bell, J. G. Cleary, and 1. H. Witten. Text compression. Prentice Hall, 1990. [2] G. E. P. Box and G. C. Tiao. Bayesian Inference in Statistical Analysis. AddisonWesley, 1973. [3] S. F. Chen and J. Goodman. An empirical study of smoothing techniques for language modeling. Technical Report TR-10-98, Center for Research in Computing Technology, Harvard University, 1998. [4] N. Friedman and Y. Singer. Efficient Bayesian parameter estimation in large discrete domains. In Neural Information Processing Systems, 1998. [5] H. Jeffreys. An invariant form for the prior probability in estimation problems. Proceedings of the Royal Society A, 186:453-461, 1946. [6] P.-S. Laplace. Philosophical Essay on Probabilities. Springer-Verlag, 1995. Originally published 1825. [7] G. Lidstone. Note on the general case of the Bayes-Laplace formula for inductive or a posteriori probabilities. Transactions of the Faculty of Actuaries, 8:182-192, 1920. [8] K. Nigam, A. K. Mccallum, S. Thrun, and T. Mitchell. Text classification fro'in labeled and unlabeled documents using EM. Machine Learning, 39:103-134, 2000. [9] W. Perks. Some observations on inverse probability, including a new indifference rule. Journal of the Institute of Actuaries, 73:285-312, 1947. [10] E. S. Ristad. A natural law ·of succession. Technical Report CS-TR-895-95, Department of Computer Science, Princeton University, 1995.
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Bayesian time series classification Peter Sykacek Department of Engineering Science University of Oxford Oxford, OX1 3PJ, UK psyk@robots.ox.ac.uk Stephen Roberts Department of Engineering Science University of Oxford Oxford, OX1 3PJ, UK sjrob@robots.ox.ac.uk Abstract This paper proposes an approach to classification of adjacent segments of a time series as being either of classes. We use a hierarchical model that consists of a feature extraction stage and a generative classifier which is built on top of these features. Such two stage approaches are often used in signal and image processing. The novel part of our work is that we link these stages probabilistically by using a latent feature space. To use one joint model is a Bayesian requirement, which has the advantage to fuse information according to its certainty. The classifier is implemented as hidden Markov model with Gaussian and Multinomial observation distributions defined on a suitably chosen representation of autoregressive models. The Markov dependency is motivated by the assumption that successive classifications will be correlated. Inference is done with Markov chain Monte Carlo (MCMC) techniques. We apply the proposed approach to synthetic data and to classification of EEG that was recorded while the subjects performed different cognitive tasks. All experiments show that using a latent feature space results in a significant improvement in generalization accuracy. Hence we expect that this idea generalizes well to other hierarchical models. 1 Introduction Many applications in signal or image processing are hierarchical in the sense that a probabilistic model is built on top of variables that are the coefficients of some feature extraction technique. In this paper we consider a particular problem of that kind, where a Gaussian and Multinomial observation hidden Markov model (GMOHMM) is used to discriminate coefficients of an Auto Regressive (AR) process as being either of classes. Bayesian inference is known to give reasonable results when applied to AR models ([RF95]). The situation with classification is similar, see for example the seminal work by [Nea96] and [Mac92]. Hence we may expect to get good results if we apply Bayesian techniques to both stages of the decision process separately. However this is suboptimal since it meant to establish a no probabilistic link between feature extraction and classification. Two arguments suggest the building of one probabilistic model which combines feature extraction and classification: Since there is a probabilistic link, the generative classifier acts as a prior for feature extraction. The advantage of using this prior is that it naturally encodes our knowledge about features as obtained from training data and other sensors. Obviously this is the only setup that is consistent with Bayesian theory ([BS94]). Since all inferences are obtained from marginal distributions, information is combined according to its certainty. Hence we expect to improve results since information from different sensors is fused in an optimal manner. 2 Methods 2.1 A Gaussian and Multinomial observation hidden Markov model As we attempt to classify adjacent segments of a time series, it is very likely that we find correlations between successive class labels. Hence our model has a hidden Markov model ([RJ86]) like architecture, with diagonal Gaussian observation models for continuous variables and Multinomial observation models for discrete variables. We call the architecture a Gaussian and Multinomial observation hidden Markov model or GMOHMM for short. Contrary to the classical approach, where each class is represented by its own trained HMM, our model has class labels which are child nodes of the hidden state variables. Figure 1 shows the directed acyclic graph (DAG) of our model. We use here the convention found in [RG97], where circular nodes are latent and square nodes are observed variables. 2.1.1 Quantities of interest We regard all variables in the DAG that represent the probabilistic model of the time series as quantities of interest. These are the hidden states, , the variables of the latent feature space, , and , the class labels, , the sufficient statistics of the AR process, , and the segments of the time series, . The DAG shows the observation model only for the
-th state. We have latent feature variables, , which represent the coefficients of the preprocessing model for of the
-th segment at sensor . The state conditional distributions, , are modeled by diagonal Gaussians. Variable is the latent model indicator which represents the model order of the preprocessing model and hence the dimension of . The corresponding observation model is a Multinomial-one distribution. The third observation, , represents the class label of the
-th segment. The observation model for is again a Multinomial-one distribution. Note that depending on whether we know the class label or not, can be a latent variable or observed. The child node of and is the observed variable , which represents a sufficient statistics of the corresponding time series segment. The proposed approach requires to calculate the likelihoods !
repeatedly. Hence using the sufficient statistics is a computational necessity. Finally we use " to represent the precision of the residual noise model. The noise level is a nuisance parameter which is integrated over. 2.1.2 Model coefficients Since we integrate over all unknown quantities, there is no conceptual difference between model coefficients and the variables described above. However there is a qualitative difference. Model parameters exist only once for the entire GMOHMM, whereas there is an individual quantity of interest for every segment
. Furthermore the model coefficients are only updated during model inference whereas all quantities of interest are updated during model inference and for prediction. We have three different prior counts, #%$ , # & and #' , which define the Dirichlet priors of the corresponding probabilities. Variable ( denotes the transition probabilities, that is )* ,+ /. (1032 . The model assumes a stationary hidden state sequence. This allows us to obtain the unconditional prior probability of states 4 from the recurrence relation 56$7 8 /. (159 8:; . The prior probability of the first hidden state, 5 $ , is therefore the normalized eigenvector of the transition probability matrix ( that corresponds to the eigenvalue . Variable represents the probabilities of class , )* . 032 , which are conditional on as well. The prior probabilities for observing the model indicator are represented by 5 . The probability )* 8 . 5 0 2 is again conditional on the state . As was mentioned above, represents the model order of the time series model. Hence another interpretation of 5 is that of state dependent prior probabilities for observing particular model orders. The observation models for are dynamic mixtures of Gaussians, with one model for each sensor . Variables and 1 represent the coefficients of all Gaussian kernels. Hence 1 8 is a variate Gaussian distribution. Another interpretation is that the discrete indicator variables 8 and determine together with and 1 a Gaussian prior over . The nodes , , , , and / define a hierarchical prior setting which is discussed below. s 1 ϕ s ϕ λ λ λ s λ δP 1 h 1 g 1 α 1 Σ β1 1 ξ 1 κ δ δ T di−1 t i T W di+1 id W 1 µ i,1 i,1 Xi,1 i,1 I i,1 P i,s Xi,s I i,s P δ P κs ξs αs µs Σs s β hs gs i,s i,s α α Figure 1: This figure illustrates the details of the proposed model as a directed acyclic graph. The graph shows the model parameters and all quantities of interest: denotes the hidden states of the HMM; are the class labels of the corresponding time series segments; are the latent coefficients of the time series model and the corresponding model indicator variables; is the precision of the residual noise. For tractable inference, we extract from the time series the sufficient statistics . All other variables denote model coefficients: ( are the transition probabilities; are the probabilities for class 3 ; and 1 are mean vectors and covariance matrices of the Gaussian observation model for sensor ; and ) are the probabilities for observing . 2.2 Likelihood and priors for the GMOHMM Suppose that we are provided with segments of training data, .
. The likelihood function of the GMOHMM parameters is then obtained by summation over all possible sequences, , of latent states, . The sums and integrals under the product make the likelihood function of Equation (1) highly nonlinear. This may be resolved by using Gibbs sampling [GG84], which uses tricks similar to those of the expectation maximization algorithm. )* /. )*
-3 " -3 -3 " (1) "! )* :
2 #$ 7 %& ' Gibbs sampling requires that we obtain full conditional distributions1 we can sample from. The conjugate priors are adopted from [RG97]. Below square brackets and index ( are used to denote a particular component of a vector or matrix. Each component mean, 032 , is given a Gaussian prior: 0 2*),+ - : , with denoting the mean and : the inverse covariance matrix. As we use diagonal covariance matrices, we may give each diagonal element an independent Gamma prior: * 0 2.( (0/ :;)1 (0/ , where denotes the shape parameter and (0/ denotes the inverse scale parameter. The hyperparameter, 2! , gets a component wise Gamma hyper prior: (0/ )31 /4(0/ . The state conditional class probabilities, 032 , get a Dirichlet prior: 032 )65 # & ' ' # & . The transition probabilities, ( 0 2 , get a Dirichlet prior: ( 0 2 )75 #$ ' ' #$ . The probabilities for observing different model orders, 5 3 0 2 , depend on the state . Their prior is Dirichlet 5 032 )85 # ' '9' # '/ . The precision gets a Jeffreys’ prior, i.e. the scale parameter :<; is set to 0. Values for are between and = , 8 is set between > ' = and and /?(0/ is typically between A@AB (?/ ! and > @AB (?/ ! , with B (0/ denoting the input range of maximum likelihood estimates for (0/
. The mean, , is the midpoint of the maximum likelihood estimates (0/
. The inverse covariance matrix A(?/ . 4@?B C(0/ ! , where B A(0/ is again the range of the estimates at sensor . We set the prior counts #& and #$ and #' to . 2.3 Sampling from the posterior During model inference we need to update all unobserved variables of the DAG, whereas for predictions we update only the variables summarized in section 2.1.1. Most of the updates are done using the corresponding full conditional distributions, which have the same functional forms as the corresponding priors. These full conditionals follow closely from what was published previously in [Syk00], with some modifications necessary (see e.g. [Rob96]), because we need to consider the Markov dependency between successive hidden states. As the derivations of the full conditionals do not differ much from previous work, we will omit them here and instead concentrate on an illustration how to update the latent feature space,
. 2.3.1 A representation of the latent feature space The AR model in Equation (2) is a linear regression model. We use :<D 2 to denote the AR coefficients, to denote the model order and E4 / to denote a sample from the noise process, which we assume to be Gaussian with precision . F / .HG 2 D : D 2 F GJI /LKE4 / (2) As is indicated by the subscript , the value of the I -th AR coefficient depends on the model order. Hence AR coefficients are not a convenient representation of the latent feature 1These are the distributions obtained when we condition on all other variables of the DAG. space. A much more convenient representation is provided by using reflection coefficients, , (statistically speaking they are partial correlation coefficients), which relate to AR coefficients via the order recursive Levinson algorithm. Below we use vector notation and the symbol 2 to denote the upside down version of the AR coefficient vector. 2 +. 2 K 2 +- 2 2 +- (3) We expect to observe only such data that was generated from dynamically stable AR processes. For such processes, the latent density is defined on G / 2
2 . This is in contrast with the proposed DAG, where we use a finite Gaussian mixture as probabilistic model for the latent variable, which is is defined on 2 . In order to avoid this mismatch, we reparameterise the space of reflection coefficients by applying ! , to obtain a more convenient representation of the latent features. . ! ;" (4) 2.3.2 Within dimensional updates The within dimensional updates can be done with a conventional Metropolis Hastings step. Integrating out , we obtain a Student t distributed likelihood function of the AR coefficients. In order to obtain likelihood ratio 1, we propose from the multivariate Student-t distribution shown below, reparameterise in terms of reflection coefficients and apply the ! transformation. $# . ! ;" /%&# (5) where # ) ' (*) with ) . + :;-, . + :; B *G , $ + :;-, = . G The proposal uses + to denote the -dimensional sample auto-covariance matrix, B is the sample variance, , . B ' '9' B 2 +/ $ is a vector of sample autocorrelations at lags to K and N denotes the number of samples of the time series . The proposal in Equation (5) gives a likelihood ratio of . The corresponding acceptance probability is : ..0/ 2134 # 65 5 587 92 7 9 2 5 5 5 5 5 5 7 2 7 2 5 5 5 :<; = ' (6) The determinant of the Jacobian arises because we transform the AR coefficients using Equations (3) and (4). 2.3.3 Updating model orders Updating model orders requires us to sample across different dimensional parameter spaces. One way of doing this is by using the reversible jump MCMC which was recently proposed in [Gre95]. We implement the reversible jump move from parameter space > 3 2 to parameter space > 2 +- as partial proposal. That is we propose a reflection coefficient from a distribution that is conditional on the AR coefficient / . Integrating out the precision of the noise model ! we obtain again a Student-t distributed likelihood. This suggests the following proposal: # . ! ; / (7) where ) ' ( with . G ! . G ! = G . G . B K = $ , K2 $ + ! . B +"! K = , $ K2 $ + * ' Equation (7) makes use of the sufficient statistics of the K -dimensional AR process, . B ' ' B +"! . We use to denote the number of observations and B to denote the estimated auto covariance at time lag to obtain , $ . B ' ' B +/ and + as dimensional sample covariance matrix. Assuming that the probability of proposing this move is independent of , the proposal from > 3 2 to > 2 +- has acceptance probability : . . / G ! ! ! :
= 1 :;! 1 ! # K G ! 8' (8) If we attempt an update from > 2 +- to > 2 , we have to invert the second argument of the .0/ operation in Equation (8). 3 Experiments Convergence of all experiments is analysed by applying the method suggested in [RL96] to the sequence of observed data likelihoods (equation (1), when filling in all variables). 3.1 Synthetic data Our first evaluation uses synthetic data. We generate a first order Markov sequence as target labels (2 state values) with 200 samples used for training and 600 used for testing. Each sample is used as label of a segment with 200 samples from an auto regressive process. If the label is , we generate data using reflection coefficients > ' G > ' > ' . If the label is = , we use the model > ' G > ' > ' . The driving noise has variance . Due to sampling effects we obtain a data set with Bayes error > . In order to make the problem more realistic, we use a second state sequence to replace =0> of the segments with white noise. These “artifacts” are not correlated with the class labels. In order to assess the effect of using a latent feature space, we perform three different tests: In the first run we use conventional feature extraction with a third order model and estimates found with maximum likelihood; In a second run we use again a third order model but integrate over feature values; Finally the third test uses the proposed architecture with a prior over model order which is “flat” between > and . When compared with conditioning on feature estimates, the latent features show increased likelihood. The likelihood gets even larger when we regard both the feature values and the model orders of the preprocessing stage as random variables. As can be seen in figure 2, this effect is also evident when we look at the generalization probabilities which become larger as well. We explain this by sharper “priors” over feature values and model orders, which are due to the information provided by temporal context2 of every segment. This reduces the variance of the observation models which in turn increases likelihoods and target probabilities. Table 1 shows that these higher probabilities correspond to a significant improvement in generalization accuracy. 50 100 150 200 250 300 350 400 450 500 550 600 0 0.5 1 Probabilities from conditioning 50 100 150 200 250 300 350 400 450 500 550 600 0 0.5 1 Probabilities from integrating over features 50 100 150 200 250 300 350 400 450 500 550 600 0 0.5 1 Probabilities from integrating over model orders and features Figure 2: This figure shows the generalization probabilities obtained with different settings. We see that the class probabilities get larger when we regard features as random variables. This effect is even stronger when both the features and the model orders are random variables. 3.2 Classification of cognitive tasks The data used in these experiments is EEG recorded from 5 young, healthy and untrained subjects while they perform different cognitive tasks. We classify 2 task pairings: auditorynavigation and left motor-right motor imagination. The recordings were taken from 3 electrode sites: T4, P4 (right tempero-parietal for spatial and auditory tasks), C3’ , C3” (left motor area for right motor imagination) and C4’ , C4” (right motor area for left motor imagination). The ground electrode was placed just lateral to the left mastoid process. The data were recorded using an ISO-DAM system (gain of and fourth order band pass filter with pass band between Hz and Hz). These signals were sampled with 384 Hz and 12 bit resolution. Each cognitive experiment was performed times for seconds. Classification uses again the same settings as with the synthetic problem. The summary in table 1 shows results obtained from fold cross validation, where one experiment is used for testing whereas all remaining data is used for training. We observe again significantly improved results when we regard features and model orders as latent variables. The values in brackets are the significance levels for comparing integration of features with conditioning and full integration with integration over feature values only. 4 Discussion We propose in this paper a novel approach to hierarchical time series processing which makes use of a latent feature representation. This understanding of features and model orders as random variables is a direct consequence of applying Bayesian theory. Empirical 2In a multi sensor setting there is spatial context as well. Table 1: Generalization accuracies of different experiments experiment conditioning marginalize features full integration synthetic L' L' = ( ' > : - ) ' ( > ' >0> = ) left vs. right motor ' <' ( ' = > : ) ' ( > ' > ) auditory vs. navigation ' = ' > ' > = ' ( = ' > : ) evaluations show that theoretical arguments are confirmed by significant improvements in generalization accuracy. The only disadvantage of having a latent feature space is that all computations get more involved, since there are additional variables that have to be integrated over. However this additional complexity does not render the method intractable since the algorithm remains polynomial in the number of segments to be classified. Finally we want to point out that the improvements observed in our results can only be attributed to the idea of using a latent feature space. This idea is certainly not limited to time series classification and should generalize well to other hierarchical architectures. Acknowledgments We want to express gratitude to Dr. Rezek, who made several valuable suggestions in the early stages of this work. We also want to thank Prof. Stokes, who provided us with the EEG recordings that were used in the experiments section. Finally we are also grateful for the valuable comments provided by the reviewers of this paper. Peter Sykacek is currently funded by grant Nr. F46/399 kindly provided by the BUPA foundation. References [BS94] J. M. Bernardo and A. F. M. Smith. Bayesian Theory. Wiley, Chichester, 1994. [GG84] S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6:721–741, 1984. [Gre95] P. J. Green. Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82:711–732, 1995. [Mac92] D. J. C. MacKay. The evidence framework applied to classification networks. Neural Computation, 4:720–736, 1992. [Nea96] R. M. Neal. Bayesian Learning for Neural Networks. Springer, New York, 1996. [RF95] J. J. K. ´O Ruanaidh and W. J. Fitzgerald. Numerical Bayesian Methods Applied to Signal Processing. Springer-Verlag, New York, 1995. [RG97] S. Richardson and P. J. Green. On Bayesian analysis of mixtures with an unknown number of components. Journal Royal Stat. Soc. B, 59:731–792, 1997. [RJ86] L. R. Rabiner and B. H. Juang. An introduction to Hidden Markov Models. IEEE ASSP Magazine, 3(1):4–16, 1986. [RL96] A. E. Raftery and S. M. Lewis. Implementing MCMC. In W.R. Gilks, S. Richardson, and D.J. Spiegelhalter, editors, Markov Chain Monte Carlo in practice, chapter 7, pages 115– 130. Chapman & Hall, London, Weinheim, New York, 1996. [Rob96] C. P. Robert. Mixtures of distributions: inference and estimation. In W. R. Gilks, S. Richardson, and D.J. Spiegelhalter, editors, Markov Chain Mont Carlo in Practice, pages 441–464. Chapman & Hall, London, 1996. [Syk00] P. Sykacek. On input selection with reversible jump Markov chain Monte Carlo sampling. In S.A. Solla, T.K. Leen, and K.-R. M¨uller, editors, Advances in Neural Information Processing Systems 12, pages 638–644, Boston, MA, 2000. MIT Press.
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Computing Time Lower Bounds for Recurrent Sigmoidal Neural Networks Michael Schmitt Lehrstuhl Mathematik und Informatik, Fakultat fUr Mathematik Ruhr-Universitat Bochum, D- 44780 Bochum, Germany mschmitt@lmi.ruhr-uni-bochum.de Abstract Recurrent neural networks of analog units are computers for realvalued functions. We study the time complexity of real computation in general recurrent neural networks. These have sigmoidal, linear, and product units of unlimited order as nodes and no restrictions on the weights. For networks operating in discrete time, we exhibit a family of functions with arbitrarily high complexity, and we derive almost tight bounds on the time required to compute these functions. Thus, evidence is given of the computational limitations that time-bounded analog recurrent neural networks are subject to. 1 Introduction Analog recurrent neural networks are known to have computational capabilities that exceed those of classical Turing machines (see, e.g., Siegelmann and Sontag, 1995; Kilian and Siegelmann, 1996; Siegelmann, 1999). Very little, however, is known about their limitations. Among the rare results in this direction, for instance, is the one of Sima and Orponen (2001) showing that continuous-time Hopfield networks may require exponential time before converging to a stable state. This bound, however, is expressed in terms of the size of the network and, hence, does not apply to fixed-size networks with a given number of nodes. Other bounds on the computational power of analog recurrent networks have been established by Maass and Orponen (1998) and Maass and Sontag (1999). They show that discretetime recurrent neural networks recognize only a subset of the regular languages in the presence of noise. This model of computation in recurrent networks, however, receives its inputs as sequences. Therefore, computing time is not an issue since the network halts when the input sequence terminates. Analog recurrent neural networks, however, can also be run as "real" computers that get as input a vector of real numbers and, after computing for a while, yield a real output value. No results are available thus far regarding the time complexity of analog recurrent neural networks with given size. We investigate here the time complexity of discrete-time recurrent neural networks that compute functions over the reals. As network nodes we allow sigmoidal units, linear units, and product units- that is, monomials where the exponents are adjustable weights (Durbin and Rumelhart, 1989). We study the complexity of real computation in the sense of Blum et aI. (1998). That means, we consider real numbers as entities that are represented exactly and processed without restricting their precision. Moreover, we do not assume that the information content of the network weights is bounded (as done, e.g., in the works of Balcazar et aI., 1997; Gavalda and Siegelmann, 1999). With such a general type of network, the question arises which functions can be computed with a given number of nodes and a limited amount of time. In the following, we exhibit a family of real-valued functions ft, l 2: 1, in one variable that is computed by some fixed size network in time O(l). Our main result is, then, showing that every recurrent neural network computing the functions ft requires at least time nW/4). Thus, we obtain almost tight time bounds for real computation in recurrent neural networks. 2 Analog Computation in Recurrent Neural Networks We study a very comprehensive type of discrete-time recurrent neural network that we call general recurrent neural network (see Figure 1). For every k, n E N there is a recurrent neural architecture consisting of k computation nodes YI , . . . ,Yk and n input nodes Xl , ... , xn . The size of a network is defined to be the number ofits computation nodes. The computation nodes form a fully connected recurrent network. Every computation node also receives connections from every input node. The input nodes play the role of the input variables of the system. All connections are parameterized by real-valued adjustable weights. There are three types of computation nodes: product units, sigmoidal units, and linear units. Assume that computation node i has connections from computation nodes weighted by Wil, ... ,Wi k and from input nodes weighted by ViI, .. . ,Vi n. Let YI (t) , . . . ,Yk (t) and Xl (t), ... ,Xn (t) be the values of the computation nodes and input nodes at time t, respectively. If node i is a product unit, it computes at time t + 1 the value (1) that is, after weighting them exponentially, the incoming values are multiplied. Sigmoidal and linear units have an additional parameter associated with them, the threshold or bias ()i . A sigmoidal unit computes the value where (J is the standard sigmoid (J(z ) = 1/ (1 + e- Z ). If node i is a linear unit, it simply outputs the weighted sum We allow the networks to be heterogeneous, that is, they may contain all three types of computation nodes simultaneously. Thus, this model encompasses a wide class of network types considered in research and applications. For instance, architectures have been proposed that include a second layer of linear computation nodes which have no recurrent connections to computation nodes but serve as output nodes (see, e.g., Koiran and Sontag, 1998; Haykin, 1999; Siegelmann, 1999). It is clear that in the definition given here, the linear units can function as these output nodes if the weights of the outgoing connections are set to O. Also very common is the use of sigmoidal units with higher-order as computation nodes in recurrent networks (see, e.g., Omlin and Giles, 1996; Gavalda and Siegelmann, 1999; Carrasco et aI., 2000). Obviously, the model here includes these higher-order networks as a special case since the computation of a higher-order sigmoidal unit can be simulated by first computing the higher-order terms using product units and then passing their computation nodes input nodes Xl I . I sigmoidal, product, and linear units Yl . Yk t Xn I Figure 1: A general recurrent neural network of size k. Any computation node may serve as output node. outputs to a sigmoidal unit. Product units, however, are even more powerful than higher-order terms since they allow to perform division operations using negative weights. Moreover, if a negative input value is weighted by a non-integer weight, the output of a product unit may be a complex number. We shall ensure here that all computations are real-valued. Since we are mainly interested in lower bounds, however, these bounds obviously remain valid if the computations of the networks are extended to the complex domain. We now define what it means that a recurrent neural network N computes a function f : ~n --+ llt Assume that N has n input nodes and let x E ~n. Given tE N, we say that N computes f(x) in t steps if after initializing at time 0 the input nodes with x and the computation nodes with some fixed values, and performing t computation steps as defined in Equations (1), (2), and (3), one of the computation nodes yields the value f(x). We assume that the input nodes remain unchanged during the computation. We further say that N computes f in time t if for every x E ~n , network N computes f in at most t steps. Note that t may depend on f but must be independent of the input vector. We emphasize that this is a very general definition of analog computation in recurrent neural networks. In particular, we do not specify any definite output node but allow the output to occur at any node. Moreover, it is not even required that the network reaches a stable state, as with attractor or Hopfield networks. It is sufficient that the output value appears at some point of the trajectory the network performs. A similar view of computation in recurrent networks is captured in a model proposed by Maass et al. (2001). Clearly, the lower bounds remain valid for more restrictive definitions of analog computation that require output nodes or stable states. Moreover, they hold for architectures that have no input nodes but receive their inputs as initial values of the computation nodes. Thus, the bounds serve as lower bounds also for the transition times between real-valued states of discrete-time dynamical systems comprising the networks considered here. Our main tool of investigation is the Vapnik-Chervonenkis dimension of neural networks. It is defined as follows (see also Anthony and Bartlett, 1999): A dichotomy of a set S ~ ~n is a partition of S into two disjoint subsets (So , Sd satisfying So U S1 = S. A class :F of functions mapping ~n to {O, I} is said to shatter S if for every dichotomy (So , Sd of S there is some f E :F that satisfies f(So) ~ {O} and f(S1) ~ {I}. The Vapnik-Chervonenkis (VC) dimension of :F is defined as 4"'+4",IL S~ 'I -1---Y-2----Y-5~1 Y5 output Y4 Figure 2: A recurrent neural network computing the functions fl in time 2l + 1. the largest number m such that there is a set of m elements shattered by F. A neural network given in terms of an architecture represents a class of functions obtained by assigning real numbers to all its adjustable parameters, that is, weights and thresholds or a subset thereof. The output of the network is assumed to be thresholded at some fixed constant so that the output values are binary. The VC dimension of a neural network is then defined as the VC dimension of the class of functions computed by this network. In deriving lower bounds in the next section, we make use of the following result on networks with product and sigmoidal units that has been previously established (Schmitt, 2002). We emphasize that the only constraint on the parameters of the product units is that they yield real-valued, that is, not complex-valued, functions. This means further that the statement holds for networks of arbitrary order, that is, it does not impose any restrictions on the magnitude of the weights of the product units. Proposition 1. (Schmitt, 2002, Theorem 2) Suppose N is a feedforward neural network consisting of sigmoidal, product, and linear units. Let k be its size and W the number of adjustable weights. The VC dimension of N restricted to real-valued functions is at most 4(Wk)2 + 20Wk log(36Wk). 3 Bounds on Computing Time We establish bounds on the time required by recurrent neural networks for computing a family of functions fl : JR -+ JR, l 2:: 1, where l can be considered as a measure of the complexity of fl. Specifically, fl is defined in terms of a dynamical system as the lth iterate of the logistic map ¢>(x) = 4x(1 - x), that is, fl(X) { ¢>(x) ¢>(fl- l (x)) l = 1, l > 2. We observe that there is a single recurrent network capable of computing every fl in time O(l). Lemma 2. There is a general recurrent neural network that computes fl in time 2l + 1 for every l. Proof. The network is shown in Figure 2. It consists of linear and second-order units. All computation nodes are initialized with 0, except Yl, which starts with 1 and outputs 0 during all following steps. The purpose of Yl is to let the input x output Figure 3: Network Nt. enter node Y2 at time 1 and keep it away at later times. Clearly, the value fl (x) results at node Y5 after 2l + 1 steps. D The network used for computing fl requires only linear and second-order units. The following result shows that the established upper bound is asymptotically almost tight, with a gap only of order four. Moreover, the lower bound holds for networks of unrestricted order and with sigmoidal units. Theorem 3. Every general recurrent neural network of size k requires at least time cl l / 4 j k to compute function fl' where c> 0 is some constant. Proof. The idea is to construct higher-order networks Nt of small size that have comparatively large VC dimension. Such a network will consist of linear and product units and hypothetical units that compute functions fJ for certain values of j. We shall derive a lower bound on the VC dimension of these networks. Assuming that the hypothetical units can be replaced by time-bounded general recurrent networks, we determine an upper bound on the VC dimension of the resulting networks in terms of size and computing time using an idea from Koiran and Sontag (1998) and Proposition 1. The comparison of the lower and upper VC dimension bounds will give an estimate of the time required for computing k Network Nt, shown in Figure 3, is a feedforward network composed of three networks • r(1) • r(2) .r(3) E h k • r(/1) 1 2 3 h l· d (/1) (/1) JVI ,JVI ,JVI . ac networ JVI ,J.L = , , , as lnput no es Xl' .. . , x I and 2l + 2 computation nodes yb/1), ... , Y~r~l (see Figure 4). There is only one adjustable parameter in Nt, denoted w, all other weights are fixed. The computation nodes are defined as follows (omitting time parameter t): for J.L = 3, for J.L = 1,2, y~/1) fll'--1 (Y~~)l) for i = 1, ... ,l and J.L = 1,2,3, y}~{ y~/1) . x~/1), for i = 1, .. . ,l and J.L = 1,2,3, (/1) (/1) (/1) c 1 2 3 Y21+l YIH + ... + Y21 lor J.L , , • The nodes Yb/1) can be considered as additional input nodes for N//1), where N;(3) gets this input from w, and N;(/1) from N;(/1+l) for J.L = 1,2. Node Y~r~l is the output node of N;(/1), and node Y~~~l is also the output node of Nt. Thus, the entire network has 3l + 6 nodes that are linear or product units and 3l nodes that compute functions h, fl' or f12. output 8 r------------' ..... L-----------, I I B B t t I x~p)1 ~ t input: w or output of N;(P+1) ----Figure 4: Network N;(p). We show that Ni shatters some set of cardinality [3, in particular, the set S = ({ ei : i = 1, . .. , [})3, where ei E {O, 1}1 is the unit vector with a 1 in position i and ° elsewhere. Every dichotomy of S can be programmed into the network parameter w using the following fact about the logistic function ¢ (see Koiran and Sontag, 1998, Lemma 2): For every binary vector b E {O, l}m, b = b1 .•. bm , there is some real number w E [0,1] such that for i = 1, ... , m E { [0,1/2) (1/2,1] if bi = 0, if bi = 1. Hence, for every dichotomy (So, Sd of S the parameter w can be chosen such that every (ei1' ei2 , ei3) E S satisfies 1/2 if (eillei2,eis) E So, 1/2 if (eillei2,eiJ E S1. Since h +i2 H i 3 .12 (w) = ¢i1 (¢i2'1 (¢i3 .12 (w))), this is the value computed by Ni on input (eill ei2' ei3), where ei" is the input given to network N;(p). (Input ei" selects the function li"'I,,-1 in N;(p).) Hence, S is shattered by Ni, implying that Ni has VC dimension at least [3. Assume now that Ii can be computed by a general recurrent neural network of size at most kj in time tj. Using an idea of Koiran and Sontag (1998), we unfold the network to obtain a feedforward network of size at most kjtj computing fj. Thus we can replace the nodes computing ft, ft, fl2 in Nz by networks of size k1t1, kltl, k12t12, respectively, such that we have a feedforward network '!J consisting of sigmoidal, product, and linear units. Since there are 3l units in Nl computing ft, ft, or fl2 and at most 3l + 6 product and linear units, the size of Nt is at most c1lkl2tl2 for some constant C1 > O. Using that Nt has one adjustable weight, we get from Proposition 1 that its VC dimension is at most c2l2kr2tr2 for some constant C2 > o. On the other hand, since Nz and Nt both shatter S, the VC dimension of Nt is at least l3. Hence, l3 ~ C2l2 kr2 tr2 holds, which implies that tl2 2: cl1/2 / kl2 for some c > 0, and hence tl 2: cl1/4 / kl. D Lemma 2 shows that a single recurrent network is capable of computing every function fl in time O(l). The following consequence of Theorem 3 establishes that this bound cannot be much improved. Corollary 4. Every general recurrent neural network requires at least time 0(ll/4 ) to compute the functions fl. 4 Conclusions and Perspectives We have established bounds on the computing time of analog recurrent neural networks. The result shows that for every network of given size there are functions of arbitrarily high time complexity. This fact does not rely on a bound on the magnitude of weights. We have derived upper and lower bounds that are rather tight- with a polynomial gap of order four- and hold for the computation of a specific family of real-valued functions in one variable. Interestingly, the upper bound is shown using second-order networks without sigmoidal units, whereas the lower bound is valid even for networks with sigmoidal units and arbitrary product units. This indicates that adding these units might decrease the computing time only marginally. The derivation made use of an upper bound on the VC dimension of higher-order sigmoidal networks. This bound is not known to be optimal. Any future improvement will therefore lead to a better lower bound on the computing time. We have focussed on product and sigmoidal units as nonlinear computing elements. However, the construction presented here is generic. Thus, it is possible to derive similar results for radial basis function units, models of spiking neurons, and other unit types that are known to yield networks with bounded VC dimension. The questions whether such results can be obtained for continuous-time networks and for networks operating in the domain of complex numbers, are challenging. A further assumption made here is that the networks compute the functions exactly. By a more detailed analysis and using the fact that the shattering of sets requires the outputs only to lie below or above some threshold, similar results can be obtained for networks that approximate the functions more or less closely and for networks that are subject to noise. Acknowledgment The author gratefully acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG). This work was also supported in part by the ESPRIT Working Group in Neural and Computational Learning II, NeuroCOLT2, No. 27150. References Anthony, M. and Bartlett, P. L. (1999). Neural Network Learning: Theoretical Foundations. Cambridge University Press, Cambridge. Balcazar, J., Gavalda, R., and Siegelmann, H. T. (1997). Computational power of neural networks: A characterization in terms of Kolmogorov complexity. IEEE Transcations on Information Theory, 43: 1175- 1183. Blum, L., Cucker, F., Shub, M., and Smale, S. (1998). Complexity and Real Computation. Springer-Verlag, New York. Carrasco, R. C., Forcada, M. L., Valdes-Munoz, M. A., and Neco, R. P. (2000). Stable encoding of finite state machines in discrete-time recurrent neural nets with sigmoid units. Neural Computation, 12:2129- 2174. Durbin, R. and Rumelhart, D. (1989). Product units: A computationally powerful and biologically plausible extension to backpropagation networks. Neural Computation, 1:133- 142. Gavalda, R. and Siegelmann, H. T. (1999). Discontinuities in recurrent neural networks. Neural Computation, 11:715- 745. Haykin, S. (1999). Neural Networks: A Comprehensive Foundation. Prentice Hall, Upper Saddle River, NJ, second edition. Kilian, J. and Siegelmann, H. T. (1996). The dynamic universality of sigmoidal neural networks. Information and Computation, 128:48- 56. Koiran, P. and Sontag, E. D. (1998). Vapnik-Chervonenkis dimension of recurrent neural networks. Discrete Applied Mathematics, 86:63- 79. Maass, W., NatschUiger, T., and Markram, H. (2001). Real-time computing without stable states: A new framework for neural computation based on perturbations. Preprint. Maass, W. and Orponen, P. (1998). On the effect of analog noise in discrete-time analog computations. Neural Computation, 10:1071- 1095. Maass, W. and Sontag, E. D. (1999). Analog neural nets with Gaussian or other common noise distributions cannot recognize arbitrary regular languages. Neural Computation, 11:771- 782. amlin, C. W. and Giles, C. L. (1996). Constructing deterministic finite-state automata in recurrent neural networks. Journal of the Association for Computing Machinery, 43:937- 972. Schmitt, M. (2002). On the complexity of computing and learning with multiplicative neural networks. Neural Computation, 14. In press. Siegelmann, H. T . (1999). Neural Networks and Analog Computation: Beyond the Turing Limit. Progress in Theoretical Computer Science. Birkhiiuser, Boston. Siegelmann, H. T. and Sontag, E. D. (1995). On the computational power of neural nets. Journal of Computer and System Sciences, 50:132- 150. Sima, J. and Orponen, P. (2001). Exponential transients in continuous-time symmetric Hopfield nets. In Dorffner, G., Bischof, H., and Hornik, K. , editors, Proceedings of the International Conference on Artificial Neural Networks ICANN 2001, volume 2130 of Lecture Notes in Computer Science, pages 806- 813, Springer, Berlin.
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Playing is believing: The role of beliefs in multi-agent learning Yu-Han Chang Artificial Intelligence Laboratory Massachusetts Institute of Technology Cambridge, Massachusetts 02139 ychang@ai.mit.edu Leslie Pack Kaelbling Artificial Intelligence Laboratory Massachusetts Institute of Technology Cambridge, Massachusetts 02139 lpk@ai.mit.edu Abstract We propose a new classification for multi-agent learning algorithms, with each league of players characterized by both their possible strategies and possible beliefs. Using this classification, we review the optimality of existing algorithms, including the case of interleague play. We propose an incremental improvementto the existing algorithms that seems to achieve average payoffs that are at least the Nash equilibrium payoffs in the longrun against fair opponents. 1 Introduction The topic of learning in multi-agent environments has received increasing attention over the past several years. Game theorists have begun to examine learning models in their study of repeated games, and reinforcement learning researchers have begun to extend their singleagent learning models to the multiple-agent case. As traditional models and methods from these two fields are adapted to tackle the problem of multi-agent learning, the central issue of optimality is worth revisiting. What do we expect a successful learner to do? Matrix games and Nash equilibrium. From the game theory perspective, the repeated game is a generalization of the traditional one-shot game, or matrix game. The matrix game is defined as a reward matrix Ri for each player, Ri : A1 × A2 →R, where Ai is the set of actions available to player i. Purely competitive games are called zero-sum games and must satisfy R1 = −R2. Each player simultaneously chooses to play a particular action ai ∈Ai, or a mixed policy µi = PD(Ai), which is a probability distribution over the possible actions, and receives reward based on the joint action taken. Some common examples of single-shot matrix games are shown in Figure 1. The traditional assumption is that each player has no prior knowledge about the other player. As is standard in the game theory literature, it is thus reasonable to assume that the opponent is fully rational and chooses actions that are in its best interest. In return, we must play a best response to the opponent’s choice of action. A best response function for player i, BRi(µ−i), is defined to be the set of all optimal policies for player i, given that the other players are playing the joint policy µ−i: BRi(µ−i) = {µ∗ i ∈Mi|Ri(µ∗ i , µ−i) ≥Ri(µi, µ−i)∀µi ∈Mi}, where Mi is the set of all possible policies for agent i. If all players are playing best responses to the other players’ strategies, µi ∈BRi(µ−i)∀i, R1 = −1 1 1 −1 R1 = 0 −1 1 1 0 −1 −1 1 0 R1 = 0 3 1 2 R1 = 2 0 3 1 R2 = −R1 R2 = −R1 R2 = 0 1 3 2 R2 = 2 3 0 1 (a) Matching pennies (b) Rock-Paper-Scissors (c) Hawk-Dove (d) Prisoner’s Dilemna Figure 1: Some common examples of single-shot matrix games. then the game is said to be in Nash equilibrium. Once all players are playing by a Nash equilibrium, no single player has an incentive to unilaterally deviate from his equilibrium policy. Any game can be solved for its Nash equilibria using quadratic programming, and a player can choose an optimal strategy in this fashion, given prior knowledge of the game structure. The only problem arises when there are multiple Nash equilibria. If the players do not manage to coordinate on one equilibrium joint policy, then they may all end up worse off. The Hawk-Dove game shown in Figure 1(c) is a good example of this problem. The two Nash equilibria occur at (1,2) and (2,1), but if the players do not coordinate, they may end up playing a joint action (1,1) and receive 0 reward. Stochastic games and reinforcement learning. Despite these problems, there is general agreement that Nash equilibrium is an appropriate solution concept for one-shot games. In contrast, for repeated games there are a range of different perspectives. Repeated games generalize one-shot games by assuming that the players repeat the matrix game over many time periods. Researchers in reinforcement learning view repeated games as a special case of stochastic, or Markov, games. Researchers in game theory, on the other hand, view repeated games as an extension of their theory of one-shot matrix games. The resulting frameworks are similar, but with a key difference in their treatment of game history. Reinforcement learning researchers focus their attention on choosing a single stationary policy µ that will maximize the learner’s expected rewards in all future time periods given that we are in time t, maxµ Eµ hPT τ=t γτ−tRτ(µ) i , where T may be finite or infinite, and µ = PD(A). In the infinite time-horizon case, we often include the discount factor 0 < γ < 1. Littman [1] analyzes this framework for zero-sum games, proving convergence to the Nash equilibrium for his minimax-Q algorithm playing against another minimax-Q agent. Claus and Boutilier [2] examine cooperative games where R1 = R2, and Hu and Wellman [3] focus on general-sum games. These algorithms share the common goal of finding and playing a Nash equilibrium. Littman [4] and Hall and Greenwald [5] further extend this approach to consider variants of Nash equilibrium for which convergence can be guaranteed. Bowling and Veloso [6] and Nagayuki et al. [7] propose to relax the mutual optimality requirement of Nash equilibrium by considering rational agents, which always learn to play a stationary best-response to their opponent’s strategy, even if the opponent is not playing an equilibrium strategy. The motivation is that it allows our agents to act rationally even if the opponent is not acting rationally because of physical or computational limitations. Fictitious play [8] is a similar algorithm from game theory. Game theoretic perspective of repeated games. As alluded to in the previous section, game theorists often take a more general view of optimality in repeated games. The key difference is the treatment of the history of actions taken in the game. Recall that in the Table 1: Summary of multi-agent learning algorithms under our new classification. B0 B1 B∞ H0 minimax-Q, Nash-Q Bully H1 Godfather H∞ Q-learning (Q0), (WoLF-)PHC, fictitious play Q1 multiplicativeweight* * assumes public knowledge of the opponent’s policy at each period stochastic game model, we took µi = PD(Ai). Here we redefine µi : H →PD(Ai), where H = S t Ht and Ht is the set of all possible histories of length t. Histories are observations of joint actions, ht = (ai, a−i, ht−1). Player i’s strategy at time t is then expressed as µi(ht−1). In essence, we are endowing our agent with memory. Moreover, the agent ought to be able to form beliefs about the opponent’s strategy, and these beliefs ought to converge to the opponent’s actual strategy given sufficient learning time. Let βi : H →PD(A−i) be player i’s belief about the opponent’s strategy. Then a learning path is defined to be a sequence of histories, beliefs, and personal strategies. Now we can define the Nash equilibrium of a repeated game in terms of our personal strategy and our beliefs about the opponent. If our prediction about the opponent’s strategy is accurate, then we can choose an appropriate best-response strategy. If this holds for all players in the game, then we are guaranteed to be in Nash equilibrium. Proposition 1.1. A learning path {(ht, µi(ht−1), βi(ht−1))|t = 1, 2, . . .} converges to a Nash equilibrium iff the following two conditions hold: • Optimization: ∀t, µi(ht−1) ∈BRi(βi(ht−1)). We always play a best-response to our prediction of the opponent’s strategy. • Prediction: limt→∞|βi(ht−1) −µ−i(ht−1)| = 0. Over time, our belief about the opponent’s strategy converges to the opponent’s actual strategy. However, Nachbar and Zame [9] shows that this requirement of simultaneous prediction and optimization is impossible to achieve, given certain assumptions about our possible strategies and possible beliefs. We can never design an agent that will learn to both predict the opponent’s future strategy and optimize over those beliefs at the same time. Despite this fact, if we assume some extra knowledge about the opponent, we can design an algorithm that approximates the best-response stationary policy over time against any opponent. In the game theory literature, this concept is often called universal consistency. Fudenburg and Levine [8] and Freund and Schapire [10] independently show that a multiplicativeweight algorithm exhibits universal consistency from the game theory and machine learning perspectives. This give us a strong result, but requires the strong assumption that we know the opponent’s policy at each time period. This is typically not the case. 2 A new classification and a new algorithm We propose a general classification that categorizes algorithms by the cross-product of their possible strategies and their possible beliefs about the opponent’s strategy, H×B. An agent’s possible strategies can be classified based upon the amount of history it has in memory, from H0 to H∞. Given more memory, the agent can formulate more complex policies, since policies are maps from histories to action distributions. H0 agents are memoryless and can only play stationary policies. Agents that can recall the actions from the previous time period are classified as H1 and can execute reactive policies. At the other extreme, H∞agents have unbounded memory and can formulate ever more complex strategies as the game is played over time. An agent’s belief classification mirrors the strategy classification in the obvious way. Agents that believe their opponent is memoryless are classified as B0 players, Bt players believe that the opponent bases its strategy on the previous tperiods of play, and so forth. Although not explicitly stated, most existing algorithms make assumptions and thus hold beliefs about the types of possible opponents in the world. We can think of each Hs × Bt as a different league of players, with players in each league roughly equal to one another in terms of their capabilities. Clearly some leagues contain less capable players than others. We can thus define a fair opponent as an opponent from an equal or lesser league. The idea is that new learning algorithms should ideally be designed to beat any fair opponent. The key role of beliefs. Within each league, we assume that players are fully rational in the sense that they can fully use their available histories to construct their future policy. However, an important observation is that the definition of full rationality depends on their beliefs about the opponent. If we believe that our opponent is a memoryless player, then even if we are an H∞player, our fully rational strategy is to simply model the opponent’s stationary strategy and play our stationary best response. Thus, our belief capacity and our history capacity are inter-related. Without a rich set of possible beliefs about our opponent, we cannot make good use of our available history. Similarly, and perhaps more obviously, without a rich set of historical observations, we cannot hope to model complex opponents. Discussion of current algorithms. Many of the existing algorithms fall within the H∞× B0 league. As discussed in the previous section, the problem with these players is that even though they have full access to the history, their fully rational strategy is stationary due to their limited belief set. A general example of a H∞× B0 player is the policy hill climber (PHC). It maintains a policy and updates the policy based upon its history in an attempt to maximize its rewards. Originally PHC was created for stochastic games, and thus each policy also depends on the current state s. In our repeated games, there is only one state. For agent i, Policy Hill Climbing (PHC) proceeds as follows: 1. Let α and δ be the learning rates. Initialize Q(s, a) ←0, µi(s, a) ← 1 |Ai|∀s ∈S, a ∈Ai. 2. Repeat, a. From state s, select action a according to the mixed policy µi(s) with some exploration. b. Observing reward r and next state s′, update Q(s, a) ←(1 −α)Q(s, a) + α(r + γ max a′ Q(s′, a′)). c. Update µ(s, a) and constrain it to a legal probability distribution: µi(s, a) ←µi(s, a) + δ if a = argmaxa′ Q(s, a′) −δ |Ai|−1 otherwise . The basic idea of PHC is that the Q-values help us to define a gradient upon which we execute hill-climbing. Bowling and Veloso’s WoLF-PHC [6] modifies PHC by adjusting δ depending on whether the agent is “winning” or “losing.” True to their league, PHC players play well against stationary opponents. At the opposite end of the spectrum, Littman and Stone [11] propose algorithms in H0×B∞ and H1 × B∞that are leader strategies in the sense that they choose a fixed strategy and hope that their opponent will “follow” by learning a best response to that fixed strategy. Their “Bully” algorithm chooses a fixed memoryless stationary policy, while “Godfather” has memory of the last time period. Opponents included normal Q-learning and Q1 players, which are similar to Q-learners except that they explicitly learn using one period of memory because they believe that their opponent is also using memory to learn. The interesting result is that “Godfather” is able to achieve non-stationary equilibria against Q1 in the repeated prisoner’s dilemna game, with rewards for both players that are higher than the stationary Nash equilibrium rewards. This demonstrates the power of having belief models. However, because these algorithms do not have access to more than one period of history, they cannot begin to attempt to construct statistical models the opponent. “Godfather” works well because it has a built-in best response to Q1 learners rather than attempting to learn a best response from experience. Finally, Hu and Wellman’s Nash-Q and Littman’s minimax-Q are classified as H0 × B0 players, because even though they attempt to learn the Nash equilibrium through experience, their play is fixed once this equilibrium has been learned. Furthermore, they assume that the opponent also plays a fixed stationary Nash equilibrium, which they hope is the other half of their own equilibrium strategy. These algorithms are summarized in Table 1. A new class of players. As discussed above, most existing algorithms do not form beliefs about the opponent beyond B0. None of these approaches is able to capture the essence of game-playing, which is a world of threats, deceits, and generally out-witting the opponent. We wish to open the door to such possibilities by designing learners that can model the opponent and use that information to achieve better rewards. Ideally we would like to design an algorithm in H∞× B∞that is able to win or come to an equilibrium against any fair opponent. Since this is impossible [9], we start by proposing an algorithm in the league H∞× B∞that plays well against a restricted class of opponents. Since many of the current algorithms are best-response players, we choose an opponent class such as PHC, which is a good example of a best-response player in H∞× B0. We will demonstrate that our algorithm indeed beats its PHC opponents and in fact does well against most of the existing fair opponents. A new algorithm: PHC-Exploiter. Our algorithm is different from most previous work in that we are explicitly modeling the opponent’s learning algorithm and not simply his current policy. In particular, we would like to model players from H∞× B0. Since we are in H∞× B∞, it is rational for us to construct such models because we believe that the opponent is learning and adapting to us over time using its history. The idea is that we will “fool” our opponent into thinking that we are stupid by playing a decoy policy for a number of time periods and then switch to a different policy that takes advantage of their best response to our decoy policy. From a learning perspective, the idea is that we adapt much faster than the opponent; in fact, when we switch away from our decoy policy, our adjustment to the new policy is immediate. In contrast, the H∞× B0 opponent adjusts its policy by small increments and is furthermore unable to model our changing behavior. We can repeat this “bluff and bash” cycle ad infinitum, thereby achieving infinite total rewards as t →∞. The opponent never catches on to us because it believes that we only play stationary policies. A good example of a H∞× B0 player is PHC. Bowling and Veloso showed that in selfplay, a restricted version of WoLF-PHC always reaches a stationary Nash equilibrium in two-player two-action games, and that the general WoLF-PHC seems to do the same in experimental trials. Thus, in the long run, a WoLF-PHC player achieves its stationary Nash equilibrium payoff against any other PHC player. We wish to do better than that by exploiting our knowledge of the PHC opponent’s learning strategy. We can construct a PHC-Exploiter algorithm for agent i that proceeds like PHC in steps 1-2b, and then continues as follows: c. Observing action at −i at time t, update our history h and calculate an estimate of the opponent’s policy: ˆµt −i(s, a) = Pt τ=t−w #(h[τ] = a) w ∀a, where w is the window of estimation and #(h[τ] = a) = 1 if the opponent’s action at time τ is equal to a, and 0 otherwise. We estimate ˆµt−w −i (s) similarly. d. Update δ by estimating the learning rate of the PHC opponent: δ ← ˆµt −i(s) −ˆµt−w −i (s) w . e. Update µi(s, a). If we are winning, i.e. P a′ µi(s, a′)Q(s, a′) > Ri(ˆµ∗ i (s), ˆµ−i(s)), then update µi(s, a) ← 1 if a = argmaxa′ Q(s, a′) 0 otherwise , otherwise we are losing, then update µi(s, a) ←µi(s, a) + δ if a = argmaxa′ Q(s, a′) −δ |Ai|−1 otherwise . Note that we derive both the opponent’s learning rate δ and the opponent’s policy ˆµ−i(s) from estimates using the observable history of actions. If we assume the game matrix is public information, then we can solve for the equilibrium strategy ˆµ∗ i (s), otherwise we can run WoLF-PHC for some finite number of time periods to obtain an estimate this equilibrium strategy. The main idea of this algorithm is that we take full advantage of all time periods in which we are winning, that is, when P a′ µi(s, a′)Q(s, a′) > Ri(ˆµ∗ i (s), ˆµ−i(s)). Analysis. The PHC-Exploiter algorithm is based upon PHC and thus exhibits the same behavior as PHC in games with a single pure Nash equilibrium. Both agents generally converge to the single pure equilibrium point. The interesting case arises in competitive games where the only equilibria require mixed strategies, as discussed by Singh et al [12] and Bowling and Veloso [6]. Matching pennies, shown in Figure 1(a), is one such game. PHC-Exploiter is able to use its model of the opponent’s learning algorithm to choose better actions. In the full knowledge case where we know our opponent’s policy µ2 and learning rate δ2 at every time period, we can prove that a PHC-Exploiter learning algorithm will guarantee us unbounded reward in the long run playing games such as matching pennies. Proposition 2.1. In the zero-sum game of matching pennies, where the only Nash equilibrium requires the use of mixed strategies, PHC-Exploiter is able to achieve unbounded rewards as t →∞against any PHC opponent given that play follows the cycle C defined by the arrowed segments shown in Figure 2. Play proceeds along Cw, Cl, then jumps from (0.5, 0) to (1,0), follows the line segments to (0.5, 1), then jumps back to (0, 1). Given a point (x, y) = (µ1(H), µ2(H)) on the graph in Figure 2, where µi(H) is the probability by which player i plays Heads, we know that our expected reward is R1(x, y) = −1 × [(x)(y) + (1 −x)(1 −y)] + 1 × [(1 −x)(y) + (x)(1 −y)]. -0.5 0 0.5 1 1.5 -0.5 0 0.5 1 1.5 Player 2 probability choosing Heads Player 1 probability choosing Heads Action distribution of the two agent system Cl Cw -0.5 0 0.5 1 1.5 -0.5 0 0.5 1 1.5 Player 2 probability choosing Heads Player 1 probability choosing Heads Action distribution of the two agent system agent1 winning agent1 losing Figure 2: Theoretical (left), Empirical (right). The cyclic play is evident in our empirical results, where we play a PHC-Exploiter player 1 against a PHC player 2. -4000 -2000 0 2000 4000 6000 8000 0 20000 40000 60000 80000 100000 total reward time period Agent 1 total reward over time Figure 3: Total rewards for agent 1 increase as we gain reward through each cycle. We wish to show that Z C R1(x, y)dt = 2 × Z Cw R1(x, y)dt + Z Cl R1(x, y)dt > 0 . We consider each part separately. In the losing section, we let g(t) = x = t and h(t) = y = 1/2 −t, where 0 ≤t ≤1/2. Then Z Cl R1(x, y)dt = Z 1/2 0 R1(g(t), h(t))dt = −1 12 . Similarly, we can show that we receive 1/4 reward over Cw. Thus, R C R1(x, y)dt = 1/3 > 0, and we have shown that we receive a payoff greater than the Nash equilibrium payoff of zero over every cycle. It is easy to see that play will indeed follow the cycle C to a good approximation, depending on the size of δ2. In the next section, we demonstrate that we can estimate µ2 and δ2 sufficiently well from past observations, thus eliminating the full knowledge requirements that were used to ensure the cyclic nature of play above. Experimental results. We used the PHC-Exploiter algorithm described above to play against several PHC variants in different iterated matrix games, including matching pennies, prisoner’s dilemna, and rock-paper-scissors. Here we give the results for the matching pennies game analyzed above, playing against WoLF-PHC. We used a window of w = 5000 time periods to estimate the opponent’s current policy µ2 and the opponent’s learning rate δ2. As shown in Figure 2, the play exhibits the cyclic nature that we predicted. The two solid vertical lines indicate periods in which our PHC-Exploiter player is winning, and the dashed, roughly diagonal, lines indicate periods in which it is losing. In the analysis given in the previous section, we derived an upper bound for our total rewards over time, which was 1/6 for each time step. Since we have to estimate various parameters in our experimental run, we do not achieve this level of reward. We gain an average of 0.08 total reward for each time period. Figure 3 plots the total reward for our PHC-Exploiter agent over time. The periods of winning and losing are very clear from this graph. Further experiments tested the effectiveness of PHC-Exploiter against other fair opponents, including itself. Against all the existing fair opponents shown in Table 1, it achieved at least its average equilibrium payoff in the long-run. Not surprisingly, it also posted this score when it played against a multiplicative-weight learner. Conclusion and future work. In this paper, we have presented a new classification for multi-agent learning algorithms and suggested an algorithm that seems to dominate existing algorithms from the fair opponent leagues when playing certain games. Ideally, we would like to create an algorithm in the league H∞× B∞that provably dominates larger classes of fair opponents in any game. Moreover, all of the discussion contained within this paper dealt with the case of iterated matrix games. We would like to extend our framework to more general stochastic games with multiple states and multiple players. Finally, it would be interesting to find practical applications of these multi-agent learning algorithms. Acknowledgements. This work was supported in part by a Graduate Research Fellowship from the National Science Foundation. References [1] Michael L. Littman. Markov games as a framework for multi-agent reinforcement learning. In Proceedings of the 11th International Conference on Machine Learning (ICML-94), 1994. [2] Caroline Claus and Craig Boutilier. The dynamics of reinforcement learning in cooperative multiaent systems. In Proceedings of the 15th Natl. Conf. on Artificial Intelligence, 1998. [3] Junling Hu and Michael P. Wellman. Multiagent reinforcement learning: Theoretical framework and an algorithm. In Proceedings of the 15th Int. Conf. on Machine Learning (ICML-98), 1998. [4] Michael L. Littman. Friend-or-foe q-learning in general-sum games. In Proceedings of the 18th Int. Conf. on Machine Learning (ICML-01), 2001. [5] Keith Hall and Amy Greenwald. Correlated q-learning. In DIMACS Workshop on Computational Issues in Game Theory and Mechanism Design, 2001. [6] Michael Bowling and Manuela Veloso. Multiagent learning using a variable learning rate. Under submission. [7] Yasuo Nagayuki, Shin Ishii, and Kenji Doya. Multi-agent reinforcement learning: An approach based on the other agent’s internal model. In Proceedings of the International Conference on Multi-Agent Systems (ICMAS-00), 2000. [8] Drew Fudenburg and David K. Levine. Consistency and cautious fictitious play. Journal of Economic Dynamics and Control, 19:1065–1089, 1995. [9] J.H. Nachbar and W.R. Zame. Non-computable strategies and discounted repeated games. Economic Theory, 1996. [10] Yoav Freund and Robert E. Schapire. Adaptive game playing using multiplicative weights. Games and Economic Behavior, 29:79–103, 1999. [11] Michael Littman and Peter Stone. Leading best-response stratgies in repeated games. In 17th Int. Joint Conf. on Artificial Intelligence (IJCAI-2001) workshop on Economic Agents, Models, and Mechanisms, 2001. [12] S. Singh, M. Kearns, and Y. Mansour. Nash convergence of gradient dynamics in general-sum games. In Proceedings of the 16th Conference on Uncertainty in Artificial Intelligence, 2000.
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Probabilistic principles in unsupervised learning of visual structure: human data and a model Shimon Edelman, Benjamin P. Hiles & Hwajin Yang Department of Psychology Cornell University, Ithaca, NY 14853 se37,bph7,hy56 @cornell.edu Nathan Intrator Institute for Brain and Neural Systems Box 1843, Brown University Providence, RI 02912 Nathan Intrator@brown.edu Abstract To find out how the representations of structured visual objects depend on the co-occurrence statistics of their constituents, we exposed subjects to a set of composite images with tight control exerted over (1) the conditional probabilities of the constituent fragments, and (2) the value of Barlow’s criterion of “suspicious coincidence” (the ratio of joint probability to the product of marginals). We then compared the part verification response times for various probe/target combinations before and after the exposure. For composite probes, the speedup was much larger for targets that contained pairs of fragments perfectly predictive of each other, compared to those that did not. This effect was modulated by the significance of their co-occurrence as estimated by Barlow’s criterion. For lone-fragment probes, the speedup in all conditions was generally lower than for composites. These results shed light on the brain’s strategies for unsupervised acquisition of structural information in vision. 1 Motivation How does the human visual system decide for which objects it should maintain distinct and persistent internal representations of the kind typically postulated by theories of object recognition? Consider, for example, the image shown in Figure 1, left. This image can be represented as a monolithic hieroglyph, a pair of Chinese characters (which we shall refer to as and ), a set of strokes, or, trivially, as a collection of pixels. Note that the second option is only available to a system previously exposed to various combinations of Chinese characters. Indeed, a principled decision whether to represent this image as , or otherwise can only be made on the basis of prior exposure to related images. According to Barlow’s [1] insight, one useful principle is tallying suspicious coincidences: two candidate fragments and should be combined into a composite object if the probability of their joint appearance
is much higher than , which is the probability expected in the case of their statistical independence. This criterion may be compared to the Minimum Description Length (MDL) principle, which has been previously discussed in the context of object representation [2, 3]. In a simplified form [4], MDL calls for representing explicitly as a whole if , just as the principle of suspicious coincidences does. While the Barlow/MDL criterion certainly indicates a suspicious coincidence, there are additional probabilistic considerations that may be used in setting the degree of association between and . One example is the possible perfect predictability of from and vice versa, as measured by
. If , then and are perfectly predictive of each other and should really be coded by a single symbol, whereas the MDL criterion may suggest merely that some association between the representation of and that of be established. In comparison, if and are not perfectly predictive of each other ( ), there is a case to be made in favor of coding them separately to allow for a maximally expressive representation, whereas MDL may actually suggest a high degree of association (if
). In this study we investigated whether the human visual system uses a criterion based on alongside MDL while learning (in an unsupervised manner) to represent composite objects. AB Figure 1: Left: how many objects are contained in image ? Without prior knowledge, a reasonable answer, which embodies a holistic bias, should be “one” (Gestalt effects, which would suggest two convex “blobs” [5], are beyond the scope of the present discussion). Right: in this set of ten images, appears five times as a whole; the other five times a fragment wholly contained in appears in isolation. This statistical fact provides grounds for considering to be composite, consisting of two fragments (call the upper one and the lower one ), because , but . To date, psychophysical explorations of the sensitivity of human subjects to stimulus statistics tended to concentrate on means (and sometimes variances) of the frequency of various stimuli (e.g., [6]. One recent and notable exception is the work of Saffran et al. [7], who showed that infants (and adults) can distinguish between “words” (stable pairs of syllables that recur in a continuous auditory stimulus stream) and non-words (syllables accidentally paired with each other, the first of which comes from one “word” and the second – from the following one). Thus, subjects can sense (and act upon) differences in transition probabilities between successive auditory stimuli. This finding has been recently replicated, with infants as young as 2 months, in the visual sequence domain, using successive presentation of simple geometric shapes with controlled transition probabilities [8]. Also in the visual domain, Fiser and Aslin [9] presented subjects with geometrical shapes in various spatial configurations, and found effects of conditional probabilities of shape co-occurrences, in a task that required the subjects to decide in each trial which of two simultaneously presented shapes was more familiar. The present study was undertaken to investigate the relevance of the various notions of statistical independence to the unsupervised learning of complex visual stimuli by human subjects. Our experimental approach differs from that of [9] in several respects. First, instead of explicitly judging shape familiarity, our subjects had to verify the presence of a probe shape embedded in a target. This objective task, which produces a pattern of response times, is arguably better suited to the investigation of internal representations involved in object recognition than subjective judgment. Second, the estimation of familiarity requires the subject to access in each trial the representations of all the objects seen in the experiment; in our task, each trial involved just two objects (the probe and the target), potentially sharpening the focus of the experimental approach. Third, our experiments tested the predictions of two distinct notions of stimulus independence: , and MDL, or Barlow’s ratio. 2 The psychophysical experiments In two experiments, we presented stimuli composed of characters such as those in Figure 1 to nearly 100 subjects unfamiliar with the Chinese script. The conditional probabilities of the appearance of individual characters were controlled. The experiments involved two types of probe conditions: PTYPE=Fragment, or (with as the reference condition), and PTYPE=Composite, or (with as reference). In this notation (see Figure 2, left), and are “familiar” fragments with controlled minimum conditional probability , and are novel (low-probability) fragments. Each of the two experiments consisted of a baseline phase, followed by training exposure (unsupervised learning), followed in turn by the test phase (Figure 2, right). In the baseline and test phases, the subjects had to indicate whether or not the probe was contained in the target (a task previously used by Palmer [5]). In the intervening training phase, the subjects merely watched the character triplets presented on the screen; to ensure their attention, the subjects were asked to note the order in which the characters appeared. probe target reference test Fragment Composite probe target V ABZ VW ABZ ABZ ABZ A AB probe target mask 1 2 3 4 baseline/test unsupervised training Figure 2: Left: illustration of the probe and target composition for the two levels of PTYPE (Fragment and Composite). For convenience, the various categories of characters that appeared in the experiment are annotated here by Latin letters: , stand for characters with controlled
, and stand for characters that appeared only once throughout an experiment. In experiment 1, the training set was constructed with for some pairs, and for others; in experiment 2, Barlow’s suspicious coincidence ratio was also controlled. Right top: the structure of a part verification trial (same for baseline and test phases). The probe stimulus was followed by the target (each presented for ; a mask was shown before and after the target). The subject had to indicate whether or not the former was contained in the latter (in this example, the correct answer is yes). A sequence consisting of 64 trials like this one was presented twice: before training (baseline phase) and after training (test phase). For “positive” trials (i.e., probe contained in target), we looked at the SPEEDUP following training, defined as
; negative trials were discarded. Right bottom: the structure of a training trial (the training phase, placed between baseline and test, consisted of 80 such trials). The three components of the stimulus appeared one by one for to make sure that the subject attended to each, then together for ! . The subject was required to note whether the sequence unfolded in a clockwise or counterclockwise order. The logic behind the psychophysical experiments rested on two premises. First, we knew from earlier work [5] that a probe is detected faster if it is represented monolithically (that is, considered to be a good “object” in the Gestalt sense). Second, we hypothesized that a composite stimulus would be treated as a monolithic object to the extent that its constituent characters are predictable from each other, as measured by a high conditional probability, , and/or by a high suspicious coincidence ratio, . The main prediction following from these premises is that the SPEEDUP (the difference in response time between baseline and test phases) for a composite probe should reflect the mutual predictability of the probe’s constituents in the training set. Thus, our hypothesis — that statistics of co-occurrence determine the constituents in terms of which structured objects are represented — would be supported if the SPEEDUP turns out to be larger for those composite probes whose constituents tend to appear together in the training set. The experiments, therefore, hinged on a comparison of the patterns of response times in the “positive” trials (in which the probe actually is embedded in the target; see Figure 2, left) before and after exposure to the training set. 0.4 0.6 0.8 1 0 100 200 300 400 speedup, ms minCP Composite Fragment 0.4 0.6 0.8 1 −0.2 −0.1 0 0.1 0.2 0.3 analog of speedup minCP Composite Fragment Figure 3: Left: unsupervised learning of statistically defined structure by human subjects, experiment 1 ( ). The dependent variable SPEED-UP is defined as the difference in
between baseline and test phases (least-squares estimates of means and standard errors, computed by the LSMEANS option of SAS procedure MIXED [10]). The SPEED-UP for composite probes (solid line) with exceeded that in the other conditions by about . Right: the results of a simulation of experiment 1 by a model derived from the one described in [4]. The model was exposed to the same 80 training images as the human subjects. The difference of reconstruction errors for probe and target served as the analog of RT; baseline measurements were conducted on half-trained networks. 2.1 Experiment 1 Fourteen subjects, none of them familiar with the Chinese writing system, participated in this experiment in exchange for course credit. Among the stimuli, two characters could be paired, in which case we had . Alternatively, could be unpaired, with , (in this experiment, we held the suspicious coincidence ratio
constant at ). For the paired the minimum conditional probability and the two characters were perfectly predictable from each other, whereas for the unpaired , and they were not. In the latter case probably should not be represented as a whole. As expected, we found the value of SPEED-UP to be strikingly different for composite probes with ( ) compared to the other three conditions (about ); see Figure 3, left. A mixed-effects repeated measures analysis of variance (SAS procedure MIXED [10]) for SPEED-UP revealed a marginal effect of PTYPE ( ! ) and a significant interaction PTYPE interaction ( ! ). This behavior conforms to the predictions of the principle: SPEEDUP was generally higher for composite probes, and disproportionately higher for composite probes with . The subjects in experiment 1 proved to be sensitive to the measure of independence in learning to associate object fragments together. Note that the suspicious coincidence ratio was the same in both cases,
. Thus, the visual system is sensitive to over and above the (constant-valued) MDLrelated criterion, according to which the propensity to form a unified representation of two fragments, and , should be determined by [1, 4]. 0.4 0.6 0.8 1 0 50 100 150 200 250 speedup, ms minCP r=1.13 0.4 0.6 0.8 1 0 50 100 150 200 250 speedup, ms minCP r=8.33 0 5 10 0 50 100 150 200 250 speedup, ms r minCP=0.5 0 5 10 0 50 100 150 200 250 speedup, ms r minCP=1.0 Figure 4: Human subjects, experiment 2 ( ). The effect of found in experiment 1 was modulated in a complicated fashion by the effect of the suspicious coincidence ratio (see text for discussion). 2.2 Experiment 2 In the second experiment, we studied the effects of varying both and together. Because these two quantities are related (through the Bayes theorem), they cannot be manipulated independently. To accommodate this constraint, some subjects saw two sets of stimuli, with and with , in the first session and other two sets, with and with , in the second session; for other subjects, the complementary combinations were used in each session. Eighty one subjects unfamiliar with the Chinese script participated in this experiment for course credit. The results (Figure 4) showed that SPEEDUP was consistently higher for composite probes. Thus, the association between probe constituents was strengthened by training in each of the four conditions. SPEEDUP was also generally higher for the high suspicious coincidence ratio case, , and disproportionately higher for composite probes in the , case, indicating a complicated synergy between the two measures of dependence, and . A mixed-effects repeated measures analysis of variance (SAS procedure MIXED [10]) for SPEED-UP revealed significant main effects of PTYPE ( ! ) and ( ! ), as well as two significant two-way interactions, ( ) and PTYPE ( ! ). There was also a marginal three-way interaction, PTYPE ( ). The findings of these two psychophysical experiments can be summarized as follows: (1) an individual complex visual shape (a Chinese character) is detected faster than a composite stimulus (a pair of such characters) when embedded in a 3-character scene, but this advantage is narrowed with practice; (2) a composite attains an “objecthood” status to the extent that its constituents are predictable from each other, as measured either by the conditional probability, , or by the suspicious coincidence ratio, ; (3) for composites, the strongest boost towards objecthood (measured by response speedup following unsupervised learning) is obtained when is high and is low, or vice versa. The nature of this latter interaction is unclear, and needs further study. 3 An unsupervised learning model and a simulated experiment The ability of our subjects to construct representations that reflect the probability of cooccurrence of complex shapes has been replicated by a pilot version of an unsupervised learning model, derived from the work of [4]. The model (Figure 5) is based on the following observation: an auto-association network fed with a sequence of composite images in which some fragment/location combinations are more likely than others develops a nonuniform spatial distribution of reconstruction errors. Specifically, smaller errors appear in those locations where the image fragments recur. This information can be used to form a spatial receptive field for the learning module, while the reconstruction error can signal its relevance to the current input [11, 12]. In the simplified pilot model, the spatial receptive field (labeled in Figure 5, left, as “relevance mask”) consists of four weights, one per quadrant: , . During the unsupervised training, the weights are updated by setting
, where is the reconstruction error in trial , and and are learning constants. In a simulation of experiment 1, a separate module with its own four-weight “receptive field” was trained for each of the composite stimuli shown to the human subjects.1 The Euclidean distance between probe and target representations at the output of the model served as the analog of response time, allowing us to compare the model’s performance with that of the humans. We found the same differential effects of for Fragment and Composite probes in the real and simulated experiments; compare Figure 3, left (humans) with Figure 3, right (model). 1The full-fledged model, currently under development, will have a more flexible receptive field structure, and will incorporate competitive learning among the modules. relevance mask (RF) error auto− associator adapt input input − reconstructed ensemble of modules erri Figure 5: Left: the functional architecture of a fragment module. The module consists of two adaptive components: a reconstruction network, and a relevance mask, which assigns different weights to different input pixels. The mask modulates the input multiplicatively, determining the module’s receptive field. Given a sequence of images, several such modules working in parallel learn to represent different categories of spatially localized patterns (fragments) that recur in those images. The reconstruction error serves as an estimate of the module’s ability to deal with the input ([11, 12]; in the error image, shown on the right, white corresponds to high values). Right: the Chorus of Fragments (CoF) is a bank of such fragment modules, each tuned to a particular shape category, appearing in a particular location [13, 4]. 4 Discussion Human subjects have been previously shown to be able to acquire, through unsupervised learning, sensitivity to transition probabilities between syllables of nonsense words [7] and between digits [14], and to co-occurrence statistics of simple geometrical figures [9]. Our results demonstrate that subjects can also learn (presumably without awareness; cf. [14]) to treat combinations of complex visual patterns differentially, depending on the conditional probabilities of the various combinations, accumulated during a short unsupervised training session. In our first experiment, the criterion of suspicious coincidence between the occurrences of and was met in both and conditions: in each case, we had . Yet, the subjects’ behavior indicated a significant holistic bias: the representation they form tends to be monolithic ( ), unless imperfect mutual predictability of the potential fragments ( and ) provides support for representing them separately. We note that a similar holistic bias, operating in a setting where a single encounter with a stimulus can make a difference, is found in language acquisition: an infant faced with an unfamiliar word will assume it refers to the entire shape of the most salient object [15]. In our second experiment, both the conditional probabilities as such, and the suspicious coincidence ratio were found to have the predicted effects, yet these two factors interacted in a complicated manner, which requires a further investigation. Our current research focuses on (1) the elucidation of the manner in which subjects process statistically structured data, (2) the development of the model of structure learning outlined in the preceding section, and (3) an exploration of the implications of this body of work for wider issues in vision, such as the computational phenomenology of scene perception [16]. References [1] H. B. Barlow. Unsupervised learning. Neural Computation, 1:295–311, 1989. [2] R. S. Zemel and G. E. Hinton. Developing population codes by minimizing description length. Neural Computation, 7:549–564, 1995. [3] E. Bienenstock, S. Geman, and D. Potter. Compositionality, MDL priors, and object recognition. In M. C. Mozer, M. I. Jordan, and T. Petsche, editors, Neural Information Processing Systems, volume 9. MIT Press, 1997. [4] S. Edelman and N. Intrator. A productive, systematic framework for the representation of visual structure. In T. K. Leen, T. G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems 13, pages 10–16. MIT Press, 2001. [5] S. E. Palmer. Hierarchical structure in perceptual representation. Cognitive Psychology, 9:441–474, 1977. [6] M. J. Flannagan, L. S. Fried, and K. J. Holyoak. Distributional expectations and the induction of category structure. Journal of Experimental Psychology: Learning, Memory and Cognition, 12:241–256, 1986. [7] J. R. Saffran, R. N. Aslin, and E. L. Newport. Statistical learning by 8-month-old infants. Science, 274:1926–1928, 1996. [8] N. Z. Kirkham, J. A. Slemmer, and S. P. Johnson. Visual statistical learning in infancy: Evidence for a domain general learning mechanism. Cognition, -:–, 2002. in press. [9] J. Fiser and R. N. Aslin. Unsupervised statistical learning of higher-order spatial structures from visual scenes. Psychological Science, 6:499–504, 2001. [10] SAS. User’s Guide, Version 8. SAS Institute Inc., Cary, NC, 1999. [11] D. Pomerleau. Input reconstruction reliability estimation. In C. L. Giles, S. J. Hanson, and J. D. Cowan, editors, Advances in Neural Information Processing Systems, volume 5, pages 279–286. Morgan Kaufmann Publishers, 1993. [12] I. Stainvas and N. Intrator. Blurred face recognition via a hybrid network architecture. In Proc. ICPR, volume 2, pages 809–812, 2000. [13] S. Edelman and N. Intrator. (Coarse Coding of Shape Fragments) + (Retinotopy) Representation of Structure. Spatial Vision, 13:255–264, 2000. [14] G. S. Berns, J. D. Cohen, and M. A. Mintun. Brain regions responsive to novelty in the absence of awareness. Science, 276:1272–1276, 1997. [15] B. Landau, L. B. Smith, and S. Jones. The importance of shape in early lexical learning. Cognitive Development, 3:299–321, 1988. [16] S. Edelman. Constraints on the nature of the neural representation of the visual world. Trends in Cognitive Sciences, 6:–, 2002. in press.
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Predictive Representations of State Michael L. Littman Richard S. Sutton AT&T Labs-Research, Florham Park, New Jersey {mlittman,sutton}~research.att.com Satinder Singh Syntek Capital, New York, New York baveja~cs.colorado.edu Abstract We show that states of a dynamical system can be usefully represented by multi-step, action-conditional predictions of future observations. State representations that are grounded in data in this way may be easier to learn, generalize better, and be less dependent on accurate prior models than, for example, POMDP state representations. Building on prior work by Jaeger and by Rivest and Schapire, in this paper we compare and contrast a linear specialization of the predictive approach with the state representations used in POMDPs and in k-order Markov models. Ours is the first specific formulation of the predictive idea that includes both stochasticity and actions (controls). We show that any system has a linear predictive state representation with number of predictions no greater than the number of states in its minimal POMDP model. In predicting or controlling a sequence of observations, the concepts of state and state estimation inevitably arise. There have been two dominant approaches. The generative-model approach, typified by research on partially observable Markov decision processes (POMDPs), hypothesizes a structure for generating observations and estimates its state and state dynamics. The history-based approach, typified by k-order Markov methods, uses simple functions of past observations as state, that is, as the immediate basis for prediction and control. (The data flow in these two approaches are diagrammed in Figure 1.) Of the two, the generative-model approach is more general. The model's internal state gives it temporally unlimited memorythe ability to remember an event that happened arbitrarily long ago--whereas a history-based approach can only remember as far back as its history extends. The bane of generative-model approaches is that they are often strongly dependent on a good model of the system's dynamics. Most uses of POMDPs, for example, assume a perfect dynamics model and attempt only to estimate state. There are algorithms for simultaneously estimating state and dynamics (e.g., Chrisman, 1992), analogous to the Baum-Welch algorithm for the uncontrolled case (Baum et al., 1970), but these are only effective at tuning parameters that are already approximately correct (e.g., Shatkay & Kaelbling, 1997). observations (and actions) (a) state 1-----1-----1..rep'n observations¢E (and actions) / state t/' rep'n 1-step --+ . delays (b) Figure 1: Data flow in a) POMDP and other recursive updating of state representation, and b) history-based state representation. In practice, history-based approaches are often much more effective. Here, the state representation is a relatively simple record of the stream of past actions and observations. It might record the occurrence of a specific subsequence or that one event has occurred more recently than another. Such representations are far more closely linked to the data than are POMDP representations. One way of saying this is that POMDP learning algorithms encounter many local minima and saddle points because all their states are equipotential. History-based systems immediately break symmetry, and their direct learning procedure makes them comparably simple. McCallum (1995) has shown in a number of examples that sophisticated history-based methods can be effective in large problems, and are often more practical than POMDP methods even in small ones. The predictive state representation (PSR) approach, which we develop in this paper, is like the generative-model approach in that it updates the state representation recursively, as in Figure l(a), rather than directly computing it from data. We show that this enables it to attain generality and compactness at least equal to that of the generative-model approach. However, the PSR approach is also like the history-based approach in that its representations are grounded in data. Whereas a history-based representation looks to the past and records what did happen, a PSR looks to the future and represents what will happen. In particular, a PSR is a vector of predictions for a specially selected set of action-observation sequences, called tests (after Rivest & Schapire, 1994). For example, consider the test U101U202, where U1 and U2 are specific actions and 01 and 02 are specific observations. The correct prediction for this test given the data stream up to time k is the probability of its observations occurring (in order) given that its actions are taken (in order) (i.e., Pr {Ok = 01, Ok+1 = 02 I A k = u1,Ak+ 1 = U2}). Each test is a kind of experiment that could be performed to tell us something about the system. If we knew the outcome of all possible tests, then we would know everything there is to know about the system. A PSR is a set of tests that is sufficient information to determine the prediction for all possible tests (a sufficient statistic). As an example of these points, consider the float/reset problem (Figure 2) consisting of a linear string of 5 states with a distinguished reset state on the far right. One action, f (float), causes the system to move uniformly at random to the right or left by one state, bounded at the two ends. The other action, r (reset), causes a jump to the reset state irrespective of the current state. The observation is always ounless the r action is taken when the system is already in the reset state, in which case the observation is 1. Thus, on an f action, the correct prediction is always 0, whereas on an r action, the correct prediction depends on how many fs there have been since the last r: for zero fS, it is 1; for one or two fS, it is 0.5; for three or four fS, it is 0.375; for five or six fs, it is 0.3125, and so on decreasing after every second f, asymptotically bottoming out at 0.2. No k-order Markov method can model this system exactly, because no limited-. .5 a) float action .5 b) reset action 1,0=1 Figure 2: Underlying dynamics of the float/reset problem for a) the float action and b) the reset action. The numbers on the arcs indicate transition probabilities. The observation is always 0 except on the reset action from the rightmost state, which produces an observation of 1. length history is a sufficient statistic. A POMDP approach can model it exactly by maintaining a belief-state representation over five or so states. A PSR, on the other hand, can exactly model the float/reset system using just two tests: rl and fOrI. Starting from the rightmost state, the correct predictions for these two tests are always two successive probabilities in the sequence given above (1, 0.5, 0.5, 0.375,...), which is always a sufficient statistic to predict the next pair in the sequence. Although this informational analysis indicates a solution is possible in principle, it would require a nonlinear updating process for the PSR. In this paper we restrict consideration to a linear special case of PSRs, for which we can guarantee that the number of tests needed does not exceed the number of states in the minimal POMDP representation (although we have not ruled out the possibility it can be considerably smaller). Of greater ultimate interest are the prospects for learning PSRs and their update functions, about which we can only speculate at this time. The difficulty of learning POMDP structures without good prior models are well known. To the extent that this difficulty is due to the indirect link between the POMDP states and the data, predictive representations may be able to do better. Jaeger (2000) introduced the idea of predictive representations as an alternative to belief states in hidden Markov models and provided a learning procedure for these models. We build on his work by treating the control case (with actions), which he did not significantly analyze. We have also been strongly influenced by the work of Rivest and Schapire (1994), who did consider tests including actions, but treated only the deterministic case, which is significantly different. They also explored construction and learning algorithms for discovering system structure. 1 Predictive State Representations We consider dynamical systems that accept actions from a discrete set A and generate observations from a discrete set O. We consider only predicting the system, not controlling it, so we do not designate an explicit reward observation. We refer to such a system as an environment. We use the term history to denote a test forming an initial stream of experience and characterize an environment by a probability distribution over all possible histories, P : {OIA}* H- [0,1], where P(Ol··· Otla1··· at) is the probability of observations 01, ... , O£ being generated, in that order, given that actions aI, ... ,at are taken, in that order. The probability of a test t conditional on a history h is defined as P(tlh) = P(ht)/P(h). Given a set of q tests Q = {til, we define their (1 x q) prediction vector, p(h) = [P(t1Ih),P(t2Ih), ... ,P(tqlh)], as a predictive state representation (PSR) if and only if it forms a sufficient statistic for the environment, Le., if and only if P(tlh) = ft(P(h)), (1) for any test t and history h, and for some projection junction ft : [0, l]q ~ [0,1]. In this paper we focus on linear PSRs, for which the projection functions are linear, that is, for which there exist a (1 x q) projection vector mt, for every test t, such that P(tlh) == ft(P(h)) =7 p(h)mf, (2) for all histories h. Let Pi(h) denote the ith component of the prediction vector for some PSR. This can be updated recursively, given a new action-observation pair a,o, by (3) .(h ) == P(t.lh ) == P(otilha) == faati(P(h)) == p(h)m'{;ati P2 ao 2 ao P(olha) faa (P(h)) p(h)mro ' where the last step is specific to linear PSRs. We can now state our main result: Theorem 1 For any environment that can be represented by a finite POMDP model, there exists a linear PSR with number of tests no larger than the number of states in the minimal POMDP model. 2 Proof of Theorem 1: Constructing a PSR from a POMDP We prove Theorem 1 by showing that for any POMDP model of the environment, we can construct in polynomial time a linear PSR for that POMDP of lesser or equal complexity that produces the same probability distribution over histories as the POMDP model. We proceed in three steps. First, we review POMDP models and how they assign probabilities to tests. Next, we define an algorithm that takes an n-state POMDP model and produces a set of n or fewer tests, each of length less than or equal to n. Finally, we show that the set of tests constitute a PSR for the POMDP, that is, that there are projection vectors that, together with the tests' predictions, produce the same probability distribution over histories as the POMDP. A POMDP (Lovejoy, 1991; Kaelbling et al., 1998) is defined by a sextuple (8, A, 0, bo,T, 0). Here, 8 is a set of n underlying (hidden) states, A is a discrete set of actions, and 0 is a discrete set of observations. The (1 x n) vector bo is an initial state distribution. The set T consists of (n x n) transition matrices Ta, one for each action a, where Tlj is the probability of a transition from state i to j when action a is chosen. The set 0 consists of diagonal (n x n) observation matrices oa,o, one for each pair of observation 0 and action a, where o~'o is the probability of observation 0 when action a is selected and state i is reached. l The state representation in a POMDP (Figure l(a)) is the belief state-the (1 x n) vector of the state-occupation probabilities given the history h. It can be computed recursively given a new action a and observation 0 by b(h)Taoa,o b(hao) = b(h)Taoa,oe;' where en is the (1 x n)-vector of all Is. Finally, a POMDP defines a probability distribution over tests (and thus histories) by P(Ol ... otlhal ... at) == b(h)Ta1oal,Ol ... Taloa£,Ole~. (4) IThere are many equivalent formulations and the conversion procedure described here can be easily modified to accommodate other POMDP definitions. We now present our algorithm for constructing a PSR for a given POMDP. It uses a function u mapping tests to (1 x n) vectors defined recursively by u(c) == en and u(aot) == (Taoa,ou(t)T)T, where c represents the null test. Conceptually, the components of u(t) are the probabilities of the test t when applied from each underlying state of the POMDP; we call u(t) the outcome vector for test t. We say a test t is linearly independent of a set of tests S if its outcome vector is linearly independent of the set of outcome vectors of the tests in S. Our algorithm search is used and defined as Q -<- search(c, {}) search(t, S): for each a E A, 0 E 0 if aot is linearly independent of S then S -<- search(aot, S U {aot}) return S The algorithm maintains a set of tests and searches for new tests that are linearly independent of those already found. It is a form of depth-first search. The algorithm halts when it checks all the one-step extensions of its tests and finds none that are linearly independent. Because the set of tests Q returned by search have linearly independent outcome vectors, the cardinality of Q is bounded by n, ensuring that the algorithm halts after a polynomial number of iterations. Because each test in Q is formed by a one-step extension to some other test in Q, no test is longer than n action-observation pairs. The check for linear independence can be performed in many ways, including Gaussian elimination, implying that search terminates in polynomial time. By construction, all one-step extensions to the set of tests Q returned by search are linearly dependent on those in Q. We now show that this is true for any test. Lemma 1 The outcome vectors of the tests in Q can be linearly combined to produce the outcome vector for any test. Proof: Let U be the (n x q) matrix formed by concatenating the outcome vectors for all tests in Q. Since, for all combinations of a and 0, the columns of Taoa,ou are linearly dependent on the columns of U, we can write Taoa,ou == UWT for some q x q matrix of weights W. If t is a test that is linearly dependent on Q, then anyone-step extension of t, aot, is linearly dependent on Q. This is because we can write the outcome vector for t as u(t) == (UwT)T for some (1 x q) weight vector w and the outcome vector for aot as u(aot) == (Taoa,ou(t)T)T == (Taoa,oUwT)T == (UWTwT)T. Thus, aot is linearly dependent on Q. Now, note that all one-step tests are linearly dependent on Q by the structure of the search algorithm. Using the previous paragraph as an inductive argument, this implies that all tests are linearly dependent on Q. 0 Returning to the float/reset example POMDP, search begins with by enumerating the 4 extensions to the null test (fO, fl, rO, and rl). Of these, only fa and rO are are linearly independent. Of the extensions of these, fOrO is the only one that is linearly independent of the other two. The remaining two tests added to Q by search are fOfOrO and fOfOfOrO. No extensions of the 5 tests in Q are linearly independent of the 5 tests in Q, so the procedure halts. We now show that the set of tests Q constitute a PSR for the POMDP by constructing projection vectors that, together with the tests' predictions, produce the same probability distribution over histories as the POMDP. For each combination of a and 0, define a q x q matrix Mao == (U+Taoa,ou)T and a 1 x q vector mao == (U+Taoa,oe;;JT, where U is the matrix of outcome vectors defined in the previous section and U+ is its pseudoinverse2 • The ith row of Mao is maoti. The probability distribution on histories implied by these projection vectors is p(h)m~101 alOl p(h)M~ol M~_10l_1 m~Ol b(h)UU+ra1 oa1,01U ... U+Tal-10al-1,Ol-1 UU+Taloal,ole; b(h)Ta10 a1,01 ... ral-l0al-t,ol-lTaloal,Ole~, Le., it is the same as that of the POMDP, as in Equation 4. Here, the last step uses the fact that UU+vT == vT for vT linearly dependent on the columns of U. This holds by construction of U in the previous section. This completes the proof of Theorem 1. Completing the float/reset example, consider the Mf,o matrix found by the process defined in this section. It derives predictions for each test in Qafter taking action f. Most of these are quite simple because the tests are so similar: the new prediction for rO is exactly the old prediction for fOrO, for example. The only non trivial test is fOfOfOrO. Its outcome can be computed from 0.250 p(rOlh) - 0.0625 p(fOrOlh) + 0.750 p(fOfOrOlh). This example illustrates that the projection vectors need not contain only positive entries. 3 Conclusion We have introduced a predictive state representation for dynamical systems that is grounded in actions and observations and shown that, even in its linear form, it is at least as general and compact as POMDPs. In essence, we have established PSRs as a non-inferior alternative to POMDPs, and suggested that they might have important advantages, while leaving demonstration of those advantages to future work. We conclude by summarizing the potential advantages (to be explored in future work): Learnability. The k-order Markov model is similar to PSRs in that it is entirely based on actions and observations. Such models can be learned trivially from data by counting-it is an open question whether something similar can be done with a PSR. Jaeger (2000) showed how to learn such a model in the uncontrolled setting, but the situation is more complex in the multiple action case since outcomes are conditioned on behavior, violating some required independence assumptions. Compactness. We have shown that there exist linear PSRs no more complex that the minimal POMDP for an environment, but in some cases the minimal linear PSR seems to be much smaller. For example, a POMDP extension of factored MDPs explored by Singh and Cohn (1998) would be cross-products of separate POMDPs and have linear PSRs that increase linearly with the number and size of the component POMDPs, whereas their minimal POMDP representation would grow as the size 2If U = A~BT is the singular value decomposition of U, then B:E+AT is the pseudoinverse. The pseudoinverse of the diagonal matrix }J replaces each non-zero element with its reciprocal. of the state space, Le., exponential in the number of component POMDPs. This (apparent) advantage stems from the PSR's combinatorial or factored structure. As a vector of state variables, capable of taking on diverse values, a PSR may be inherently more powerful than the distribution over discrete states (the belief state) of a POMDP. We have already seen that general PSRs can be more compact than POMDPs; they are also capable of efficiently capturing environments in the diversity representation used by Rivest and Schapire (1994), which is known to provide an extremely compact representation for some environments. Generalization. There are reasons to think that state variables that are themselves predictions may be particularly useful in learning to make other predictions. With so many things to predict, we have in effect a set or sequence of learning problems, all due to the same environment. In many such cases the solutions to earlier problems have been shown to provide features that generalize particularly well to subsequent problems (e.g., Baxter, 2000; Thrun & Pratt, 1998). Powerful, extensible representations. PSRs that predict tests could be generalized to predict the outcomes of multi-step options (e.g., Sutton et al., 1999). In this case, particularly, they would constitute a powerful language for representing the state of complex environments. AcknowledgIllents: We thank Peter Dayan, Lawrence Saul, Fernando Pereira and Rob Schapire for many helpful discussions of these and related ideas. References Baum, L. E., Petrie, T., Soules, G., & Weiss, N. (1970). A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Annals of Mathematical Statistics, 41, 164-171. Baxter, J. (2000). A model of inductive bias learning. Journal of Artificial Intelligence Research, 12, 149-198. Chrisman, L. (1992). Reinforcement learning with perceptual aliasing: The perceptual distinctions approach. Proceedings of the Tenth National Conference on Artificial Intelligence (pp. 183-188). San Jose, California: AAAI Press. Jaeger, H. (2000). Observable operator models for discrete stochastic time series. Neural Computation, 12, 1371-1398. Kaelbling, L. P., Littman, M. L., & Cassandra, A. R. (1998). Planning and acting in ' partially observable stochastic domains. Artificial Intelligence, 101, 99-134. Lovejoy, W. S. (1991). A survey of algorithmic methods for partially observable Markov decision processes. Annals of Operations Research, 28, 47-65. McCallum, A. K. (1995). Reinforcement learning with selective perception and hidden state. Doctoral diss.ertation, Department of Computer Science, University of Rochester. Rivest, R. L., & Schapire, R. E. (1994). Diversity-based inference of finite automata. Journal of the ACM, 41, 555-589. Shatkay, H., & Kaelbling, L. P. (1997). Learning topological maps with weak local odometric information~ Proceedings of Fifteenth International Joint Conference on Artificial Intelligence (IJCAI-91) (pp. 920-929). Singh, S., & Cohn, D. (1998). How to dynamically merge Markov decision processes. Advances in Neural and Information Processing Systems 10 (pp. 1057-1063). Sutton, R. S., Precup, D., & Singh, S. (1999). Between MDPs and semi-MDPs: A framework for temporal abstraction in reinforcement learning. Artificial Intelligence, 181-211. Thrun, S., & Pratt, L. (Eds.). (1998). Learning to learn. Kluwer Academic Publishers.
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Discriminative Direction for Kernel Classifiers Polina Golland Artificial Intelligence Lab Massachusetts Institute of Technology Cambridge, MA 02139 polina@ai.mit.edu Abstract In many scientific and engineering applications, detecting and understanding differences between two groups of examples can be reduced to a classical problem of training a classifier for labeling new examples while making as few mistakes as possible. In the traditional classification setting, the resulting classifier is rarely analyzed in terms of the properties of the input data captured by the discriminative model. However, such analysis is crucial if we want to understand and visualize the detected differences. We propose an approach to interpretation of the statistical model in the original feature space that allows us to argue about the model in terms of the relevant changes to the input vectors. For each point in the input space, we define a discriminative direction to be the direction that moves the point towards the other class while introducing as little irrelevant change as possible with respect to the classifier function. We derive the discriminative direction for kernel-based classifiers, demonstrate the technique on several examples and briefly discuss its use in the statistical shape analysis, an application that originally motivated this work. 1 Introduction Once a classifier is estimated from the training data, it can be used to label new examples, and in many application domains, such as character recognition, text classification and others, this constitutes the final goal of the learning stage. The statistical learning algorithms are also used in scientific studies to detect and analyze differences between the two classes when the “correct answer” is unknown, and the information we have on the differences is represented implicitly by the training set. Example applications include morphological analysis of anatomical organs (comparing organ shape in patients vs. normal controls), molecular design (identifying complex molecules that satisfy certain requirements), etc. In such applications, interpretation of the resulting classifier in terms of the original feature vectors can provide an insight into the nature of the differences detected by the learning algorithm and is therefore a crucial step in the analysis. Furthermore, we would argue that studying the spatial structure of the data captured by the classification function is important in any application, as it leads to a better understanding of the data and can potentially help in improving the technique. This paper addresses the problem of translating a classifier into a different representation that allows us to visualize and study the differences between the classes. We introduce and derive a so called discriminative direction at every point in the original feature space with respect to a given classifier. Informally speaking, the discriminative direction tells us how to change any input example to make it look more like an example from another class without introducing any irrelevant changes that possibly make it more similar to other examples from the same class. It allows us to characterize differences captured by the classifier and to express them as changes in the original input examples. This paper is organized as follows. We start with a brief background section on kernelbased classification, stating without proof the main facts on kernel-based SVMs necessary for derivation of the discriminative direction. We follow the notation used in [3, 8, 9]. In Section 3, we provide a formal definition of the discriminative direction and explain how it can be estimated from the classification function. We then present some special cases, in which the computation can be simplified significantly due to a particular structure of the kernel. Section 4 demonstrates the discriminative direction for different kernels, followed by an example from the problem of statistical analysis of shape differences that originally motivated this work. 2 Basic Notation Given a training set of l pairs {(xk, yk)}l k=1, where xk ∈Rn are observations and yk ∈{−1, 1} are corresponding labels, and a kernel function K : Rn × Rn 7→R, (with its implied mapping function ΦK : Rn 7→F), the Support Vector Machines (SVMs) algorithm [8] constructs a classifier by implicitly mapping the training data into a higher dimensional space and estimating a linear classifier in that space that maximizes the margin between the classes (Fig. 1a). The normal to the resulting separating hyperplane is a linear combination of the training data: w = X k αkykΦK(xk), (1) where the coefficients αk are computed by solving a constrained quadratic optimization problem. The resulting classifier fK(x) = ⟨x · w⟩+b = X k αkyk ⟨ΦK(x) · ΦK(xk)⟩+b = X k αkykK(x, xk)+b (2) defines a nonlinear separating boundary in the original feature space. 3 Discriminative Direction Equations (1) and (2) imply that the classification function fK(x) is directly proportional to the signed distance from the input point to the separating boundary computed in the higher dimensional space defined by the mapping ΦK. In other words, the function output depends only on the projection of vector ΦK(x) onto w and completely ignores the component of ΦK(x) that is perpendicular to w. This suggests that in order to create a displacement of ΦK(x) that corresponds to the differences between the two classes, one should change the vector’s projection onto w while keeping its perpendicular component the same. In the linear case, we can easily perform this operation, since we have access to the image vectors, ΦK(x) = x. This is similar to visualization techniques typically used in linear generative modeling, where the data variation is captured using PCA, and new samples are generated by changing a single principal component at a time. However, this approach is infeasible in the non-linear case, because we do not have access to the image vectors ΦK(x). Furthermore, the resulting image vector might not even have a source in the original feature space, i.e., there might be no vector in the original space Rn that maps into the resulting vector in the space F. Our solution is to search for the direction around (a) Φ w (b) dx z x d z p e w Φ Figure 1: Kernel-based classification (a) and the discriminative direction (b). the feature vector x in the original space that minimizes the divergence of its image ΦK(x) from the direction of the projection vector w1. We call it a discriminative direction, as it represents the direction that affects the output of the classifier while introducing as little irrelevant change as possible into the input vector. Formally, as we move from x to x + dx in Rn, the image vector in the space F changes by dz = ΦK(x + dx) −ΦK(x) (Fig. 1b). This displacement can be thought of as a vector sum of its projection onto w and its deviation from w: p = ⟨dz · w⟩ ⟨w · w⟩w and e = dz −p = dz −⟨dz · w⟩ ⟨w · w⟩w. (3) The discriminative direction minimizes the divergence component e, leading to the following optimization problem: minimize E(dx) = ∥e∥2 = ⟨dz · dz⟩−⟨dz · w⟩2 ⟨w · w⟩ (4) s.t. ∥dx∥2 = ϵ. (5) Since the cost function depends only on dot products of vectors in the space F, it can be computed using the kernel function K: ⟨w · w⟩ = X k,m αkαmykymK(xk, xm), (6) ⟨dz · w⟩ = ∇fK(x)dx, (7) ⟨dz · dz⟩ = dxT HK(x)dx, (8) where ∇fK(x) is the gradient of the classifier function fK evaluated at x and represented by a row-vector and matrix HK(x) is one of the (equivalent) off-diagonal quarters of the Hessian of K, evaluated at (x, x): HK(x)[i, j] = ∂2K(u, v) ∂ui∂vj (u=x,v=x) . (9) Substituting into Equation (4), we obtain minimize E(dx) = dxT HK(x) −∥w∥−2∇f T K(x)∇fK(x) dx (10) s.t. ∥dx∥2 = ϵ. (11) 1A similar complication arises in kernel-based generative modeling, e.g., kernel PCA [7]. Constructing linear combinations of vectors in the space F leads to a global search in the original space [6, 7]. Since we are interested in the direction that best approximates w, we use infinitesimal analysis that results in a different optimization problem. The solution to this problem is the smallest eigenvector of matrix QK(x) = HK(x) −∥w∥−2∇f T K(x)∇fK(x). (12) Note that in general, the matrix QK(x) and its smallest eigenvector are not the same for different points in the original space and must be estimated separately for every input vector x. Furthermore, each solution defines two opposite directions in the input space, corresponding to the positive and the negative projections onto w. We want to move the input example towards the opposite class and therefore assign the direction of increasing function values to the examples with label −1 and the direction of decreasing function values to the examples with label 1. Obtaining a closed-form solution of this minimization problem could be desired, or even necessary, if the dimensionality of the input space is high and computing the smallest eigenvector is computationally expensive and numerically challenging. In the next section, we demonstrate how a particular form of the matrix HK(x) leads to an analytical solution for a large family of kernel functions2. 3.1 Analytical Solution for Discriminative Direction It is easy to see that if HK(x) is a multiple of the identity matrix, HK(x) = cI, then the smallest eigenvector of the matrix QK(x) is equal to the largest eigenvector of the matrix ∇f T K(x)∇fK(x), namely the gradient of the classifier function ∇f T K(x). We will show in this section that both for the linear kernel and, more surprisingly, for RBF kernels, the matrix HK(x) is of the right form to yield an analytical solution of this form. It is well known that to achieve the fastest change in the value of a function, one should move along its gradient. In the case of the linear and the RBF kernels, the gradient also corresponds to the direction that distinguishes between the two classes while ignoring inter-class variability. Dot product kernels, K(u, v) = k(⟨u · v⟩). For any dot product kernel, ∂2K(u, v) ∂ui∂vj (u=x,v=x) = k′(∥x∥2)δij + k′′(∥x∥2)xixj, (13) and therefore HK(x) = cI for all x if and only if k′′(∥x∥2) ≡0, i.e., when k is a linear function. Thus the linear kernel is the only dot product kernel for which this simplification is relevant. In the linear case, HK(x) = I, and the discriminative direction is defined as dx∗= ∇f T K(x) = w = X αkykxk; E(dx∗) = 0. (14) This is not entirely surprising, as the classifier is a linear function in the original space and we can move precisely along w. Polynomial kernels are a special case of dot product kernels. For polynomials of degree d ≥2, ∂2K(u, v) ∂ui∂vj (u=x,v=x) = d(1 + ∥x∥2)d−1δij + d(d −1)(1 + ∥x∥2)d−2xixj. (15) HK(x) is not necessarily diagonal for all x, and we have to solve the general eigenvector problem to identify the discriminative direction. 2While a very specialized structure of HK(x) in the next section is sufficient for simplifying the solution significantly, it is by no means necessary, and other kernel families might exist for which estimating the discriminative direction does not require solving the full eigenvector problem. Distance kernels, K(u, v) = k(∥u −v∥2). For a distance kernel, ∂2K(u, v) ∂ui∂vj (u=x,v=x) = −2k′(0)δij, (16) and therefore the discriminative direction can be determined analytically: dx∗= ∇f T K(x); E(dx∗) = −2k′(0) −∥w∥−2∥∇f T K(x)∥ 2. (17) The Gaussian kernels are a special case of the distance kernel family, and yield a closed form solution for the discriminative direction: dx∗= −2/γ X k αkyke−∥x−xk∥2 γ (x−xk); E(dx∗) = 2/γ −∥∇f T K(x)∥ 2/∥w∥2. (18) Unlike the linear case, we cannot achieve zero error, and the discriminative direction is only an approximation. The exact solution is unattainable in this case, as it has no corresponding direction in the original space. 3.2 Geometric Interpretation We start by noting that the image vectors ΦK(x)’s do not populate the entire space F, but rather form a manifold of lower dimensionality whose geometry is fully defined by the kernel function K (Fig. 1). We will refer to this manifold as the target manifold in this discussion. We cannot explicitly manipulate elements of the space F, but can only explore the target manifold through search in the original space. We perform the search in the original space by considering all points on an infinitesimally small sphere centered at the original input vector x. In the range space of the mapping function ΦK, the images of points x + dx form an ellipsoid defined by the quadratic form dzT dz = dxT HK(x)dx. For HK(x) ∼I, the ellipsoid becomes a sphere, all dz’s are of the same length, and the minimum of error in the displacement vector dz corresponds to the maximum of the projection of dz onto w. Therefore, the discriminative direction is parallel to the gradient of the classifier function. If HK(x) is of any other form, the length of the displacement vector dz changes as we vary dx, and the minimum of the error in the displacement is not necessarily aligned with the direction that maximizes the projection. As a side note, our sufficient condition, HK(x) ∼I, implies that the target manifold is locally flat, i.e., its Riemannian curvature is zero. Curvature and other properties of target manifolds have been studied extensively for different kernel functions [1, 4]. In particular, one can show that the kernel function implies a metric on the original space. Similarly to the natural gradient [2] that maximizes the change in the function value under an arbitrary metric, we minimize the changes that do not affect the function under the metric implied by the kernel. 3.3 Selecting Inputs Given any input example, we can compute the discriminative direction that represents the differences between the two classes captured by the classifier in the neighborhood of the example. But how should we choose the input examples for which to compute the discriminative direction? We argue that in order to study the differences between the classes, one has to examine the input vectors that are close to the separating boundary, namely, the support vectors. Note that this approach is significantly different from the generative modeling, where a “typical” representative, often constructed by computing the mean of the training data, is used for analysis and visualization. In the discriminative framework, we are more interested in the examples that lie close to the opposite class, as they define the differences between the two classes and the optimal separating boundary. (a) −3 −2 −1 0 1 2 3 4 5 6 7 8 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 (b) −3 −2 −1 0 1 2 3 4 5 6 7 8 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 (c) −3 −2 −1 0 1 2 3 4 5 6 7 8 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 Figure 2: Discriminative direction for linear (a), quadratic (b) and Gaussian RBF (c) classifiers. The background is colored using the values of the classifier function. The black solid line is the separating boundary, the dotted lines indicate the margin corridor. Support vectors are indicated using solid markers. The length of the vectors is proportional to the magnitude of the classifier gradient. Support vectors define a margin corridor whose shape is determined by the kernel type used for training. We can estimate the distance from any support vector to the separating boundary by examining the gradient of the classification function for that vector. Large gradient indicates that the support vector is close to the separating boundary and therefore can provide more information on the spatial structure of the boundary. This provides a natural heuristic for assigning importance weighting to different support vectors in the analysis of the discriminative direction. 4 Simple Example We first demonstrate the the proposed approach on a simple example. Fig. 2 shows three different classifiers, linear, quadratic and Gaussian RBF, for the same example training set that was generated using two Gaussian densities with different means and covariance matrices. We show the estimated discriminative direction for all points that are close to the separating boundary, not just support vectors. While the magnitude of discriminative direction vector is irrelevant in our infinitesimal analysis, we scaled the vectors in the figure according to the magnitude of the classifier gradient to illustrate importance ranking. Note that for the RBF support vectors far away from the boundary (Fig. 2c), the magnitude of the gradient is so small (tenth of the magnitude at the boundary), it renders the vectors Normal Control Patient Figure 3: Right hippocampus in schizophrenia study. First support vector from each group is shown, four views per shape (front, medial, back, lateral). The color coding is used to visualize the amount and the direction of the deformation that corresponds to the discriminative direction, changing from blue (moving inwards) to green (zero deformation) to red (moving outwards). too short to be visible in the figure. We can see that in the areas where there is enough evidence to estimate the boundary reliably, all three classifiers agree on the boundary and the discriminative direction (lower cluster of arrows). However, if the boundary location is reconstructed based on the regularization defined by the kernel, the classifiers suggest different answers (the upper cluster of arrows), stressing the importance of model selection for classification. The classifiers also provide an indication of the reliability of the differences represented by each arrow, which was repeatedly demonstrated in other experiments we performed. 5 Morphological Studies Morphological studies of anatomical organs motivated the analysis presented in this paper. Here, we show the results for the hippocampus study in schizophrenia. In this study, MRI scans of the brain were acquired for schizophrenia patients and a matched group of normal control subjects. The hippocampus structure was segmented (outlined) in all of the scans. Using the shape information (positions of the outline points), we trained a Gaussian RBF classifier to discriminate between schizophrenia patients and normal controls. However, the classifier in its original form does not provide the medical researchers with information on how the hippocampal shape varies between the two groups. Our goal was to translate the information captured by the classifier into anatomically meaningful terms of organ development and deformation. In this application, the coordinates in the input space correspond to the surface point locations for any particular example shape. The discriminative direction vector corresponds to displacements of the surface points and can be conveniently represented by a deformation of the original shape, yielding an intuitive description of shape differences for visualization and further analysis. We show the deformation that corresponds to the discriminative direction, omitting the details of shape extraction (see [5] for more information). Fig. 3 displays the first support vector from each group with the discriminative direction “painted” on it. Each row shows four snapshots of the same shape form different viewpoints3. The color at every node of the surface encodes the corresponding component of the discriminative direction. Note that the deformation represented by the two vectors is very similar in nature, but of opposite signs, as expected from the analysis in Section 3.3. We can see that the main deformation represented by this pair of vectors is localized in the bulbous “head” of 3An alternative way to visualize the same information is to actually generate the animation of the example shape undergoing the detected deformation. the structure. The next four support vectors in each group represent a virtually identical deformation to the one shown here. Starting with such visualization, the medical researchers can explore the organ deformation and interaction caused by the disease. 6 Conclusions We presented an approach to quantifying the classifier’s behavior with respect to small changes in the input vectors, trying to answer the following question: what changes would make the original input look more like an example from the other class without introducing irrelevant changes? We introduced the notion of the discriminative direction, which corresponds to the maximum changes in the classifier’s response while minimizing irrelevant changes in the input. For kernel-based classifiers the discriminative directions is determined by minimizing the divergence of the infinitesimal displacement vector and the normal to the separating hyperplane in the higher dimensional kernel space. The classifier interpretation in terms of the original features in general, and the discriminative direction in particular, is an important component of the data analysis in many applications where the statistical learning techniques are used to discover and study structural differences in the data. Acknowledgments. Quadratic optimization was performed using PR LOQO optimizer written by Alex Smola. This research was supported in part by NSF IIS 9610249 grant. References [1] S. Amari and S. Wu. Improving Support Vector Machines by Modifying Kernel Functions. Neural Networks, 783-789, 1999. [2] S. Amari. Natural Gradient Works Efficiently in Learning. Neural Comp., 10:251-276, 1998. [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. Geometry and Invariance in Kernel Based Methods. In Adv. in Kernel Methods: Support Vector Learning, Eds. Sch¨olkopf, Burges and Smola, MIT Press, 89-116, 1999. [5] P. Golland et al. Small Sample Size Learning for Shape Analysis of Anatomical Structures. In Proc. of MICCAI’2000, LNCS 1935:72-82, 2000. [6] B. Sch¨olkopf et al. Input Space vs. Feature Space in Kernel-Based Methods. IEEE Trans. on Neural Networks, 10(5):1000-1017, 1999. [7] B. Sch¨olkopf, A. Smola, and K.-R. M¨uller. Nonlinear Component Analysis as a Kernel Eigenvalue Problem. Neural Comp., 10:1299-1319, 1998. [8] V. N. Vapnik. The Nature of Statistical Learning Theory. Springer, 1995. [9] V. N. Vapnik. Statistical Learning Theory. John Wiley & Sons, 1998.
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Contextual Modulation of Target Saliency Antonio Torralba Dept. of Brain and Cognitive Sciences MIT, Cambridge, MA 02139 torralba@ai.mit. edu Abstract The most popular algorithms for object detection require the use of exhaustive spatial and scale search procedures. In such approaches, an object is defined by means of local features. fu this paper we show that including contextual information in object detection procedures provides an efficient way of cutting down the need for exhaustive search. We present results with real images showing that the proposed scheme is able to accurately predict likely object classes, locations and sizes. 1 Introduction Although there is growing evidence of the role of contextual information in human perception [1], research in computational vision is dominated by object-based representations [5,9,10,15]. In real-world scenes, intrinsic object information is often degraded due to occlusion, low contrast, and poor resolution. In such situations, the object recognition problem based on intrinsic object representations is ill-posed. A more comprehensive representation of an object should include contextual information [11,13]: Obj. representatian == {intrisic obj. model, contextual obj. model}. In this representation, an object is defined by 1) a model of the intrinsic properties of the object and 2) a model of the typical contexts in which the object is immersed. Here we show how incorporating contextual models can enhance target object saliency and provide an estimate of its likelihood and intrinsic properties. 2 Target saliency and object likelihood Image information can be partitioned into two sets of features: local features, VL, that are intrinsic to an object, and contextual features, rUe which encode structural properties of the background. In a statistical framework, object detection requires evaluation of the likelihood function (target saliency function): P(O IVL, va) which provides the probability of presence of the object 0 given a set of local and contextual measurements. 0 is the set of parameters that define an object immersed in a scene: 0 == {on, x, y, i} with on==object class, (x,y)==location in image coordinates (1) and bobject appearance parameters. By applying Bayes rule we can write: P(O IVL, va) = P(vL11 va) P(VL 10, va)P(O Iva) Those three factors provide a simplified framework for representing three levels of attention guidance when looking for a target: The normalization factor, 1/P(VL Iva), does not depend on the target or task constraints, and therefore is a bottom-up factor. It provides a measure of how unlikely it is to find a set of local measurements VL within the context va. We can define local saliency as S(x,y) == l/P(vL(x,y) Iva). Saliency is large for unlikely features ina' scene. The second factor, P(VL 10, va), gives the likelihood of the local measurements VL when the object is present at such location in a particular context. We can write P(VL 10, va) ~ P(VL 10), which is a convenient approximation when the aspect of the target object is fully determined by the parameters given by the description O. This factor represents the top-down knowledge of the target· appearance and how it contributes to the search. Regions of the image with features unlikely to belong to the target object are vetoed. and regions with attended features are enhanced. The third factor, the PDF P(O Iva), provides context-based priors on object class, location and scale. It is of capital importance for insuring reliable inferences in situations where the local image measurements VL produce ambiguous interpretations. This factor does not depend on local measurements and target models [8,13]. Therefore, the term P(O Iva) modulates the saliency of local image properties when looking for an object of the class On. Contextual priors become more evident if we apply Bayes rule successively in order to split the PDF P(0 Iva) into three factors that model three kinds of context priming on object search: (2) According to this decomposition of the PDF, the contextual modulation of target saliency is a function of three main factors: Object likelihood: P(on Iva) provides the probability of presence of the object class On in the scene. If P(On Iva) is very small, then object search need not be initiated (we do not need to look for cars in a living room). Contextual control of focus of attention: P(x, y I On, va)· This PDDF gives the most likely locations for the presence of object On given context information, and it allocates computational resources into relevant scene regions. Contextual selection of local target appearance: P(tl.va, on). This gives the likely (prototypical) shapes (point of views, size, aspect ratio, object aspect) of the object On in the context Va- Here t == {a, p}, with a==scale and p==aspect ratio. Other parameters describing the appearance of an object in an image can be added. The image features most commonly used for describing local structures are the energy outputs of oriented band-pass filters, as they have been shown to be relevant for the task of object detection [9,10] and scene recognition [2,4,8,12]~ Therefore, the local image representation at the spatial location (x) is given by the vector VL(X) == {v(X,k)}k==l,N with: (3) 1 1 1 ",.-..... '0 ",.-..... u u ;> ;> ;> -a -a -a 0 0 0 '0: P: 0: 2 3 4 o 1 2 3 4 o 1 2 3 4 Figure 1: Contextual object prImIng of four objects categories (I-people, 2furniture, 3-vehicles and 4-trees) where i(x) is the input image and gk(X) are oriented band-pass filters defined by gk(i) == e-llxI12/u~e27fj<f~,x>. In such a representation [8], v(i,k) is the output magnitude- at the location i of a complex Gabor filter tuned to the spatial frequency f~. The variable k indexes filters tuned to different spatial frequencies and orientations. On the other ,hand, contextual features have to summarize the structure of the whole image. It has been shown that a holistic low-dimensional encoding of the local image features conveys enough information for a semantic categorization of the scene/context [8] and can be used for contextual priming in object recognition tasks [13]. Such a representation can be achieved by decomposing the image features into the basis functions provided by PCA: an == L L v{x, k) 1/ln{x, k) x k N v(x, k) ~ L an1/ln(x, k) n=l (4) We propose to use the decomposition coefficients vc == {an}n=l,N as context features. The functions 1/ln are the eigenfunctions of the covariance operator given by v(x, k). By using only a reduced set of components (N == 60 for the rest of the paper), the coefficients {an}n=l,N encode the main spectral characteristics of the scene with a coarse description of their spatial arrangement. In essence, {an}n=l,N is a holistic representation as all the regions of the image contribute to all the coefficients, and objects are not encoded individually [8]. In the rest of the paper we show the efficacy of this set of features in context modeling for object detection tasks. 3 Contextual object priming The PDF P(On Iva) gives the probability of presence of the object class On given contextual information. In other words, the PDF P{on Ive) evaluates the consistency of the object On with the context vc. For instance, a car has a high probability of presence in a highway scene but it is inconsistent with an indoor environment. The goal of P(on Ive) is to cut down the number of possible object categories to deal with before- expending computational resources in the object recognition process. The learning of the PDF P(on Ive) == P(ve IOn)P(on)/p(ve) with p(vo) == P(vc IOn)P{on) + P(vc l-,on)P(-,on) is done by approximating the in-class and out-of-class PDFs by a mixture of Gaussians: L P(ve IOn) == L bi,nG(VC;Vi,n, Vi,n) i=l (5) Figure 2: Contextual control of focus of attention when the algorithm is looking for cars (upper row) or heads (bottom row). The model parameters (bi,n, Vi,n, Vi,n) for the object class On are obtained using the EM algorithm [3]. The learning requires the use of few Gaussian clusters (L == 2 provides very good performances). For the learning, the system is trained with a set of examples manually annotated with the .presence/absence of four objects categories (i-people, 2-furniture, 3-vehicles and 4-trees). Fig. 1 shows some typical results from the priming model on the four superordinate categories of objects defined. Note. that the probability function P(on Ive) provides information about the probable presence of one object without scanning the picture. IfP(On Ive) > 1th then we can predict that the target is present. On the other hand, if P(On Ive) < th we can predict that the object is likely to be absent before exploring the image. The number of scenes in which the system may be able to take high confidence decisions will depend on different factors such as: the strength of the relationship between the target object and its context and the ability of ve for efficiently characterizing the context. Figure 1 shows some typical results from the priming model for a set of super-ordinate categories of objects. When forcing the model to take binary decisions in all the images (by selecting an acceptance threshold of th == 0.5) the presence/absence of the objects was correctly predicted by the model on 81%of the scenes of the test set. For each object category, high confidence predictions (th == .1) were made in at least 50% of the tested scene pictures and the presence/absence of each object class was correctly predicted by the model on 95% of those images. Therefore, for those images, we do not need to use local image analysis to decide about the presence/absence of the object. 4 Contextual control of focus of attention One of the strategies that biological visual systems use to deal with the analysis of real-world scenes is to focus attention (and, therefore, computational resources) onto the important image regions while neglecting others. Current computational models of visual attention (saliency maps anQ target detection) rely exclusively on local information or intrinsic object models [6,7,9,14,16]. The control of the focus of attention by contextual information that we propose. here is both task driven (looking for object on) and context driven (given global context information: ve). However, it does riot include any model of the target object at this stage. In our framework, the problem of contextual control of the focus of attention involves the CARS. 1 ~ CARS o ••• I P; •• "'0 0: • ~ :.S: _ filii •• \ • .~\.:.. ~ .\. fill tI':,._.: •• ••\ ~ .tto II 0.4 tre. • • 1,---~_"""""""-----R_eal_sc_al--..Je .: oReal pose 1 10 pixels 100 0.4 1 100 Q.) ~ ~ Q.) HEADS •••• 11 1.8 100 ] •• ~ "'0 .~.~. E t. ,.,:-.,,, • S•• , -: • 10 .~ ... ,••-=- •• ~ •• fIlIIe·': ":I':·.? 1 ~ , Real scale 1 10 pixels 100 Figure 3: Estimation results of object scale and pose based on contextual features. (6) evaluation of the PDF P(xlon,vo). For the learning, the joint PDF is modeled as a sum of gaussian clusters. Each cluster is decomposed into the product of two gaussians modeling respectively the distribution of object locations and the distribution of contextual features for each cluster: L P(x, vol on) == L bi,n G(x; Xi,n, Xi,n)G(VO; Vi,n, Vi,n) i==l The training set used for the learning of the PDF P(x, vol on) is a subset of'the pictures that contain the object On. The training data is {Vt}t==l,Nt and {Xt}t==l,Nt where Vt are the contextual features of the picture t of the training set and Xt is the location of object On in the image. The model parameters are obtained using the EM algorithm [3,13]. We used 1200 pictures for training and a separate set of 1200 pictures for testing. The success of the PDF in narrowing the region of the focus of attention will depend on the consistency of the relationship between the object and the context. Fig. 2 shows several examples of images and the selected regions based on contextual features when looking for cars and faces. From the PDF P(x, Vo IOn) we selected the region with the highest probability (33% of the image size on average). 87% of the heads present in the test pictures were inside the selected regions. 5 Contextual selection of object appearance models One major problem for computational approaches to object detection is the large variability in object appearance. The classical solution is to explore the space of possible shapes looking for the best match. The main sources of variability in object appearance are size, pose and intra-class shape variability (deformations, style, etc.). We show here that including contextual information can reduce at le.ast the first two sources of variability. For instance, the expected size of people in an image differs greatly between an indoor environment and a perspective view of a street. Both environments produce different patterns of contextual features vo [8]. For the second factor, pose, in the case of cars, there is a strong relationship between the possible orientations of the object and the scene configuration. For instance, looking down a highway, we expect to see the back of the cars, however, in a street view, looking towards the buildings, lateral views of cars are more likely. The expected scale and pose of the target object can be estimated by a regression procedure. The training database used for building the regression is a set of 1000 images in which the target object On is present. For each training image the target Figure 4: Selection of prototypical object appearances based on contextual cues. object was selected by cropping a rectangular window. For faces and cars we define the u == scale as the height of the selected window and the P == pose as the ratio between the horizontal and vertical dimensions of the window (~y/~x). On average, this definition of pose provides a good estimation of the orientation for cars but not for heads. Here we used regression using a mixture of gaussians for estimating the conditional PDFs between scale, pose and contextual features: P(u IVa, on) and PCP Iva, on). This yields the next regression procedures [3]: (j == Ei Ui,nbi,nG(Va; Vi,n, Vi,n) Ei bi,nG(vO; Vi,n, Vi,n) _ EiPi,nbi,nG(VO;Vi,n, Vi,n) P == Ei bi,nG(VC;Vi,n, Vi,n) (7) The results summarized in fig. 3 show that context is a strong cue for scale selection for the face detection task but less important for the car detection task. On the other hand, context introduces strong constraints on the prototypical point of views of cars but not at all for heads. Once the two parameters (pose and scale) have been estimated, we can build a prototypical model of the target object. In the case of a view-based object representation, the model of the object will consist of a collection of templates that correspond to the possible aspects of the target. For each image the system produces a collection of views, selected among a database of target examples that have the scale and pose given by eqs. (7). Fig. 4 shows some results from this procedure. In the statistical framework, the object detection requires the evaluation of the function P(VL 10, va). We can approximate Input image (target = cars) Object priming and Contextual control Target model selection of focus of attention 1 Integration of local features Target saliency Figure 5: Schematic layout of the model for object detection (here cars) by integration of contextual and local information. The bottom example is an error in detection due to incorrect context identification. P(VL 10, va) ~ P(VL IOn' (J", p). Fig. 5 and 6 show the complete chain of operations and some detection results using a simple correlation technique between the image and the generated object models (100 exemplars) at only one scale. The last image of each row shows the total object likelihood obtained by multiplying the object saliency maps (obtained by the correlation) and the contextual control of the focus of attention. The result shows how the use of context helps reduce false alarms. This results in good detection performances despite the simplicity of the matching procedure used. 6 Conclusion The contextual schema of a scene provides the likelihood of presence, typical locations and appearances of objects within the scene. We have proposed a model for incorporating such contextual cues in the task of object detection. The main aspects of our approach are: 1) Progressive reduction of the window of focus of attention: the system reduces the size of the focus of attention by first integrating contextual information and then local information. 2) Inhibition of target like patterns that are in inconsistent locations. 3) Faster detection of correctly scaled targets that have a pose in agreement with the context. 4) No requirement of parsing a scene into individual objects. Furthermore, once one object has been detected, it can introduce new contextual information for analyzing the rest of the scene. Acknowledglllents The author wishes to thank Dr. Pawan Sinha, Dr. Aude Oliva and Prof. Whitman Richards for fruitful discussions. References [1] Biederman, I., Mezzanotte, R.J., & Rabinowitz, J.C. (1982). Scene perception: detecting and judging objects undergoing relational violations. Cognitive Psychology, 14:143177. Feature maps \ I V t---HXJ---+l . . . . ~ Figure 6: Schema for object detection (e.g. cars) integrating local and giobal information. [2] Carson, C., Belongie, S., Greenspan, H., and Malik, J. (1997). Region-based image querying. Proc. IEEE W. on Content-Based Access of Image and Video Libraries, pp: 42-49. [3] Gershnfeld, N. The nature of mathematical modeling. Cambridge university press, 1999. [4] Gorkani, M. M., Picard, R. W. (1994). Texture orientation for sorting photos 'at a glance'. Proc. Int. Conf. Pat. Rec., Jerusalem, Vol. I: 459-464. [5] Heisle, B., T. Serre, S. Mukherjee and T. Poggio. (2001) Feature Reduction and Hierarchy of Classifiers for Fast Object Detection in Video Images. In: Proceedings of 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, IEEE Computer Society Press, Jauai, Hawaii. [6] Itti, L., Koch, C., & Niebur, E. (1998). A model of saliency-based visual attention for rapid scene analysis. IEEE Trans. Pattern Analysis and Machine Vision, 20(11):1254. [7] Moghaddam, B., & Pentland, A. (1997). Probabilistic Visual Learning for Object Representation. IEEE Trans. Pattern Analysis and Machine Vision, 19(7):696-710. [8] Oliva, A., & Torralba, A. (2001). Modeling the Shape of the Scene: A holistic representation of the spatial envelope. Int. Journal of Computer Vision, 42(3):145-175. [9] Rao, R.P.N., Zelinsky, G.J., Hayhoe, M.M., & Ballard, D.H. (1996). Modeling saccadic targeting in visual search. NIPS 8. Cambridge, MA: MIT Press. [10] Schiele, B., Crowley, J. L. (2000) Recognition without Correspondence using Multidimensional Receptive Field Histograms, Int. Journal of Computer Vision, Vol. 36(1):31-50. [11] Strat, T. M., & Fischler, M. A. (1991). Context-based vision: recognizing objects using information from both 2-D and 3-D imagery. IEEE trans. on Pattern Analysis and Machine Intelligence, 13(10): 1050-1065. [12] Szummer, M., and Picard, R. W. (1998). Indoor-outdoor image classification. In IEEE intl. workshop on Content-based Access of Image and Video Databases, 1998. [13] Torralba, A., & Sinha, P. (2001). Statistical context priming for object detection. IEEE Proc. Of Int. Conf in Compo Vision. [14] Treisman, A., & Gelade, G. (1980). A feature integration theory of attention. Cognitive Psychology, Vol. 12:97-136. [15] Viola, P. and Jones, M. (2001). Rapid object detection using a boosted cascade of simple features. In: Proceedings of 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2001), IEEE Computer Society Press, Jauai, Hawaii. [16] Wolfe, J. M. (1994). Guided search 2.0. A revised model of visual search. Psychonomic Bulletin and Review, 1:202-228
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On Kernel-Target Alignment N ello Cristianini BIOwulf Technologies nello@support-vector. net Andre Elisseeff BIOwulf Technologies andre@barnhilltechnologies.com John Shawe-Taylor Royal Holloway, University of London john@cs.rhul.ac.uk Jaz Kandola Royal Holloway, University of London jaz@cs.rhul.ac.uk Abstract We introduce the notion of kernel-alignment, a measure of similarity between two kernel functions or between a kernel and a target function. This quantity captures the degree of agreement between a kernel and a given learning task, and has very natural interpretations in machine learning, leading also to simple algorithms for model selection and learning. We analyse its theoretical properties, proving that it is sharply concentrated around its expected value, and we discuss its relation with other standard measures of performance. Finally we describe some of the algorithms that can be obtained within this framework, giving experimental results showing that adapting the kernel to improve alignment on the labelled data significantly increases the alignment on the test set, giving improved classification accuracy. Hence, the approach provides a principled method of performing transduction. Keywords: Kernels, alignment, eigenvectors, eigenvalues, transduction 1 Introduction Kernel based learning algorithms [1] are modular systems formed by a generalpurpose learning element and by a problem specific kernel function. It is crucial for the performance of the system that the kernel function somehow fits the learning target, that is that in the feature space the data distribution is somehow correlated to the label distribution. Several results exist showing that generalization takes place only when such correlation exists (nofreelunch; luckiness), and many classic estimators of performance (eg the margin) can be understood as estimating this relation. In other words, selecting a kernel in this class of systems amounts to the classic feature and model selection problems in machine learning. Measuring the similarity between two kernels, or the degree of agreement between a kernel and a given target function, is hence an important problem both for conceptual and for practical reasons. As an example, it is well known that one can obtain complex kernels by combining or manipulating simpler ones, but how can one predict whether the resulting kernel is better or worse than its components? What a kernel does is to virtually map data into a feature space so that their relative positions in that space are what matters. The degree of clustering achieved in that space, and the relation between the clusters and the labeling to be learned, should be captured by such an estimator. Alternatively, one could regard kernels as 'oracles' or 'experts' giving their opinion on whether two given points belong to the same class or not. In this case, the correlation between experts (seen as random variables) should provide an indication of their similarity. We will argue that - if one were in possess of this information - the ideal kernel for a classification target y(x) would be K(x, z) = y(x)y(z). One way of estimating the extent to which the kernel achieves the right clustering is to compare the sum of the within class distances with the sum of the between class distances. This will correspond to the alignment between the kernel and the ideal kernel y(x)y(z). By measuring the similarity of this kernel with the kernel at hand - on the training set - one can assess the degree of fitness of such kernel. The measure of similarity that we propose, 'kernel alignment' would give in this way a reliable estimate of its expected value, since it is sharply concentrated around its mean. In this paper we will motivate and introduce the notion of Alignment (Section 2); prove its concentration (Section 3); discuss its implications for the generalisation of a simple classifier (Section 4) and deduce some simple algorithms (Section 5) to optimize it and finally report on some experiments (Section 6). 2 Alignment Given an (unlabelled) sample 8 = {Xl, ... ,xm }, we use the following inner product between Gram matrices, (K1,K2)F = 2:7,'j=l K 1(Xi,Xj)K2(Xi,Xj) Definition 1 Alignment The (empirical) alignment of a kernel k1 with a kernel k2 with respect to the sample 8 is the quantity A(8 k k) = (K1,K2 )F , 1, 2 J(K1,K1)F(K2, K2)F' where Ki is the kernel matrix for the sample 8 using kernel ki. This can also be viewed as the cosine of the angle between two bi-dimensional vectors K1 and K2, representing the Gram matrices. If we consider K2 = yyl, where y is the vector of { -1, + I} labels for the sample, then A(8 K I) (K, yyl)F (K, yyl)F . / 1 I) 2 , ,yy =. / / K K) / 1 I) . / / K K) , smce \yy ,yy F = m y \, F\YY ,yy F my \, F We will occasionally omit the arguments K or y when these are understood from the context or when y forms part of the sample. In the next section we will see how this definition provides with a method for selecting kernel parameters and also for combining kernels. 3 Concentration The following theorem shows that the alignment is not too dependent on the training set 8. This result is expressed in terms of 'concentration'. Concentration means that the probability of an empirical estimate deviating from its mean can be bounded as an exponentially decaying function of that deviation. This will have a number of implications for the application and optimisation of the alignment. For example if we optimise the alignment on a random sample we can expect it to remain high on a second sample. Furthermore we will show in the next section that if the expected value of the alignment is high, then there exist functions that generalise well. Hence, the result suggests that we can optimise the alignment on a training set and expect to keep high alignment and hence good performance on a test set. Our experiments will demonstrate that this is indeed the case. The theorem makes use of the following result due to McDiarmid. Note that lEs is the expectation operator under the selection of the sample. TheoreIll 2 (McDiarmid!4}) Let Xl, ... ,Xn be independent random variables taking values in a set A, and assume that f : An -+ m. satisfies for 1 ::::; i ::::; n then for all f > 0, TheoreIll 3 The sample based estimate of the alignment is concentrated around its expected value. For a kernel with feature vectors of norm 1, we have that pm{s: 1.4(S) - A(y)1 ::::: €} ::::; 8 where € = C(S)V8ln(2/8)/m, (1) for a non-trivial function C (S) and value A(y). Proof: Let A 1 ~ A 1 ~ 2 lEs[.41(S)] A1(S) = m 2 .~ Yiyjk(Xi,Xj),A2(S) = m 2 .~ k(xi,Xj) , and A(y) = / A • ',J=l ',J=l ylES [A2(S)] First note that .4(S) = .41(S)/) .42(S). Define Al = lES[A1(S)] and A2 = lES[A2(S)], First we make use of McDiarmid's theorem to show that Ai(S) are concentrated for i = 1,2. Consider the training set S' = S \ {(Xi, Yi)} U {(X~, y~)}. We must bound the difference A A 1 4 IAj(S) - Aj(S')1 ::::; -2 (2(m - 1)2) < -, m m for j = 1,2. Hence, we have Ci = 4/m for all i and we obtain from an application of McDiarmid's Theorem for j = 1 and 2, < 2exp ( f;m) Setting f = V8ln(2/8)/m, the right hand sides are less than or equal to 8/2. Hence, with probability at least 1 - 8, we have for j = 1, 2 1 Aj (S) - Aj 1 < f. But whenever these two inequalities hold, we have < < Remark. We could also define the true Alignment, based on the input distribution P, as follows: given functions f,g : X 2 --+ JR, we define (j,g)p = IX2 f(x, z)g(x, z)dP(x)dP(z), Then the alignment of a kernel k1 with a kernel k2 is the quantity A(k1' k2) = J (kl,k2)P . (kl ,kl) P (k2 ,k2) P Then it is possible to prove that asymptotically as m tends to infinity the empirical alignment as defined above converges to the true alignment. However if one wants to obtain unbiased convergence it is necessary to slightly modify its definition by removing the diagonal, since for finite samples it biases the expectation by receiving too large a weight. With this modification A(y) in the statement of the theorem becomes the true alignment. We prefer not to pursue this avenue further for simplicity in this short article, we just note that the change is not significant. 4 Generalization In this section we consider the implications of high alignment for the generalisation of a classifier. By generalisation we mean the test error err(h) = P(h(x) ¥- y). Our next observation relates the generalisation of a simple classification function to the value of the alignment. The function we consider is the expected Parzen window estimator hex) = sign(f(x)) = sign (lE(XI ,v') [y'k(x', x)]). This corresponds to thresholding a linear function f in the feature space. We will show that if there is high alignment then this function will have good generalisation. Hence, by optimising the alignment we may expect Parzen window estimators to perform well. We will demonstrate that this prediction does indeed hold good in experiments. Theorem 4 Given any 8 > O. With probability 1 - 8 over a randomly drawn training set S, the generalisation accuracy of the expected Parzen window estimator h(x) = sign (lE(XI ,yl) [y' k(X', x)]) is bounded from above by err(h(x)) ::::: 1- A(S) + E + (mJ A2(S)) - 1, where E = C(S)V! ln~. Proof: (sketch) We assume throughout that the kernel has been normalised so that k(x,x) = 1 for all x. First observe that by Theorem 3 with probability greater than 1- 8/2, IA(y) - A(S)I ::::: E. The result will follow if we show that with probability greater than 1- 8/2 the generalisation error of hS\(xl,y,) can be upper bounded by 1 - A(y) + ~. Consider the quantity A(y) from Theorem 3. m A2(S) A(y) But lEs [~L:Z;= 1 Yiyjk(xi,xj)] lEs [~2 L:Z;=1 k(Xi,Xj)2] I mC-ml f(x) I IlE [2] < V (x,y) y lEs [~L:#j Yiyjk(xi,xj)] + ~ C (m -1)2 I 2 C2m 2 lE(XI,yl) [k(x, x ) ] < 1 Hence, if E P(f(x) ¥y) and a P(f(x) y), we have lEs [C~2 L:#j YiYj k(Xi' Xj)] ::::: 1 x a + 0 x E = a and E = 1 - a ::::: 1 - A(y) + c~, D An empirical estimate of the function f would be the Parzen window function. The expected margin of the empirical function is concentrated around the expected margin of the expected Parzen window. Hence, with high probability we can bound the error of j in terms of the empirically estimated alignment A(S). This is omitted due to lack of space. The concentration of j is considered in [3]. 5 Algorithms The concentration of the alignment can be directly used for tuning a kernel family to the particular task, or for selecting a kernel from a set, with no need for training. The probability that the level of alignment observed on the training set will be out by more than € from its expectation for one of the kernels is bounded by 6, where € is given by equation (1) for E = J ~ (InINI + lnj), where INI is the size of the set from which the kernel has been chosen. In fact we will select from an infinite family of kernels. Providing a uniform bound for such a class would require covering numbers and is beyond the scope of this paper. One of the main consequences of the definition of kernel alignment is in providing a practical criterion for combining kernels. We will justify the intuitively appealing idea that two kernels with a certain alignment with a target that are not aligned to each other, will give rise to a more aligned kernel combination. In particular we have that This shows that if two kernels with equal alignment to a given target yare also completely aligned to each other, then IIKI + K211F = IIKlllF + IIK211F and the alignment of the combined kernel remains the same. If on the other hand the kernels are not completely aligned, then the alignment of the combined kernel is correspondingly increased. To illustrate the approach we will take to optimising the kernel, consider a kernel that can be written in the form k(x, Xl) = l:.k I-tk(yk(x)yk(xl)) , where all the yk are orthogonal with respect to the inner product defined on the training set S, (y, yl)S = l:.:l YiYj. Assume further that one of them yt is the true label vector. We can now evaluate the alignment as A(y) ~ I-tt/v'l:.kl-t% . In terms of the Gram matrix this is written as Kij = l:.k I-tkyfyj where yf is the i-th label of the k-th classification. This special case is approximated by the decomposition into eigenvectors of the kernel matrix K = l:. Aiviv~, where Vi denotes the transpose of v and Vi is the i-th eigenvector with eigenvalue Ai. In other words, the more peaked the spectrum the more aligned (specific) the kernel can be. If by chance the eigenvector of the largest eigenvalue Al corresponds to the target labeling, then we will give to that labeling a fraction Ad v'l:.i AT of the weight that we can allocate to different possible labelings. The larger the emphasis of the kernel on a given target, the higher its alignment. In the previous subsection we observed that combining non-aligned kernels that are aligned with the target yields a kernel that is more aligned to the target. Consider the base kernels Ki = ViV~ where Vi are the eigenvectors of K, the kernel matrix for both labeled and unlabeled data. Instead of choosing only the most aligned ones, one could use a linear combination, with the weights proportional to their alignment (to the available labels): k = l:.i f(ai)viv~ where ai is the alignment of the kernel K i , and f(a) is a monotonically increasing function (eg. the identity or an exponential). Note that a recombination of these rank 1 kernels was made in so-called latent semantic kernels [2]. The overall alignment of the new kernel with the labeled data should be increased, and the new kernel matrix is expected also to be more aligned to the unseen test labels (because of the concentration, and the assumption that the split was random). Moreover, in general one can set up an optimization problem, aimed at finding the optimal a, that is the parameters that maximize the alignment of the combined kernel with the available labels. Given K = Li aiviv~ , using the orthonormality of the Vi and that (v v' ,uu') F = (v, u)}, the alignment can be written as A.(y) = (K, yy')F Li ai(vi, y)} mJLij aiaj(viv~, VjVj)F J(yy', yY')FJLi a;· Hence we have the following optimization problem: maximise W (a) (2) Setting derivatives to zero we obtain ~:. (Vi,Y)} - A2ai = 0 and hence ai (X (Vi,Y)}, giving the overall alignment A.(y) = JL,i~i'Y)j". This analysis suggests the following transduction algorithm. Given a partially labelled set of examples optimise its alignment by adapting the full kernel matrix by recombining its rank one eigenmatrices ViV~ using the coefficients ai determined by measuring the alignment between Vi and y on the labelled examples. Our results suggest that we should see a corresponding increase in the alignment on the unlabelled part of the set, and hence a reduction in test error when using a Parzen window estimator. Results of experiments testing these predictions are given in the next section. 6 Experiments We applied the transduction algorithm designed to take advantage of our results by optimizing alignment with the labeled part of the dataset using the spectral method described above. All of the results are averaged over 20 random splits with the standard deviation given in brackets. Table 1 shows the alignments of the Train Align Test Align Train Align Test Align 0.076 (0.007) 0.092 (0.029) 0.207 (0.020) 0.240 (0.083 0.228 ~0.012) 0.219 ~0.041) 0.240 ~0.016) 0.257 ~0.059) K50 0.075 ~0.016) 0.084 ~0.017) 0.210 ~0.031) 0.216 ~0.033) G50 0.242 (0.023) 0.181 (0.043) 0.257 (0.023) 0.202 (0.015) K 20 0.072 ~0.022) 0.081 ~0.006) 0.227 ~0.057) 0.210 ~0.015) G20 0.273 ~0.037) 0.034 ~0.046) 0.326 ~0.023) 0.118 ~0.017) Table 1: Mean and associated standard deviation alignment values using a linear kernel on the Breast (left two columns) and Ionosphere (right two columns). Gram matrices to the label matrix for different sizes of training set. The index indicates the percentage of training points. The K matrices are before adaptation, while the G matrices are after optimisation of the alignment using equation (2). The results on the left are for Breast Cancer data using a linear kernel, while the results on the right are for Ionosphere data. The left two columns of Table 2 shows the alignment values for Breast Cancer data using a Gaussian kernel together with the performance of an SVM classifier trained Table 2: Breast alignment (cols 1,2) and SVM error for a Gaussian kernel (sigma = 6) (col 3), Parzen window error for Breast (col 4) and Ionosphere (col 5) with the given gram matrix in the third column. The right two columns show the performance of the Parzen window classifier on the test set for Breast linear kernel (left column) and Ionosphere (right column). The results clearly show that optimising the alignment on the training set does indeed increase its value in all but one case by more than the sum of the standard deviations. Furthermore, as predicted by the concentration this improvement is maintained in the alignment measured on the test set with both linear and Gaussian kernels in all but one case (20% train with the linear kernel). The results for Ionosphere are less conclusive. Again as predicted by the theory the larger the alignment the better the performance that is obtained using the Parzen window estimator. The results of applying an SVM to the Breast Cancer data using a Gaussian kernel show a very slight improvement in the test error for both 80% and 50% training sets. 7 Conclusions We have introduced a measure of performance of a kernel machine that is much easier to analyse than standard measures (eg the margin) and that provides much simpler algorithms. We have discussed its statistical and geometrical properties, demonstrating that it is a well motivated and formally useful quantity. By identifying that the ideal kernel matrix has a structure of the type yy', we have been able to transform a measure of similarity between kernels into a measure of fitness of a given kernel. The ease and reliability with which this quantity can be estimated using only training set information prior to training makes it an ideal tool for practical model selection. We have given preliminary experimental results that largely confirm the theoretical analysis and augur well for the use of this tool in more sophisticated model (kernel) selection applications. References [1] N. Cristianini and J. Shawe-Taylor. An Introduction to Support Vector Machines. Cambridge University Press, 2000. See also the web site www.supportvector.net. [2] Nello Cristianini, Huma Lodhi, and John Shawe-Taylor. Latent semantic kernels for feature selection. Technical Report NC-TR-00-080, NeuroCOLT Working Group, http://www.neurocolt.org, 2000. [3] L. Devroye, L. Gyorfi, and G. Lugosi. A Probabilistic Theory of Pattern Recognition. Number 31 in Applications of mathematics. Springer, 1996. [4] C. McDiarmid. On the method of bounded differences. In Surveys in Combinatorics 1989, pages 148-188. Cambridge University Press, 1989.
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Eye movements and the maturation of cortical orientation selectivity Michele Rucci and Antonino Casile Department of Cognitive and Neural Systems, Boston University, Boston, MA 02215. Scuola Superiore S. Anna, Pisa, Italy Abstract Neural activity appears to be a crucial component for shaping the receptive fields of cortical simple cells into adjacent, oriented subregions alternately receiving ON- and OFF-center excitatory geniculate inputs. It is known that the orientation selective responses of V1 neurons are refined by visual experience. After eye opening, the spatiotemporal structure of neural activity in the early stages of the visual pathway depends both on the visual environment and on how the environment is scanned. We have used computational modeling to investigate how eye movements might affect the refinement of the orientation tuning of simple cells in the presence of a Hebbian scheme of synaptic plasticity. Levels of correlation between the activity of simulated cells were examined while natural scenes were scanned so as to model sequences of saccades and fixational eye movements, such as microsaccades, tremor and ocular drift. The specific patterns of activity required for a quantitatively accurate development of simple cell receptive fields with segregated ON and OFF subregions were observed during fixational eye movements, but not in the presence of saccades or with static presentation of natural visual input. These results suggest an important role for the eye movements occurring during visual fixation in the refinement of orientation selectivity. 1 Introduction Cortical orientation selectivity, i.e. the preference to edges with specific orientations exhibited by most cells in the primary visual cortex of different mammal species, is one of the most investigated characteristics of neural responses. Although the essential elements of cortical orientation selectivity seem to develop before the exposure to patterned visual input, visual experience appears essential both for refining orientation selectivity, and maintaining the normal response properties of cortical neurons. The precise mechanisms by which visually-induced activity contribute to the maturation of neural responses are not known. A number of experimental findings support the hypothesis that the development of orientation selective responses relies on Hebbian/covariance mechanisms of plasticity. According to this hypothesis, the stabilization of synchronously firing afferents onto common postsynaptic neurons may account for the segregation of neural inputs observed in the receptive fields of simple cells, where the adjacent oriented excitatory and inhibitory subregions receive selective input from geniculate ON- and OFF-center cells in the same retinotopic positions. Modeling studies [10, 9] have shown the feasibility of this proposal assuming suitable patterns of spontaneous activity in the LGN before eye opening. After eye opening, the spatiotemporal structure of LGN activity depends not only on the characteristics of the visual input, but also on the movements performed by the animal while exploring its environment. It may be expected that changes in the visual input induced by these movements play an important role in shaping the responses of neurons in the visual system. In this paper we focus on how visual experience and eye movements might jointly influence the refinement of orientation selectivity under the assumption of a Hebbian mechanism of synaptic plasticity. As illustrated in Fig. 1, a necessary requirement of the Hebbian hypothesis is a consistency between the correlated activity of thalamic afferents and the organization of simple cell receptive fields. Synchronous activation is required among geniculate cells of the same type (ON- or OFF-center) with receptive fields located at distances smaller than the width of a simple cell subregion, and among cells of opposite polarity with receptive fields at distances comparable to the separation between adjacent subregions. We have analyzed the second order statistical structure of neural activity in a model of cat LGN when natural visual input was scanned so as to replicate the oculomotor behavior of the cat. Patterns of correlated activity were compared to the structure of simple cell receptive fields at different visual eccentricities. 2 The model Modeling the activity of LGN cells LGN cells were modeled as linear elements with quasi-separable spatial and temporal components as proposed by [3]. This model, derived using the reverse-correlation technique, has been shown to produce accurate estimates of the activity of different types of LGN cells. Changes in the instantaneous firing rates with respect to the level of spontaneous activity were generated by evaluating the spatiotemporal convolution of the input image with the receptive field kernel
(1) where is the symbol for convolution, and are the spatial and temporal variables, and the operator indicates rectification ( !
"$#&% if &'(%! ) otherwise). For each cell, the kernel consisted of two additive components, representing the center ( * ) and the periphery ( + ) of the receptive field respectively. Each of these two contributions was separable in its spatial ( , ) and temporal ( - ) elements: .
,0/ ! -1/ 0# ,02 ! -12 The spatial receptive fields of both center and surround were modeled as two-dimensional Gaussians, with a common space constant for both dimensions. Spatial parameters varied with eccentricity following neurophysiological measurements. As in [3], the temporal profile of the response was given by the difference of two gamma functions, with the temporal function for the periphery equal to that for the center and delayed by 3 ms. Modeling eye movements Modeled eye movements included saccades (both large-scale saccades and microsaccades), ocular drift, and tremor. Saccades— Voluntary saccadic eye movements, the fast shifts of gaze among fixation points, were modeled by assuming a generalized exponential distribution of fixation times. The amplitude and direction of a saccade were randomly selected among all possible saccades that would keep the point of fixation on the image. Following data described in the literature, the duration of each saccade was proportional to its amplitude. A modulation of geniculate activity was present in correspondence of each saccade [7]. Neural activity around the time of a saccade was multiplied by a gain function so that an initial suppression of activity with a peak of 10%, gradually reversed to a 20% facilitation with peak occurring 100 ms after the end of the saccade. Fixational eye movements— Small eye movements included fixational saccades, ocular drift and tremor. Microsaccades were modeled in a similar way to voluntary saccades, with amplitude randomly selected from a uniform distribution between 1 and 10 minutes of arc. No modulation of LGN activity was present in the case of microsaccades. Ocular drift and tremor were modeled together by approximating their power spectrum by means of a Poisson process filtered by a second order eye plant transfer function over the frequency range 0-40 Hz where the power declines as . This term represents the irregular discharge rate of motor units for frequency less than 40 Hz. Parameters were adjusted so as to give a mean amplitude of and a mean velocity equal to /s, which are the values measured in the cat [11]. 3 Results We simulated the activity of geniculate cells with receptive fields in different positions of the visual field, while receiving visual input in the presence of different types of eye movements. The relative level of correlation between units of the same and different types at positions and
in the LGN was measured by means of the correlation difference, D
ONON # ONOFF , where the two terms are the correlation coefficients evaluated between the two ON units at positions and
, and between the ON unit at position and the OFF unit at position
respectively. D is positive when the activity of units of the same type covary more strongly than that of units of different types, and is negative when the opposite occurs. The average relative levels of correlation between units with receptive fields at different distances in the visual field were examined by means of the function D
D ' , which evaluates the average correlation difference D among all pairs of cells at positions and
at distance from each other. For simplicity, in the following we refer to D as the correlation difference, implicitly assuming that a spatial averaging has taken place. The correlation difference is a useful tool for predicting the emerging patterns of connectivity in the presence of a Hebbian mechanism of synaptic plasticity. The average separation at which D changes sign is a key element in determining the spatial extent of the different subfields within the receptive fields of simple cells. Fig. 1 ( ) provides an example of application of the correlation difference function to quantify the correlated activity of LGN cells. In this example we have measured the level of correlation between pairs of cells with receptive fields at different separations when a spot of light was presented as input. An important element in the resulting level of correlation is the polarity of the two cells (i.e. whether they are ON- or OFF-center). As shown in Fig. 1 ( ), since geniculate cells tend to be coactive when the ON and OFF subregions of their receptive fields overlap, the correlation between pairs of cells of the same type decreases when the separation between their receptive fields is increased, while pairs of cells of opposite types tend to become more correlated. As a consequence, the correlation difference function, D , is positive at small separations, and negative at large ones. Fig. 2 shows the measured correlated activity for LGN cells located around 17 deg. of visual eccentricity in the presence of two types of visual input: retinal spontaneous activity and natural visual stimulation. Spontaneous activity was simulated on the basis of Matronarde’s data on the correlated firing of ganglion cells in the cat retina [8]. As illustrated by the graph, a close correspondence is present between the measured D and the response profile of an average cortical simple cell at this eccentricity, indicating that a ... ...
LGN OFF LGN ON V1 V1 RF (a) OFF ON 0 20 40 60 80 100 distance (min.) −1 0 1 2 normalized correlation ON−ON, OFF−OFF ON−OFF, OFF−ON difference (b) Figure 1: ( ) Patterns of correlated activity required by a Hebbian mechanism of synaptic plasticity to produce a segregation of geniculate afferents. On average ON- and OFF-center LGN cells overlapping excitatory and inhibitory subregions in the receptive field of a simple cell must be simultaneously active. ( ) Example of application of the correlation difference function, D . The icons on the top of each graph represent the positions of the receptive fields of the two cells at the corresponding separations along the axis. The bright dot marks the center of the spot of light. The three curves represent the correlation coefficients for pairs of units of the same type 2 (continuous thin line), units of opposite types (dashed line), and the correlation difference function D
2 # (bold line). Positive (negative) values of D indicate that the activity of LGN cells of the same (opposite) type covary more closely than the activity of cells of opposite (same) types. Hebbian mechanism of synaptic plasticity can well account for the structure of simple cell receptive fields before eye opening. What happens in the presence of natural visual input? We evaluated the correlation difference function on a database of 30 images of natural scenes. The mean power spectrum of our database was best approximated by !
"! $# %'& , which is consistent with the results of several studies investigating the power spectrum of natural images. The mean correlation difference function measured when the input images were analyzed statically is marked by dark triangles in the left panel of Fig. 2. Due to the wide spatial correlations of natural visual input, the estimated correlation difference did not change sign within the receptive field of a typical simple cell. That is, LGN cells of the same type were found to covary more closely than cells of opposite types at all separations within the receptive field of a simple cell. This result is not consistent with the putative role of a direct Hebbian/covariance model in the refinement of orientation selectivity after eye opening. A second series of simulations was dedicated to analyze the effects of eye movements on the structure of correlated activity. In these simulations the images of natural scenes were scanned so as to replicate cat oculomotor behavior. As shown in right panel of Fig. 2, significantly different patterns of correlated neural activity were found in the LGN in the presence of different types of eye movements. In the presence of large saccades, levels of correlations among the activity of geniculate cells were similar to the case of static presentation of natural visual input, and they did not match the structure of simple cell receptive fields. The dark triangles in Fig. 2 represent the correlation difference function evaluated over a window of observation of 100 ms in the presence of both large saccades and fixation eye movements. In contrast, when our analysis was restricted to the periods of visual fixation during which microscopic eye movements occurred, strong covariances were measured between cells of the same type located nearby and between cells of opposite types at distances compatible with the separation between different subregions in the receptive fields of simple cells. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 distance (deg.) −0.3 0.2 0.7 normalized correlation cortical RF spontaneous activity natural visual input 0.0 0.5 1.0 1.5 2.0 distance (deg.) −0.3 0.2 0.7 normalized Cd Saccade + Fixation Cortical RF Fixation Figure 2: Analysis of the correlated activity of LGN units in different experimental conditions. In both graphs, the curve marked by white circles is the average receptive field of a simple cell, as measured by Jones and Palmer (1987) shown here for comparison. (Left) Static analysis: patterns of correlated activity in the presence of spontaneous activity and when natural visual input was analyzed statically. (Right) Effects of eye movements: correlation difference functions measured when natural images were scanned with sequence or saccades or fixational eye movements. Fig. 3 shows the results of a similar analysis for LGN cells at different visual eccentricities. The white circles in the panels of Fig. 3 represent the width of the largest subfield in the receptive field of cortical simple cells as measured by [13]. The other curves on the left panel represent the widths of the central lobe of the correlation difference functions (the spatial separation over which cells of the same type possess correlated activity, measured as the double of the point in which the correlation difference function intersects the zero axis) in the cases of spontaneous activity and static presentation of natural visual input. As in Fig. 2, (1) a close correspondence was present between the experimental data and the subregion widths predicted by the correlation difference function in the case of spontaneous activity; and (2) a significant deviation between the two measurements was present in the case of static examination of natural visual input. The right panel in Fig. 3 shows the correlation difference functions obtained at different visual eccentricities in the presence of fixational eye movements. The minimum separation between receptive fields necessary for observing strong levels of covariance between cells with opposite polarity increased with eccentricity, as illustrated by the increase in the central lobe of the estimated correlation functions at the different visual eccentricities. As for the case of spontaneous activity, a close correspondence is now present between the spatiotemporal characteristics of LGN activity and the organization of simple cell receptive fields. 4 Discussion In this paper we have used computer modeling to study the correlated activity of LGN cells when images of natural scenes were scanned so as to replicate cat eye movements. In the absence of eye movements, when a natural visual environment was observed statically, similar to the way it is examined by animals with their eyes paralyzed, we found that the simulated responses of geniculate cells of the same type at any separation smaller than the receptive field of a simple cell were strongly correlated. These spatial patterns of covarying geniculate activity did not match the structure of simple cell receptive fields. A similar result was obtained when natural scenes were scanned through saccades. Conversely, in 0 10 20 30 eccentricity (deg.) 0 2 5 8 10 central width (deg) Wilson & Sherman, 1976 spontaneous activity natural input (static) 0 10 20 30 eccentricity (deg.) 0 2 5 8 10 central width (deg) Wilson & Sherman, 1976 visual fixation Figure 3: Analysis of the correlated activity of LGN units at different visual eccentricities. The width of the larger subfield in the receptive field of simple cells at different eccentricities as measured by Wilson and Sherman (1976) (white circles) is compared to the width of the central lobe of the correlation difference functions measured in different conditions (Left) Static analysis: results obtained in the presence of spontaneous activity and when natural visual input was analyzed statically. (Right) Case of fixational eye movements and natural visual input. the case of micromovements, including both microsaccades and the combination of ocular drift and tremor, strong correlations were measured among cells of the same type located nearby and among cells of opposite types at distances compatible with the separation between different subregions in the receptive fields of simple cells. These results suggest a developmental role for the small eye movements that occur during visual fixation. Although the role of visual experience in the development of orientation selectivity has been extensively investigated, relatively few studies have focused on whether eye movements contribute to the development of the responses of cortical cells. Yet, experiments in which kittens were raised with their eyes paralyzed have shown basic deficiencies in the development of visually-guided behavior [6], as well as impairments in ocular dominance plasticity [4, 12]. In addition, it has been shown that eye movements are necessary for the reestablishment of cortical orientation selectivity in dark-reared kittens exposed to visual experience within the critical period [2, 5]. This indicates that simultaneous experience of visual input and eye movements (and/or eye movement proprioception) may be necessary for the refinement of orientation selectivity [1]. Our finding that the patterns of LGN activity with static presentation of natural images did not match the spatial structure of the receptive fields of simple cells is in agreement with the hypothesis that exposure to pattern vision per se is not sufficient to account for a normal visual development. A main assumption of this study is that the refinement and maintenance of orientation selectivity after eye opening is mediated by a Hebbian/covariance process of synaptic plasticity. The term Hebbian is used here with a generalized meaning to indicate the family of algorithms in which modifications of synaptic efficacies occur on the basis of the patterns of input covariances. While no previous theoretical study has investigated the influence of eye movements on the development of orientation selectivity, some models have shown that schemes of synaptic modifications based on the correlated activity of thalamic afferents can account well for the segregation of ON- and OFF-center inputs before eye opening in the presence of suitable patterns of spontaneous activity [10, 9]. By showing that, during fixation, the spatiotemporal structure of visually-driven geniculate activity is compatible with the structure of simple cell receptive fields, the results of the present study extend the plausibility of such schemes to the period after eye opening in which exposure to pattern vision occurs. Ocular movements are a common feature of the visual system of different species. It should not come as a surprise that a trace of their existence can be found even in some of the most basic properties of neurons in the early stages of the visual system, such as orientation selectivity. Further studies are needed to investigate whether similar traces can be found in other features of visual neural responses. References [1] P. Buisseret. Influence of extraocular muscle proprioception on vision. Physiol. Rev., 75(2):323–338, 1995. [2] P. Buisseret, E. Gary-Bobo, and M. Imbert. Ocular motility and recovery of orientational properties of visual cortical neurons in dark-reared kittens. Nature, 272:816– 817, 1978. [3] D. Cai, G. C. DeAngelis, and R. D. Freeman. Spatiotemporal receptive field organization in the lateral geniculate nucleus of cats and kitten. J. Neurophysiol., 78(2):1045– 61, 1997. [4] R. D. Freeman and A. B. Bonds. Cortical plasticity in monocularly deprived immobilized kittens depends on eye movement. Science, 206:1093–1095, 1979. [5] E. Gary-Bobo, C. Milleret, and P. Buisseret. Role of eye movements in developmental process of orientation selectivity in the kitten visual cortex. Vision Res., 26(4):557– 567, 1986. [6] A. Hein, F. Vital-Durand, W. Salinger, and R. Diamond. Eye movements initiate visual-motor development in the cat. Science, 204:1321–1322, 1979. [7] D. Lee and J. G. Malpeli. Effect of saccades on the activity of neurons in the cat lateral geniculate nucleus. J. Neurophysiol., 79:922–936, 1998. [8] D. N. Mastronarde. Correlated firing of cat retinal ganglion cells. I spontaneously active inputs to X and Y cells. J. Neurophysiol., 49(2):303–323, 1983. [9] K. D. Miller. A model of the development of simple cell receptive fields and the ordered arrangement of orientation columns through activity-dependent competition between ON- and OFF- center inputs. J. Neurosci., 14(1):409–441, 1994. [10] M. Miyashita and S. Tanaka. A mathematical model for the self-organization of orientation columns in visual cortex. Neuroreport, 3:69–72, 1992. [11] E. Olivier, A. Grantyn, M. Chat, and A. Berthoz. The control of slow orienting eye movements by tectoreticulospinal neurons in the cat: behavior, discharge patterns and underlying connections. Exp. Brain Res., 93:435–449, 1993. [12] W. Singer and J. Raushecker. Central-core control of developmental plasticity in the kitten visual cortex II. Electrical activation of mesencephalic and diencephalic projections. Exp. Brain Res., 47:22–233, 1982. [13] J. R. Wilson and S. M. Sherman. Receptive-field characteristics of neurons in the cat striate cortex: changes with visual field eccentricity. J. Neurophysiol., 39(3):512–531, 1976.
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Escaping the Convex Hull with Extrapolated Vector Machines. Patrick Haffner AT&T Labs-Research, 200 Laurel Ave, Middletown, NJ 07748 haffner@research.att.com Abstract Maximum margin classifiers such as Support Vector Machines (SVMs) critically depends upon the convex hulls of the training samples of each class, as they implicitly search for the minimum distance between the convex hulls. We propose Extrapolated Vector Machines (XVMs) which rely on extrapolations outside these convex hulls. XVMs improve SVM generalization very significantly on the MNIST [7] OCR data. They share similarities with the Fisher discriminant: maximize the inter-class margin while minimizing the intra-class disparity. 1 Introduction Both intuition and theory [9] seem to support that the best linear separation between two classes is the one that maximizes the margin. But is this always true? In the example shown in Fig.(l), the maximum margin hyperplane is Wo; however, most observers would say that the separating hyperplane WI has better chances to generalize, as it takes into account the expected location of additional training sam••••••••••• f\J:- •••••••••• •• .., --.Q. •• ~".' ~, _ .. ................ ~x~... W 1 --------------~--------------·K~ ··········· '''-,,- /0- 0 00 00 00 0'\ "OW-0_ o __ oo- o::-o . .. . ....... (} ............ . Figure 1: Example of separation where the large margin is undesirable. The convex hull and the separation that corresponds to the standard SVM use plain lines while the extrapolated convex hulls and XVMs use dotted lines. pIes. Traditionally, to take this into account, one would estimate the distribution of the data. In this paper, we just use a very elementary form of extrapolation ("the poor man variance") and show that it can be implemented into a new extension to SVMs that we call Extrapolated Vector Machines (XVMs). 2 Adding Extrapolation to Maximum Margin Constraints This section states extrapolation as a constrained optimization problem and computes a simpler dual form. Take two classes C+ and C_ with Y+ = +1 and Y_ = -1 1 as respective targets. The N training samples {(Xi, Yi); 1 ::::; i ::::; N} are separated with a margin p if there exists a set of weights W such that Ilwll = 1 and Vk E {+, -}, Vi E Ck, Yk(w,xi+b) 2: p (1) SVMs offer techniques to find the weights W which maximize the margin p. Now, instead of imposing the margin constraint on each training point, suppose that for two points in the same class Ck, we require any possible extrapolation within a range factor 17k 2: 0 to be larger than the margin: Vi,j E Ck, V)" E [-17k, l+17k], Yk (W.()"Xi + (l-)")Xj) + b) 2: P (2) It is sufficient to enforce the constraints at the end of the extrapolation segments, and (3) Keeping the constraint over each pair of points would result in N 2 Lagrange multipliers. But we can reduce it to a double constraint applied to each single point. If follows from Eq.(3) that: (4) (5) We consider J.Lk = max (Yk(W.Xj)) and Vk = min (Yk(W.Xj)) as optimization varilEC. lEC. abIes. By adding Eq.(4) and (5), the margin becomes 2p = L ((17k+1)vk- 17kJ.Lk) = L (Vk -17dJ.Lk - Vk)) (6) k k Our problem is to maximize the margin under the double constraint: Vi E Ck, Vk ::::; Yk(W.Xi) ::::; J.Lk In other words, the extrapolated margin maximization is equivalent to squeezing the points belonging to a given class between two hyperplanes. Eq.(6) shows that p is maximized when Vk is maximized while J.Lk - Vk is minimized. Maximizing the margin over J.Lk , Vk and W with Lagrangian techniques gives us the following dual problem: (7) lIn this paper, it is necessary to index the outputs y with the class k rather than the more traditional sample index i, as extrapolation constraints require two examples to belong to the same class. The resulting equations are more concise, but harder to read. Compared to the standard SVM formulation, we have two sets of support vectors. Moreover, the Lagrange multipliers that we chose are normalized differently from the traditional SVM multipliers (note that this is one possible choice of notation, see Section.6 for an alternative choice). They sum to 1 and allow and interesting geometric interpretation developed in the next section. 3 Geometric Interpretation and Iterative Algorithm For each class k, we define the nearest point to the other class convex hull along the direction of w: Nk = I:iECk f3iXi. Nk is a combination of the internal support vectors that belong to class k with f3i > O. At the minimum of (7), because they correspond to non zero Lagrange multipliers, they fallon the internal margin Yk(W,Xi) = Vk; therefore, we obtain Vk = Ykw.Nk· Similarly, we define the furthest point Fk = I:i ECk ~i Xi' Fk is a combination of the external support vectors, and we have flk = Ykw.Fk. The dual problem is equivalent to the distance minimization problem min IILYk ((1Jk+I)Nk _1Jk F k)11 2 Nk ,Fk EHk k where 1{k is the convex hull containing the examples of class k. It is possible to solve this optimization problem using an iterative Extrapolated Convex Hull Distance Minimization (XCHDM) algorithm. It is an extension of the Nearest Point [5] or Maximal Margin Percept ron [6] algorithms. An interesting geometric interpretation is also offered in [3]. All the aforementioned algorithms search for the points in the convex hulls of each class that are the nearest to each other (Nt and No on Fig.I), the maximal margin weight vector w = Nt - No-' XCHDM look for nearest points in the extrapolated convex hulls (X+ I and X-I on Fig.I). The extrapolated nearest points are X k = 1JkNk - 1JkFk' Note that they can be outside the convex hull because we allow negative contribution from external support vectors. Here again, the weight vector can be expressed as a difference between two points w = X+ - X - . When the data is non-separable, the solution is trivial with w = O. With the double set of Lagrange multipliers, the description of the XCHDM algorithm is beyond the scope of this paper. XCHDM with 1Jk = 0 are simple SVMs trained by the same algorithm as in [6]. An interesting way to follow the convergence of the XCHDM algorithm is the following. Define the extrapolated primal margin 1'; = 2p = L ((1Jk+ I )vk- 1Jkflk) k and the dual margin 1'; = IIX+ - X-II Convergence consists in reducing the duality gap 1'~ -1'; down to zero. In the rest of the paper, we will measure convergence with the duality ratio r = 1'~ . 1'2 To determine the threshold to compute the classifier output class sign(w.x+b) leaves us with two choices. We can require the separation to happen at the center of the primal margin, with the primal threshold (subtract Eq.(5) from Eq.(4)) 1 bl = -2" LYk ((1Jk+ I )vk-1JkJ.lk) k or at the center of the dual margin, with the dual threshold b2 = - ~w. 2:)(T}k+1)Nk - T}kFk) = - ~ (IIx+ 112 -lix-in k Again, at the minimum, it is easy to verify that b1 = b2 . When we did not let the XCHDM algorithm converge to the minimum, we found that b1 gave better generalization results. Our standard stopping heuristic is numerical: stop when the duality ratio gets over a fixed value (typically between 0.5 and 0.9). The only other stopping heuristic we have tried so far is based on the following idea. Define the set of extrapolated pairs as {(T}k+1)Xi -T}kXj; 1 :S i,j :S N}. Convergence means that we find extrapolated support pairs that contain every extrapolated pair on the correct side of the margin. We can relax this constraint and stop when the extrapolated support pairs contain every vector. This means that 12 must be lower than the primal true margin along w (measured on the non-extrapolated data) 11 = y+ + Y -. This causes the XCHDM algorithm to stop long before 12 reaches Ii and is called the hybrid stopping heuristic. 4 Beyond SVMs and discriminant approaches. Kernel Machines consist of any classifier of the type f(x) = L:i Yi(XiK(x, Xi). SVMs offer one solution among many others, with the constraint (Xi > O. XVMs look for solutions that no longer bear this constraint. While the algorithm described in Section 2 converges toward a solution where vectors act as support of margins (internal and external), experiments show that the performance of XVMs can be significantly improved if we stopped before full convergence. In this case, the vectors with (Xi =/: 0 do not line up onto any type of margin, and should not be called support vectors. The extrapolated margin contains terms which are caused by the extrapolation and are proportional to the width of each class along the direction of w. We would observe the same phenomenon if we had trained the classifier using Maximum Likelihood Estimation (MLE) (replace class width with variance). In both MLE and XVMs, examples which are the furthest from the decision surface play an important role. XVMs suggest an explanation why. Note also that like the Fisher discriminant, XVMs look for the projection that maximizes the inter-class variance while minimizing the intra-class variances. 5 Experiments on MNIST The MNIST OCR database contains 60,000 handwritten digits for training and 10,000 for testing (the testing data can be extended to 60,000 but we prefer to keep unseen test data for final testing and comparisons). This database has been extensively studied on a large variety of learning approaches [7]. It lead to the first SVM "success story" [2], and results have been improved since then by using knowledge about the invariance of the data [4]. The input vector is a list of 28x28 pixels ranging from 0 to 255. Before computing the kernels, the input vectors are normalized to 1: x = II~II' Good polynomial kernels are easy to define as Kp(x, y) = (x.y)P. We found these normalized kernels to outperform the unnormalized kernels Kp(x, y) = (a(x.y)+b)P that have been traditionally used for the MNIST data significantly. For instance, the baseline error rate with K4 is below 1.2%, whereas it hovers around 1.5% for K4 (after choosing optimal values for a and b)2. We also define normalized Gaussian kernels: Kp(x, y) = exp ( - ~ Ilx - y112) = [exp (x.y- 1)JP. (8) Eq.(8) shows how they relate to normalized polynomial kernels: when x.y « 1, Kp and Kp have the same asymptotic behavior. We observed that on MNIST, the performance with Kp is very similar to what is obtained with unnormalized Gaussian kernels Ku(x, y) = exp _(X~Y)2. However, they are easier to analyze and compare to polynomial kernels. MNIST contains 1 class per digit, so the total number of classes is M=10. To combine binary classifiers to perform multiclass classifications, the two most common approaches were considered . • In the one-vs-others case (lvsR) , we have one classifier per class c, with the positive examples taken from class c and negative examples form the other classes. Class c is recognized when the corresponding classifier yields the largest output . • In the one-vs-one case (lvs1), each classifier only discriminates one class from another: we need a total of (MU:;-l) = 45 classifiers. Despite the effort we spent on optimizing the recombination of the classifiers [8] 3, 1 vsR SVMs (Table 1) perform significantly better than 1 vs1 SVMs (Table 2). 4 For each trial, the number of errors over the 10,000 test samples (#err) and the total number of support vectors( #SV) are reported. As we only count SV s which are shared by different classes once, this predicts the test time. For instance, 12,000 support vectors mean that 20% of the 60,000 vectors are used as support. Preliminary experiments to choose the value of rJk with the hybrid criterion show that the results for rJk = 1 are better than rJk = 1.5 in a statistically significant way, and slightly better than rJk = 0.5. We did not consider configurations where rJ+ f; rJ-; however, this would make sense for the assymetrical 1 vsR classifiers. XVM gain in performance over SVMs for a given configuration ranges from 15% (1 vsR in Table 3) to 25% (1 vs1 in Table 2). 2This may partly explain a nagging mystery among researchers working on MNIST: how did Cortes and Vapnik [2] obtain 1.1% error with a degree 4 polynomial ? 3We compared the Max Wins voting algorithm with the DAGSVM decision tree algorithm and found them to perform equally, and worse than 1 vsR SVMs. This is is surprising in the light of results published on other tasks [8], and would require further investigations beyond the scope of this paper. 4Slightly better performance was obtained with a new algorithm that uses the incremental properties of our training procedure (this is be the performance reported in the tables). In a transductive inference framework, treat the test example as a training example: for each of the M possible labels, retrain the M among (M(":-l) classifiers that use examples with such label. The best label will be the one that causes the smallest increase in the multiclass margin p such that it combines the classifier margins pc in the following manner ~= ,,~ 2 ~ 2 P c~M Pc The fact that this margin predicts generalization is "justified" by Theorem 1 in [8]. Duality Ratio stop Kernel 0.40 0.75 0.99 #err #SV #err #SV # err #SV K3 136 8367 136 11132 132 13762 K4 127 8331 117 11807 119 15746 K5 125 8834 119 12786 119 17868 Kg 136 13002 137 18784 141 25953 [(2 147 9014 128 11663 131 13918 [(4 125 8668 119 12222 117 16604 K5 125 8944 125 12852 125 18085 Table 1: SVMs on MNIST with 10 1vsR classifiers Kernel SVM/ratio at 0.99 XVM/Hybrid # err #SV # err #SV K3 138 11952 117 17020 K4 135 13526 110 16066 K5 191 13526 114 15775 Table 2: SVMjXVM on MNIST with 45 1 vs1 classifiers The 103 errors obtained with K4 and r = 0.5 in Table 3 represent only about 1% error: this is the lowest error ever reported for any learning technique without a priori knowledge about the fact that the input data corresponds to a pixel map (the lowest reproducible error previously reported was 1.2% with SVMs and polynomials of degree 9 [4], it could be reduced to 0.6% by using invariance properties of the pixel map). The downside is that XVMs require 4 times as many support vectors as standards SVMs. Table 3 compares stopping according to the duality ratio and the hybrid criterion. With the duality ratio, the best performance is most often reached with r = 0.50 (if this happens to be consistently true, validation data to decide when to stop could be spared). The hybrid criterion does not require validation data and yields errors that, while higher than the best XVM, are lower than SVMs and only require a few more support vectors. It takes fewer iterations to train than SVMs. One way to interpret this hybrid stopping criterion is that we stop when interpolation in some (but not all) directions account for all non-interpolated vectors. This suggest that interpolation is only desirable in a few directions. XVM gain is stronger in the 1 vs 1 case (Table 2). This suggests that extrapolating on a convex hull that contains several different classes (in the 1 vsR case) may be undesirable. Duality Ratio stop Hybrid. Kernel 0.40 0.50 0.75 Stop Crit. # err #SV # err #SV # err #SV # err #SV K3 118 46662 111 43819 116 50216 125 20604 K4 112 40274 103 43132 110 52861 107 18002 K5 109 36912 106 44226 110 49383 107 17322 Kg 128 35809 126 39462 131 50233 125 19218 K2 114 43909 114 46905 114 53676 119 20152 [(4 108 36980 111 40329 114 51088 108 16895 Table 3: XVMs on MNIST with 10 1 vsR classifiers 6 The Soft Margin Case MNIST is characterized by the quasi-absence of outliers, so to assume that the data is fully separable does not impair performance at all. To extend XVMs to non-separable data, we first considered the traditional approaches of adding slack variables to allow margin constraints to be violated. The most commonly used approach with SVMs adds linear slack variables to the unitary margin. Its application to the XVM requires to give up the weight normalization constraint, so that the usual unitary margin can be used in the constraints [9] . Compared to standard SVMs, a new issue to tackle is the fact that each constraint corresponds to a pair of vectors: ideally, we should handle N 2 slack variables ~ij. To have linear constraints that can be solved with KKT, we need to have the decomposition ~ij = ('T}k+1)~i+'T}k~; (factors ('T}k+1) and 'T}k are added here to ease later simplifications). Similarly to Eq.(3), the constraint on the extrapolation from any pair of points is Vi,j E Ck, Yk (w. (('T}k+1)xi - 'T}kXj) +b) 2: 1 - ('T}k+1)~i - 'T}k~; with ~i'~; 2: 0 (9) Introducing J.tk = max (Yk(w,xj+b) ~;) and Vk = min (Yk(W,Xi+b) + ~i)' we obJECk .ECk tain the simpler double constraint Vi E Ck, Vk -~i ~ Yk(W,Xi+b) ~ J.tk+~; with ~i'~; 2: 0 (10) It follows from Eq.(9) that J.tk and Vk are tied through (l+'T}k)vk = l+'T}kJ.tk If we fix J.tk (and thus Vk) instead of treating it as an optimization variable, it would amount to a standard SVM regression problem with {-I, + I} outputs, the width of the asymmetric f-insensitive tube being J.tk-Vk = (~~~;)' This remark makes it possible for the reader to verify the results we reported on MNIST. Vsing the publicly available SVM software SVMtorch [1] with C = 10 and f = 0.1 as the width of the f-tube yields a 10-class error rate of 1.15% while the best performance using SVMtorch in classification mode is 1.3% (in both cases, we use Gaussian kernels with parameter (J = 1650). An explicit minimization on J.tk requires to add to the standard SVM regression problem the following constraint over the Lagrange multipliers (we use the same notation as in [9]): Yi=l Yi=- l Yi=l Yi=- l Note that we still have the standard regression constraint I: ai = I: ai This has not been implemented yet, as we question the pertinence of the ~; slack variables for XVMs. Experiments with SVMtorch on a variety of tasks where non-zero slacks are required to achieve optimal performance (Reuters, VCI/Forest, VCI/Breast cancer) have not shown significant improvement using the regression mode while we vary the width of the f-tube. Many experiments on SVMs have reported that removing the outliers often gives efficient and sparse solutions. The early stopping heuristics that we have presented for XVMs suggest strategies to avoid learning (or to unlearn) the outliers, and this is the approach we are currently exploring. 7 Concluding Remarks This paper shows that large margin classification on extrapolated data is equivalent to the addition of the minimization of a second external margin to the standard SVM approach. The associated optimization problem is solved efficiently with convex hull distance minimization algorithms. A 1 % error rate is obtained on the MNIST dataset: it is the lowest ever obtained without a-priori knowledge about the data. We are currently trying to identify what other types of dataset show similar gains over SVMs, to determine how dependent XVM performance is on the facts that the data is separable or has invariance properties. We have only explored a few among the many variations the XVM models and algorithms allow, and a justification of why and when they generalize would help model selection. Geometry-based algorithms that handle potential outliers are also under investigation. Learning Theory bounds that would be a function of both the margin and some form of variance of the data would be necessary to predict XVM generalization and allow us to also consider the extrapolation factor 'TJ as an optimization variable. References [1] R. Collobert and S. Bengio. Support vector machines for large-scale regression problems. Technical Report IDIAP-RR-00-17, IDIAP, 2000. [2] C. Cortes and V. Vapnik. Support vector networks. Machine Learning, 20:1- 25, 1995. [3] D. Crisp and C.J.C. Burges. A geometric interpretation of v-SVM classifiers. In Advances in Neural Information Processing Systems 12, S. A. Solla, T. K. Leen, K.-R. Mller, eds, Cambridge, MA, 2000. MIT Press. [4] D. DeCoste and B. Schoelkopf. Training invariant support vector machines. Machine Learning, special issue on Support Vector Machines and Methods, 200l. [5] S.S. Keerthi, S.K. Shevade, C. Bhattacharyya, and K.R.K. Murthy. A fast iterative nearest point algorithm for support vector machine classifier design. IEEE transactions on neural networks, 11(1):124 - 136, jan 2000. [6] A. Kowalczyk. Maximal margin perceptron. In Advances in Large Margin Classifiers, Smola, Bartlett, Schlkopf, and Schuurmans, editors, Cambridge, MA, 2000. MIT Press. [7] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. proceedings of the IEEE, 86(11), 1998. [8] J. Platt, N. Christianini, and J. Shawe-Taylor. Large margin dags for multiclass classification. In Advances in Neural Information Processing Systems 12, S. A. Solla, T. K. Leen, K.-R. Mller, eds, Cambridge, MA, 2000. MIT Press. [9] V. N. Vapnik. Statistical Learning Theory. John Wiley & Sons, New-York, 1998.
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Algorithmic Luckiness Ralf Herbrich Microsoft Research Ltd. CB3 OFB Cambridge United Kingdom rherb@microsoft·com Robert C. Williamson Australian National University Canberra 0200 Australia Bob. Williamson@anu.edu.au Abstract In contrast to standard statistical learning theory which studies uniform bounds on the expected error we present a framework that exploits the specific learning algorithm used. Motivated by the luckiness framework [8] we are also able to exploit the serendipity of the training sample. The main difference to previous approaches lies in the complexity measure; rather than covering all hypotheses in a given hypothesis space it is only necessary to cover the functions which could have been learned using the fixed learning algorithm. We show how the resulting framework relates to the VC, luckiness and compression frameworks. Finally, we present an application of this framework to the maximum margin algorithm for linear classifiers which results in a bound that exploits both the margin and the distribution of the data in feature space. 1 Introduction Statistical learning theory is mainly concerned with the study of uniform bounds on the expected error of hypotheses from a given hypothesis space [9, 1]. Such bounds have the appealing feature that they provide performance guarantees for classifiers found by any learning algorithm. However, it has been observed that these bounds tend to be overly pessimistic. One explanation is that only in the case of learning algorithms which minimise the training error it has been proven that uniformity of the bounds is equivalent to studying the learning algorithm's generalisation performance directly. In this paper we present a theoretical framework which aims at directly studying the generalisation error of a learning algorithm rather than taking the detour via the uniform convergence of training errors to expected errors in a given hypothesis space. In addition, our new model of learning allows the exploitation of the fact that we serendipitously observe a training sample which is easy to learn by a given learning algorithm. In that sense, our framework is a descendant of the luckiness framework of Shawe-Taylor et al. [8]. In the present case, the luckiness is a function of a given learning algorithm and a given training sample and characterises the diversity of the algorithms solutions. The notion of luckiness allows us to study given learning algorithms at many different perspectives. For example, the maximum margin algorithm [9] can either been studied via the number of dimensions in feature space, the margin of the classifier learned or the sparsity of the resulting classifier. Our main results are two generalisation error bounds for learning algorithms: one for the zero training error scenario and one agnostic bound (Section 2). We shall demonstrate the usefulness of our new framework by studying its relation to the VC framework, the original luckiness framework and the compression framework of Littlestone and Warmuth [6] (Section 3). Finally, we present an application of the new framework to the maximum margin algorithm for linear classifiers (Section 4). The detailed proofs of our main results can be found in [5]. We denote vectors using bold face, e.g. x = (Xl, ... ,xm ) and the length of this vector by lxi, i.e. Ixl = m. In order to unburden notation we use the shorthand notation Z[i:jJ := (Zi,"" Zj) for i :::; j. Random variables are typeset in sans-serif font. The symbols Px, Ex [f (X)] and IT denote a probability measure over X, the expectation of f (.) over the random draw of its argument X and the indicator function, respectively. The shorthand notation Z(oo) := U;;'=l zm denotes the union of all m- fold Cartesian products of the set Z. For any mEN we define 1m C {I, ... , m } m as the set of all permutations of the numbers 1, ... ,m, 1m := {(il , ... ,im) E {I, ... ,m}m I'v'j f:- k: ij f:- id . Given a 2m- vector i E hm and a sample z E z2m we define Wi : {I, ... , 2m} -+ {I, ... , 2m} by Wi (j) := ij and IIdz) by IIi (z) := (Z7ri(l), ... , Z7ri(2m))' 2 Algorithmic Luckiness Suppose we are given a training sample z = (x, y) E (X x y)m = zm of size mEN independently drawn (iid) from some unknown but fixed distribution PXy = Pz together with a learning algorithm A : Z( 00) -+ yX . For a predefined loss l : y x y -+ [0,1] we would like to investigate the generalisation error Gl [A, z] := Rl [A (z)] - infhEYx Rl [h] of the algorithm where the expected error Rl [h] of his defined by Rl [h] := Exy [l (h (X) ,Y)] . Since infhEYx Rl [h] (which is also known as the Bayes error) is independent of A it suffices to bound Rl [A (z)]. Although we know that for any fixed hypothesis h the training error ~ 1 Rdh,z]:=~ L l(h(xi),Yi) (X i ,Yi) Ez is with high probability (over the random draw of the training sample z E Z(oo)) close to Rl [h], this might no longer be true for the random hypothesis A (z). Hence we would like to state that with only small probability (at most 8), the expected error Rl [A (z)] is larger than the training error HI [A (z), z] plus some sample and algorithm dependent complexity c (A, z, 8), Pzm (Rl [A (Z)] > HI [A (Z), Z] + c (A, Z,8)) < 8. (1) In order to derive such a bound we utilise a modified version of the basic lemma of Vapnik and Chervonenkis [10]. Lemma 1. For all loss functions l : y x y -+ [0,1], all probability measures Pz, all algorithms A and all measurable formulas Y : zm -+ {true, false}, if mc2 > 2 then Pzm (( RdA (Z)] > HdA (Z) , Z] + c) /\ Y (Z)) < 2PZ2m ((HI [A (Z[l:m]) ,Z[(m+l):2mJJ > HI [A (Z[l:mJ) ,Z[l:mJJ + ~) /\ Y (Z[l:m])) . , .I V J(Z) Proof (Sketch). The probability on the r.h.s. is lower bounded by the probability of the conjunction of event on the l.h.s. and Q (z) = Rl [A (Z[l:mj)] Rl [A (Z[l:mj) ,Z(m+1):2m] < ~. Note that this probability is over z E z2m. If we now condition on the first m examples, A (Z[l:mj) is fixed and therefore by an application of Hoeffding's inequality (see, e.g. [1]) and since m€2 > 2 the additional event Q has probability of at least ~ over the random draw of (Zm+1, ... , Z2m). 0 Use of Lemma 1 which is similar to the approach of classical VC analysis reduces the original problem (1) to the problem of studying the deviation of the training errors on the first and second half of a double sample z E z2m of size 2m. It is of utmost importance that the hypothesis A (Z[l:mj) is always learned from the first m examples. Now, in order to fully exploit our assumptions of the mutual independence of the double sample Z E z2m we use a technique known as symmetrisation by permutation: since PZ2~ is a product measure, it has the property that PZ2»> (J (Z)) = PZ2~ (J (ITi (Z))) for any i E hm. Hence, it suffices to bound the probability of permutations Jri such that J (ITi (z)) is true for a given and fixed double sample z. As a consequence thereof, we only need to count the number of different hypotheses that can be learned by A from the first m examples when permuting the double sample. Definition 1 (Algorithmic luckiness). Any function L that maps an algorithm A : Z(oo) -+ yX and a training sample z E Z(oo) to a real value is called an algorithmic luckiness. For all mEN, for any z E z2m , the lucky set HA (L , z) ~ yX is the set of all hypotheses that are learned from the first m examples (Z7ri(1),···, Z7ri(m)) when permuting the whole sample z whilst not decreasing the luckiness, i.e. (2) where Given a fixed loss function 1 : y x y -+ [0,1] the induced loss function set £1 (HA (L,z)) is defined by £1 (HA (L,z)) := {(x,y) r-+ 1 (h(x) ,y) I h E HA (L,z)} . For any luckiness function L and any learning algorithm A , the complexity of the double sample z is the minimal number N1 (T, £1 (HA (L, z)) ,z) of hypotheses h E yX needed to cover £1 (HA (L , z)) at some predefined scale T, i.e. for any hypothesis hE HA (L, z) there exists a h E yX such that (4) To see this note that whenever J (ITi (z)) is true (over the random draw of permutations) then there exists a function h which has a difference in the training errors on the double sample of at least ~ + 2T. By an application of the union bound we see that the number N 1 (T, £1 (HA (L, z)) , z) is of central importance. Hence, if we are able to bound this number over the random draw of the double sample z only using the luckiness on the first m examples we can use this bound in place of the worst case complexity SUPzEZ2~ N1 (T, £1 (HA (L, z)) ,z) as usually done in the VC framework (see [9]). Definition 2 (w- smallness of L). Given an algorithm A : Z ( 00) -+ yX and a loss l : y x y -+ [a, 1] the algorithmic luckiness function Lis w- small at scale T E jR+ if for all mEN, all J E (a, 1] and all Pz PZ2~ (Nl (T, £"1 (1iA (L, Z)), Z) > w (L (A, Z[l:ml) ,l, m, J,T)) < J. , " v S(Z) Note that if the range of l is {a, I} then N 1 (2~ ' £"1 (1iA (L, z)) , z) equals the number of dichotomies on z incurred by £"1 (1iA (L,z)). Theorem 1 (Algorithmic luckiness bounds). Suppose we have a learning algorithm A : Z( oo) -+ yX and an algorithmic luckiness L that is w-small at scale T for a loss function l : y X Y -+ [a, 1]. For any probability measure Pz, any dEN and any J E (a, 1], with probability at least 1 - J over the random draw of the training sample z E zm of size m, if w (L (A, z) ,l, m, J/4, T) :::; 2d then Rz[A (z)] :::; Rz[A (z), z] + ! (d + 10g2 (~) ) + 4T. (5) Furthermore, under the above conditions if the algorithmic luckiness L is wsmall at scale 2~ for a binary loss function l (".) E {a, I} and Rl [A (z), z] = a then (6) Proof (Compressed Sketch). We will only sketch the proof of equation (5) ; the proof of (6) is similar and can be found in [5]. First, we apply Lemma 1 with Y (z) == w (L (A,z) ,l,m,J/4,T) :::; 2d. We now exploit the fact that PZ2~ (J (Z)) :Z 2~ (J (Z) 1\ S (Z) ), +PZ 2~ (J (Z) 1\ ...,S (Z)) v :::: P Z 2~ (S(Z)) J < 4 + PZ2~ (J (Z) I\...,S (Z)) , which follows from Definition 2. Following the above-mentioned argument it suffices to bound the probability of a random permutation III (z) that J (III (z)) 1\ ...,S (III (z)) is true for a fixed double sample z. Noticing that Y (z) 1\ ...,S (z) => Nl (T,£"l (1iA (L,z)) ,z) :::; 2d we see that we only consider swappings Jri for which Nl (T,£"l (1iA (L,IIi (z))) ,IIi (z)) :::; 2d. Thus let us consider such a cover of size not more than 2. By (4) we know that whenever J (IIi (z)) 1\ ...,S (IIi (z)) is true for a swapping i then there exists a hypothesis h E yX in the cover such that Rl [h, (III (z))[(m+1):2ml] - Rl [h, (III (z))[l:ml] > ~ + 2T. Using the union bound and Hoeffding's inequality for a particular choice of PI shows that PI (J (III (z)) 1\ ...,S (III (z))) :::; £ which finalises the proof. D A closer look at (5) and (6) reveals that the essential difference to uniform bounds on the expected error is within the definition of the covering number: rather than covering all hypotheses h in a given hypothesis space 1i ~ yX for a given double sample it suffices to cover all hypotheses that can be learned by a given learning algorithm from the first half when permuting the double sample. Note that the usage of permutations in the definition of (2) is not only a technical matter; it fully exploits all the assumptions made for the training sample, namely the training sample is drawn iid. 3 Relationship to Other Learning Frameworks In this section we present the relationship of algorithmic luckiness to other learning frameworks (see [9, 8, 6] for further details of these frameworks). VC Framework If we consider a binary loss function l (".) E {a, I} and assume that the algorithm A selects functions from a given hypothesis space H ~ yX then L (A, z) = - VCDim (H) is a w- smallluckiness function where ( 1) (2em) -Lo w Lo,l,m,8, 2m :S -Lo . (7) This can easily be seen by noticing that the latter term is an upper bound on maxzEZ2", I{ (l (h (Xl) ,yI) , ... ,l (h (X2m), Y2m)) : h E H}I (see also [9]). Note that this luckiness function neither exploits the particular training sample observed nor the learning algorithm used. Luckiness Framework Firstly, the luckiness framework of Shawe-Taylor et al. [8] only considered binary loss functions l and the zero training error case. In this work, the luckiness £ is a function of hypothesis and training samples and is called wsmall if the probability over the random draw of a 2m sample z that there exists a hypothesis h with w(£(h, (Zl, ... ,zm)), 8) < J'--h (2;" {(X, y) t--+ l (g (x) ,y) 1£ (g, z) ::::: £ (h, Z)}, z), is smaller than 8. Although similar in spirit, the classical luckiness framework does not allow exploitation of the learning algorithm used to the same extent as our new luckiness. In fact, in this framework not only the covering number must be estimable but also the variation of the luckiness £ itself. These differences make it very difficult to formally relate the two frameworks. Compression Framework In the compression framework of Littlestone and Warmuth [6] one considers learning algorithms A which are compression schemes, i.e. A (z) = :R (e (z)) where e (z) selects a subsample z ~ z and :R : Z(oo) -+ yX is a permutation invariant reconstruction function. For this class of learning algorithms, the luckiness L(A,z) = -le(z)1 is w- small where w is given by (7). In order to see this we note that (3) ensures that we only consider permutations 7ri where e (IIi (z)) :S Ie (z)l, i.e. we use not more than -L training examples from z E z2m. As there are exactly e;;) distinct choices of d training examples from 2m examples the result follows by application of Sauer's lemma [9]. Disregarding constants, Theorem 1 gives exactly the same bound as in [6]. 4 A New Margin Bound For Support Vector Machines In this section we study the maximum margin algorithm for linear classifiers, i.e. A : Z(oo) -+ Hcp where Hcp := {x t--+ (¢ (x), w) I wE }C} and ¢ : X -+ }C ~ £~ is known as the feature mapping. Let us assume that l (h (x) ,y) = lO-l (h (x) ,y) := lIyh(x)::;o, Classical VC generalisation error bounds exploit the fact that VCDim (Hcp) = nand (7). In the luckiness framework of Shawe-Taylor et al. [8] it has been shown that we can use fat1i.p h'z (w)) :S h'z (W))-2 (at the price of an extra 10g2 (32m) factor) in place of VCDim (Hcp) where "(z (w) = min(xi,Yi)Ez Yi (¢ (Xi) , w) / Ilwll is known as the margin. Now, the maximum margin algorithm finds the weight vector WMM that maximises "(z (w). It is known that WMM can be written as a linear combination of the ¢ (Xi). For notational convenience, we shall assume that A: Z(oo) -+ 1R(00) maps to the expansion coefficients 0: such that Ilwall = 1 where Wa := 2:1~ 1 (XicfJ(Xi). Our new margin bound follows from the following theorem together with (6). Theorem 2. Let fi (x) be the smallest 10 > 0 such that {cfJ (Xl) , ... , cfJ (Xm) } can be covered by at most i balls of radius less than or equal f. Let f z (w) be d fi d b f ( ) .. Yi (4)(Xi),W) D th l l d e ne y z W . mm(Xi,Yi)Ez 114>(Xi)II.llwll. ror e zero-one oss 0-1 an the maximum margin algorithm A , the luckiness function L(A ) =_ . {. ",,-T .> (fi (X)2:7=1 IA(Z)jl) 2} ,Z mIn ~ E 1'1 ~ _ ( ) , fz W A(z) (8) is w-small at scale 112m w.r.t. the function ( 1) (2em)- 2LO w Lo,l,m,8, 2m = -Lo (9) Proof (Sketch). First we note that by a slight refinement of a theorem of Makovoz [7] we know that for any Z E zm there exists a weight vector w = 2::1 iiicfJ (Xi) such that (10) and a E ]Rm has no more than - L (A, z) non-zero components. Although only WA(z) is of unit length, one can show that (10) implies that (WA(z), wi IIwll) ~ )1- f; (WA(z»). Using equation (10) of [4] this implies that w correctly classifies Z E zm. Consider a fixed double sample Z E z2m and let ko := L (A, (Zl , ... , zm)). By virtue of (3) and the aforementioned argument we only need to consider permutations tri such that there exists a weight vector w = 2:;:1 iijcfJ (Xj) with no more than ko non-zero iij. As there are exactly (2;;) distinct choices of dE {I, ... , ko} training examples from the 2m examples Z there are no more than (2emlko)kO different subsamples to be used in w. For each particular subsample z ~ Z the weight vector w is a member of the class of linear classifiers in a ko (or less) dimensional space. Thus, from (7) it follows that for the given subsample z there are no more (2emlko)kO different dichotomies induced on the double sample Z E z2m. As this holds for any double sample, the theorem is proven. D There are several interesting features about this margin bound. Firstly, observe that 2:;:1 IA (Z)j I is a measure of sparsity of the solution found by the maximum margin algorithm which, in the present case, is combined with margin. Note that for normalised data, i.e. IlcfJ Oil = constant, the two notion of margins coincide, i.e. f z (w) = I Z (w). Secondly, the quantity fi (x) can be considered as a measure of the distribution of the mapped data points in feature space. Note that for all i E N, fi (x) :S 101 (x) :S maxjE{l, ... ,m} IlcfJ (xj)ll. Supposing that the two classconditional probabilities PX1Y=y are highly clustered, 102 (x) will be very small. An extension of this reasoning is useful in the multi-class case; binary maximum margin classifiers are often used to solve multi-class problems [9]. There appears to be also a close relationship of fi (x) with the notion of kernel alignment recently introduced in [3]. Finally, one can use standard entropy number techniques to bound fi (x) in terms of eigenvalues of the inner product matrix or its centred variants. It is worth mentioning that although our aim was to study the maximum margin algorithm the above theorem actually holds for any algorithm whose solution can be represented as a linear combination of the data points. 5 Conclusions In this paper we have introduced a new theoretical framework to study the generalisation error of learning algorithms. In contrast to previous approaches, we considered specific learning algorithms rather than specific hypothesis spaces. We introduced the notion of algorithmic luckiness which allowed us to devise data dependent generalisation error bounds. Thus we were able to relate the compression framework of Littlestone and Warmuth with the VC framework. Furthermore, we presented a new bound for the maximum margin algorithm which not only exploits the margin but also the distribution of the actual training data in feature space. Perhaps the most appealing feature of our margin based bound is that it naturally combines the three factors considered important for generalisation with linear classifiers: margin, sparsity and the distribution of the data. Further research is concentrated on studying Bayesian algorithms and the relation of algorithmic luckiness to the recent findings for stable learning algorithms [2]. Acknowledgements This work was done while RCW was visiting Microsoft Research Cambridge. This work was also partly supported by the Australian Research Council. RH would like to thank Olivier Bousquet for stimulating discussions. References [1) M. Anthony and P. Bartlett. A Theory of Learning in Artificial Neural Networks. Cambridge University Press, 1999. [2) O. Bousquet and A. Elisseeff. Algorithmic stability and generalization performance. In T. K. Leen, T. G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems 13, pages 196- 202. MIT Press, 2001. [3) N. Cristianini, A. Elisseeff, and J. Shawe-Taylor. On optimizing kernel alignment. Technical Report NC2-TR-2001-087, NeuroCOLT, http://www.neurocolt.com. 2001. [4) R. Herbrich and T . Graepel. A PAC-Bayesian margin bound for linear classifiers: Why SVMs work. In T. K. Leen, T. G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems 13, pages 224- 230, Cambridge, MA, 2001. MIT Press. [5) R. Herbrich and R. C. Williamson. Algorithmic luckiness. Technical report, Microsoft Research, 2002. [6) N. Littlestone and M. Warmuth. Relating data compression and learnability. Technical report, University of California Santa Cruz, 1986. [7) Y. Makovoz. Random approximants and neural networks. Journal of Approximation Theory, 85:98- 109, 1996. [8) J. Shawe-Taylor, P. L. Bartlett, R. C. Williamson, and M. Anthony. Structural risk minimization over data-dependent hierarchies. IEEE Transactions on Information Theory, 44(5):1926- 1940, 1998. [9) V. Vapnik. Statistical Learning Theory. John Wiley and Sons, New York, 1998. [10) V. N. Vapnik and A. Y. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications, 16(2):264- 281, 1971.
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Optimising Synchronisation Times for Mobile Devices Neil D. Lawrence Department of Computer Science, Regent Court, 211 Portobello Road, Sheffield, Sl 4DP, U.K. neil~dcs.shef . ac.uk Antony 1. T. Rowstron Christopher M . Bishop Michael J. Taylor Microsoft Research 7 J. J. Thomson A venue Cambridge, CB3 OFB, U.K. {antr,cmbishop,mitaylor}~microsoft.com Abstract With the increasing number of users of mobile computing devices (e.g. personal digital assistants) and the advent of third generation mobile phones, wireless communications are becoming increasingly important. Many applications rely on the device maintaining a replica of a data-structure which is stored on a server, for example news databases, calendars and e-mail. ill this paper we explore the question of the optimal strategy for synchronising such replicas. We utilise probabilistic models to represent how the data-structures evolve and to model user behaviour. We then formulate objective functions which can be minimised with respect to the synchronisation timings. We demonstrate, using two real world data-sets, that a user can obtain more up-to-date information using our approach. 1 Introduction As the available bandwidth for wireless devices increases, new challenges are presented in the utilisation of such bandwidth. Given that always up connections are generally considered infeasible an important area of research in mobile devices is the development of intelligent strategies for communicating between mobile devices and servers. ill this paper we consider the scenario where we are interested in maintaining, on a personal digital assistant (PDA) with wireless access, an up-to-date replica of some, perhaps disparate, data-structures which are evolving in time. The objective is to make sure our replica is not 'stale'. We will consider a limited number of connections or synchronisations. Each synchronisation involves a reconciliation between the replica on the mobile device and the data-structures of interest on the server. Later in the paper we shall examine two typical examples of such an application,an internet news database and a user's e-mail messages. Currently the typical strategy! for performing such reconciliations is to synchronise every M minutes, lSee, for example, AvantGo http://vvv. avantgo. com. where M is a constant, we will call this strategy the uniformly-spaced strategy. We will make the timings of the synchronisations adaptable by developing a cost function that can be optimised with respect to the timings, thereby improving system performance. 2 Cost Function We wish to minimise the staleness of the replica, where we define staleness as the time between an update of a portion of the data-structure on the server and the time of the synchronisation of that update with the PDA. For simplicity we shall assume that each time the PDA synchronises all the outstanding updates are transferred. Thus, after synchronisation the replica on the mobile device is consistent with the master copy on the server. Therefore, if skis the time of the kth synchronisation in a day, and updates to the data-structure occur at times Uj then the average staleness of the updates transferred during synchronisation Sk will be (1) As well as staleness, we may be interested in optimising other criteria. For example, mobile phone companies may seek to equalise demand across the network by introducing time varying costs for the synchronisations, c(t). Additionally one could argue that there is little point in keeping the replica fresh during periods when the user is unlikely to check his PDA, for example when he or she is sleeping. We might therefore want to minimise the time between the user's examination of the PDA and the last synchronisation. If the user looks at the PDA at times ai then we can express this as (2) Given the timings Uj and ai, the call cost schedule c(t) and K synchronisations, the total cost function may now be written K C = L (-aFk + fJSk + C(Sk)) ' (3) k=l where a and fJ are constants with units of ~~~:y which express the relative importance of the separate parts of the cost function. Unfortunately, of course, whilst we are likely to have knowledge of the call cost schedule, c(t), we won't know the true timings {Uj} and {ai} and the cost function will be a priori incomputable. If, though, we have historic data2 relating to these times, we can seek to make progress by modelling these timings probabilistically. Then, rather than minimising the actual cost function, we can look to minimise the expectation of the cost function under these probabilistic models. 3 Expected Cost There are several different possibilities for our modelling strategy. A sensible assumption is that there is independence between different parts of the data-structure (i.e. e-mail and business news can be modelled separately), however, there may be dependencies between update times which occur within the same part. The 2When modelling user ru:cess times, if historic data is not available, models could also be constructed by querying the user about their likely ru:tivities. periodicity of the data may be something we can take advantage of, but any nonstationarity in the data may cause problems. There are various model classes we could consider; for this work however, we restrict ourselves to stationary models, and ones in which updates arrive independently and in a periodic fashion . . Let us take T to be the largest period of oscillation in the data arrivals, for a particular portion of a data-structure. We model this portion with a probability distribution, Pu(t). Naturally more than one update may occur in that interval, therefore our probability distribution really specifies a distribution over time given one that one update (or user access) has occurred. To fully specify the model we also are required to store the expected number of updates, Ju , (or accesses, Ja ) that occur in that interval. The expected value of Sk may now be written, (4) where Op(:v) is an expectation under the distribution p(x), Au(t) = JuPu(t) can be viewed as the rate at which updates are occurring and So = SK - T. We can model the user access times, ai, in a similar manner, which leads us to the expected value of the freshness, (Fk)Pa(t) = J:k k +l Aa(t)(t - sk)dt, where Aa(t) = JaPa(t) The overall expected cost, which we will utilise as our objective function, may therefore be written K (C) = L (Sk)p" - (Fk)Pa + C(Sk)) . (5) k=l 3.1 Probabilistic Models. We now have an objective function which is a function of the variables we wish to optimise, the synchronisation times, but whilst we have mentioned some characteristics of the models Pu(t) and Pa(t) we have not yet fully specified their form. We have decreed that the models should be periodic and that they may consider each datum to occur independently. In effect we are modelling data which is mapped to a circle. Various options are available for handling such models; for this work, we constrain our investigations to kernel density estimates (KDE). In order to maintain periodicity, we must select a basis function for our KDE which represents a distribution on a circle, one simple way of achieving this aim is to wrap a distribution that is defined along a line to the circle (Mardia, 1972). A traditional density which represents a distribution on the line, p(t), may be wrapped around a circle of circumference T to give us a distribution defined on the circle, p( 0), where 0 = t mod T. This means a basis function with its centre at T - 8, that will typically have probability mass when u > T, wraps around to maintain a continuous density at T. The wrapped Gaussian distribution3 that we make use of takes the form (6) The final kernel density estimate thus consists of mapping the data points tn -t On 3In practice we must approximate the wrapped distribution by restricting the number of terms in the sum. Thu Fri Sat Sun Thu Fri Sat Sun "-' ~60 .§ ~40 "-' .S 1)520 ~ OJ H Co> OJ ""0 ~ -20 Figure 1: Left: part of the KDE developed for the business category together with a piecewise constant approximation. Middle: the same portion of the KDE for the FA Carling Premiership data. Right: percent decrease in staleness vs number of synchronisations per day for e-mail data. and obtaining a distribution 1 N p(()) = N L W N(()I()n, (}"2), n=l (7) where N is the number of data-points and the width parameters, (}", can be set through cross validation. Models of this type may be made use of for both Pu(t) and Pa(t). 3.2 Incorporating Prior Knowledge. The underlying component frequencies of the data will clearly be more complex than simply a weekly or daily basis. Ideally we should be looking to incorporate as much of our prior knowledge about these component frequencies as possible. IT we were modelling financial market's news, for example, we would expect weekdays to have similar characteristics to each other, but differing characteristics from the weekend. For this work, we considered four different scenarios of this type. For the first scenario, we took T = 1 day and placed no other constraints on the model. For the second we considered the longest period to be one week, T = 1 week, and placed no further constraints on the model. For the remaining two though we also considered T to be one week, but we implemented further assumptions about the nature of the data. Firstly we split the data into weekdays and weekends. We then modelled these two categories separately, making sure that we maintained a continuous function for the whole week by wrapping basis functions between weekdays and weekends. Secondly we split the datainto weekdays, Saturdays and Sundays, modelling each category separately and again wrapping basis functions across the days. 3.3 Model Selection. To select the basis function widths, and to determine which periodicity assumption best matched the data, we · utilised ten fold cross validation. For each different periodicity we used cross validation to first select the basis function width. We then compared the average likelihood across the ten validation sets, selecting the periodicity with the highest associated value. 4 Optimising the Synchronisation Times Given that our user model, Pa(t), and our data model, Pu(t) will be a KDE based on wrapped Gaussians, we should be in a position to compute the required integrals '" C/J t{360 t{360 60 >=1 >=1 Q) Q) x <il X ]40 + + + .....,40 X !I 40 j C/J + )I( C/J X ~ C/J .S .S C/J ! i Q) ~ I~! >=1 S520 S520 .$20 ~ ~ ~ Q) Q) ....., !-< !-< C/J U U .S X Q) Q) "0 4 "0 2 2 12 24 X Q) ~ ~ C/J X X ~ X -20 -20 ~20 X X Figure 2: Results from the news database tests. Left: u xx Q) X February /March based models tested on April. Middle: "0 ~40 X X xx March/ April testing on May. Right: April/May testing on X X June. The results are in the form of box plots. The lower xXx line of the box represents the 25th percentile of the data, the -60 upper line the 75th percentile and the central line the median. The 'whiskers' represent the maximum extent of the data up to 1.5 x (75th percentile - 25th percentile). Data which lies outside the whiskers is marked with crosses. in (5) and evaluate our objective function and derivatives thereof. First though, we must give some attention to the target application for the algorithm. A known disadvantage of the standard kernel density estimate is the high storage requirements of the end model. The model requires that N floating point numbers must be stored, where N is the quantity of training data. Secondly, integrating across the cost function results in an objective function which is dependent on a large number of evaluations of the cumulative Gaussian distribution. Given that we envisage that such optimisations could be occurring within a PDA or mobile phone, it would seem prudent to seek a simpler approach to the required minimisation. An alternative approach that we explored is to approximate the given distributions with a functional form which is more amenable to the integration. For example, a piecewise constant approximation to the KDE simplifies the integral considerably. It leads to a piecewise constant approximation for Aa(t) and Au(t). Integration over which simply leads to a piecewise linear function which may be computed in a straightforward manner. Gradients may also be computed. We chose to reduce the optimisation to a series of one-dimensional line minimisations. This can be achieved in the following manner. First, note that the objective function, as a function of a particular synchronisation time Sk, may be written: (8) In other words, each synchronisation is only dependent on that of its neighbours. We may therefore perform the optimisation by visiting each synchronisation time, Sk, in a random order and optimising its position between its neighbours, which involves a one dimensional line minimisation of (8). This process, which is guaranteed to find a (local) minimum in our objective function, may be repeated until convergence. 5 Results In this section we mainly explore the effectiveness of modelling the data-structures of interest. We will briefly touch upon the utility of modelling the cost evolution and user accesses in Section 5.2 but we leave a more detailed exploration of this area to later works. 5.1 Modelling Data Structures To determine the effectiveness of our approach, we utilised two different sources of data: a news web-site and e-mail on a mail server. The news database data-set was collected from the BBC News web site4 . This site maintains a database of articles which are categorised according to subject, for example, UK News, Business News, Motorsport etc .. We had six months of data from February to July 2000 for 24 categories of the database. We modelled the data by decomposing it into the different categories and modelling each separately. This allowed us to explore the periodicity of each category independently. This is a sensible approach given that the nature of the data varies considerably across the categories. . Two extreme examples are Business news and FA Carling Premiership news5, Figure 1. Business news predominantly arrives during the week whereas FA Carling Premiership news arrives typically just after soccer games finish on a Saturday. Business news was best modelled on a Weekday/Weekend basis, and FA Carling Premiership news was best modelled on a Weekday /Saturday /Sunday basis. To evaluate the feasibility of our approach, we selected three consecutive months of data. The inference step consisted of constructing our models on data from the first two months. To restrict our investigations to the nature of the data evolution only, user access frequency was taken to be uniform and cost of connection was considered to be constant. For the decision step we considered 1 to 24 synchronisations a day. The synchronisation times were optimised for each category separately, they were initialised with a uniformly-spaced strategy, optimisation of the timings then proceeded as described in Section 4. The staleness associated with these timings was then computed for the third month. This value was compared with the staleness resulting from the uniformly-spaced strategy containing the same number of synchronisations6 . The percentage decrease in staleness is shown in figures 2 and 3 in the form of box-plots. 60 x x x 40 x x 20 12 -20 X X Xx 00 x x ~XX ~ g}40 ~,(¢<: ~ § ~x x '"@ ++ \xx t260 + )( .,« .S + x x -120 -140 -160 -180 -200 x + x + + + + + + x + + + + + + + + Figure 3: May/June based models tested on July. + signifies the FA Carling Premiership Generally an improvement in performance is observed, Stream. however, we note that in Figure 3 the performance for several categories is extremely 4http://news.bbc.co.uk. 5The FA Carling Premiership is England's premier division soccer. 6The uniformly-spaced strategy's staleness varies with the timing of the first of the K synchronisations. This figure was therefore an average of the staleness from all possible starting points taken at five minute intervals. poor. In particular the FA Carling Premiership stream in Figure 3. The poor performance is caused by the soccer season ending in May. As a result relatively few articles are written in July, most of them concerning player transfer speculation, and the timing of those articles is very different from those in May. In other words the data evolves in a non-stationary manner which we have not modelled. The other poor performers are also sports related categories exhibiting non-stationarities. The e-mail data-set was collected by examining the logs of e-mail arrival times for 9 researchers from Microsoft's Cambridge research lab. This data was collected for January and February 2001. We utilised the January data to build the probabilistic models and the February data to evaluate the average reduction in staleness. Figure 1 shows the results obtained. In practice, a user is more likely to be interested in a combination of different categories of data. Perhaps several different streams of news and his e-mail. Therefore, to recreate a more realistic situation where a user has a combination of interests, we also collected e-mail arrivals for three users from February, March and April 2000. We randomly generated user profiles by sampling, without replacement, five categories from the available twenty-seven, rejecting samples where more than one e-mail stream was selected. We then modelled the users' interests by constructing an unweighted mixture of the five categories and proceeded to optimise the synchronisation times based on this model. This was performed one hundred times. The average staleness for the different numbers of synchronisations per day is shown in Figure 4. Note that the performance for the combined categories is worse than it is for each individually. This is to be expected as the entropy of the combined model will always be greater than that of its constituents, we therefore have less information about arrival times, and as a result there are less gains to be made over the uniformlyspaced strategy7. 5.2 Affect of Cost and User Model In the previous sections we focussed on modelling the evolution of the databases. Here we now briefly turn our attention to the other portions of the system, user behaviour and connection cost. For this preliminary study, it proved difficult to obtain high quality data representing user access times. We therefore artificially generated a model which represents a user who accesses there device frequently at breakfast, lunchtime and during the evening, and rarely at night. Figure 4 simply shows the user model, Pa(t), along with the result of optimising the cost function for uniform data arrivals and fixed cost under this user model. Note how synchronisation times are suggested just before high periods of user activity are about to occur. Also in Figure 4 is the effect of a varying cost, c(t), under uniform Pa(t) and Pa(t). Currently most mobile internet access providers appear to be charging a flat fee for call costs (typically in the U.K. about 15 cents per minute). However, when demand on their systems rise they may wish to incorporate a varying cost to flatten peak demands. This cost could be an actual cost for the user, or alternatively a 'shadow price' specified by service provider for controlling demand (Kelly, 2000). We give a simple example of such a call cost in Figure 4. For this we considered user access and data update rates to be constant. Note how the times move away from periods of high cost. 7The uniformly-spaced strategy can be shown to be optimal when the entropy of the underlying distribution is maximised (a uniform distribution across the interval). "-' gj 60 >=1 Cl) 0.3 7Lo "-' X 0.25 1200 .S ....., 0.2 ~ 900 ~20 0.15 U '" Cl) :::::::: 600 .... 0.1 '" U U Cl) 0.05 300 "'0 ~ X 0 00:00 08:00 00:00 08:00 16:00 00:00 -20 Figure 4: Left: change in synchronisation times for variable user access rates. x shows the initialisation points, + the end points. Middle: change in synchronisation times for a variable cost. Right: performance improvements for the combination of news and e-mail. 6 Discussion The optimisation strategy we suggest could be sensitive to local minima, we did not try a range of different initialisations to explore this phenomena. However, by initialising with the uniformly-spaced strategy we ensured that we increased the objective function relative to the standard strategy. The month of July showed how a non-stationarity in the data structure can dramatically affect our performance. We are currently exploring on-line Bayesian models which we hope will track such non-stationarities. The system we have explored in this work assumed that the data replicated on the mobile device was only modified on the server. A more general problem is that of mutable replicas where the data may be modified on the server or the client. Typical applications of such technology include mobile databases, where sales personnel modify portions of the database whilst on the road, and a calendar application on a PDA, where the user adds appointments on the PDA. Finally there are many other applications of this type of technology beyond mobile devices. Web crawlers need to estimate when pages are modified to maintain a representative cache (eho and Garcia-Molina, 2000). Proxy servers could also be made to intelligent maintain their caches of web-pages up-to-date (Willis and Mikhailov, 1999; Wolman et al., 1999) . References Cho, J. and H. Garcia-Molina (2000). Synchronizing a database to improve freshness. In Proceedings 2000 ACM International Conference on Management of Data (SIGMOD). Kelly, F. P. (2000). Models for a self-managed internet. Philosophical Transactions of the Royal Society A358, 2335-2348. Mardia, K. V. (1972). Statistics of Directional Data. London: Academic Press. Rowstron, A. 1. T., N. D. Lawrence, and C. M. Bishop (2001). Probabilistic modelling of replica divergence. In Proceedings of the 8th Workshop on Hot Topics in Operating Systems HOTOS (VIII). Willis, C. E. and M. Mikhailov (1999). Towards a better understanding of web resources and server responses for improved caching. In Proceedings of the 8th International World Wide Web Conference, pp. 153-165. Wolman, A., G. M. Voelker, N. Sharma, N. Cardwell, A. Karlin, and H. M. Levy (1999). On the scale and performance of co-operative web proxy caching. In 17th ACM Symposium Operating System Principles (SOSP'99), pp. 16-3l. Yu, H. and A. Vahdat (2000). Design and evaluation of a continuous consistency model for replicated services. 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Spike timing and the coding of naturalistic sounds in a central auditory area of songbirds Brian D. Wright, Kamal Sen, William Bialek and Allison J. Doupe Sloan–Swartz Center for Theoretical Neurobiology Departments of Physiology and Psychiatry University of California at San Francisco, San Francisco, California 94143–0444 NEC Research Institute, 4 Independence Way, Princeton, New Jersey 08540 Department of Physics, Princeton University, Princeton, New Jersey 08544 bdwright/kamal/ajd @phy.ucsf.edu, wbialek@princeton.edu Abstract In nature, animals encounter high dimensional sensory stimuli that have complex statistical and dynamical structure. Attempts to study the neural coding of these natural signals face challenges both in the selection of the signal ensemble and in the analysis of the resulting neural responses. For zebra finches, naturalistic stimuli can be defined as sounds that they encounter in a colony of conspecific birds. We assembled an ensemble of these sounds by recording groups of 10-40 zebra finches, and then analyzed the response of single neurons in the songbird central auditory area (field L) to continuous playback of long segments from this ensemble. Following methods developed in the fly visual system, we measured the information that spike trains provide about the acoustic stimulus without any assumptions about which features of the stimulus are relevant. Preliminary results indicate that large amounts of information are carried by spike timing, with roughly half of the information accessible only at time resolutions better than 10 ms; additional information is still being revealed as time resolution is improved to 2 ms. Information can be decomposed into that carried by the locking of individual spikes to the stimulus (or modulations of spike rate) vs. that carried by timing in spike patterns. Initial results show that in field L, temporal patterns give at least
% extra information. Thus, single central auditory neurons can provide an informative representation of naturalistic sounds, in which spike timing may play a significant role. 1 Introduction Nearly fifty years ago, Barlow [1] and Attneave [2] suggested that the brain may construct a neural code that provides an efficient representation for the sensory stimuli that occur in the natural world. Slightly earlier, MacKay and McCulloch [3] emphasized that neurons that could make use of spike timing—rather than a coarser “rate code”—would have available a vastly larger capacity to convey information, although they left open the question of whether this capacity is used efficiently. Theories for timing codes and efficient representation have been discussed extensively, but the evidence for these attractive ideas remains tenuous. A real attack on these issues requires (at least) that we actually measure the information content and efficiency of the neural code under stimulus conditions that approximate the natural ones. In practice, constructing an ensemble of “natural” stimuli inevitably involves compromises, and the responses to such complex dynamic signals can be very difficult to analyze. At present the clearest evidence on efficiency and timing in the coding of naturalistic stimuli comes from central invertebrate neurons [4, 5] and from the sensory periphery [6, 7] and thalamus [8, 9] of vertebrates. The situation for central vertebrate brain areas is much less clear. Here we use the songbird auditory system as an accessible test case for these ideas. The set of songbird telencephalic auditory areas known as the field L complex is analogous to mammalian auditory cortex and contains neurons that are strongly driven by natural sounds, including the songs of birds of the same species (conspecifics) [10, 11, 12, 13]. We record from the zebra finch field L, using naturalistic stimuli that consist of recordings from groups of 10-40 conspecific birds. We find that single neurons in field L show robust and well modulated responses to playback of long segments from this song ensemble, and that we are able to maintain recordings of sufficient stability to collect the large data sets that are required for a model independent information theoretic analysis. Here we give a preliminary account of our experiments. 2 A naturalistic ensemble Auditory processing of complex sounds is critical for perception and communication in many species, including humans, but surprisingly little is known about how high level brain areas accomplish this task. Songbirds provide a useful model for tackling this issue, because each bird within a species produces a complex individualized acoustic signal known as a song, which reflects some innate information about the species’ song as well as information learned from a “tutor” in early life. In addition to learning their own song, birds use the acoustic information in songs of others to identify mates and group members, to discriminate neighbors from intruders, and to control their living space [14]. Consistent with how ethologically critical these functions are, songbirds have a large number of forebrain auditory areas with strong and increasingly specialized responses to songs [11, 15, 16]. The combination of a rich set of behaviorally relevant stimuli and a series of high-level auditory areas responsive to those sounds provides an opportunity to reveal general principles of central neural encoding of complex sensory stimuli. Many prior studies have chosen to study neural responses to individual songs or altered versions thereof. In order to make the sounds studied increasingly complex and natural, we have made recordings of the sounds encountered by birds in our colony of zebra finches. To generate the sound ensemble that was used in this study we first created long records of the vocalizations of groups of 10-40 zebra finches in a soundproof acoustic chamber with a directional microphone above the bird cages. The group of birds generated a wide variety of vocalizations including songs and a variety of different types of calls. Segments of these sounds were then joined to create the sounds presented in the experiment. One of the segments that was presented (
sec) was repeated in alternation with different segments. We recorded the neural responses in field L of one of the birds from the group to the ensemble of natural sounds played back through a speaker, at an intensity approximately equal to that in the colony recording. This bird was lightly anesthetized with urethane. We used a single electrode to record the neural response waveforms and sorted single units offline. Further details concerning experimental techniques can be found in Ref. [13]. 50 Hz 500 ms A B C D Figure 1: A. Spike raster of 4 seconds of the responses of a single neuron in field L to a 30 second segment of a natural sound ensemble of zebra finch sounds. The stimulus was repeated 80 times. B. Peri-stimulus time histogram (PSTH) with 1 ms bins. C. Sound pressure waveform for the natural sound ensemble. D. Blowup of segment shown in the box in A. The scale bar is 50 ms. 3 Information in spike sequences The auditory telencephalon of birds consists of a set of areas known as the field L complex, which receive input from the auditory thalamus and project to increasingly selective auditory areas such as NCM, cHV and NIf [12, 17] and ultimately to the brain areas specialized for the bird’s own song. Field L neurons respond to simple stimuli such as tone bursts, and are organized in a roughly tonotopic fashion [18], but also respond robustly to many complex sounds, including songs. Figure 1 shows 4 seconds of the responses of a cell in field L to repeated presentations of a 30 sec segment from the natural ensemble described above. Averaging over presentations, we see that spike rates are well modulated. Looking at the responses on a finer time resolution we see that aspects of the spike train are reproducible on at least a
ms time scale. This encourages us to measure the information content of these responses over a range of time scales, down to millisecond resolution. Our approach to estimating the information content of spike trains follows Ref. [4]. At some time (defined relative to the repeating stimulus) we open a window of size to look at the response. Within this window we discretize the spike arrival times with resolution so that the response becomes a “word” with letters. If the time resolution is very small, the allowed letters are only 1 and 0, but as becomes larger one must keep track of multiple spikes within each bin. Examining the whole experiment, we sample 0 0.01 0.02 0.03 0.04 0.05 0.06 0 5 10 15 20 25 30 35 40 1/Nrepeats Information Rate (bits/sec) Total Entropy Noise Entropy Mutual Info Figure 2: Mutual information rate for the spike train is shown as a function of data size for ms and ms. the probability distribution of words, , and the entropy of this distribution sets the capacity of the code to convey information about the stimulus:
!" #%$'&)(+* (1) where the notation reminds us that the entropy depends both on the size of the words that we consider and on the time resolution with which we classify the responses. We can think of this entropy as measuring the size of the neuron’s vocabulary. Because the whole experiment contributes to defining the vocabulary size, estimating the distribution , and hence the total entropy is not significantly limited by the problems of finite sample size. This can be seen in Fig. 2 in the stability of the total entropy with changing the number of repeats used in the analysis. Here we show the total entropy as a rate in bits per second by dividing the entropy by the time window . While the capacity of the code is limited by the total entropy, to convey information particular words in the vocabulary must be associated, more or less reliably, with particular stimulus features. If we look at one time relative to the (long) stimulus, and examine the words generated on repeated presentations, we sample the conditional distribution -/. . This distribution has an entropy that quantifies the noise in the response at time , and averaging over all times we obtain the average noise entropy, 10 )2 34 5 7689 "/. : ' ; "/. =<,>?#%$'&)(+* (2) where > indicates a time average (in general, denotes an average over the variable ). Technically, the above average should be an average over stimuli , however, for a sufficiently long and rich stimulus, the ensemble average over can be replaced by a time average. For the noise entropy, the problem of sampling is much more severe, since each distribution /. is estimated from a number of examples given by the number of repeats. Still, as shown in Fig. 2, we find that the dependence of our estimate on sample size is simple and regular; specifically, we find 5
4 4 3 5 4 4 3 (3) This is what we expect for any entropy estimate if the distribution is well sampled, and if we make stronger assumptions about the sampling process (independence of trials etc.) we can even estimate the correction coefficient [19]. In systems where much larger data sets are available this extrapolation procedure has been checked, and the observation of a good fit to Eq. (3) is a strong indication that larger sample sizes will be consistent with 5 ; further, this extrapolation can be tested against bounds on the entropy that are derived from more robust quantities [4]. Most importantly, failure to observe Eq. (3) means that we are in a regime where sampling is not sufficient to draw reliable conclusions without more sophisticated arguments, and we exclude these regions of and from our discussion. Ideally, to measure the spike train total and noise entropy rates, we want to go to the limit of infinite word duration. A true entropy is extensive, which here means that it grows linearly with spike train word duration , so that the entropy rate is constant. For finite word duration however, words sampled at neighboring times will have correlations between them due, in part, to correlations in the stimulus (for birdsong these stimulus autocorrelation time scales can extend up to
ms). Since the word samples are not completely independent, the raw entropy rate is an overestimate of the true entropy rate. The effect is larger for smaller word duration and the leading dependence of the raw estimate is 5 * (4) where and we have already taken the infinite data size limit. We cannot directly take the large limit, since for large word lengths we eventually reach a data sampling limit beyond which we are unable to reliably compute the word distributions. On the other hand, if there is a range of for which the distributions are sufficiently well sampled, the behavior in Eq. (4) should be observed and can be used to extrapolate to infinite word size [4]. We have checked that our data shows this behavior and that it sets in for word sizes below the limit where the data sampling problem occurs. For example, in the case of the noise entropy, for ms, it applies for below the limit of ms (above this we run into sampling problems). The total entropy estimate is nearly perfectly extensive. Finally, we combine estimates of total and noise entropies to obtain the information that words carry about the sensory stimulus,
5 0 2 3 4 5 #%$'&)( (5) Figure 2 shows the total and noise entropy rates as well as the mutual information rate for a time window ms and time resolution ms. The error bars on the raw entropy and information rates were estimated to be approximately ! bits/sec using a simple bootstrap procedure over the repeated trials. The extrapolation to infinite data size is shown for the mutual information rate estimate (error bars in the extrapolated values will be "#! bits/sec) and is consistent with the prediction of Eq. (3). Since the total entropy is nearly extensive and the noise entropy rate decreases with word duration due to subextensive corrections as described above, the mutual information rate shown in Fig. 2 grows with word duration. We find that there is an upward change in the mutual information 0 5 10 15 20 25 30 35 1.5 2 2.5 3 3.5 4 4.5 5 ∆τ (ms) Information Rate (bits/sec) Independent Events Spike Train Figure 3: Information rates for the spike train ( ms) and single spike events as a function of time resolution of the spike rasters, corrected for finite data size effects. rate (computed at ms and ms) of
%, in the large limit. For simplicity in the following, we shall look at a fixed word duration ms that is in the well-sampled region for all time resolutions considered. The mutual information rate measures the rate at which the spike train removes uncertainty about the stimulus. However, the mutual information estimate does not depend on identifying either the relevant features of the stimulus or the relevant features of the response, which is crucial in analyzing the response to such complex stimuli. In this sense, our estimates of information transmission and efficiency are independent of any model for the code, and provide a benchmark against which such models could be tested. One way to look at the information results is to fix our time window and ask what happens as we change our time resolution . When , the “word” describing the response is nothing but the number of spikes in the window, so we have a rate or counting code. As we decrease , we gradually distinguish more and more detail in the arrangement of spikes in the window. We chose a range of values from ms in our analyses to cover previously observed response windows for field L neurons and to probe the behaviorally relevant time scale (
ms) of individual song syllables or notes. For ms, we show the results (extrapolated to infinite data size) in the upper curve of Fig. 3. The spike train mutual information shows a clear increase as the timing resolution is improved. In addition, Fig. 3 shows that roughly half of the information is accessible at time resolutions better than ms and additional information is still being revealed as time resolution is improved to 2 ms. 4 Information in rate modulation Knowing the mutual information between the stimulus and the spike train (defined in the window ), we would like to ask whether this can be accounted for by the information in single spike events or whether there is some additional information conveyed by the patterns of spikes. In the latter case, we have precisely what we mean by a temporal or timing code: there is information beyond that attributable to the probability of single spike events occurring at time relative to the onset of the stimulus. By event at time , we mean that the event occurs between time and time , where is the resolution at which we are looking at the spike train. This probability is simply proportional to the firing rate (or peri-stimulus time histogram (PSTH)) at time normalized by the mean firing rate . Specifically if the duration of each repeated trial is 4 4 we have spk @ . 4 4 * (6) where denotes the stimulus history ( " ). The probability of a spike event at , a priori of knowing the stimulus history, is flat: spk @ 4 4 . Thus, the mutual information between the stimulus and the single spike events is [20]:
(%$ spk @ spk @ .
6 <-> #%$'&)(+* (7) where is the PSTH binned to resolution and the stimulus average in the first expression is replaced by a time average in the second (as discussed in the calculation of the noise entropy in spike train words in the previous section). We find that this information is approximately bit for ms. Supposing that the individual spike events are independent (i.e. no intrinsic spike train correlations), the information rate in single spike events is obtained by multiplying the mutual information per spike (Eq. 7) by the mean firing rate of the neuron (
Hz). This gives an upper bound to the single spike event contribution to the information rate and is shown in the lower curve of Fig. 3 (error bars are again " ! bits/sec). Comparing with the spike train information (upper curve), we see that at a resolution of 8 ms, there is at least
% of the total information in the spike train that cannot be attributable to single spike events. Thus there is some pattern of spikes that is contributing synergistically to the mutual information. The fact discussed, in the previous section, that the spike train information rate grows subextensively with the the word duration out to the point where data sampling becomes problematic is further confirmation of the synergy from spike patterns. Thus we have shown model-independent evidence for a temporal code in the neural responses. 5 Conclusion Until now, few experiments on neural responses in high level, central vertebrate brain areas have measured the information that these responses provide about dynamic, naturalistic sensory signals. As emphasized in earlier work on invertebrate systems, information theoretic approaches have the advantage that they require no assumptions about the features of the stimulus to which neurons respond. Using this method in the songbird auditory forebrain, we found that patterns of spikes seem to be special events in the neural code of these neurons, since they carry more information than expected by adding up the contributions of individual spikes. It remains to be determined what these spike patterns are, what stimulus features they may encode, and what mechanisms may be responsible for reading such codes at even higher levels of processing. Acknowledgments Work at UCSF was supported by grants from the NIH (NS34835) and the Sloan-Swartz Center for Theoretical Neurobiology. BDW and KS supported by NRSA grants from the NIDCD. We thank Katrin Schenk and Robert Liu for useful discussions. References 1. Barlow, H.B. (1961). Possible principles underlying the transformation of sensory messages. In Sensory Communication, W.A. Rosenblith, ed., pp. 217–234 (MIT Press, Cambridge, MA). 2. Attneave, F. (1954). Some informational aspects of visual perception. Psychol. Rev. 61, 183–193. 3. MacKay, D. and McCulloch, W.S. (1952). The limiting information capacity of a neuronal link. Bull. Math. Biophys. 14, 127–135. 4. Strong, S.P., Koberle, R., de Ruyter van Steveninck, R. and Bialek, W. (1998). Entropy and information in neural spike trains, Phys. Rev. Lett. 80, 197–200. 5. Lewen, G.D., Bialek, W. and de Ruyter van Steveninck, R.R. (2001). Neural coding of naturalistic motion stimuli. Network 12, 317–329. 6. Rieke, F., Bodnar, D.A. and Bialek, W. (1995). Naturalistic stimuli increase the rate and efficiency of information transmission by primary auditory afferents. Proc. R. Soc. Lond. B 262, 259–265. 7. Berry II, M.J., Warland, D.K. and Meister, M. (1997). The structure and precision of retinal spike trains. Proc. Nat. Acad. Sci. (USA) 94, 5411–5416. 8. Reinagel, P. and Reid, R.C. (2000). Temporal coding of visual information in the thalamus. J. Neurosci. 20, 5392–5400. 9. Liu, R.C., Tzonev, S., Rebrik, S. and Miller, K.D. (2001). Variability and information in a neural code of the cat lateral geniculate nucleus. J. Neurophysiol. 86, 2789–2806. 10. Scheich, H., Langner, G. and Bonke, D. (1979). Responsiveness of units in the auditory neostriatum of the guinea fowl (Numida meleagris) to species-specific calls and synthetic stimuli II. Discrimination of Iambus-Like Calls. J. Comp. Physiol. A 132, 257–276. 11. Lewicki, M.S. and Arthur, B.J. (1996). Hierarchical organization of auditory temporal context sensitivity. J. Neurosci. 16(21), 6987–6998. 12. Janata, P. and Margoliash, D. (1999). Gradual emergence of song selectivity in sensorimotor structures of the male zebra finch song system. J. Neurosci. 19(12), 5108–5118. 13. Theunissen, F.E., Sen, K. and Doupe, A.J. (2000). Spectral temporal receptive fields of nonlinear auditory neurons obtained using natural sounds. J. Neurosci. 20(6), 2315–2331. 14. Searcy, W.A. and Nowicki, S. (1999). In The Design of Animal Communication, M.D. Hauser and M. Konishi, eds., pp. 577–595 (MIT Press, Cambridge, MA). 15. Margoliash, D. (1983). Acoustic parameters underlying the responses of song-specific neurons in the white-crowned sparrow. J. Neurosci. 3(5), 1039–1057. 16. Sen, K., Theunissen, F.E. and Doupe, A.J. (2001). Feature analysis of natural sounds in the songbird auditory forebrain. J. Neurophysiol. 86, 1445–1458. 17. Stripling, R., Kruse, A.A. and Clayton, D.F. (2001). Development of song responses in the zebra finch caudomedial neostriatum: role of genomic and electrophysiological activities. J. Neurobiol. 48, 163–180. 18. Zaretsky, M.D. and Konishi, M. (1976). Tonotopic organization in the avian telencephalon. Brain Res. 111, 167–171. 19. Treves, A. and Panzeri, S. (1995). The upward bias in measures of information derived from limited data samples. Neural Comput., 7, 399–407. 20. Brenner, N., Strong, S., Koberle, R. and Bialek, W. (2000). Synergy in a neural code, Neural Comput. 12, 1531–1552.
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A Variational Approach to Learning Curves D¨orthe Malzahn Manfred Opper Neural Computing Research Group School of Engineering and Applied Science Aston University, Birmingham B4 7ET, United Kingdom. [malzahnd,opperm]@aston.ac.uk Abstract We combine the replica approach from statistical physics with a variational approach to analyze learning curves analytically. We apply the method to Gaussian process regression. As a main result we derive approximative relations between empirical error measures, the generalization error and the posterior variance. 1 Introduction Approximate expressions for generalization errors for finite dimensional statistical data models can be often obtained in the large data limit using asymptotic expansions. Such methods can yield approximate relations for empirical and true errors which can be used to assess the quality of the trained model see e.g. [1]. Unfortunately, such an approximation scheme does not seem to be easily applicable to popular non-parametric models like Gaussian process (GP) models and Support Vector Machines (SVMs). We apply the replica approach of statistical physics to asses the average case learning performance of these kernel machines. So far, the tools of statistical physics have been successfully applied to a variety of learning problems [2]. However, this elegant method suffers from the drawback that data averages can be performed exactly only under very idealistic assumptions on the data distribution in the ”thermodynamic” limit of infinite data space dimension. We try to overcome these limitations by combining the replica method with a variational approximation. For Bayesian models, our method allows us to express useful data averaged a-posteriori expectations by means of an approximate measure. The derivation of this measure requires no assumptions about the data density and no assumptions about the input dimension. The main focus of this article is Gaussian process regression where we demonstrate the various strengths of the presented method. It solves some of the problems stated at the end of our previous NIPS paper [3] which was based on a simpler somewhat unmotivated truncation of a cumulant expansion. For Gaussian process models we show that our method does not only give explicit approximations for generalization errors but also of their sample fluctuations. Furthermore, we show how to compute corrections to our theory and demonstrate the possibility of deriving approximate universal relations between average empirical and true errors which might be of practical interest. An earlier version of our approach, which was still restricted to the assumption of idealized data distributions appeared in [4]. 2 Setup and Notation We assume that a class of elementary predictors (neural networks, regressors etc.) is given by functions . In a Bayesian formulation, we have a prior distribution over this class of functions . Assuming that a set of observations
is conditionally independent given inputs
, we assign a likelihood term of the form
to each observation. Posterior expectations (denoted by angular brackets) of any functional !#" $ are expressed in the form % !#" $ &(' ) * +-,.!#" $
, / 10 2 3 5464 (1) where the partition function * normalizes the posterior and + denotes the expectation with respect to the prior. We are interested in computing averages 7 % !#" $ &58 9 of posterior expectations over different drawings of training data sets : ' " ;
# <$ were all data examples are independently generated from the same distribution. In the next section we will show how to derive a measure which enables us to compute analytically approximate combined data and posterior averages. 3 A Grand-Canonical Approach We utilize the statistical mechanics approach to the analysis of learning. Our aim is to compute the so-called averaged ”free energy” 7 >=1? * 8@9 which serves as a generating function for suitable data averages of posterior expectations. The partition function * is * ' + , / 10 . 3 4 (2) To perform the average 7 =A? * 8B9 we use the replica trick 7 =1? * 8B9C'C=AD1EGFHJILK.M NPO Q.R SUTWV K F , where 7 * F 8B9 is computed for integer X and the continuation is performed at the end [5]. We obtain * FYZ[ ' 7 * F 8B9\' + F^]_G` , F / a 0 a . 54Jbc1de fg hi (3) where + F denotes the expectation over the replicated prior measure. Eq.(3) can be transformed into a simpler form by introducing the ”grand canonical” partition function j F kl j F kl 'nm / 0YIpo q Zsr * F Z>t' + F vu F (4) with the Hamiltonian uwFx'y o q ` , F / a 0 l a z4Jbc1de fg (5) The density o|{} R evaluates all X replicas
~F of the predictor at the same data point and the expectation 7W
8 c1de fg is taken with respect to the true data density . The ”grand canonical” partition function j Fkl represents a ”poissonized” version of the original model with fluctuating number of examples. The ”chemical potential” k determines the expected value of Z' K.M N R c q g K q which yields simply k'
=A?Z for X . For sufficiently large Z , we can replace the sum in Eq. (4) by its dominating term =A? * FY Z>t=A? j FYkllZs =1?Z ) UZk (6) thereby neglecting relative fluctuations. We recover the original (canonical) free energy as K M NQ R c g K F K6M N R c M N g K F . 4 Variational Approximation For most interesting cases, the partition function j FYkl can not be computed in closed form for given X . Hence, we use a variational approach to approximate u F by a different tractable Hamiltonian u I F . It is easy to write down the first terms in an expansion of the ”grand canonical” free energy with respect to the difference u F u I F >=A? j FYkl ' >=1? + F o {} R % uwFYu I F &5I ) % u#Fu I F &5I; % uwFYu I F & I
(7) The brackets %
&zI denote averages with respect to the effective measure which is induced by the prior and o;{.} R and acts in the space of replicated variables. As is well known, the first two leading terms in Eq.(7) present an upper bound [6] to >=A? j FYkl . Although differentiating the bound with respect to X will usually not preserve the inequality, we still expect 1 that an optimization with respect to u I F is a sensible thing to do [7]. 4.1 Variational Equations The grand-canonical ensemble was chosen such that Eq.(5) can be rewritten as an integral over a local quantity in the input variable , i.e. in the form u F ' Z
| " a $ with s " a .<$ '
, F / a 0 l a z4 (8) We will now specialize to Gaussian priors over , for which a local quadratic expression u I F '
| / a ) a . a .l / a a a . (9) is a suitable trial Hamiltonian, leading to Gaussian averages %
& I . The functions a . and a . are variational parameters to be optimized. It is important to have an explicit dependence on the input variable in order to take a non uniform input density into account. To perform the variation of the first two terms in Eq.(7) we note that the locality of Eq.(8) makes the ”variational free energy” >=A? + F zvu I F % uwF u I F &zI an explicit function of the first two local moments a . ' % a . .&5I a . ' % a &5I (10) Hence, a straightforward variation yields Z
% s &zI
a . ' a Z
% . &zI
a . ' a (11) To extend the variational solutions to non-integer values of X , we assume that for all the optimal parameters are replica symmetric, ie. a ' . as well as a .' . for ' and aa ' I| . We also use a corresponding notation for a . and a . 1Guided by the success of the method in physical applications, for instance in polymer physics. 4.2 Interpretation of u I F Note, that our approach is not equivalent to a variational approximation of the original posterior. In contrast, uwI contains the full information of the statistics of the training data. We can use the distribution induced by the prior and o {.} R in order to compute approximate combined data and posterior averages. As an example, we first consider the expected local posterior variance . ' 7 % .& % .& 8 9 . Following the algebra of the replica method (see [5]) this is approximated within the variational replica approach as ' =AD1E F|H I % a & I % a . & I ' I .t . (12) Second, we consider the noisy local mean square prediction error of the posterior mean predictor ' % & which is given by ' 7 .t 8@9 . In this case ' =AD1E FH I % a . &5I % a .&5I ' U
( (13) We can also calculate fluctuations with respect to the data average, for example 7 .t | B @8 9 ' =AD1E FH I a 0 e a ( 0 e B @
I (14) 5 Regression with Gaussian Processes This statistical model assumes that data are generated as ' 2 3
, where is Gaussian white noise with variance { . The prior over functions has zero mean and covariance t' + 7 . 58 . Hence, we have l t' . . Using the definitions Eqs.(12,13), we get % " a .<$ &5I X 'y . =1? ) | .
. { (15) which yields the set of variational equations (11). They become particularly easy when the regression model uses a translationally invariant kernel and the input distribution is homogeneous in a finite interval. The variational equations (11) can then be solved in terms of the eigenvalues of the Gaussian process kernel. [8, 9] studied learning curves for Gaussian process regression which are not only averaged over the data but also over the data generating process using a Gaussian process prior on . Applying these averages to our theory and adapting the notation of [9] simply replaces in Eq.(15) the term
. by . while . . . 5.1 Learning Curves and Fluctuations Practical situations differ from this ”typical case” analysis. The data generating process is unknown but assumed to be fixed. The resulting learning curve is then conditioned on this particular ”teacher” . The left panel of Fig.1 shows an example. Displayed are the mean square prediction error (circle and solid line) and its sample fluctuations (error bars) with respect to the data average (cross and broken line). The target was a random but fixed realization from a Gaussian process prior with a periodic Radial Basis Function kernel p' "! z $# &% ' , ' ' ) . We keep the example simple, e.g the Gaussian process regression model used the same kernel and noise { ' ' ) . The inputs are one dimensional, independent and uniformly distributed )( 7 ) 8 . Symbols represent simulation data. A typical property of our theory (lines) is that it becomes very accurate for sufficiently large number of example data. 0 50 100 150 200 Number m of Example Data 10 −4 10 −3 10 −2 10 −1 10 0 Generalization Error ε, Fluctuation ∆ε ε ∆ε Theory: Lines Simulation: Symbols 0 20 40 60 80 100 Number m of Example Data −4 −3 −2 −1 0 Correction of Free Energy β −1=0.25 β −1=0.01 β −1=0.0001 Figure 1: Gaussian process regression using a periodic Radial Basis Function kernel, input dimension d=1, ( 7 ) 8 , and homogeneous input density. Left: Generalization error and fluctuations for data noise ' { ' ) . Right: Correction of the free energy. Symbols: We subtracted the first two contributions to Eq.(7) from the true value of the free energy. The latter was obtained by simulations. Lines show the third contribution of Eq.(7). The value of the noise variance { decreases from top to bottom. All y-data was set equal to zero. 5.2 Corrections to the Variational Approximation It is a strength of our method that the quality of the variational approximation Eq.(7) can be characterized and systematically improved. In this paper, we restrict ourself to a characterization and consider the case where all -data is set equal to zero. Since the posterior variance is independent of the data this is still an interesting model from which the posterior variance can be estimated. We consider the third term in the expansion to the free energy Eq.(7). It is a correction to the variational free energy and evaluates to =ADAE FH I X ) % u F u I F & I % u F u I F & I 'y[) 7 I . I B ; B58 d e d Z ` =A? , ) l^ { 4Jb d e d Z ` I { b d e d (16) with ' =1DAE FH I % a . a p a . & I . Eq.(16) is shown by lines in the right panel of Fig.1 for different values of the model noise { . We considered a homogeneous input density, the input dimension is one and the regression model uses a periodic RBF kernel. The symbols in Fig.1 show the difference between the true value of the free energy which is obtained by simulations and the first two terms of Eq.(7). The correction term is found to be qualitatively accurate and emphasizes a discrepancy between free energy and the first two terms of the expansion Eq.(7) for a medium amount of example data. The calculated learning curves inherit this behaviour. 5.3 Universal Relations We can relate the training error and the empirical posterior variance ' ) Z ` / A0 b 9 ' ) Z ` / 10 ; b 9 (17) 0 0.2 0.4 0.6 0.8 1 βσT 2 0 0.2 0.4 0.6 0.8 1 [βσ 2(x)/(βσ 2(x)+1)]x Theory d=1, periodic d=2, periodic d=3, periodic d=2+2, non-periodic 0 0.2 0.4 0.6 0.8 1 βεT 0 0.2 0.4 0.6 0.8 1 [βε(x,y)/(βσ 2(x)+1) 2](x,y) Theory 1d, periodic 2d, periodic 3d, periodic Figure 2: Illustration of relation Eq.(19) (left) and Eq.(20) (right). All error measures are scaled with . Symbols show simulation results for Radial Basis Function (RBF) regression and a homogeneous input distribution in
x' ) dimensions (square, circle, diamond). The RBF kernel was periodic. Additionally, the left figure shows an example were the inputs lie on a quasi two-dimensional manifold which is embedded in
' dimensions (cross). In this case the RBF kernel was non-periodic. to the free energy 7 =A? * 8 9 ' . Using Eqs.(6,7) and the stationarity of the grand-canonical free energy with respect to the variational parameters we obtain the following relation
| 7 >=1? * 8B9 yZ
| % " a $ &5I X (18) We use the fact that the posterior variance is independent of the -data and simply estimate it from the model where all -data is set equal to zero. In this case, Eq.(18) yields '
) ^ . (19) which relates the empirical posterior variance to the local posterior variance . at test inputs . Similarly, we can derive an expression for the training error by using Eqs.(15,18) in combination with Eq.(19) '
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) . (20) It is interesting to note, that the relations (19,20) contain no assumptions about the data generating process. They hold in general for Gaussian process models with a Gaussian likelihood. An illustration of Eqs.(19,20) is given by Fig.2 for the example of Gaussian process regression with a Radial Basis Function kernel. In the left panel of Fig.2, learning starts in the upper right corner as the rescaled empirical posterior variance is initially one and decreases with increasing number of example data. For the right panel of Fig.2, learning starts in the lower left corner. The rescaled training error on the noisy data set is initially zero and increases to one with increasing number of example data. The theory (line) holds for a sufficiently large number of example data and its accuracy increases with the input dimension. Eqs.(19,20) can also be tested on real data. For common benchmark sets such as Abalone and Boston Housing data we find that Eqs.(19,20) hold well even for small and medium sizes of the training data set. 6 Outlook One may question if our approximate universal relations are of any practical use as, for example, the relation between training error and generalization error involves also the unknown posterior variance . . Nevertheless, this relation could be useful for cases, where a large number of data inputs without output labels are available. Since for regression, the posterior variance is independent of the output labels, we could use these extra input points to estimate . The application of our technique to more complicated models is possible and technically more involved. For example, replacing o { by # .. ) in Eq.(1) and further rescaling the kernel t' % of the Gaussian process prior gives a model for hard margin Support Vector Machine Classification with SVM kernel . The condition of maximum margin classification will be ensured by the limes . Of particular interest is the computation of empirical estimators that can be used in practice for model selection as well as the calculation of fluctuations (error bars) for such estimators. A prominent example is an efficient approximate leave-one-out estimator for SVMs. Work on these issues is in progress. Acknowledgement We would like to thank Peter Sollich for may inspiring discussions. The work was supported by EPSRC grant GR/M81601. References [1] N. Murata, S. Yoshizawa, S. Amari, IEEE Transactions on Neural Networks 5, p. 865-872, (1994). [2] A. Engel, C. Van den Broeck, Statistical Mechanics of Learning, Cambridge University Press (2001). [3] D. Malzahn, M. Opper, Neural Information Processing Systems 13, p. 273, T. K. Leen, T. G. Dietterich and V. Tresp, eds., MIT Press, Cambridge MA (2001). [4] D. Malzahn, M. Opper, Lecture Notes in Computer Science 2130, p. 271, G. Dorffner, H. Bischof and K. Hornik, eds., Springer, Berlin (2001). [5] M. M´ezard, G. Parisi, M. Virasoro, Spin Glass Theory and Beyond, World Scientific, Singapore, (1987). [6] R. P. Feynman and A. R. Hibbs, Quantum mechanics and path integrals, Mc GrawHill Inc., (1965). [7] T. Garel, H. Orland, Europhys. Lett. 6, p. 307 (1988). [8] P. Sollich, Neural Information Processing Systems 11, p. 344, M. S. Kearns, S. A. Solla and D. A. Cohn, eds., MIT Press, Cambridge MA (1999). [9] P. Sollich, Neural Information Processing Systems 14, T. G. Dietterich, S. Becker, Z. Ghahramani, eds., MIT Press (2002).
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Why neuronal dynamics should control synaptic learning rules Jesper Tegner Stockholm Bioinformatics Center Dept. of Numerical Analysis & Computing Science Royal Institute for Technology S-10044 Stockholm, Sweden jespert@nada.kth.se Adam Kepecs Volen Center for Complex Systems Brandeis University Waltham, MA 02454 kepecs@brandeis.edu Abstract Hebbian learning rules are generally formulated as static rules. Under changing condition (e.g. neuromodulation, input statistics) most rules are sensitive to parameters. In particular, recent work has focused on two different formulations of spike-timing-dependent plasticity rules. Additive STDP [1] is remarkably versatile but also very fragile, whereas multiplicative STDP [2, 3] is more robust but lacks attractive features such as synaptic competition and rate stabilization. Here we address the problem of robustness in the additive STDP rule. We derive an adaptive control scheme, where the learning function is under fast dynamic control by postsynaptic activity to stabilize learning under a variety of conditions. Such a control scheme can be implemented using known biophysical mechanisms of synapses. We show that this adaptive rule makes the additive STDP more robust. Finally, we give an example how meta plasticity of the adaptive rule can be used to guide STDP into different type of learning regimes. 1 Introduction Hebbian learning rules are widely used to model synaptic modification shaping the functional connectivity of neural networks [4, 5]. To ensure competition between synapses and stability of learning, constraints have to be added to correlational Hebbian learning rules [6]. Recent experiments revealed a mode of synaptic plasticity that provides new possibilities and constraints for synaptic learning rules [7, 8, 9]. It has been found that synapses are strengthened if a presynaptic spike precedes a postsynaptic spike within a short (::::: 20 ms) time window, while the reverse spike order leads to synaptic weakening. This rule has been termed spike-timing dependent plasticity (STDP) [1]. Computational models highlighted how STDP combines synaptic strengthening and weakening so that learning gives rise to synaptic competition in a way that neuronal firing rates are stabilized. Recent modeling studies have, however, demonstrated that whether an STDP type rule results in competition or rate stabilization depends on exact formulation of the weight update scheme [3, 2]. Sompolinsky and colleagues [2] introduced a distinction between additive and multiplicative weight updating in STDP. In the additive version of an STDP update rule studied by Abbott and coworkers [1, 10], the magnitude of synaptic change is independent on synaptic strength. Here, it is necessary to add hard weight bounds to stabilize learning. For this version of the rule (aSTDP), the steady-state synaptic weight distribution is bimodal. In sharp contrast to this, using a multiplicative STDP rule where the amount of weight increase scales inversely with present weight size produces neither synaptic competition nor rate normalization [3, 2]. In this multiplicative scenario the synaptic weight distribution is unimodal. Activity-dependent synaptic scaling has recently been proposed as a separate mechanism to ensure synaptic competition operating on a slow (days) time scale [3]. Experimental data as of today is not yet sufficient to determine the circumstances under which the STDP rule is additive or multiplicative. In this study we examine the stabilization properties of the additive STDP rule. In the first section we show that the aSTDP rule normalizes postsynaptic firing rates only in a limited parameter range. The critical parameter of aSTDP becomes the ratio (0;) between the amount of synaptic depression and potentiation. We show that different input statistics necessitate different 0; ratios for aSTDP to remain stable. This lead us to consider an adaptive version of aSTDP in order to create a rule that is both competitive as well as rate stabilizing under different circumstances. Next, we use a Fokker-Planck formalism to clarify what determines when an additive STDP rule fails to stabilize the postsynaptic firing rate. Here we derive the requirement for how the potentiation to depression ratio should change with neuronal activity. In the last section we provide a biologically realistic implementation of the adaptive rule and perform numerical simulations to show the how different parameterizations of the adaptive rule can guide STDP into differentially rate-sensitive regimes. 2 Additive STDP does not always stabilize learning First, we numerically simulated an integrate-and-fire model receiving 1000 excitatory and 250 inhibitory afferents. The weights of the excitatory synapses were updated according to the additive STDP rule. We used the model developed by Song et al, 2000 [1]. The learning kernel L(T) is A+exp(T/T+) if T < 0 or -A_ exp( -T/L) if T > 0 where A_ / A+ denotes the amplitude of depression/potentiation respectively. Following [1] we use T + = T _ = 20 ms for the time window of learning. The integral over the temporal window of the synaptic learning function (L) is always negative. Synaptic weights change according to dWi J ill = L(T)Spre(t + T)Spost(T)dT , Wi E[O,Wmax ] (1) where s(t) denotes a delta function representing a spike at time t. Correlations between input rates were generated by adding a common bias rate in a graded manner across synapses so that the first afferent is has zero while the last afferent has the maximal correlation, Cmax . We first examine how the depression/potentiation ratio (0; = LTD / LT P) [2] controls the dependence of the output firing rate on the synaptic input rate, here referred to as the effective neuronal gain. Provided that 0; is sufficiently large, the STDP rule controls the postsynaptic firing rate (Fig. 1A). The stabilizing effect of the STDP rule is therefore equivalent to having weak a neuronal gain. 600 500 100 10 ~ ~ ~ W W M 00 00 Inpul Rate (liz) B 250 ;; mD , -; Increasing j:: I'lP"'C'~~ 50~ % m w w w ~ ~ Input Rale(Hz) c ,---~--------~ 250 200 150 Increasing t.05 Reference Ratio LTDlt.TPratios I ': I~ 0 -o ~ W 00 00 100 Input Rate (Hz) Figure I: A STDP controls neuronal gain. The slope of the dependence of the postsynaptic output rate on the presynaptic input rate is referred to as the effective neuronal gain. The initial firing rate is shown by the upper curve while the lower line displays the final postsynaptic firing rate. The gain is reduced provided that the depression/potentiation ratio (0: = 1.05 here) is large enough. The input is uncorrelated. B Increasing input correlations increases neuronal gain. When the synaptic input is strongly correlated the postsynaptic neuron operates in a high gain mode characterized by a larger slope and larger baseline rate. Input correlations were uniformly distributed between 0 and a maximal value, Cm a x . The maximal correlation increases in the direction of the arrow: 0.0; 0.2; 0.3; 0.4; 0.5; 0.6; 0.7. The 0: ratio is 1.05. Note that for further increases in the presynaptic rates, postsynaptic firing can increase to over 1000 Hz. C The depression/potentiation ratio sets the neuronal gain. The 0: ratios increase in the direction of arrow:1.025;1.05;1.075;1.1025;1.155;1.2075. Cm a x is 0.5. We find that the neuronal gain is extremely sensitive to the value of 0: as well as to the amount of afferent input correlations. Figure IB shows that increasing the amount of input correlations for a given 0: value increases the overall firing rate and the slope of the input-output curve, thus leading to larger effective gain. Increasing the amount of correlations between the synaptic afferents could therefore be interpreted as increasing the effective neuronal gain. Note that the baseline firing at a presynaptic drive of 20Hz is also increased. Next, we examined how neuronal gain depends on the value of 0: in the STDP rule (Figure IC). The high gain and high rate mode induced by strong input correlations was reduced to a lower gain and lower rate mode by increasing 0: (see arrow in Figure IC). Note, however, that there is no correct 0: value as it depends on both the input statistics as well as the desired input/output relationship. 3 Conditions for an adaptive additive STDP rule Here we address how the learning ratio, 0:, should depend on the input rate in order to produce a given neuronal input-output relationship. Using this functional form we will be able to formulate constraints for an adaptive additive STDP rule. This will guide us in the derivation of a biophysical implementation of the adaptive control scheme. The problem in its generality is to find (i) how the learning ratio should depend on the postsynaptic rate and (ii) how the postsynaptic rate depends on the input rate and the synaptic weights. By performing self-consistent calculations using a Fokker-Planck formulation, the problem is reduced to finding conditions for how the learning ratio should depend on the input rates only. Let 0: denote depression/potentiation ratio 0: = LTD/LTP as before. Now we meanw A 30 ouput rate B ,-------------~ 0.6,-------------~ 25 20 15 D o. 1 ,-----,----==:=::=--,---~ 0.05 T···· • • ••• • •••• •••••••••• • • •••• • ••• . . . . °0L--2~0---4~0--6~0---8~0 input rate 0.5 0.4 0.3 0.2 0.1 . ~ . .•.... . . . . . . •.... . . . . . .. . ... . . ... •....... . . .. . . -- ... - . . . . : .. . . . . 0.5 w C WTOT 40,-------------~ 35 30 25 0.5 w Figure 2: Self consistent Fokker-Planck calculations. Conditions for zero neuronal gain. U A The output rate does not depend on the input rate. Zero neuronal gain. B Dependence of the mean synaptic weight on input rates. C W tot ex: Tpre < W >, see text. D The dependence of j3 = a - 1 on input rate. E,F A( w) and P( w) are functions of the synaptic strength and depend on the input rate .. Note that eight different input rates are used but only traces 1, 3, 5, 7 are shown for A(w) and pew) in which the dashed line correspond to the case with the lowest presynaptic rate. determine how the parameter fJ = 0: - 1 should scale with presynaptic rates in order to control the neuronal gain. The Fokker-Planck formulation permits an analytic calculation of the steady state distribution of synaptic weights [3]. The competition parameter for N excitatory afferents is given by Wtot = twrpreN < w > where the time window tw is defined as the probability for depression (Pd = tw/tisi) that a synaptic event occurs within the time window (tw < tisi ). The amount of potentiation and depression for the additive STDP yields in the steady-state, neglecting the exponential timing dependence, the following expression for the drift term A(w) A(w) = PdA-[W/Wtot - (1 - 1/0:)] (2) A( w) represents the net weight "force field" experienced by an individual synapse. Thus, A( w) determines whether a given synapse (w) will increase or decrease as a function of its synaptic weight. The steepness of the A( w) function determines the degree of synaptic competition. The w /Wtot is a competition term whereas the (1 - 1/0:) provides a destabilizing force. When Wmax > (1 - l/o:)Wtot the synaptic weight distribution is bimodal. The steady state distribution reads P(w) = Ke[(-w(1-1 /a) +w 2 /(2 Wt ot ))/(A _ )] (3) where K normalizes the P(w) distribution [3]. Now, equations (2-3), with appropriate definitions of the terms, constitute a selfconsistent system. Using these equations one can calculate how the parameter fJ should scale with the presynaptic input rate in order to produce a given postsynaptic firing rate. For a given presynaptic rate, equations (2-3) can be iterated in until a self-consistent solution is found. At that point, the postsynaptic firing rate can be calculated. Here, instead we impose a fixed postsynaptic output rate for a given input rate and search for a self-consistent solution using (3 as a free parameter. Performing this calculation for a range of input rates provides us with the desired dependency of (3 on the presynaptic firing rate. Once a solution is reached we also examine the resulting steady state synaptic weight distribution (P(w)) and the corresponding drift term A( w) as a function of the presynaptic input rate. The results of such a calculation are illustrated in Figure 2. The neuronal gain, the ratio between the postsynaptic firing rate and the input rate is set to be zero (Fig 2A). To normalize postsynaptic firing rates the average synaptic weight has to decrease in order to compensate for the increasing presynaptic firing rate. This can be seen in (Fig 2B). The condition for a zero neuronal gain is that the average synaptic weight should decrease as 1 j r pre. This makes Wtot constant as shown in Fig 2C. For these values, (3 has to increase with input rate as shown in Fig 2D. Note that this curve is approximately linear. The dependence of A( w) and the synaptic weight distribution P( w) on different presynaptic rates is illustrated in Fig 2E and F. As the presynaptic rates increase, the A(w) function is lowered (dashed line indicates the smallest presynaptic rate), thus pushing more synapses to smaller values since they experience a net negative "force field". This is also reflected in the synaptic weight distribution which is pushed to the lower boundary as the input rates increase. When enforcing a different neuronal gain, the dependence of the (3 term on the presynaptic rates remains approximately linear but with a different slope (not shown). 4 Derivation of an adaptive learning rule with biophysical components The key insight from the above calculations is the observed linear dependence of (3 on presynaptic rates. However, when implementing an adaptive rule with biophysical elements it is very likely that individual components will have a non-linear dependence on each other. The Fokker-Planck analysis suggests that the non-linearities should effectively cancel. Why should the system be linear? Another way to see from where the linearity requirement comes is that the (w jWtot - (3) term in expression for A(w) (valid for small (3) has to be appropriately balanced when the input rates increases. The linearity of (3(rpre ) follows from Wtot being linear in r pre . Now, how could (3 depend on presynaptic rates? A natural solution would be to use postsynaptic calcium to measure the postsynaptic firing and therefore indirectly the presynaptic firing rate. Moreover, the asymmetry ((3) of the learning ratio could depend on the level of postsynaptic calcium. It is known that increased resting calcium levels inhibit NMDA channels and thus calcium influx due to synaptic input. Additionally, the calcium levels required for depression are easier to reach. Both of these effects in turn increase the probability of LTD induction. Incorporating these intermediate steps gives the following scheme: (3 q c p h +-'-+ a t-=--+ r po st +-'---+ r pr e This scheme introduces parameters (p and q) and a function Ut} to control for the linearity jnon-linearity between the variables. The global constraint from the Fokker-Planck is that the effective relation between (3 and r pre should be linear. A biophysical formulation of the above scheme is the following 200 150 ~l oo 5 o 50 No Adaptive Tracking Adaptive Tracking 20 40 60 80 input rat. i-2:WlUlliWWU] ~-40 > -60 0~----------~5~00~--------~10~0~0----------~1~500 '·'r ~A_ .~ 1 ~'ll V'~ "1 100 '0 500 1000 1500 Time (ms) Figure 3: Left Steady-state response with (squares) or without (circles) the adaptive tracking scheme. When the STDP rule is extended with an adaptive control loop, the output rates are normalized in the presence of correlated input. Right Fast adaptive tracking. Since (3 tracks changes in intracellular calcium on a rapid time-scale, every spike experiences a different learning ratio, 0:. Note that the adaptive scheme approximates the learning ratio (0: = 1.05) used in [1]. d(3 T(3 = - (3 + [Ca]q dt (4) (5) The parameter p determines how the calcium concentration scales with the postsynaptic firing rate (delta spikes r5 above) and q controls the learning sensitivity. "( controls the rise of steady-state calcium with increasing postsynaptic rates (rpost). The time constants TCa and T(3 determine the calcium dynamics and the time course of the adaptive rule respectively. Note that we have not specified the neuronal transfer function, it. To ensure a linear relation between (3 and r pre it follows from the Fokker-Planck analysis that [it (rpre)]pq is approximately linear in r pre . The neuronal gain can now be independently be controlled by the parameter T Moreover, the drift term A( w) becomes (6) for (3 < < 1. A( w) can be written in this form since we use that Wd - A_ = -A+CI: = -A+(l + [TCa"(r~ost]q). The w/Wtot is a competition term whereas the [TCa"(r~ost ]q provides a destabilizing force. Note also, that when W max > [TCa"(r~ost ]qWtot there is a bimodal synaptic weight distribution and synaptic competition is preserved. A complete stability analysis is beyond the scope of the present study. A B C .-.. 75 75 75 ...• ~. N :EO) 50 50 50 1\1 ... - 25 25 .... ~ . ; 25 :::J Co :::J 0 0 0 0 0 25 50 75 100 0 25 50 75 100 0 25 50 75 100 Input rate (Hz) Input rate (Hz) Input rate (Hz) Figure 4: Full numerical simulation of the adaptive additive STDP rule. Parameters: p = q = 1. Tea = 10ms, T f! = lOOms. A I = 1.25. B I = 0.25. C Input correlations are Cmax = 0, 0.3, 0.6 5 Numerical simulations Next, we examine whether the theory of adaptive normalization carryover to a full scale simulation of the integrate-and-fire model with the STDP rule and the biophysical adaptive scheme as described above. First, we studied the neuronal gain (cf. Figure 1) when the inputs were strongly correlated. Driving a neuron with increasing input rates increases the output rate significantly when there is no adaptive scheme (squares, Figure 3 Left) as observed previously (cf. Figure IB). Adding the adaptive loop normalizes the output rates (circles, Figure 3 Left). This simulation shows that the average postsynaptic firing rate is regulated by the adaptive tracking scheme. This is expected since the Fokker-Planck analysis is based on the steady-state synaptic weight distribution. To further gain insight into the operation of the adaptive loop we examined the spike-to-spike dependence of the tracking scheme. Figure 3 (Right) displays the evolution of the membrane potential (top) and the learning ratio 0: = 1 + (3 (bottom) . The adaptive rule tracks fast changes in firing by adjusting the learning ratio for each spike. Thus, the strength plasticity is different for every spike. Interestingly, the learning ratio (0:) fluctuates around the value 1.05 which was used in previous studies [1] . Our fast, spike-to-spike tracking scheme is in contrast to other homeostatic mechanisms operating on the time-scale of hours to days [11, 12, 13, 14]. In our formulation, the learning ratio, via (3, tracks changes in intra-cellular calcium, which in turn reflects the instantaneous firing rate. Slower homeostatic mechanisms are unable to detect these rapid changes in firing statistics. Because this fast adaptive scheme depends on recent neuronal firing, pairing several spikes on the time-scale comparable to the calcium dynamics introduces non-linear summation effects. Neurons with this adaptive STDP control loop can detect changes in the input correlation while being only weakly dependent on the presynaptic firing rate. Figure 4a and 4b show two different regimes corresponding to two different values of the parameter , . In the high , regime (Fig. 4a) the neuronal gain is zero. The neuronal gain increased when , decreased (Fig. 4b) as expected from the theory. In a different regime where we introduce increasing correlations between the synaptic inputs [1] we find that the neuronal gain is changed little with increasing input rates but increases substantially with increasing input correlations (Fig 4c) . Thus, the adaptive aSTDP rule can normalize the mean postsynaptic rate even when the input statistics change. With other adaptive parameters we also found learning regimes where the responses to input correlations were affected differentially (not shown). 6 Discussion Synaptic learning rules have to operate under widely changes conditions such as different input statistics or neuromodulation. How can a learning rule dynamically guide a network into functionally similar operating regime under different conditions? We have addressed this issue in the context of spike-timing-dependent plasticity (STDP) [1, 10J. We found that STDP is very sensitive to the ratio of synaptic strengthening to weakening, (t, and requires different values for different input statistics. To correct for this, we proposed an adaptive control scheme to adjust the plasticity rule. This adaptive mechanisms makes the learning rule more robust to changing input conditions while preserving its interesting properties, such as synaptic competition. We suggested a biophysically plausible mechanism that can implement the adaptive changes consistent with the requirements derived using the Fokker-Planck analysis. Our adaptive STDP rule adjusts the learning ratio on a millisecond time-scale. This in contrast to other, slow homeostatic controllers considered previously [11, 12, 13, 14, 3J. Because the learning rule changes rapidly, it is very sensitive the input statistics. Furthermore, the synaptic weight changes add non-linearly due to the rapid self-regulation. In recent experiments similar non-linearities have been detected (Y. Dan, personal communication) which might have roles in making synaptic plasticity adaptive. Finally, the new set of adaptive parameters could be independently controlled by meta-plasticity to bring the neuron into different operating regimes. Acknowledgments We thank Larry Abbott, Mark van Rossum, and Sen Song for helpful discussions. J.T. was supported by the Wennergren Foundation, and grants from Swedish Medical Research Foundation, and The Royal Academy for Science. A.K. was supported by the NIH Grant 2 ROI NS27337-12 and 5 ROI NS27337-13. Both A.K. and J.T. thank the Sloan Foundation for support. References [1] Song, S., Miller, K , & Abbott, L. Nature Neuroscience, 3:919-926, 2000. [2] Rubin, J., Lee, D., & Sompolinsky, H. Physical Review Letter, 86:364-367, 200l. [3] van Rossum, M., G-Q, B. , & Thrrigiano, G. J Neurosci, 20:8812- 8821, 2000. [4] Sejnowski, T. J Theoretical Biology, 69:385- 389, 1997. [5] Abbott, L. & Nelson, S. Nature Neuroscience, 3:1178- 1183, 2000. [6] Miller, K & MacKay, D. Neural Computation, 6:100- 126, 1994. [7] Markram, H., Lubke, J., Frotscher, M., & Sakmann, B. Science, 275:213- 215, 1997. [8] Bell, C., Han, V., Sugawara, Y., & Grant, K Nature, 387:278- 81, 1997. [9] Bi, G.-Q. & Poo, M. J Neuroscience, 18:10464- 10472, 1998. [10] Kempter, R., Gerstner, W., & van Hemmen, J. Neural Computation, 13:2709- 2742, 200l. [11] Bell, A. In Moody, J., Hanson, S., & Lippmann, R., editors, Advances in Neural Information Processing Systems, volume 4. Morgan-Kaufmann, 1992. [12] LeMasson, G., Marder, E., & Abbott, L. Science, 259:1915- 7, 1993. [13] Thrrigiano, G. , Leslie, K , Desai, N., Rutherford, L., & Nelson, S. Nature, 391:892- 6, 1998. [14] Thrrigiano, G. & Nelson, S. Curr Opin Neurobiol, 10:358- 64, 2000.
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Latent Dirichlet Allocation David M. Blei, Andrew Y. Ng and Michael I. Jordan University of California, Berkeley Berkeley, CA 94720 Abstract We propose a generative model for text and other collections of discrete data that generalizes or improves on several previous models including naive Bayes/unigram, mixture of unigrams [6], and Hofmann's aspect model, also known as probabilistic latent semantic indexing (pLSI) [3]. In the context of text modeling, our model posits that each document is generated as a mixture of topics, where the continuous-valued mixture proportions are distributed as a latent Dirichlet random variable. Inference and learning are carried out efficiently via variational algorithms. We present empirical results on applications of this model to problems in text modeling, collaborative filtering, and text classification. 1 Introduction Recent years have seen the development and successful application of several latent factor models for discrete data. One notable example, Hofmann's pLSI/aspect model [3], has received the attention of many researchers, and applications have emerged in text modeling [3], collaborative filtering [7], and link analysis [1]. In the context of text modeling, pLSI is a "bag-of-words" model in that it ignores the ordering of the words in a document. It performs dimensionality reduction, relating each document to a position in low-dimensional "topic" space. In this sense, it is analogous to PCA, except that it is explicitly designed for and works on discrete data. A sometimes poorly-understood subtlety of pLSI is that, even though it is typically described as a generative model, its documents have no generative probabilistic semantics and are treated simply as a set of labels for the specific documents seen in the training set. Thus there is no natural way to pose questions such as "what is the probability of this previously unseen document?". Moreover, since each training document is treated as a separate entity, the pLSI model has a large number of parameters and heuristic "tempering" methods are needed to prevent overfitting. In this paper we describe a new model for collections of discrete data that provides full generative probabilistic semantics for documents. Documents are modeled via a hidden Dirichlet random variable that specifies a probability distribution on a latent, low-dimensional topic space. The distribution over words of an unseen document is a continuous mixture over document space and a discrete mixture over all possible topics. 2 Generative models for text 2.1 Latent Dirichlet Allocation (LDA) model To simplify our discussion, we will use text modeling as a running example throughout this section, though it should be clear that the model is broadly applicable to general collections of discrete data. In LDA, we assume that there are k underlying latent topics according to which documents are generated, and that each topic is represented as a multinomial distribution over the IVI words in the vocabulary. A document is generated by sampling a mixture of these topics and then sampling words from that mixture. More precisely, a document of N words w = (W1,'" ,W N) is generated by the following process. First, B is sampled from a Dirichlet(a1,'" ,ak) distribution. This means that B lies in the (k I)-dimensional simplex: Bi 2': 0, 2:i Bi = 1. Then, for each of the N words, a topic Zn E {I, ... ,k} is sampled from a Mult(B) distribution p(zn = ilB) = Bi. Finally, each word Wn is sampled, conditioned on the znth topic, from the multinomial distribution p(wlzn). Intuitively, Bi can be thought of as the degree to which topic i is referred to in the document. Written out in full, the probability of a document is therefore the following mixture: p(w) = Ie (11 z~/(wnl zn; ,8)P(Zn IB») p(B; a)dB, (1) where p(B; a) is Dirichlet, p(znIB) is a multinomial parameterized by B, and p( Wn IZn;,8) is a multinomial over the words. This model is parameterized by the kdimensional Dirichlet parameters a = (a1,' .. ,ak) and a k x IVI matrix,8, which are parameters controlling the k multinomial distributions over words. The graphical model representation of LDA is shown in Figure 1. As Figure 1 makes clear, this model is not a simple Dirichlet-multinomial clustering model. In such a model the innermost plate would contain only W n ; the topic node would be sampled only once for each document; and the Dirichlet would be sampled only once for the whole collection. In LDA, the Dirichlet is sampled for each document, and the multinomial topic node is sampled repeatedly within the document. The Dirichlet is thus a component in the probability model rather than a prior distribution over the model parameters. We see from Eq. (1) that there is a second interpretation of LDA. Having sampled B, words are drawn iid from the multinomial/unigram model given by p(wIB) = 2::=1 p(wlz)p(zIB). Thus, LDA is a mixture model where the unigram models p(wIB) are the mixture components, and p(B; a) gives the mixture weights. Note that unlike a traditional mixture of unigrams model, this distribution has an infinite o 1'0 '. I Zn Wn Nd D Figure 1: Graphical model representation of LDA. The boxes are plates representing replicates. The outer plate represents documents, while the inner plate represents the repeated choice of topics and words within a document. Figure 2: An example distribution on unigram models p(wIB) under LDA for three words and four topics. The triangle embedded in the x-y plane is the 2-D simplex over all possible multinomial distributions over three words. (E.g., each of the vertices of the triangle corresponds to a deterministic distribution that assigns one of the words probability 1; the midpoint of an edge gives two of the words 0.5 probability each; and the centroid of the triangle is the uniform distribution over all 3 words). The four points marked with an x are the locations of the multinomial distributions p(wlz) for each of the four topics, and the surface shown on top of the simplex is an example of a resulting density over multinomial distributions given by LDA. number of continuously-varying mixture components indexed by B. The example in Figure 2 illustrates this interpretation of LDA as defining a random distribution over unigram models p(wIB). 2.2 Related models The mixture of unigrams model [6] posits that every document is generated by a single randomly chosen topic: (2) This model allows for different documents to come from different topics, but fails to capture the possibility that a document may express multiple topics. LDA captures this possibility, and does so with an increase in the parameter count of only one parameter: rather than having k - 1 free parameters for the multinomial p(z) over the k topics, we have k free parameters for the Dirichlet. A second related model is Hofmann's probabilistic latent semantic indexing (pLSI) [3], which posits that a document label d and a word ware conditionally independent given the hidden topic z : p(d, w) = L~= l p(wlz)p(zld)p(d). (3) This model does capture the possibility that a document may contain multiple topics since p(zld) serve as the mixture weights of the topics. However, a subtlety of pLSIand the crucial difference between it and LDA-is that d is a dummy index into the list of documents in the training set. Thus, d is a multinomial random variable with as many possible values as there are training documents, and the model learns the topic mixtures p(zld) only for those documents on which it is trained. For this reason, pLSI is not a fully generative model and there is no clean way to use it to assign probability to a previously unseen document. Furthermore, the number of parameters in pLSI is on the order of klVl + klDI, where IDI is the number of documents in the training set. Linear growth in the number of parameters with the size of the training set suggests that overfitting is likely to be a problem and indeed, in practice, a "tempering" heuristic is used to smooth the parameters of the model. 3 Inference and learning Let us begin our description of inference and learning problems for LDA by examining the contribution to the likelihood made by a single document. To simplify our notation, let w~ = 1 iff Wn is the jth word in the vocabulary and z~ = 1 iff Zn is the ith topic. Let j3ij denote p(wj = Ilzi = 1), and W = (WI, ... ,WN), Z = (ZI, ... ,ZN). Expanding Eq. (1), we have: (4) This is a hypergeometric function that is infeasible to compute exactly [4]. Large text collections require fast inference and learning algorithms and thus we have utilized a variational approach [5] to approximate the likelihood in Eq. (4). We use the following variational approximation to the log likelihood: logp(w; a, 13) log r :Ep(wlz; j3)p(zIB)p(B; a) q~:, z:" ~~ dB le z q ,Z", > Eq[logp(wlz;j3) +logp(zIB) +logp(B;a) -logq(B,z; , ,¢)], where we choose a fully factorized variational distribution q(B, z;" ¢) q(B; ,) fIn q(Zn; ¢n) parameterized by , and ¢n, so that q(B; ,) is Dirichlet({), and q(zn; ¢n) is MUlt(¢n). Under this distribution, the terms in the variational lower bound are computable and differentiable, and we can maximize the bound with respect to, and ¢ to obtain the best approximation to p(w;a,j3). Note that the third and fourth terms in the variational bound are not straightforward to compute since they involve the entropy of a Dirichlet distribution, a (k - I)-dimensional integral over B which is expensive to compute numerically. In the full version of this paper, we present a sequence of reductions on these terms which use the log r function and its derivatives. This allows us to compute the integral using well-known numerical routines. Variational inference is coordinate ascent in the bound on the probability of a single document. In particular, we alternate between the following two equations until the objective converges: (5) ,i ai + 2:~=1 ¢ni (6) where \]i is the first derivative of the log r function. Note that the resulting variational parameters can also be used and interpreted as an approximation of the parameters of the true posterior. In the current paper we focus on maximum likelihood methods for parameter estimation. Given a collection of documents V = {WI' ... ' WM}, we utilize the EM algorithm with a variational E step, maximizing a lower bound on the log likelihood: M logp(V) 2:: l:= Eqm [logp(B, z, w)]- Eqm [logqm(B, z)]. (7) m=l The E step refits qm for each document by running the inference step described above. The M step optimizes Eq. (7) with respect to the model parameters a and (3. For the multinomial parameters (3ij we have the following M step update equation: M Iwml (3ij ex: l:= l:= ¢>mniwtnn· (8) m=l n=l The Dirichlet parameters ai are not independent of each other and we apply N ewton-Raphson to optimize them: The variational EM algorithm alternates between maximizing Eq. (7) with respect to qm and with respect to (a, (3) until convergence. 4 Experiments and Examples We first tested LDA on two text corpora.1 The first was drawn from the TREC AP corpus, and consisted of 2500 news articles, with a vocabulary size of IVI = 37,871 words. The second was the CRAN corpus, consisting of 1400 technical abstracts, with IVI = 7747 words. We begin with an example showing how LDA can capture multiple-topic phenomena in documents. By examining the (variational) posterior distribution on the topic mixture q(B; ')'), we can identify the topics which were most likely to have contributed to many words in a given document; specifically, these are the topics i with the largest ')'i. Examining the most likely words in the corresponding multinomials can then further tell us what these topics might be about. The following is an article from the TREC collection. The William Randolph Hearst Foundation will give $1.25 million to Lincoln Center, Metropolitan Opera Co., New York Philharmonic and Juilliard School. "Our board felt that we had a real opportunity to make a mark on the future of the performing arts with these grants an act every bit as important as our traditional areas of support in health, medical research, education and the social services," Hearst Foundation President Randolph A. Hearst said Monday in announcing the grants. Lincoln Center's share will be $200,000 for its new building, which will house young artists and provide new public facilities. The Metropolitan Opera Co. and New York Philharmonic will receive $400,000 each. The Juilliard School, where music and the performing arts are taught, will get $250,000. The Hearst Foundation, a leading supporter of the Lincoln Center Consolidated Corporate Fund, will make its usual annual $100,000 donation, too. Figure 3 shows the Dirichlet parameters of the corresponding variational distribution for those topics where ')'i > 1 (k = 100), and also lists the top 15 words (in iTo enable repeated large scale comparison of various models on large corpora, we implemented our variational inference algorithm on a parallel computing cluster. The (bottleneck) E step is distributed across nodes so that the qm for different documents are calculated in parallel. Topic 1 Topic 2 Topic 3 Topic 4 Topic 5 SCHOOL MILLION SAID SAID SAID SAID YEAR AIDS NEW NEW STUDENTS SAID HEALTH PRESIDENT MUSIC BOARD SALES DISEASE CHIEF YEAR SCHOOLS BILLION VIRUS CHAIRMAN THEATER STUDENT TOTAL CHILDREN EXECUTIVE MUSICAL TEACHER SHARE BLOOD VICE BAND POLICE EARNINGS PATIENTS YEARS PLAY PROGRAM PROFIT TREATMENT COMPANY WON TEACHERS QUARTER STUDY YORK TWO MEMBERS ORDERS IMMUNE SCHOOL AVAILABLE YEAROLD LAST CANCER TWO AWARD GANG DEC PEOPLE TODAY OPERA I" DEPARTMENT REVENUE PERCENT COLUMBIA BEST Figure 3: The Dirichlet parameters where Ii > 1 (k = 100), and the top 15 words from the corresponding topics, for the document discussed in the text. __ LDA -x- pLSI .. pLSI(00 lemper) ~ MIx1Un;grams v · ram -><--------------k (number of topics) woo ' .~ 4500 ',.. _ .l\ ! k (number of topiCS) Figure 4: Perplexity results on the CRAN and AP corpora for LDA, pLSI, mixture of unigrams, and the unigram model. order) from these topics. This document is mostly a combination of words about school policy (topic 4) and music (topic 5). The less prominent topics reflect other words about education (topic 1), finance (topic 2), and health (topic 3). 4.1 Formal evaluation: Perplexity To compare the generalization performance of LDA with other models, we computed the perplexity of a test set for the AP and CRAN corpora. The perplexity, used by convention in language modeling, is monotonically decreasing in the likelihood of the test data, and can be thought of as the inverse of the per-word likelihood. More formally, for a test set of M documents, perplexity(Vtest ) = exp (-l:m logp(wm)/ l:m Iwml}. We compared LDA to both the mixture of unigrams and pLSI described in Section 2.2. We trained the pLSI model with and without tempering to reduce overfitting. When tempering, we used part of the test set as the hold-out data, thereby giving it a slight unfair advantage. As mentioned previously, pLSI does not readily generate or assign probabilities to previously unseen documents; in our experiments, we assigned probability to a new document d by marginalizing out the dummy training set indices2 : pew ) = l:d(rr: =1l:z p(wn lz)p(zld))p(d) . 2 A second natural method, marginalizing out d and z to form a unigram model using the resulting p(w)'s, did not perform well (its performance was similar to the standard unigram model). 1-:- ~Dc.~UrUg,ams I ~ W' • M" x NaiveBaes k (number of topics) k (number of topics) Figure 5: Results for classification (left) and collaborative filtering (right) Figure 4 shows the perplexity for each model and both corpora for different values of k. The latent variable models generally do better than the simple unigram model. The pLSI model severely overfits when not tempered (the values beyond k = 10 are off the graph) but manages to outperform mixture of unigrams when tempered. LDA consistently does better than the other models. To our knowledge, these are by far the best text perplexity results obtained by a bag-of-words model. 4.2 Classification We also tested LDA on a text classification task. For each class c, we learn a separate model p(wlc) of the documents in that class. An unseen document is classified by picking argmaxcp(Clw) = argmaxcp(wlc)p(c). Note that using a simple unigram distribution for p(wlc) recovers the traditional naive Bayes classification model. Using the same (standard) subset of the WebKB dataset as used in [6], we obtained classification error rates illustrated in Figure 5 (left). In all cases, the difference between LDA and the other algorithms' performance is statistically significant (p < 0.05). 4.3 Collaborative filtering Our final experiment utilized the EachMovie collaborative filtering dataset. In this dataset a collection of users indicates their preferred movie choices. A user and the movies he chose are analogous to a document and the words in the document (respectively) . The collaborative filtering task is as follows. We train the model on a fully observed set of users. Then, for each test user, we are shown all but one of the movies that she liked and are asked to predict what the held-out movie is. The different algorithms are evaluated according to the likelihood they assign to the held-out movie. More precisely define the predictive perplexity on M test users to be exp(- ~~=llogP(WmNd lwml' ... ,Wm(Nd-l))/M). With 5000 training users, 3500 testing users, and a vocabulary of 1600 movies, we find predictive perplexities illustrated in Figure 5 (right). 5 Conclusions We have presented a generative probabilistic framework for modeling the topical structure of documents and other collections of discrete data. Topics are represented explicitly via a multinomial variable Zn that is repeatedly selected, once for each word, in a given document. In this sense, the model generates an allocation of the words in a document to topics. When computing the probability of a new document, this unknown allocation induces a mixture distribution across the words in the vocabulary. There is a many-to-many relationship between topics and words as well as a many-to-many relationship between documents and topics. While Dirichlet distributions are often used as conjugate priors for multinomials in Bayesian modeling, it is preferable to instead think of the Dirichlet in our model as a component of the likelihood. The Dirichlet random variable e is a latent variable that gives generative probabilistic semantics to the notion of a "document" in the sense that it allows us to put a distribution on the space of possible documents. The words that are actually obtained are viewed as a continuous mixture over this space, as well as being a discrete mixture over topics.3 The generative nature of LDA makes it easy to use as a module in more complex architectures and to extend it in various directions. We have already seen that collections of LDA can be used in a classification setting. If the classification variable is treated as a latent variable we obtain a mixture of LDA models, a useful model for situations in which documents cluster not only according to their topic overlap, but along other dimensions as well. Another extension arises from generalizing LDA to consider Dirichlet/multinomial mixtures of bigram or trigram models, rather than the simple unigram models that we have considered here. Finally, we can readily fuse LDA models which have different vocabularies (e.g., words and images); these models interact via a common abstract topic variable and can elegantly use both vocabularies in determining the topic mixture of a given document. Acknowledgments A. Ng is supported by a Microsoft Research fellowship. This work was also supported by a grant from Intel Corporation, NSF grant IIS-9988642, and ONR MURI N 00014-00-1-0637. References [1] D. Cohn and T. Hofmann. The missing link- A probabilistic model of document content and hypertext connectivity. In Advances in Neural Information Processing Systems 13, 2001. [2] P.J. Green and S. Richardson. Modelling heterogeneity with and without the Dirichlet process. Technical Report, University of Bristol, 1998. [3] T. Hofmann. Probabilistic latent semantic indexing. Proceedings of the Twenty-Second Annual International SIGIR Conference, 1999. [4] T. J. Jiang, J. B. Kadane, and J. M. Dickey. Computation of Carlson's multiple hypergeometric functions r for Bayesian applications. Journal of Computational and Graphical Statistics, 1:231- 251, 1992. [5] M. I. Jordan, Z. Ghahramani, T. S. Jaakkola, and L. K. Saul. Introduction to variational methods for graphical models. Machine Learning, 37:183- 233, 1999. [6] K. Nigam, A. Mccallum, S. Thrun, and T. Mitchell. Text classification from labeled and unlabeled documents using EM. Machine Learning, 39(2/3):103- 134, 2000. [7] A. Popescul, L. H. Ungar, D. M. Pennock, and S. Lawrence. Probabilistic models for unified collaborative and content-based recommendation in sparse-data environments. In Uncertainty in Artificial Intelligence, Proceedings of the Seventeenth Conference, 2001. 3These remarks also distinguish our model from the Bayesian Dirichlet/Multinomial allocation model (DMA)of [2], which is a finite alternative to the Dirichlet process. The DMA places a mixture of Dirichlet priors on p(wlz) and sets O i = 00 for all i.
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A Bayesian Network for Real-Time Musical Accompaniment Christopher Raphael Department of Mathematics and Statistics, University of Massachusetts at Amherst, Amherst, MA 01003-4515, raphael~math.umass.edu Abstract We describe a computer system that provides a real-time musical accompaniment for a live soloist in a piece of non-improvised music for soloist and accompaniment. A Bayesian network is developed that represents the joint distribution on the times at which the solo and accompaniment notes are played, relating the two parts through a layer of hidden variables. The network is first constructed using the rhythmic information contained in the musical score. The network is then trained to capture the musical interpretations of the soloist and accompanist in an off-line rehearsal phase. During live accompaniment the learned distribution of the network is combined with a real-time analysis of the soloist's acoustic signal, performed with a hidden Markov model, to generate a musically principled accompaniment that respects all available sources of knowledge. A live demonstration will be provided. 1 Introduction We discuss our continuing work in developing a computer system that plays the role of a musical accompanist in a piece of non-improvisatory music for soloist and accompaniment. The system begins with the musical score to a given piece of music. Then, using training for the accompaniment part as well as a series of rehearsals, we learn a performer-specific model for the rhythmic interpretation of the composition. In performance, the system takes the acoustic signal of the live player and generates the accompaniment around this signal, in real-time, while respecting the learned model and the constraints imposed by the score. The accompaniment played by our system responds both flexibly and expressively to the soloist's musical interpretation. Our system is composed of two high level tasks we call "Listen" and "Play." Listen takes as input the acoustic signal of the soloist and, using a hidden Markov model, performs a real-time analysis of the signal. The output of Listen is essentially a running commentary on the acoustic input which identifies note boundaries in the solo part and communicates these events with variable latency. The HMM framework is well-suited to the listening task and has several attributes we regard as indispensable to any workable solution: 1. The HMM allows unsupervised training using the Baum-Welch algorithm. Thus we can automatically adapt to changes in solo instrument, microphone placement, ambient noise, room acoustics, and the sound of the accompaniment instrument. 2. Musical accompaniment is inherently a real-time problem. Fast dynamic programming algorithms provide the computational efficiency necessary to process the soloist's acoustic signal at a rate consistent with the real-time demands of our application. 3. Musical signals are occasionally ambiguous locally in time, but become easier to parse when more context is considered. Our system owes much of its accuracy to the probabilistic formulation of the HMM. This formulation allows one to compute the probability that an event is in the past. We delay the estimation of the precise location of an event until we are reasonably confident that it is, in fact, past. In this way our system achieves accuracy while retaining the lowest latency possible in the identification of musical events. Our work on the Listen component is documented thoroughly in [1] and we omit a more detailed discussion here. The heart of our system, the Play component, develops a Bayesian network consisting of hundreds of Gaussian random variables including both observable quantities, such as note onset times, and unobservable quantities, such as local tempo. The network can be trained during a rehearsal phase to model both the soloist's and accompanist's interpretations of a specific piece of music. This model then forms the backbone of a principled real-time decision-making engine used in performance. We focus here on the Play component which is the most challenging part of our system. A more detailed treatment of various aspects of this work is given in [2- 4]. 2 Knowledge Sources A musical accompaniment requires the synthesis of a number of different knowledge sources. From a modeling perspective, the fundamental challenge of musical accompaniment is to express these disparate knowledge sources in terms of a common denominator. We describe here the three knowledge sources we use. 1. We work with non-improvisatory music so naturally the musical score, which gives the pitches and relative durations of the various notes, as well as points of synchronization between the soloist and accompaniment, must figure prominently in our model. The score should not be viewed as a rigid grid prescribing the precise times at which musical events will occur; rather, the score gives the basic elastic material which will be stretched in various ways to to produce the actual performance. The score simply does not address most interpretive aspects of performance. 2. Since our accompanist must follow the soloist, the output of the Listen component, which identifies note boundaries in the solo part, constitutes our second knowledge source. While most musical events, such as changes between neighboring diatonic pitches, can be detected very shortly after the change of note, some events, such as rearticulations and octave slurs, are much less obvious and can only be precisely located with the benefit of longer term hindsight. With this in mind, we feel that any successful accompaniment system cannot synchronize in a purely responsive manner. Rather it must be able to predict the future using the past and base its synchronization on these predictions, as human musicians do. 3. While the same player's performance of a particular piece will vary from rendition to rendition, many aspects of musical interpretation are clearly established with only a few repeated examples. These examples, both of solo performances and human (MIDI) performances of the accompaniment part constitute the third knowledge source for our system. The solo data is used primarily to teach the system how to predict the future evolution of the solo part. The accompaniment data is used to learn the musicality necessary to bring the accompaniment to life. We have developed a probabilistic model, a Bayesian network, that represents all of these knowledge sources through a jointly Gaussian distribution containing hundreds of random variables. The observable variables in this model are the estimated soloist note onset times produced by Listen and the directly observable times for the accompaniment notes. Between these observable variables lies a layer of hidden variables that describe unobservable quantities such as local tempo, change in tempo, and rhythmic stress. 3 A Model for Rhythmic Interpretation We begin by describing a model for the sequence of note onset times generated by a monophonic (single voice) musical instrument playing a known piece of music. For each of the notes, indexed by n = 0, . . . ,N, we define a random vector representing the time, tn, (in seconds) at which the note begins, and the local "tempo," Sn, (in secs. per measure) for the note. We model this sequence ofrandom vectors through a random difference equation: (1) n = 0, ... , N - 1, where in is the musical length of the nth note, in measures, and the {(Tn' CTnY} and (to, so)t are mutually independent Gaussian random vectors. The distributions of the {CTn} will tend concentrate around ° expressing the notion that tempo changes are gradual. The means and variances of the {CT n} show where the soloist is speeding-up (negative mean), slowing-down (positive mean), and tell us if these tempo changes are nearly deterministic (low variance), or quite variable (high variance). The { Tn} variables describe stretches (positive mean) or compressions (negative mean) in the music that occur without any actual change in tempo, as in a tenuto or agogic accent. The addition of the {Tn} variables leads to a more musically plausible model, since not all variation in note lengths can be explained through tempo variation. Equally important, however, the {Tn} variables stabilize the model by not forcing the model to explain, and hence respond to, all note length variation as tempo variation. Collectively, the distributions of the (Tn' CTn)t vectors characterize the solo player's rhythmic interpretation. Both overall tendencies (means) and the repeatability of these tendencies (covariances) are captured by these distributions. 3.1 Joint Model of Solo and Accompaniment In modeling the situation of musical accompaniment we begin with the our basic rhythm model of Eqn. 1, now applied to the composite rhythm. More precisely, Listen Update Composite Accomp Figure 1: A graphical description of the dependency structure of our model. The top layer of the graph corresponds to the solo note onset times detected by Listen. The 2nd layer of the graph describes the ( Tn, 0" n) variables that characterize the rhythmic interpretation. The 3rd layer of the graph is the time-tempo process {(Sn, tn)}. The bottom layer is the observed accompaniment event times. let mo, ... , mivs and mg, ... , m'Na denote the positions, in measures, of the various solo and accompaniment events. For example, a sequence of quarter notes in 3/ 4 time would lie at measure positions 0, 1/ 3, 2/ 3, etc. We then let mo, ... , mN be the sorted union of these two sets of positions with duplicate times removed; thus mo < ml < ... < mN· We then use the model of Eqn. 1 with In = mn+1 - m n, n = 0, . . . , N - 1. A graphical description of this model is given in the middle two layers of Figure 1. In this figure, the layer labeled "Composite" corresponds to the time-tempo variables, (tn, sn)t, for the composite rhythm, while the layer labeled "Update" corresponds to the interpretation variables ( Tn, 0" n) t. The directed arrows of this graph indicate the conditional dependency structure of our model. Thus, given all variables "upstream" of a variable, x, in the graph, the conditional distribution of x depends only on the parent variables. Recall that the Listen component estimates the times at which solo notes begin. How do these estimates figure into our model? We model the note onset times estimated by Listen as noisy observations of the true positions {tn}. Thus if m n is a measure position at which a solo note occurs, then the corresponding estimate from Listen is modeled as an = tn + an where an rv N(O, 1I2). Similarly, if m n is the measure position of an accompaniment event, then we model the observed time at which the event occurs as bn = tn + f3n where f3n rv N(O, ",2). These two collections of observable variables constitute the top layer of our figure, labeled "Listen," and the bottom layer, labeled "Accomp." There are, of course, measure positions at which both solo and accompaniment events should occur. If n indexes such a time then an and bn will both be noisy observations of the true time tn. The vectors/ variables {(to, so)t, (Tn ' O"n)t, an, f3n} are assumed to be mutually independent. 4 Training the Model Our system learns its rhythmic interpretation by estimating the parameters of the (Tn,O"n) variables. We begin with a collection of J performances of the accompaniment part played in isolation. We refer to the model learned from this accompaniment data as the "practice room" distribution since it reflects the way the accompanist plays when the constraint of following the soloist is absent. For each Listen Update Composite Accomp Figure 2: Conditioning on the observed accompaniment performance (darkened circles), we use the message passing algorithm to compute the conditional distributions on the unobservable {Tn' O"n} variables. such performance, we treat the sequence of times at which accompaniment events occur as observed variables in our model. These variables are shown with darkened circles in Figure 2. Given an initial assignment of of means and covariances to the (Tn ,O"n) variables, we use the "message passing" algorithm of Bayesian Networks [8,9] to compute the conditional distributions (given the observed performance) of the (Tn,O"n) variables. Several such performances lead to several such estimates, enabling us to improve our initial estimates by reestimating the (Tn' O"n) parameters from these conditional distributions. More specifically, we estimate the (Tn,O"n) parameters using the EM algorithm, as follows, as in [7]. We let J-L~, ~~ be our initial mean and covariance matrix for the vector ( Tn, 0" n). The conditional distribution of ( Tn, 0" n) given the jth accompaniment performance, and using {J-L~ , ~~} , has a N(m;,n, S~ ) distribution where the m;,n and S~ parameters are computed using the message passing algorithm. We then update our parameter estimates by 1 J . } Lmj,n j = l ~ i+ l n The conventional wisdom of musicians is that the accompaniment should follow the soloist. In past versions of our system we have explicitly modeled the asymmetric roles of soloist and accompaniment through a rather complicated graph structure [2- 4]. At present we deal with this asymmetry in a more ad hoc, however, perhaps more effective, manner, as follows. Training using the accompaniment performances allows our model to learn some of the musicality these performances demonstrate. Since the soloist's interpretation must take precedence, we want to use this accompaniment interpretation only to the extent that it does not conflict with that of the soloist. We accomplish this by first beginning with the result of the accompaniment training described above. We use the practice room distributions, (the distributions on the {(Tn, O"n)} learned from the accompaniment data) , as the initial distributions, {J-L~ , ~~} . We then run the EM algorithm as described above now treating the currently available collection of solo performances as the observed data. During this phase, only those parameters relevant to the soloist's rhythmic interpretation will be modified significantly. Parameters describing the interpretation of a musical segment in which the soloist is mostly absent will be largely unaffected by the second training pass. Listen Update Composite Accomp Figure 3: At any given point in the performance we will have observed a collection of solo note times estimated estimated by Listen, and the accompaniment event times (the darkened circles). We compute the conditional distribution on the next unplayed accompaniment event, given these observations. This solo training actually happens over the course of a series of rehearsals. We first initialize our model to the practice room distribution by training with the accompaniment data. Then we iterate the process of creating a performance with our system, (described in the next section), extracting the sequence of solo note onset times in an off-line estimation process, and then retraining the model using all currently available solo performances. In our experience, only a few such rehearsals are necessary to train a system that responds gracefully and anticipates the soloist's rhythmic nuance where appropriate generally less than 10. 5 Real Time Accompaniment The methodological key to our real-time accompaniment algorithm is the computation of (conditional) marginal distributions facilitated by the message-passing machinery of Bayesian networks. At any point during the performance some collection of solo notes and accompaniment notes will have been observed, as in Fig. 3. Conditioned on this information we can compute the distribution on the next unplayed accompaniment. The real-time computational requirement is limited by passing only the messages necessary to compute the marginal distribution on the pending accompaniment note. Once the conditional marginal distribution of the pending accompaniment note is calculated we schedule the note accordingly. Currently we schedule the note to be played at the conditional mean time, given all observed information, however other reasonable choices are possible. Note that this conditional distribution depends on all of the sources of information included in our model: The score information, all currently observed solo and accompaniment note times, and the rhythmic interpretations demonstrated by both the soloist and accompanist captured during the training phase. The initial scheduling of each accompaniment note takes place immediately after the previous accompaniment note is played. It is possible that a solo note will be detected before the pending accompaniment is played; in this event the pending accompaniment event is rescheduled by recomputing the its conditional distribution using the newly available information. The pending accompaniment note is rescheduled each time an additional solo note is detected until its currently scheduled time arrives, at which time it is finally played. In this way our accompaniment makes use of all currently available information. Does our system pass the musical equivalent of the Turing Test? We presume no more objectivity in answering this question than we would have in judging the merits of our other children. However, we believe that the level of musicality attained by our system is truly surprising, while the reliability is sufficient for live demonstration. We hope that the interested reader will form an independent opinion, even if different from ours, and to this end we have made musical examples demonstrating our progress available on the web page: http://fafner.math.umass.edu/musicplus_one. Acknowledgments This work supported by NSF grants IIS-998789 and IIS-0113496. References [1] Raphael C. (1999), "Automatic Segmentation of Acoustic Musical Signals Using Hidden Markov Models," IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 21, No.4, pp. 360-370. [2] Raphael C. (2001), "A Probabilistic Expert System for Automatic Musical Accompaniment," Journal of Computational and Graphical Statistics, vol. 10 no. 3, 487-512. [3] Raphael C. (2001), "Can the Computer Learn to Play Expressively?" Proceedings of Eighth International Workshop on Artificial Intelligence and Statistics, 113-120, Morgan Kauffman. [4] Raphael C. (2001), "Synthesizing Musical Accompaniments with Bayesian Belief Networks," Journal of New Music Research, vol. 30, no. 1, 59-67. [5] Spiegelhalter D., Dawid A. P., Lauritzen S., Cowell R. (1993), "Bayesian Analysis in Expert Systems," Statistical Science, Vol. 8, No.3, pp. 219-283. [6] Cowell R., Dawid A. P., Lauritzen S., Spiegelhalter D. (1999), "Probabilistic Networks and Expert Systems," Springer, New York. [7] Lauritzen S. L. (1995), "The EM Algorithm for Graphical Association Models with Missing Data," Computational Statistics and Data Analysis, Vol. 19, pp. 191-20l. [8] Lauritzen S. L. (1992), "Propagation of Probabilities, Means, and Variances in Mixed Graphical Association Models," Journal of the American Statistical Association, Vol. 87, No. 420, (Theory and Methods), pp. 1098-1108. [9] Lauritzen S. L. and F. Jensen (1999), "Stable Local Computation with Conditional Gaussian Distributions," Technical Report R-99-2014, Department of Mathematic Sciences, Aalborg University.
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Efficiency versus Convergence of Boolean Kernels for On-Line Learning Algorithms Roni Khardon Tufts University Medford, MA 02155 roni@eecs.tufts.edu Dan Roth University of Illinois Urbana, IL 61801 danr@cs.uiuc.edu Rocco Servedio Harvard University Cambridge, MA 02138 rocco@deas.harvard.edu Abstract We study online learning in Boolean domains using kernels which capture feature expansions equivalent to using conjunctions over basic features. We demonstrate a tradeoff between the computational efficiency with which these kernels can be computed and the generalization ability of the resulting classifier. We first describe several kernel functions which capture either limited forms of conjunctions or all conjunctions. We show that these kernels can be used to efficiently run the Perceptron algorithm over an exponential number of conjunctions; however we also prove that using such kernels the Perceptron algorithm can make an exponential number of mistakes even when learning simple functions. We also consider an analogous use of kernel functions to run the multiplicative-update Winnow algorithm over an expanded feature space of exponentially many conjunctions. While known upper bounds imply that Winnow can learn DNF formulae with a polynomial mistake bound in this setting, we prove that it is computationally hard to simulate Winnow’s behavior for learning DNF over such a feature set, and thus that such kernel functions for Winnow are not efficiently computable. 1 Introduction The Perceptron and Winnow algorithms are well known learning algorithms that make predictions using a linear function in their feature space. Despite their limited expressiveness, they have been applied successfully in recent years to several large scale real world classification problems. The SNoW system [7, 2], for example, has successfully applied variations of Perceptron [6] and Winnow [4] to problems in natural language processing. The system first extracts Boolean features from examples (given as text) and then runs learning algorithms over restricted conjunctions of these basic features. There are several ways to enhance the set of features after the initial extraction. One idea is to expand the set of basic features using conjunctions such as and use these expanded higher-dimensional examples, in which each conjunction plays the role of a basic feature, for learning. This approach clearly leads to an increase in expressiveness and thus may improve performance. However, it also dramatically increases the number of features (from
to if all conjunctions are used) and thus may adversely affect both the computation time and convergence rate of learning. This paper studies the computational efficiency and convergence of the Perceptron and Winnow algorithms over such expanded feature spaces of conjunctions. Specifically, we study the use of kernel functions to expand the feature space and thus enhance the learning abilities of Perceptron and Winnow; we refer to these enhanced algorithms as kernel Perceptron and kernel Winnow. 1.1 Background: Perceptron and Winnow Throughout its execution Perceptron maintains a weight vector which is initially Upon receiving an example
the algorithm predicts according to the linear threshold function If the prediction is and the label is (false positive prediction) then the vector is set to , while if the prediction is and the label is (false negative) then is set to No change is made if the prediction is correct. The famous Perceptron Convergence Theorem [6] bounds the number of mistakes which the Perceptron algorithm can make: Theorem 1 Let be a sequence of labeled examples with ! "# $ $%& and (') +* for all , . Let -. /10 be such that -1 "/ for all , Then Perceptron makes at most 243658795:3 ; 3 mistakes on this example sequence. The Winnow algorithm [4] has a very similar structure. Winnow maintains a hypothesis vector <# which is initially >= Winnow is parameterized by a promotion factor ? and a threshold @ 0 BA upon receiving an example C' +*D Winnow predicts according to the threshold function ( @ If the prediction is and the label is then for all , such that =E the value of is set to GF ? ; this is a demotion step. If the prediction is and the label is then for all , such that =H the value of is set to ?I ; this is a promotion step. No change is made if the prediction is correct. For our purposes the following mistake bound, implicit in [4], is of interest: Theorem 2 Let the target function be a J -literal monotone disjunction K = 8L M NN M PO For any sequence of examples in ' +*D labeled according to K the number of prediction mistakes made by Winnow ? @ is at most Q QBR S J ?TU VXWZY\[ Q @ 1.2 Our Results Our first result in Section 2 shows that it is possible to efficiently run the kernel Perceptron algorithm over an exponential number of conjunctive features: Theorem 3 There is an algorithm that simulates Perceptron over the -dimensional feature space of all conjunctions of
basic features. Given a sequence of ] labeled examples in ' ^* the prediction and update for each example take poly
] time steps. This result is closely related to one of the main open problems in learning theory: efficient learnability of disjunctions of conjunctions, or DNF (Disjunctive Normal Form) expressions.1 Since linear threshold elements can represent disjunctions (e.g. M !_ M is true iff !_ ), Theorems 1 and 3 imply that kernel Perceptron can be used to learn DNF. However, in this framework the values of ` and & in Theorem 1 can be exponentially large, and hence the mistake bound given by Theorem 1 is exponential rather than polynomial in
The question thus arises whether, for kernel Perceptron, the exponential 1Angluin [1] proved that DNF expressions cannot be learned efficiently using hypotheses which are themselves DNF expressions from equivalence queries and thus also in the mistake bound model which we are considering here. However this result does not preclude the efficient learnability of DNF using a different class of hypotheses such as those generated by the kernel Perceptron algorithm. upper bound implied by Theorem 1 is essentially tight. We give an affirmative answer, thus showing that kernel Perceptron cannot efficiently learn DNF: Theorem 4 There is a monotone DNF K over and a sequence of examples labeled according to K which causes the kernel Perceptron algorithm to make mistakes. Turning to Winnow, an attractive feature of Theorem 2 is that for suitable ? @ the bound is logarithmic in the total number of features ` (e.g. ?= and @ =H` ). Therefore, as noted by several researchers [5], if a Winnow analogue of Theorem 3 could be obtained this would imply efficient learnability of DNF. We show that no such analogue can exist: Theorem 5 There is no polynomial time algorithm which simulates Winnow over exponentially many monotone conjunctive features for learning monotone DNF, unless every problem in #P can be solved in polynomial time. We observe that, in contrast to Theorem 5, Maass and Warmuth have shown that the Winnow algorithm can be simulated efficiently over exponentially many conjunctive features for learning some simple geometric concept classes [5]. While several of our results are negative, in practice one can achieve good performance by using kernel Perceptron (if
is small) or the limited-conjunction kernel described in Section 2 (if
is large). This is similar to common practice with polynomial kernels2 where typically a small degree is used to aid convergence. These observations are supported by our preliminary experiments in an NLP domain which are not reported here. 2 Theorem 3: Kernel Perceptron with Exponentially Many Features It is easily observed, and well known, that the hypothesis of the Perceptron algorithm is a sum of the previous examples on which prediction mistakes were made. If we let "' ^* denote the label of example , then =
where is the set of examples on which the algorithm made a mistake. Thus the prediction of Perceptron on is 1 iff U =
B =
. For an example ' +* let denote its transformation into an enhanced feature space such as the space of all conjunctions. To run the Perceptron algorithm over the enhanced space we must predict iff where is the weight vector in the enhanced space; from the above discussion this holds iff
B B . Denoting = B this holds iff
. Thus we never need to construct the enhanced feature space explicitly; we need only be able to compute the kernel function efficiently. This is the idea behind all so-called kernel methods, which can be applied to any algorithm (such as support vector machines) whose prediction is a function of inner products of examples; see e.g. [3] for a discussion. The result in Theorem 3 is simply obtained by presenting a kernel function capturing all conjunctions. We also describe kernels for all monotone conjunctions which allow no negative literals, and kernels capturing all (monotone) conjunctions of up to J literals. The general case: When includes all conjunctions (with positive and negative literals) must compute the number of conjunctions which are true in both and . Clearly, any literal in such a conjunction must satisfy both and and thus the corresponding bit in must have the same value. Counting all such conjunctions gives = "!$# &%' ( where )+*-,/. is the number of original features that have the same value in and . This kernel has been obtained independently by [8]. 2Our Boolean kernels are different than standard polynomial kernels in that all the conjunctions are weighted equally. While expressive power does not change, convergence and behavior, do. Monotone Monomials: In some applications the total number
of basic features may be very large but in any one example only a small number of features take value 1. In other applications the number of features
may not be known in advance (e.g. due to unseen words in text domains). In these cases it may be useful to consider only monotone monomials. To express all monotone monomials we take = +! # &%' ( where )+*,/.!Y) is the number of active features common to both and . A parameterized kernel: In general, one may want to trade off expressivity against number of examples and convergence time. Thus we consider a parameterized kernel which captures all conjunctions of size at most J for some J
The number of such conjunctions that satisfy both and is =
"!$# &%' ( . This kernel is reported also in [10]. For monotone conjunctions of size at most J we have = +! # &%' ( . 3 Theorem 4: Kernel Perceptron with Exponentially Many Mistakes We describe a monotone DNF target function and a sequence of labeled examples which cause the monotone kernel Perceptron algorithm to make exponentially many mistakes. For C' +* we write to denote the number of 1’s in and . to denote )+*,/.!Y) We use the following lemma (constants have not been optimized): Lemma 6 There is a set of
-bit strings =E' * ' +* with ] = such that \=
F for % , % ] and ! #" %
F%$ for % ,&(' % ] Proof: The proof uses the probabilistic method. For each ,V= ] let ! X' +* be chosen by independently setting each bit to with probability 1/10. For any , it is clear that )+*, =
F A a Chernoff bound implies that .0/%*1 !
F % R 2 and thus the probability that any satisfies 3
F is at most ]4 R 2 Similarly, for any ,5 =6' we have )+*1 7 #" -#=
F DBA a Chernoff bound implies that .8/%*1 9 #" 0
F:$ % R 2 and thus the probability that any " with ,;5 =6' satisfies " 0
F%$ is at most _ R <2 For ] == the value of _ R 3 <2 ]4 R 2 is less than 1. Thus for some choice of we have each
F and # #" %
F:$ For any which has 0
F we can set ^
F of the 1s to 0s, and the lemma is proved. The target DNF is very simple: it is the single conjunction !_ While the original Perceptron algorithm over the
features makes at most poly
mistakes for this target function, we now show that the monotone kernel Perceptron algorithm which runs over all monotone monomials can make > mistakes. Recall that at the beginning of the Perceptron algorithm’s execution all coordinates of are 0. The first example is the negative example A since = Perceptron incorrectly predicts 1 on this example. The resulting update causes the coefficient ? corresponding to the empty monomial (satisfied by any example ) to become but all other coordinates of remain 0. The next example is the positive example For this example we have = so Perceptron incorrectly predicts Since all monotone conjunctions are satisfied by this example the resulting update causes ? to become 0 and all other coordinates of to become 1. The next examples are the vectors described in Lemma 6. Since each such example has )=
F each example is negative; however as we now show the Perceptron algorithm will predict on each of these examples. Fix any value % , % and consider the hypothesis vector just before example is received. Since !=
F the value of is a sum of the _ different coordinates which correspond to the monomials satisfied by ! More precisely we have =
) where contains the monomials which are satisfied by and #" for some '65 =E, and contains the monomials which are satisfied by but no #" with '5 =, We lower bound the two sums separately. Let be any monomial in By Lemma 6 any H contains at most
F%$ variables and thus there can be at most 2 _ monomials in Using the well known bound Q
" ; " % Q where ? % F and ? is the binary entropy function there can be at most terms in Moreover the value of each must be at least 34 since decreases by at most 1 for each example, and hence
; 34 0 _ On the other hand, for any we clearly have = By Lemma 6 for any 0
F:$ every -variable monomial satisfied by must belong to and hence
3_ 2 _ 0 Combining these inequalities we have _ < 0 and hence the Perceptron prediction on is 1. 4 Theorem 5: Learning DNF with Kernel Winnow is Hard In this section, for .' ^* denotes the -element vector whose coordinates are all nonempty monomials (monotone conjunctions) over A sequence of labeled examples is monotone consistent if it is consistent with some monotone function, i.e. % " for all J =
implies % " If is monotone consistent and has ] labeled examples then clearly there is a monotone DNF formula consistent with which contains at most ] conjunctions. We consider the following problem: KERNEL WINNOW PREDICTION ? @ (KWP) Instance: Monotone consistent sequence C= of labeled examples with each .' ^* and each .') +* A unlabeled example ' +* Question: Is @ where is the ` = . -dimensional hypothesis vector generated by running Winnow ? @ on the example sequence ? In order to run Winnow over all
. nonempty monomials to learn monotone DNF, one must be able to solve KWP efficiently. The main result of this section is proved by showing that KWP is computationally hard for any parameter settings which yield a polynomial mistake bound for Winnow via Theorem 2. Theorem 7 Let ` =
U and ? 0 @ be such that , * Q QBR S ? > WZY\[ Q @ = poly ! Then KWP ? @ is #P-hard. Proof of Theorem 7: For ` ? and @ as described above it can easily be verified that poly ? poly ! and _#" poly @ poly The proof of the theorem is a reduction from the following #P-hard problem [9]: (See [9] also for details on #P.) MONOTONE 2-SAT (M2SAT) Instance: Monotone 2-CNF Boolean formula $ =&% (' % _)' *' % with % = 8L4M 3 and each ,+ ' * A integer such that % % Question: Is $ R i.e. does $ have at least satisfying assignments in ' +* ? 4.1 High-Level Idea of the Proof The high level idea of the proof is simple: let $ be an instance of M2SAT where $ is defined over variables The Winnow algorithm maintains a weight for each monomial over variables We define a 1-1 correspondence between these monomials and truth assignments ' +* for $ and we give a sequence of examples for Winnow which causes . if $ = and = if $ = The value of is thus related to $ R A some additional work ensures that @ if and only if $ R In more detail, let =
WZYD[ Q W YD[ ? = Q =
_ Q and =
C
_ We describe a polynomial time transformation which maps an
-variable instance $ of M2SAT to an -variable instance of KWP ? @ where X=E is monotone consistent, each and belong to ' ^* and @ if and only if $ R The Winnow variables are divided into three sets and where = ' * = '
* and = '
* The unlabeled example is
R R
i.e. all variables in and are set to 1 and all variables in are set to 0. We thus have = ( where = ? = ? and ( = ' ? ' ? We refer to monomials 5 = as type monomials, monomials 5 = as type monomials, and monomials ! 5 = 5 = as type monomials. The example sequence is divided into four stages. Stage 1 results in $ R A as described below the
variables in correspond to the
variables in the CNF formula $ Stage 2 results in ?" $ R for some positive integer # Stages 3 and 4 together result in ( @ ? " Thus the final value of is approximately @# ? " $ R ^ so we have @ if and only if $ R Since all variables in are 0 in if includes a variable in then the value of does not affect The variables in are “slack variables” which (i) make Winnow perform the correct promotions/demotions and (ii) ensure that is monotone consistent. 4.2 Details of the Proof Stage 1: Setting $&%('*),+.-0/214365) . We define the following correspondence between truth assignments ' +* and monomials 7 &8 = if and only if is not present in For each clause L M 3 in $ Stage 1 contains negative examples such that 8L = 3 = and = for all other Assuming that (1) Winnow makes a false positive prediction on each of these examples and (2) in Stage 1 Winnow never does a promotion on any example which has any variable in set to 1, then after Stage 1 we will have that = if $ = and < % ? R9 if $ = Thus we will have = $ R ;: for some !: ? R9 _ We now show how the Stage 1 examples cause Winnow to make a false positive prediction on negative examples which have 8L = 3 = and = for all other , in as described above. For each such negative example in Stage 1 six new slack variables =< =< ! are used as follows: Stage 1 has PW YD[ Q @ F repeated instances of the positive example which has >< = =< _ = and all other bits 0. These examples cause promotions which result in @ % %@?BA L %@?BA 3 %@?BA L %@?BA 3 ? @ and hence %@?BA L @ F Two other groups of similar examples (the first with >< = =< = the second with =< C = =< = ) cause %@?BAD @ F and %@?BAE @ F The next example in is the negative example which has 8L = 3 = = for all other in =< = =< = =< C = and all other bits 0. For this example 0 %@?BA L %@?BAD ( %@?BAE @ so Winnow makes a false positive prediction. Since $ has at most
_ clauses and there are negative examples per clause, this construction can be carried out using
_ slack variables
9 3 Stage 2: Setting $&%'GFIHJ) +.-0/61K365) . The first Stage 2 example is a positive example with = for all ,
9 3 = and all other bits 0. Since each of the monomials which contain
9 3 and are satisfied by this example have = we have - = $ R 6;: Since @ 0 F poly ! 0 after the resulting promotion we have = ? $ R : U?$ Let #=PW YD[ Q @ F so ? " U@ % ? " Stage 2 consists of # repeated instances of the positive example described above. After these promotions we have = ?" $ R N;: "? " @ Since $ R N;: we also have ? " =>? " $ R N : U? " U@ F Stage 3: Setting $ ' . At the start of Stage 3 each type and type monomial has = There are
variables in and variables in so at the start of Stage 2 we have =
and ( =
Since no example in Stages 3 or 4 satisfies any in at the end of Stage 4 will still be ? " $ R 6 : and ( will still be .
. Thus ideally at the end of Stage 4 the value of would be @B
? " : since this would imply that 9 =>@) ? " $ R which is at least @ if and only if $ R However it is not necessary for to assume this exact value; since $ R must be an integer and : _ as long as @
.? " % U@
( .? " _ we will have that ( @ if and only if $ R For ease of notation let denote @9
? " We now describe the examples in Stages 3 and 4 and show that they will cause to satisfy % E _ ? " Let %
= PW YD[ Q so ? " R % ? " and hence there is a unique smallest integer such that % ? " R ? " The Stage 3 examples will result in Using the definition of and the fact that % it can be verified that ? " R % ? " R H ? " % @ ? " % ? " ? " R =H? " R ? Hence we have % ? % Q
=
We use the following lemma: Lemma 8 For all for all % % there is a monotone CNF $ ' over Boolean variables which has at most clauses, has exactly satisfying assignments in ' +* and can be constructed from and in poly time. Proof: The proof is by induction on . For the base case (= we have = and $ ' = Assuming the lemma is true for =< J we now prove it for = J 8 If % % then the desired CNF is $ ' = ' $ -' Since $ -' has at most J clauses $ ' has at most J " clauses. If " % % then the desired CNF is $ ' = M $ -' R _ O By distributing over each clause of $ -' R _ O we can write $ ' as a CNF with at most J clauses. If T= then $ ' = Let $
' be an -clause monotone CNF formula over the variables in which has satisfying assignments. Similar to Stage 1, for each clause of $
' , Stage 3 has negative examples corresponding to that clause, and as in Stage 1 slack variables in are used to ensure that Winnow makes a false positive prediction on each such negative example. Thus the examples in Stage 3 cause = : _ where : _
? R Since six slack variables in are used for each negative example and there are % negative examples, the slack variables
9 3 _ R _ are sufficient for Stage 3. Stage 4: Setting $ $7% ' FIH . All that remains is to perform #1&% promotions on examples which have each in set to 1. This will cause % ? " R ;: _ ? " R = C ? " : _ ? " R E _ ? " which is as desired. It can be verified from the definitions of and % that
R Q % The first #
R Q examples in are all the same positive example which has each in set to 1 and R =< The first time this example is received =
.: _
It can be verified that
"@ so Winnow performs a promotion; after #
R Q occurrences of this example - =<? " R
: _ ? " R
% ? " >@ and =>? " R
;: _ The remaining examples in Stage 4 are
R Q % repetitions of the positive example which has each in set to 1 and = If promotions occurred on each repetition of this example then we would have = ?
R
? " R
: _ so we need only show that this is less than @ We reexpress this quantity as ?
R
? " R I: _ We have ? " R I: _ ? " R ? " R % @ ? " ? " U@9 _ ? " Some easy manipulations show that ?
R
% Q _ _
R _ ? " so indeed U@ Finally, we observe that by construction the example sequence is monotone consistent. Since = poly
and contains poly
examples the transformation from M2SAT to KWP ? @ is polynomial-time computable and the theorem is proved. (Theorem 7) 5 Conclusion It is necessary to expand the feature space if linear learning algorithms are to learn expressive functions. This work explores the tradeoff between computational efficiency and convergence (i.e. generalization ability) when using expanded feature spaces. We have shown that additive and multiplicative update algorithms differ significantly in this respect; we believe that this fact could have significant practical implications. Future directions include the utilization of the kernels developed here and studying convergence issues of Boolean-kernel Perceptron and Support Vector Machines in the PAC model. Acknowledgements: R. Khardon was supported by NSF grant IIS-0099446. D. Roth was supported by NSF grants ITR-IIS-00-85836 and IIS-9984168 and by EPSRC grant GR/N03167 while visiting University of Edinburgh. R. Servedio was supported by NSF grant CCR-98-77049 and by a NSF Mathematical Sciences Postdoctoral Fellowship. References [1] D. Angluin. Negative results for equivalence queries. Machine Learning, 2:121–150, 1990. [2] A. Carlson, C. Cumby, J. Rosen, and D. Roth. The SNoW learning architecture. Technical Report UIUCDCS-R-99-2101, UIUC Computer Science Department, May 1999. [3] N. Cristianini and J. Shaw-Taylor. An Introduction to Support Vector Machines. Cambridge Press, 2000. [4] N. Littlestone. Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. Machine Learning, 2:285–318, 1988. [5] W. Maass and M. K. Warmuth. Efficient learning with virtual threshold gates. Information and Computation, 141(1):378–386, 1998. [6] A. Novikoff. On convergence proofs for perceptrons. In Proceeding of the Symposium on the Mathematical Theory of Automata, volume 12, pages 615–622, 1963. [7] D. Roth. Learning to resolve natural language ambiguities: A unified approach. In Proc. of the American Association of Artificial Intelligence, pages 806–813, 1998. [8] K. Sadohara. Learning of boolean functions using support vector machines. In Proc. of the Conference on Algorithmic Learning Theory, pages 106–118. Springer, 2001. LNAI 2225. [9] L. G. Valiant. The complexity of enumeration and reliability problems. SIAM Journal of Computing, 8:410–421, 1979. [10] C. Watkins. Kernels from matching operations. Technical Report CSD-TR-98-07, Computer Science Department, Royal Holloway, University of London, 1999.
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Prodding the ROC Curve: Constrained Optimization of Classifier Performance Michael C. Mozer*+, Robert Dodier*, Michael D. Colagrosso*+, César Guerra-Salcedo*, Richard Wolniewicz* * Advanced Technology Group + Department of Computer Science Athene Software University of Colorado 2060 Broadway Campus Box 430 Boulder, CO 80302 Boulder, CO 80309 Abstract When designing a two-alternative classifier, one ordinarily aims to maximize the classifier’s ability to discriminate between members of the two classes. We describe a situation in a real-world business application of machine-learning prediction in which an additional constraint is placed on the nature of the solution: that the classifier achieve a specified correct acceptance or correct rejection rate (i.e., that it achieve a fixed accuracy on members of one class or the other). Our domain is predicting churn in the telecommunications industry. Churn refers to customers who switch from one service provider to another. We propose four algorithms for training a classifier subject to this domain constraint, and present results showing that each algorithm yields a reliable improvement in performance. Although the improvement is modest in magnitude, it is nonetheless impressive given the difficulty of the problem and the financial return that it achieves to the service provider. When designing a classifier, one must specify an objective measure by which the classifier’s performance is to be evaluated. One simple objective measure is to minimize the number of misclassifications. If the cost of a classification error depends on the target and/ or response class, one might utilize a risk-minimization framework to reduce the expected loss. A more general approach is to maximize the classifier’s ability to discriminate one class from another class (e.g., Chang & Lippmann, 1994). An ROC curve (Green & Swets, 1966) can be used to visualize the discriminative performance of a two-alternative classifier that outputs class posteriors. To explain the ROC curve, a classifier can be thought of as making a positive/negative judgement as to whether an input is a member of some class. Two different accuracy measures can be obtained from the classifier: the accuracy of correctly identifying an input as a member of the class (a correct acceptance or CA), and the accuracy of correctly identifying an input as a nonmember of the class (a correct rejection or CR). To evaluate the CA and CR rates, it is necessary to pick a threshold above which the classifier’s probability estimate is interpreted as an “accept,” and below which is interpreted as a “reject”—call this the criterion. The ROC curve plots CA against CR rates for various criteria (Figure 1a). Note that as the threshold is lowered, the CA rate increases and the CR rate decreases. For a criterion of 1, the CA rate approaches 0 and the CR rate 1; for a criterion of 0, the CA rate approaches 1 and the CR rate 0. Thus, the ROC curve is anchored at (0,1) and (1,0), and is monotonically nonincreasing. The degree to which the curve is bowed reflects the discriminative ability of the classifier. The dashed curve in Figure 1a is therefore a better classifier than the solid curve. The degree to which the curve is bowed can be quantified by various measures such as the area under the ROC curve or d’, the distance between the positive and negative distributions. However, training a classifier to maximize either the ROC area or d’ often yields the same result as training a classifier to estimate posterior class probabilities, or equivalently, to minimize the mean squared error (e.g., Frederick & Floyd, 1998). The ROC area and d’ scores are useful, however, because they reflect a classifier’s intrinsic ability to discriminate between two classes, regardless of how the decision criterion is set. That is, each point on an ROC curve indicates one possible CA/CR trade off the classifier can achieve, and that trade off is determined by the criterion. But changing the criterion does not change the classifier’s intrinsic ability to discriminate. Generally, one seeks to optimize the discrimination performance of a classifier. However, we are working in a domain where overall discrimination performance is not as critical as performance at a particular point on the ROC curve, and we are not interested in the remainder of the ROC curve. To gain an intuition as to why this goal should be feasible, consider Figure 1b. Both the solid and dashed curves are valid ROC curves, because they satisfy the monotonicity constraint: as the criterion is lowered, the CA rate does not decrease and the CR rate does not increase. Although the bow shape of the solid curve is typical, it is not mandatory; the precise shape of the curve depends on the nature of the classifier and the nature of the domain. Thus, it is conceivable that a classifier could produce a curve like the dashed one. The dashed curve indicates better performance when the CA rate is around 50%, but worse performance when the CA rate is much lower or higher than 50%. Consequently, if our goal is to maximize the CR rate subject to the constraint that the CA rate is around 50%, or to maximize the CA rate subject to the constraint that the CR rate is around 90%, the dashed curve is superior to the solid curve. One can imagine that better performance can be obtained along some stretches of the curve by sacrificing performance along other stretches of the curve. Note that obtaining a result such as the dashed curve requires a nonstandard training algorithm, as the discrimination performance as measured by the ROC area is worse for the dashed curve than for the solid curve. In this paper, we propose and evaluate four algorithms for optimizing performance in a certain region of the ROC curve. To begin, we explain the domain we are concerned with and why focusing on a certain region of the ROC curve is important in this domain. 20 40 60 80 100 0 20 40 60 80 100 0 correct acceptance rate correct rejection rate 20 40 (b) 60 80 100 0 correct acceptance rate correct rejection rate 20 40 60 80 100 0 (a) FIGURE 1. (a) two ROC curves reflecting discrimination performance; the dashed curve indicates better performance. (b) two plausible ROC curves, neither of which is clearly superior to the other. 1 OUR DOMAIN Athene Software focuses on predicting and managing subscriber churn in the telecommunications industry (Mozer, Wolniewicz, Grimes, Johnson, & Kaushansky, 2000). “Churn” refers to the loss of subscribers who switch from one company to the other. Churn is a significant problem for wireless, long distance, and internet service providers. For example, in the wireless industry, domestic monthly churn rates are 2–3% of the customer base. Consequently, service providers are highly motivated to identify subscribers who are dissatisfied with their service and offer them incentives to prevent churn. We use techniques from statistical machine learning—primarily neural networks and ensemble methods—to estimate the probability that an individual subscriber will churn in the near future. The prediction of churn is based on various sources of information about a subscriber, including: call detail records (date, time, duration, and location of each call, and whether call was dropped due to lack of coverage or available bandwidth), financial information appearing on a subscriber’s bill (monthly base fee, additional charges for roaming and usage beyond monthly prepaid limit), complaints to the customer service department and their resolution, information from the initial application for service (contract details, rate plan, handset type, credit report), market information (e.g., rate plans offered by the service provider and its competitors), and demographic data. Churn prediction is an extremely difficult problem for several reasons. First, the business environment is highly nonstationary; models trained on data from a certain time period perform far better with hold-out examples from that same time period than examples drawn from successive time periods. Second, features available for prediction are only weakly related to churn; when computing mutual information between individual features and churn, the greatest value we typically encounter is .01 bits. Third, information critical to predicting subscriber behavior, such as quality of service, is often unavailable. Obtaining accurate churn predictions is only part of the challenge of subscriber retention. Subscribers who are likely to churn must be contacted by a call center and offered some incentive to remain with the service provider. In a mathematically principled business scenario, one would frame the challenge as maximizing profitability to a service provider, and making the decision about whether to contact a subscriber and what incentive to offer would be based on the expected utility of offering versus not offering an incentive. However, business practices complicate the scenario and place some unique constraints on predictive models. First, call centers are operated by a staff of customer service representatives who can contact subscribers at a fixed rate; consequently, our models cannot advise contacting 50,000 subscribers one week, and 50 the next. Second, internal business strategies at the service providers constrain the minimum acceptable CA or CR rates (above and beyond the goal of maximizing profitability). Third, contracts that Athene makes with service providers will occasionally call for achieving a specific target CA and CR rate. These three practical issues pose formal problems which, to the best of our knowledge, have not been addressed by the machine learning community. The formal problems can be stated in various ways, including: (1) maximize the CA rate, subject to the constraint that a fixed percentage of the subscriber base is identified as potential churners, (2) optimize the CR rate, subject to the constraint that the CA rate should be αCA, (3) optimize the CA rate, subject to the constraint that the CR rate should be αCR, and finally—what marketing executives really want—(4) design a classifier that has a CA rate of αCA and a CR rate of αCR. Problem (1) sounds somewhat different than problems (2) or (3), but it can be expressed in terms of a lift curve, which plots the CA rate as a function of the total fraction of subscribers identified by the model. Problem (1) thus imposes the constraint that the solution lies at one coordinate of the lift curve, just as problems (2) and (3) place the constraint that the solution lies at one coordinate of the ROC curve. Thus, a solution to problems (2) or (3) will also serve as a solution to (1). Although addressing problem (4) seems most fanciful, it encompasses problems (2) and (3), and thus we focus on it. Our goal is not altogether unreasonable, because a solution to problem (4) has the property we characterized in Figure 1b: the ROC curve can suffer everywhere except in the region near CA αCA and CR αCR. Hence, the approaches we consider will trade off performance in some regions of the ROC curve against performance in other regions. We call this prodding the ROC curve. 2 FOUR ALGORITHMS TO PROD THE ROC CURVE In this section, we describe four algorithms for prodding the ROC curve toward a target CA rate of αCA and a target CR rate of αCR. 2.1 EMPHASIZING CRITICAL TRAINING EXAMPLES Suppose we train a classifier on a set of positive and negative examples from a class— churners and nonchurners in our domain. Following training, the classifier will assign a posterior probability of class membership to each example. The examples can be sorted by the posterior and arranged on a continuum anchored by probabilities 0 and 1 (Figure 2). We can identify the thresholds, θCA and θCR, which yield CA and CR rates of αCA and αCR, respectively. If the classifier’s discrimination performance fails to achieve the target CA and CR rates, then θCA will be lower than θCR, as depicted in the Figure. If we can bring these two thresholds together, we will achieve the target CA and CR rates. Thus, the first algorithm we propose involves training a series of classifiers, attempting to make classifier n+1 achieve better CA and CR rates by focusing its effort on examples from classifier n that lie between θCA and θCR; the positive examples must be pushed above θCR and the negative examples must be pushed below θCA. (Of course, the thresholds are specific to a classifier, and hence should be indexed by n.) We call this the emphasis algorithm, because it involves placing greater weight on the examples that lie between the two thresholds. In the Figure, the emphasis for classifier n+1 would be on examples e5 through e8. This retraining procedure can be iterated until the classifier’s training set performance reaches asymptote. In our implementation, we define a weighting of each example i for training classifier n, . For classifier 1, . For subsequent classifiers, if example i is not in the region of emphasis, or otherwise, where κe is a constant, κe > 1. 2.2 DEEMPHASIZING IRRELEVANT TRAINING EXAMPLES The second algorithm we propose is related to the first, but takes a slightly different perspective on the continuum depicted in Figure 2. Positive examples below θCA—such as e2—are clearly the most dif ficult positive examples to classify correctly. Not only are they the most difficult positive examples, but they do not in fact need to be classified correctly to achieve the target CA and CR rates. Threshold θCR does not depend on examples such as e2, and threshold θCA allows a fraction (1–αCA) of the positive examples to be classified incorrectly. Likewise, one can argue that negative examples above θCR—such as e10 and e11—need not be of concern. Essentially , the second algorithm, which we term thedeemphasis algorithm, is like the emphasis algorithm in that a series of classifiers are trained, but when training classifier n+1, less weight is placed on the examples whose correct clasFIGURE 2. A schematic depiction of all training examples arranged by the classifier’s posterior. Each solid bar corresponds to a positive example (e.g., a churner) and each grey bar corresponds to a negative example (e.g., a nonchurner). 0 1 churn probability θCA θCR e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 λi n λi 1 1 = λi n 1 + λi n = λi n 1 + κeλi n = sification is unnecessary to achieve the target CA and CR rates for classifier n. As with the emphasis algorithm, the retraining procedure can be iterated until no further performance improvements are obtained on the training set. Note that the set of examples given emphasis by the previous algorithm is not the complement of the set of examples deemphasized by the current algorithm; the algorithms are not identical. In our implementation, we assign a weight to each example i for training classifier n, . For classifier 1, . For subsequent classifiers, if example i is not in the region of deemphasis, or otherwise, where κd is a constant, κd<1. 2.3 CONSTRAINED OPTIMIZATION The third algorithm we propose is formulated as maximizing the CR rate while maintaining the CA rate equal to αCA. (We do not attempt to simultaneously maximize the CA rate while maintaining the CR rate equal to αCR.) Gradient methods cannot be applied directly because the CA and CR rates are nondifferentiable, but we can approximate the CA and CR rates with smooth differentiable functions: , where P and N are the set of positive and negative examples, respectively, f(x,w) is the model posterior for input x, w is the parameterization of the model, t is a threshold, and σβ is a sigmoid function with scaling parameter β: . The larger β is, the more nearly step-like the sigmoid is and the more nearly equal the approximations are to the model CR and CA rates. We consider the problem formulation in which CA is a constraint and CR is a figure of merit. We convert the constrained optimization problem into an unconstrained problem by the augmented Lagrangian method (Bertsekas, 1982), which involves iteratively maximizing an objective function with a fixed Lagrangian multiplier, ν, and then updating ν following the optimization step: , where and are the values found by the optimization step. We initialize and fix and and iterate until ν converges. 2.4 GENETIC ALGORITHM The fourth algorithm we explore is a steady-state genetic search over a space defined by the continuous parameters of a classifier (Whitley, 1989). The fitness of a classifier is the reciprocal of the number of training examples falling between the θCA and θCR thresholds. Much like the emphasis algorithm, this fitness function encourages the two thresholds to come together. The genetic search permits direct optimization over a nondifferentiable criterion, and therefore seems sensible for the present task. 3 METHODOLOGY For our tests, we studied two large data bases made available to Athene by two telecommunications providers. Data set 1 had 50,000 subscribers described by 35 input features and a churn rate of 4.86%. Data set 2 had 169,727 subscribers described by 51 input features and a churn rate of 6.42%. For each data base, the features input to the classifier were obtained by proprietary transformations of the raw data (see Mozer et al., 2000). We chose these two large, real world data sets because achieving gains with these data sets should be more difficult than with smaller, less noisy data sets. Plus, with our real-world data, we can evaluate the cost savings achieved by an improvement in prediction accuracy. We performed 10-fold cross-validation on each data set, preserving the overall churn/nonchurn ratio in each split. In all tests, we chose and , values which, based on our past experience in this domain, are ambitious yet realizable targets for data sets such as λi n λi 1 1 = λi n 1 + λi n = λi n 1 + κdλi n = CA w t, ( ) 1 P -----σβ f xi w , ( ) t – ( ) i P ∈∑ = CR w t, ( ) 1 N ------σβ t f xi w , ( ) – ( ) i N ∈∑ = σβ y ( ) 1 exp βy – ( ) + ( ) 1 – = A w t, ( ) CR w t, ( ) ν CA w t, ( ) αCA – µ 2--- CA w t, ( ) αCA – 2 + + = ν ν µ CA w* t* , ( ) αCA – + ← w* t* ν 1 = µ 1 = β 10 = αCR 0.90 = αCA 0.50 = these. We used a logistic regression model (i.e., a no hidden unit neural network) for our studies, believing that it would be more difficult to obtain improvements with such a model than with a more flexible multilayer perceptron. For the emphasis and deemphasis algorithms, models were trained to minimize mean-squared error on the training set. We chose κe = 1.3 and κd = .75 by quick exploration. Because the weightings are cumulative over training restarts, the choice of κ is not critical for either algorithm; rather, the magnitude of κ controls how many restarts are necessary to reach asymptotic performance, but the results we obtained were robust to the choice of κ. The emphasis and deemphasis algorithms were run for 100 iterations, which was the number of iterations required to reach asymptotic performance on the training set. 4 RESULTS Figure 3 illustrates training set performance for the emphasis algorithm on data set 1. The graph on the left shows the CA rate when the CR rate is .9, and the graph on the right show the CR rate when the CA rate is .5. Clearly, the algorithm appears to be stable, and the ROC curve is improving in the region around (αCA, αCR). Figure 4 shows cross-validation performance on the two data sets for the four prodding algorithms as well as for a traditional least-squares training procedure. The emphasis and deemphasis algorithms yield reliable improvements in performance in the critical region of the ROC curve over the traditional training procedure. The constrained-optimization and genetic algorithms perform well on achieving a high CR rate for a fixed CA rate, but neither does as well on achieving a high CA rate for a fixed CR rate. For the constrained-optimization algorithm, this result is not surprising as it was trained asymmetrically, with the CA rate as the constraint. However, for the genetic algorithm, we have little explanation for its poor performance, other than the difficulty faced in searching a continuous space without gradient information. 5 DISCUSSION In this paper, we have identified an interesting, novel problem in classifier design which is motivated by our domain of churn prediction and real-world business considerations. Rather than seeking a classifier that maximizes discriminability between two classes, as measured by area under the ROC curve, we are concerned with optimizing performance at certain points along the ROC curve. We presented four alternative approaches to prodding the ROC curve, and found that all four have promise, depending on the specific goal. Although the magnitude of the gain is small—an increase of about .01 in the CR rate given a target CA rate of .50—the impro vement results in significant dollar savings. Using a framework for evaluating dollar savings to a service provider, based on estimates of subscriber retention and costs of intervention obtained in real world data collection (Mozer et 0.365 0.37 0.375 0.38 0.385 0.39 0.395 0.4 0 5 10 15 20 25 30 35 40 45 50 CA rate Iteration 0.81 0.815 0.82 0.825 0.83 0.835 0.84 0.845 0 5 10 15 20 25 30 35 40 45 50 CR rate Iteration FIGURE 3. Training set performance for the emphasis algorithm on data set 1. (a) CA rate as a function of iteration for a CR rate of .9; (b) CR rate as a function of iteration for a CA rate of .5. Error bars indicate +/–1 standard error of the mean. al., 2000), we obtain a savings of $11 per churnable subscriber when the (CA, CR) rates go from (.50, .80) to (.50, .81), which amounts to an 8% increase in profitability of the subscriber intervention effort. These figures are clearly promising. However, based on the data sets we have studied, it is difficult to know whether another algorithm might exist that achieves even greater gains. Interestingly, all algorithms we proposed yielded roughly the same gains when successful, suggesting that we may have milked the data for whatever gain could be had, given the model class evaluated. Our work clearly illustrate the difficulty of the problem, and we hope that others in the NIPS community will be motivated by the problem to suggest even more powerful, theoretically grounded approaches. 6 ACKNOWLEDGEMENTS No white males were angered in the course of conducting this research. We thank Lian Yan and David Grimes for comments and assistance on this research. This research was supported in part by McDonnell-Pew grant 97-18, NSF award IBN-9873492, and NIH/IFOPAL R01 MH61549–01A1. 7 REFERENCES Bertsekas, D. P. (1982). Constrained optimization and Lagrange multiplier methods. NY: Academic. Chang, E. I., & Lippmann, R. P. (1994). Figure of merit training for detection and spotting. In J. D. Cowan, G. Tesauro, & J. Alspector (Eds.), Advances in Neural Information Processing Systems 6 (1019–1026). San Mateo, CA: Morgan Kaufmann. Frederick, E. D., & Floyd, C. E. (1998). Analysis of mammographic findings and patient history data with genetic algorithms for the prediction of breast cancer biopsy outcome. Proceedings of the SPIE, 3338, 241–245. Green, D. M., & Swets, J. A. (1966). Signal detection theory and psychophysics. New York: Wiley. Mozer, M. C., Wolniewicz, R., Grimes, D., Johnson, E., & Kaushansky, H. (2000). Maximizing revenue by predicting and addressing customer dissatisfaction. IEEE Transactions on Neural Networks, 11, 690–696. Whitley, D. (1989). The GENITOR algorithm and selective pressure: Why rank-based allocation of reproductive trials is best. In D. Schaffer (Ed.), Proceedings of the Third International Conference on Genetic Algorithms (pp. 116–121). San Mateo, CA: Morgan Kaufmann. ISP Test Set 0.350 0.355 0.360 0.365 0.370 0.375 0.380 0.385 0.390 std emph deemph constr GA CA rate 0.800 0.805 0.810 0.815 0.820 0.825 0.830 0.835 0.840 std emph deemph constr GA CR rate Wireless Test Set 0.300 0.325 0.350 0.375 std emph deemph constr GA CA rate 0.800 0.825 0.850 0.875 0.900 std emph deemph constr GA CR rate FIGURE 4. Cross-validation performance on the two data sets for the standard training procedure (STD), as well as the emphasis (EMPH), deemphasis (DEEMPH), constrained optimization (CONSTR), and genetic (GEN) algorithms. The left column shows the CA rate for CR rate .9; the right column shows the CR rate for CA rate .5. The error bar indicates one standard error of the mean over the 10 data splits. Data set 1 Data set 2
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Orientation-Selective aVLSI Spiking Neurons Shih-Chii Liu, J¨org Kramer, Giacomo Indiveri, Tobias Delbr¨uck, and Rodney Douglas Institute of Neuroinformatics University of Zurich and ETH Zurich Winterthurerstrasse 190 CH-8057 Zurich, Switzerland Abstract We describe a programmable multi-chip VLSI neuronal system that can be used for exploring spike-based information processing models. The system consists of a silicon retina, a PIC microcontroller, and a transceiver chip whose integrate-and-fire neurons are connected in a soft winner-take-all architecture. The circuit on this multi-neuron chip approximates a cortical microcircuit. The neurons can be configured for different computational properties by the virtual connections of a selected set of pixels on the silicon retina. The virtual wiring between the different chips is effected by an event-driven communication protocol that uses asynchronous digital pulses, similar to spikes in a neuronal system. We used the multi-chip spike-based system to synthesize orientation-tuned neurons using both a feedforward model and a feedback model. The performance of our analog hardware spiking model matched the experimental observations and digital simulations of continuous-valued neurons. The multi-chip VLSI system has advantages over computer neuronal models in that it is real-time, and the computational time does not scale with the size of the neuronal network. 1 Introduction The sheer number of cortical neurons and the vast connectivity within the cortex are difficult to duplicate in either hardware or software. Simulations of a network consisting of thousands of neurons with a connectivity that is representative of cortical neurons can take minutes to hours on a fast Pentium, particularly if spiking behavior is simulated. The simulation time of the network increases as the size of the network increases. We have taken initial steps in mitigating the simulation time of neuronal networks by developing a multi-chip VLSI system that can support spike-based cortical processing models. The connectivity between neurons on different chips and between neurons on the same chip are reconfigurable. The receptive fields are effected by appropriate mapping of the spikes from source neurons to target neurons. A significant advantage of these hardware simulation systems is their real-time property; the simulation time of these systems does not increase with the size of the network. In this work, we show how we synthesized orientation-tuned spiking neurons using the multi-chip system in Figure 1. The virtual connection from a selected set of neurons on Silicon retina Router (White matter) Network of neurons Orientationselective neurons Figure 1: Block diagram of a neuromorphic multi-chip system in which virtual connections from a set of neurons on a silicon retina onto another set of neurons on a transceiver chip are effected by a microcontroller. The retina communicates through the AER protocol to the PIC when it has an active pixel. The PIC communicates with the multi-neuron chip if the retina address falls into one of its stored templates. The address from the PIC is decoded by the multi-neuron transceiver. The address of the active neuron on this array can also be communicated off-chip to another receiver/transceiver. the retina to the target neurons on the multi-neuron transceiver chip is achieved with a PIC microcontroller and an asynchronous event-driven communication protocol. The circuit on this multi-neuron chip approximates a cortical microcircuit (Douglas and Martin, 1991). We explored different models that have been proposed for the generation of orientation tuning in neurons of the V1 cortical area. There have been earlier attempts to use multichip systems for creating orientation-selective neurons (Boahen et al., 1997; Whatley et al., 1997). In the present work, the receptive fields are created in a manner similar to that described in (Whatley et al., 1997). However we extend their work and quantify the tuning curves of different models. Visual cortical neurons receive inputs from the lateral geniculate nucleus (LGN) neurons which are not orientation-selective. Models for the emergence of orientation-selectivity in cortical neurons can be divided into two groups; feedforward models and feedback models. In a feedforward model, the orientation selectivity of a cortical neuron is conferred by the spatial alignment of the LGN neurons that are presynaptic to the cortical neuron (Hubel and Wiesel, 1962). In a feedback model, a weak orientation bias provided by the LGN input is sharpened by the intracortical excitatory and/or inhibitory feedback (Somers et al., 1995; Ben-Yishai et al., 1995; Douglas et al., 1995). In this work, we quantify the tuning curves of neurons created using a feedforward model and a feedback model with global inhibition. 2 System Architecture The multi-chip system (Figure 1) in this work consists of a 16 16 silicon ON/OFF retina, a PIC microcontroller, and a transceiver chip with a ring of 16 integrate-and-fire neurons and a global inhibitory neuron. All three modules communicate using the address event representation (AER) protocol (Lazzaro et al., 1993; Boahen, 1996). The communication channel signals consist of the address bits, the REQ signal, and the ACK signal. The PIC and the multi-neuron chip are both transceivers: They can both receive events and send events (Liu et al., 2001). The retina with an on-chip arbiter can only send events. Each pixel is composed of an adaptive photoreceptor that has a rectifying temporal differentiator (Kramer, 2001) in its feedback loop as shown in Figure 2. Positive temporal irradiance transients (dark-to-bright or ON transitions) and negative irradiance transients (bright-to-dark or OFF transitions) appear at two different outputs of the pixel. The outputs are then coded in the form of asynchronous binary pulses by two neurons within the pixel. These asynchronous pulses M1 temporal differentiator ON neuron ON REQ ON ACK OFF Arbiter OFF REQ OFF ACK neuron M2 bias M3 Figure 2: Pixel of the transient imager. The circuit contains a photodiode with a transistor in a a source-follower configuration with a high-gain inverting amplifier ( , ) in a negative feedback loop. A rectifying temporal differentiator in the feedback loop extracts transient ON and OFF signals. These signals go to individual neurons that generate the REQ signals to the arbiter. In this schematic, we only show the REQ and ACK signals to the X-arbiter. The duration of the ACK signal from the X-arbiter is extended within the pixel by a global refractory bias. This duration sets the refractory period of the neuron. are the request signals to the AER communication interface. A global parameter sets the minimum time (or refractory period) between subsequent pulses from the same output. Hence, the pixel can respond either with one pulse or multiple pulses to a transient. The pixels are arranged on a rectangular grid. The position of a pixel is encoded with a 4-bit column address (X address) and a 4-bit row address (Y address) as shown in Figure 3. An active neuron makes a request to the on-chip arbiter. If the neuron is selected by the arbiter, then the X and Y addresses which code the location of this neuron are placed on the output address bus of the chip. The retina then handshakes with the PIC microcontroller. The multi-neuron chip has an on-chip address decoder for the incoming events and an onchip arbiter to send events. The X address to the chip codes the identity of the neuron and the Y address codes the input synapse used to stimulate the neuron. Each neuron can be stimulated externally through an excitatory synapse or an inhibitory synapse. The excitatory neurons of this array are mutually connected via hard-wired excitatory synapses. These excitatory neurons also excite a global inhibitory neuron which in turn inhibits all the excitatory neurons. The membrane potentials of the neurons can be monitored by an on-chip scanner and the output spikes of the neurons can be monitored by the chip’s AER output. The address on the output bus codes the active neuron. In this work, the excitatory neurons on the multi-neuron chip model the orientation tuning properties of simple cells in the visual cortex and the global inhibitory neuron models an inhibitory interneuron in the visual cortex. The receptive fields of the neurons are created by configuring the connections from a subset of the source pixels on the retina onto the appropriate target neurons on the multi-neuron transceiver chip through a PIC 16C74 microcontroller. The subsets of retina pixels are determined by user-supplied templates. The microcontroller filters each retinal event to 0 deg template 90 deg template X 15 0 Y 0 15 Inhibitory synapse Excitatory synapse Figure 3: Spikes from a selected set of neurons within the two rectangular regions on the retina are mapped by the PIC onto the corresponding orientation-selective neurons on the transceiver chip. The light-shaded triangles mark the somas of the excitatory neurons and the dark-shaded triangle marks the soma of the global inhibitory neuron. Only two neurons, which are mapped for orthogonal orientations, were used in this experiment. decide if it lies in one or more of the receptive fields (RFs) of the neurons on the receiver. If it does, an event is transmitted to the appropriate receiver neuron. The typical transmission time from a spike from the sender to the receiver is about 15 s. This cycle time can be reduced by using a faster processor in place of the PIC. The retina and transceiver chips can handle handshaking cycle times on the order of 100 ns. 3 Neuron Circuit The circuit of a neuron and an excitatory synapse on the transceiver chip is shown in Figure 4. The synapse circuit (M1–M4) in the left box of the figure was originally described in (Boahen, 1996). The presynaptic spike drives the transistor M4, which acts like a switch. The bias voltages and set the the strength and the dynamics of the synapse. The circuit in the right box of Figure 4 implements a linear threshold integrate and fire neuron with an adjustable voltage threshold, spike pulse width and refractory period. The synaptic current charges up the capacitance of the membrane
. When the membrane potential
exceeds a threshold voltage , the output of the transconductance amplifier M5–M9 switches to a voltage close to . The output of the two inverters (M10– M12 and M13–M15), , also switches to . The bias voltage, , limits the current through the transconductance amplifier and the first inverter. The capacitors "!# and $
implement a capacitive divider that provides positive feedback to
. This feedback speeds up the circuit’s response and provides hysteresis to ensure that small fluctuations of
around % do not make & switch erratically. When & is high, (' is discharged through transistors M20 and M21 at a rate that is dependent on ) . This bias voltage controls the spike’s pulse width. Once
is below % , the transconductance amplifier switches to ground. The first inverter then switches to but does not immediately go to zero; it decreases linearly at a rate set by *+!# . In this way, transistor M21 is kept on, even after ,
has decreased below % . As long as the gate voltage of M21 is sufficiently high, the neuron is in its refractory period. Once transistor M21 is turned off, a new spike is generated in a time that is inversely proportional to the magnitude of - & . The spike output of the circuit is taken from the output of the first inverter. Iinj Ve Vw Vspk Vdd Vrfr Vpw Vout Vpb Cfb Cr Vmem Vdd Vdd Vdd Vtau Vthr Vqua M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M20 M21 M16 M17 M18 Cca M19 M5 M1 M2 M3 M4 Vdd Cm Cs Figure 4: Circuit diagram of an excitatory synapse (left box) connected to a linear threshold integrate-and-fire neuron (right box). Transistors M16–M19 implement a spike frequency adaptation mechanism (Boahen, 1996). A fixed amount of charge (set by ) is dumped onto the capacitor with every output spike. The resulting charge on sets the current that is subtracted from the input current, and the neuron’s output frequency decreases accordingly. The voltages and are used to set the gain and dynamics of the integrator. 4 System Responses A rotating drum with a black and white strip was placed in front of the retina. The spike addresses and spike times generated by the retina and the multi-neuron chip at an image speed of 7.9 mm/s (or 89 pixels/s) of the rotating stimulus were recorded using a logic analyzer. The orientations of the stimuli are defined in Figure 3. Each pixel of the retina responded with only one spike to the transition of an edge of the stimulus because the refractory period of the pixel was set to 500 s. The spike addresses during the time of travel of the OFF edge of a 0 deg oriented stimulus through the entire array (Figure 5(a)) indicates that almost all the pixels along a row transmitted their addresses sequentially as the edge passed by. This sequential ordering can be seen because the stimulus was oriented slightly different from 0 deg. If the stimulus was perfectly at 0 deg, then there would be a random ordering of the pixel addresses within each row. The same observation can be made for the OFF-transient spikes recorded in response to a 90 deg oriented stimulus (Figure 5(b)). The receptive fields of two orientation-selective neurons were synthesized by mapping the OFF transient outputs of a selected set of pixels on the retina as shown in Figure 3. These two neurons have orthogonal preferred orientations. The local excitatory coupling between the neurons was disabled. There is no self excitation to each neuron so we explored only a feedforward model and a feedback model using global inhibition. We varied the size and aspect ratio of the receptive fields of the neurons by changing the template size used in the mapping of the retina spikes to the transceiver chip. The template size and aspect ratio determine the orientation responses of the neurons. The orientation response of these neurons also depends on the time constant of the neuron. On this multi-neuron chip, we do not have an explicit transistor that allows us to control the time constant. Instead, we generated a leak current through in Figure 4 by controlling the source voltage of , 7.88 7.9 7.92 7.94 7.96 7.98 8 8.02 0 50 100 150 200 250 Time (s) Retina address for OFF spikes (a) −50 0 50 100 150 0 50 100 150 200 250 Time (ms) Retina address for OFF spikes (b) Figure 5: The spike addresses from the retina were recorded when a 0 deg (a) and a 90 deg (b) oriented stimulus moved across the retina. The figure shows the time progression of the stimulated pixels (OFF spikes are marked with circles) as the 0 deg oriented stimulus (see Figure 3 for the orientation definition) passed over each row in (a). The address on the ordinate is defined as 16Y + X. A similar observation is true of (b) for the ordering of the OFF-transient spikes when each column on the retina was stimulated by the 90 deg oriented stimulus. & . By increasing , we decrease the time constant of the neuron. Because the neuron charges up to threshold through the summation of the incoming EPSPs, it can only spike if the ISIs of the incoming spikes are small enough. The synaptic weight determines the number of EPSPs needed to drive the neuron above threshold. We first investigated the feedforward model by using a template size of 5 7 (3 deg 4.2 deg) for one neuron and 7 5 (4.2 deg 3 deg) for the second neuron. The aspect ratio of this template was 1.4. (We have repeated the following experiments using smaller template sizes (3 5 and 1 3) and the experimental results were pretty much the same.) The time constant of the neuron and synaptic gain and strength were adjusted so that both neurons responded optimally to the stimulus. The connection from the global inhibitory neuron to the two excitatory neurons was disabled. Data was collected from the multi-neuron chip for different orientations of the drum (and hence of the stimulus). The stimulus was presented approximately 500–1000 times to the retina. Since the orientation-selective neurons responded with only 1–3 spikes every time the stimulus moved over the retina, we normalized the total number of spikes collected in these experiments to the number of stimulus presentations. The results are shown as a polar plot in Figure 6(a) for the two neurons that are sensitive to orthogonal orientations. Each neuron was more sensitive to a stimulus at its preferred orientation than the nonpreferred orientations. The neuron responded more to the orthogonal orientation than to the in-between orientations because there were a small number of retina spikes that arrived with a small ISI when the orthogonally-oriented stimulus moved across the template space of the retina (see Figure 3). We used an orientation-selective (OS) index to quantify the orientation selectivity of the neuron. This index is defined as % !#
& ! % !# & ! where R() is the response of the neuron. As an example, R(preferred) for neuron 5, which is sensitive to vertical orientations, is R(90)+R(270) and R(nonpreferred) is R(0)+R(180). We next investigated the feedback model. In the presence of global inhibition, the multi30 210 60 240 90 270 120 300 150 330 180 0 (a) 30 210 60 240 90 270 120 300 150 330 180 0 (b) Figure 6: Orientation tuning curves of the two neurons in the (a) absence and (b) presence of global inhibition. The responses of the neurons were measured by the number of spikes collected per stimulus presentation. The radius of the polar plot is normalized to the maximum response of both neurons. The data was collected for stimulus orientations spaced at 30 deg intervals. The neuron that responded preferably to a 90 deg oriented stimulus (solid curve) also had a small response to a stimulus at 0 deg orientation (OS=0.428). The same observation is true for the other neuron (dashed curve) (OS=0.195). In the presence of global inhibition, each neuron responded less to the non-preferred orientation due to the suppression from the other neuron (cross-orientation inhibition). The output firing rates were also lower in this case (approximately half of the firing rates in the absence of inhibition). The OS indices are 0.546 (solid curve) and 0.497 (dashed curve) respectively. neuron system acts like a soft winner-take-all circuit. We tuned the coupling strengths between the excitatory neurons and the inhibitory neuron so that we obtained the optimal response to the same stimulus presentations as in the feedforward case. The new tuning curves are plotted in Figure 6(b). The non-preferred response of a neuron was suppressed by the other neuron through the recurrent inhibition (cross-orientation inhibition). 5 Conclusion We demonstrated a programmable multi-chip VLSI system that can be used for exploring spike-based processing models. This system has advantages over computer neuronal models in that it is real-time and the computational time does not scale with the size of the neuronal network. The spiking neurons can be configured for different computational properties. Interchip and intrachip connectivity between neurons can be programmed using the AER protocol. In this work, we created receptive fields for orientation-tuned spiking neurons by mapping the transient spikes from a silicon retina onto the neurons using a microcontroller. We have not mapped onto all the neurons on the transceiver chip because the PIC microcontroller we used is not fast enough to create receptive fields for more neurons without distorting the ISI distribution of the incoming retina spikes. We evaluated the responses of the orientation-tuned spiking neurons for different receptive field sizes and aspect ratios and also in the absence and presence of feedback inhibition. In a feedforward model, the aVLSI spiking neurons show orientation selectivity similar to digital simulations of continuous-valued neurons. Adding inhibition increased the selectivity of the spiking neurons between orthogonal orientations. We can extend the multi-chip VLSI system in this work to a more sophisticated system that supports multiple senders and multiple receivers. Such a system can be used, for example, to implement multi-scale cortical models. The success of the system in this work opens up the way for more elaborate spike-based emulations in the future. 6 Acknowledgements We acknowledge T. Horiuchi for the original design of the transceiver chip and David Lawrence for the software driver development in this work. This work was supported in part by the Swiss National Foundation Research SPP grant and the K¨obler Foundation. References Ben-Yishai, R., Bar-Or, R. L., and Sompolinsky, H. (1995). Theory of orientation tuning in visual cortex. P. Natl. Acad. Sci. USA, 92(9):3844–3848. Boahen, K. A. (1996). A retinomorphic vision system. IEEE Micro, 16(5):30–39. Boahen, K. A., Andreou, A., Hinck, T., Kramer, J., and Whatley, A. (1997). Computation- and memory-based projective field processors. In Sejnowski, T., Koch, C., and Douglas, R., editors, Telluride NSF workshop on neuromorphic engineering, Telluride, CO. Douglas, R., Koch, C., Mahowald, M., Martin, K., and Suarez, H. (1995). Recurrent excitation in neocortical circuits. Science, 269(5226):981–985. Douglas, R. and Martin, K. (1991). A functional microcircuit for cat visual cortex. J. Physiol., 440:735–769. Hubel, D. and Wiesel, T. (1962). Receptive fields, binocular interaction and functional architecture. J. of Physio.(Lond), 160:106–154. Kramer, J. (2001). An integrated optical transient sensor. Submitted for publication. Lazzaro, J., Wawrzynek, J., Mahowald, M., Sivilotti, M., and Gillespie, D. (1993). Silicon auditory processors as computer peripherals. IEEE Transactions on Neural Networks, 4(3):523–528. Liu, S.-C., Kramer, J., Indiveri, G., Delbruck, T., Burg, T., and Douglas, R. (2001). Orientation-selective aVLSI spiking neurons. Neural Networks, 14(6/7):629–643. Special Issue on Spiking Neurons in Neuroscience and Technology. Somers, D., Nelson, S., and Sur, M. (1995). An emergent model of orientation selectivity in cat visual cortex simple cells. Journal of Neuroscience, 15(8):5448–5465. Whatley, A., Kramer, J., and Douglas, R. (1997). ON/OFF retina to silicon cortex. In Sejnowski, T., Koch, C., and Douglas, R., editors, Telluride NSF workshop on neuromorphic engineering, Telluride, CO.
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Natural Language Grammar Induction using a Constituent-Context Model Dan Klein and Christopher D. Manning Computer Science Department Stanford University Stanford, CA 94305-9040 {klein, manning}@cs.stanford.edu Abstract This paper presents a novel approach to the unsupervised learning of syntactic analyses of natural language text. Most previous work has focused on maximizing likelihood according to generative PCFG models. In contrast, we employ a simpler probabilistic model over trees based directly on constituent identity and linear context, and use an EM-like iterative procedure to induce structure. This method produces much higher quality analyses, giving the best published results on the ATIS dataset. 1 Overview To enable a wide range of subsequent tasks, human language sentences are standardly given tree-structure analyses, wherein the nodes in a tree dominate contiguous spans of words called constituents, as in figure 1(a). Constituents are the linguistically coherent units in the sentence, and are usually labeled with a constituent category, such as noun phrase (NP) or verb phrase (VP). An aim of grammar induction systems is to figure out, given just the sentences in a corpus S, what tree structures correspond to them. In this sense, the grammar induction problem is an incomplete data problem, where the complete data is the corpus of trees T, but we only observe their yields S. This paper presents a new approach to this problem, which gains leverage by directly making use of constituent contexts. It is an open problem whether entirely unsupervised methods can produce linguistically accurate parses of sentences. Due to the difficulty of this task, the vast majority of statistical parsing work has focused on supervised learning approaches to parsing, where one uses a treebank of fully parsed sentences to induce a model which parses unseen sentences [7, 3]. But there are compelling motivations for unsupervised grammar induction. Building supervised training data requires considerable resources, including time and linguistic expertise. Investigating unsupervised methods can shed light on linguistic phenomena which are implicit within a supervised parser’s supervisory information (e.g., unsupervised systems often have difficulty correctly attaching subjects to verbs above objects, whereas for a supervised parser, this ordering is implicit in the supervisory information). Finally, while the presented system makes no claims to modeling human language acquisition, results on whether there is enough information in sentences to recover their structure are important data for linguistic theory, where it has standardly been assumed that the information in the data is deficient, and strong innate knowledge is required for language acquisition [4]. S NP NN1 Factory NNS payrolls VP VBD fell PP IN in NN2 September Node Constituent Context S NN NNS VBD IN NN ⋄– ⋄ NP NN NNS ⋄– VBD VP VBD IN NN NNS – ⋄ PP IN NN VBD – ⋄ NN1 NN ⋄– NNS NNS NNS NN – VBD VBD VBD NNS – IN IN IN VBD – NN NN2 NNS IN – ⋄ Empty Context ϵ0 ⋄– NN ϵ1 NN – NNS ϵ2 NNS – VBD ϵ3 VBD – IN ϵ4 IN – NN ϵ5 NN – ⋄ Figure 1: Example parse tree with the constituents and contexts for each tree node. 2 Previous Approaches One aspect of grammar induction where there has already been substantial success is the induction of parts-of-speech. Several different distributional clustering approaches have resulted in relatively high-quality clusterings, though the clusters’ resemblance to classical parts-of-speech varies substantially [9, 15]. For the present work, we take the part-ofspeech induction problem as solved and work with sequences of parts-of-speech rather than words. In some ways this makes the problem easier, such as by reducing sparsity, but in other ways it complicates the task (even supervised parsers perform relatively poorly with the actual words replaced by parts-of-speech). Work attempting to induce tree structures has met with much less success. Most grammar induction work assumes that trees are generated by a symbolic or probabilistic context-free grammar (CFG or PCFG). These systems generally boil down to one of two types. Some fix the structure of the grammar in advance [12], often with an aim to incorporate linguistic constraints [2] or prior knowledge [13]. These systems typically then attempt to find the grammar production parameters 2 which maximize the likelihood P(S|2) using the inside-outside algorithm [1], which is an efficient (dynamic programming) instance of the EM algorithm [8] for PCFGs. Other systems (which have generally been more successful) incorporate a structural search as well, typically using a heuristic to propose candidate grammar modifications which minimize the joint encoding of data and grammar using an MDL criterion, which asserts that a good analysis is a short one, in that the joint encoding of the grammar and the data is compact [6, 16, 18, 17]. These approaches can also be seen as likelihood maximization where the objective function is the a posteriori likelihood of the grammar given the data, and the description length provides a structural prior. The “compact grammar” aspect of MDL is close to some traditional linguistic argumentation which at times has argued for minimal grammars on grounds of analytical [10] or cognitive [5] economy. However, the primary weakness of MDL-based systems does not have to do with the objective function, but the search procedures they employ. Such systems end up growing structures greedily, in a bottom-up fashion. Therefore, their induction quality is determined by how well they are able to heuristically predict what local intermediate structures will fit into good final global solutions. A potential advantage of systems which fix the grammar and only perform parameter search is that they do compare complete grammars against each other, and are therefore able to detect which give rise to systematically compatible parses. However, although early work showed that small, artificial CFGs could be induced with the EM algorithm [12], studies with large natural language grammars have generally suggested that completely unsupervised EM over PCFGs is ineffective for grammar acquisition. For instance, Carroll and Charniak [2] describe experiments running the EM algorithm from random starting points, which produced widely varying learned grammars, almost all of extremely poor quality.1 1We duplicated one of their experiments, which used grammars restricted to rules of the form x →x y | y x, where there is one category x for each part-of-speech (such a restricted CFG is isomorphic to a dependency grammar). We began reestimation from a grammar with uniform rewrite It is well-known that EM is only locally optimal, and one might think that the locality of the search procedure, not the objective function, is to blame. The truth is somewhere in between. There are linguistic reasons to distrust an ML objective function. It encourages the symbols and rules to align in ways which maximize the truth of the conditional independence assumptions embodied by the PCFG. The symbols and rules of a natural language grammar, on the other hand, represent syntactically and semantically coherent units, for which a host of linguistic arguments have been made [14]. None of these have anything to do with conditional independence; traditional linguistic constituency reflects only grammatical regularities and possibilities for expansion. There are expected to be strong connections across phrases (such as dependencies between verbs and their selected arguments). It could be that ML over PCFGs and linguistic criteria align, but in practice they do not always seem to. Experiments with both artificial [12] and real [13] data have shown that starting from fixed, correct (or at least linguistically reasonable) structure, EM produces a grammar which has higher log-likelihood than the linguistically determined grammar, but lower parsing accuracy. However, we additionally conjecture that EM over PCFGs fails to propagate contextual cues efficiently. The reason we expect an algorithm to converge on a good PCFG is that there seem to be coherent categories, like noun phrases, which occur in distinctive environments, like between the beginning of the sentence and the verb phrase. In the inside-outside algorithm, the product of inside and outside probabilities α j(p, q)β j(p, q) is the probability of generating the sentence with a j constituent spanning words p through q: the outside probability captures the environment, and the inside probability the coherent category. If we had a good idea of what VPs and NPs looked like, then if a novel NP appeared in an NP context, the outside probabilities should pressure the sequence to be parsed as an NP. However, what happens early in the EM procedure, when we have no real idea about the grammar parameters? With randomly-weighted, complete grammars over a symbol set X, we have observed that a frequent, short, noun phrase sequence often does get assigned to some category x early on. However, since there is not a clear overall structure learned, there is only very weak pressure for other NPs, even if they occur in the same positions, to also be assigned to x, and the reestimation process goes astray. To enable this kind of constituent-context pressure to be effective, we propose the model in the following section. 3 The Constituent-Context Model We propose an alternate parametric family of models over trees which is better suited for grammar induction. Broadly speaking, inducing trees like the one shown in figure 1(a) can be broken into two tasks. One is deciding constituent identity: where the brackets should be placed. The second is deciding what to label the constituents. These tasks are certainly correlated and are usually solved jointly. However, the task of labeling chosen brackets is essentially the same as the part-of-speech induction problem, and the solutions cited above can be adapted to cluster constituents [6]. The task of deciding brackets, is the harder task. For example, the sequence DT NN IN DT NN ([the man in the moon]) is virtually always a noun phrase when it is a constituent, but it is only a constituent 66% of the time, because the IN DT NN is often attached elsewhere ([we [sent a man] [to the moon]]). Figure 2(a) probabilities. Figure 4 shows that the resulting grammar (DEP-PCFG) is not as bad as conventional wisdom suggests. Carroll and Charniak are right to observe that the search spaces is riddled with pronounced local maxima, and EM does not do nearly so well when randomly initialized. The need for random seeding in using EM over PCFGs is two-fold. For some grammars, such as one over a set X of non-terminals in which any x1 →x2 x3, xi ∈X is possible, it is needed to break symmetry. This is not the case for dependency grammars, where symmetry is broken by the yields (e.g., a sentence noun verb can only be covered by a noun or verb projection). The second reason is to start the search from a random region of the space. But unless one does many random restarts, the uniform starting condition is better than most extreme points in the space, and produces superior results. −1.5 −1 −0.5 0 0.5 1 −3 −2 −1 0 1 2 NP VP PP −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 Usually a Constituent Rarely a Constituent (a) (b) Figure 2: The most frequent examples of (a) different constituent labels and (b) constituents and non-constituents, in the vector space of linear contexts, projected onto the first two principal components. Clustering is effective for labeling, but not detecting constituents. shows the 50 most frequent constituent sequences of three types, represented as points in the vector space of their contexts (see below), projected onto their first two principal components. The three clusters are relatively coherent, and it is not difficult to believe that a clustering algorithm could detect them in the unprojected space. Figure 2(a), however, shows 150 sequences which are parsed as constituents at least 50% of the time along with 150 which are not, again projected onto the first two components. This plot at least suggests that the constituent/non-constituent classification is less amenable to direct clustering. Thus, it is important that an induction system be able to detect constituents, either implicitly or explicitly. A variety of methods of constituent detection have been proposed [11, 6], usually based on information-theoretic properties of a sequence’s distributional context. However, here we rely entirely on the following two simple assumptions: (i) constituents of a parse do not cross each other, and (ii) constituents occur in constituent contexts. The first property is self-evident from the nature of the parse trees. The second is an extremely weakened version of classic linguistic constituency tests [14]. Let σ be a terminal sequence. Every occurrence of σ will be in some linear context c(σ) = x σ y, where x and y are the adjacent terminals or sentence boundaries. Then we can view any tree t over a sentence s as a collection of sequences and contexts, one of each for every node in the tree, plus one for each inter-terminal empty span, as in figure 1(b). Good trees will include nodes whose yields frequently occur as constituents and whose contexts frequently surround constituents. Formally, we use a conditional exponential model of the form: P(t|s, 2) = exp(P (σ,c)∈t λσ fσ + λc fc) P t:yield(t)=s exp(P (σ,c)∈t λσ fσ + λc fc) We have one feature fσ (t) for each sequence σ whose value on a tree t is the number of nodes in t with yield σ, and one feature fc(t) for each context c representing the number of times c is the context of the yield of some node in the tree.2 No joint features over c and σ are used, and, unlike many other systems, there is no distinction between constituent types. We model only the conditional likelihood of the trees, P(T |S, 2), where 2 = {λσ, λc}. We then use an iterative EM-style procedure to find a local maximum P(T |S, 2) of the completed data (trees) T (P(T |S, 2) = Q t∈T,s=yield(t) P(t|s, 2)). We initialize 2 such that each λ is zero and initialize T to any arbitrary set of trees. In alternating steps, we first fix the parameters 2 and find the most probable single tree structure t∗for each sentence s according to P(t|s, 2), using a simple dynamic program. For any 2 this produces the 2So, for the tree in figure 1(a), P(t|s) ∝exp(λNN NNS + λVBD IN NN + λIN NN + λ⋄−VBD + λNNS−⋄+ λVBD−⋄+ λ⋄−NNS + λNN−VBD + λNNS−IN + λVBD−NN + λIN−⋄). set of parses T ∗which maximizes P(T |S, 2). Since T ∗maximizes this quantity, if T ′ is the former set of trees, P(T ∗|S, 2) ≥P(T ′|S, 2). Second, we fix the trees and estimate new parameters 2. The task of finding the parameters 2∗which maximize P(T |S, 2) is simply the well-studied task of fitting our exponential model to maximize the conditional likelihood of the fixed parses. Running, for example, a conjugate gradient (CG) ascent on 2 will produce the desired 2∗. If 2′ is the former parameters, then we will have P(T |S, 2∗) ≥P(T |S, 2′). Therefore, each iteration will increase P(T |S, 2) until convergence.3 Note that our parsing model is not a generative model, and this procedure, though clearly related, is not exactly an instance of the EM algorithm. We merely guarantee that the conditional likelihood of the data completions is increasing. Furthermore, unlike in EM where each iteration increases the marginal likelihood of the fixed observed data, our procedure increases the conditional likelihood of a changing complete data set, with the completions changing at every iteration as we reparse. Several implementation details were important in making the system work well. First, tiebreaking was needed, most of all for the first round. Initially, the parameters are zero, and all parses are therefore equally likely. To prevent bias, all ties were broken randomly. Second, like so many statistical NLP tasks, smoothing was vital. There are features in our model for arbitrarily long yields and most yield types occurred only a few times. The most severe consequence of this sparsity was that initial parsing choices could easily become frozen. If a λσ for some yield σ was either ≫0 or ≪0, which was usually the case for rare yields, σ would either be locked into always occurring or never occurring, respectively. Not only did we want to push the λσ values close to zero, we also wanted to account for the fact that most spans are not constituents.4 Therefore, we expect the distribution of the λσ to be skewed towards low values.5 A greater amount of smoothing was needed for the first few iterations, while much less was required in later iterations. Finally, parameter estimation using a CG method was slow and difficult to smooth in the desired manner, and so we used the smoothed relative frequency estimates λσ = count( fσ )/(count(σ) + M) and λc = count( fc)/(count(c) + N). These estimates ensured that the λ values were between 0 and 1, and gave the desired bias towards non-constituency. These estimates were fast and surprisingly effective, but do not guarantee non-decreasing conditional likelihood (though the conditional likelihood was increasing in practice).6 4 Results In all experiments, we used hand-parsed sentences from the Penn Treebank. For training, we took the approximately 7500 sentences in the Wall Street Journal (WSJ) section which contained 10 words or fewer after the removal of punctuation. For testing, we evaluated the system by comparing the system’s parses for those same sentences against the supervised parses in the treebank. We consider each parse as a set of constituent brackets, discarding all trivial brackets.7 We calculated the precision and recall of these brackets against the treebank parses in the obvious way. 3In practice, we stopped the system after 10 iterations, but final behavior was apparent after 4–8. 4In a sentence of length n, there are (n + 1)(n + 2)/2 total (possibly size zero) spans, but only 3n constituent spans: n −1 of size ≥2, n of size 1, and n + 1 empty spans. 5Gaussian priors for the exponential model accomplish the former goal, but not the latter. 6The relative frequency estimators had a somewhat subtle positive effect. Empty spans have no effect on the model when using CG fitting, as all trees include the same empty spans. However, including their counts improved performance substantially when using relative frequency estimators. This is perhaps an indication that a generative version of this model would be advantageous. 7We discarded both brackets of length one and brackets spanning the entire sentence, since all of these are impossible to get incorrect, and hence ignored sentences of length ≤2 during testing. S NP DT The NN screen VP VBD was NP NP DT a NN sea PP IN of NP NN red σ σ σ DT The NN screen VBD was σ σ DT a NN sea σ IN of NN red VBD VBD DT DT The NN screen VBD was DT DT DT DT a NN sea IN of NN red (a) (b) (c) Figure 3: Alternate parse trees for a sentence: (a) the Penn Treebank tree (deemed correct), (b) the one found by our system CCM, and (c) the one found by DEP-PCFG. Method UP UR F1 NP UR PP UR VP UR LBRANCH 20.5 24.2 22.2 28.9 6.3 0.6 RANDOM 29.0 31.0 30.0 42.8 23.6 26.3 DEP-PCFG 39.5 42.3 40.9 69.7 44.1 22.8 RBRANCH 54.1 67.5 60.0 38.3 44.5 85.8 CCM 60.1 75.4 66.9 83.8 71.6 66.3 UBOUND 78.2 100.0 87.8 100.0 100.0 100.0 System UP UR F1 CB EMILE 51.6 16.8 25.4 0.84 ABL 43.6 35.6 39.2 2.12 CDC-40 53.4 34.6 42.0 1.46 RBRANCH 39.9 46.4 42.9 2.18 CCM 54.4 46.8 50.3 1.61 (a) (b) Figure 4: Comparative accuracy on WSJ sentences (a) and on the ATIS corpus (b). UR = unlabeled recall; UP = unlabeled precision; F1 = the harmonic mean of UR and UP; CB = crossing brackets. Separate recall values are shown for three major categories. To situate the results of our system, figure 4(a) gives the values of several parsing strategies. CCM is our constituent-context model. DEP-PCFG is a dependency PCFG model [2] trained using the inside-outside algorithm. Figure 3 shows sample parses to give a feel for the parses the systems produce. We also tested several baselines. RANDOM parses randomly. This is an appropriate baseline for an unsupervised system. RBRANCH always chooses the right-branching chain, while LBRANCH always chooses the left-branching chain. RBRANCH is often used as a baseline for supervised systems, but exploits a systematic right-branching tendency of English. An unsupervised system has no a priori reason to prefer right chains to left chains, and LBRANCH is well worse than RANDOM. A system need not beat RBRANCH to claim partial success at grammar induction. Finally, we include an upper bound. All of the parsing strategies and systems mentioned here give fully binary-branching structures. Treebank trees, however, need not be fully binary-branching, and generally are not. As a result, there is an upper bound UBOUND on the precision and F1 scores achievable when structurally confined to binary trees. Clearly, CCM is parsing much better than the RANDOM baseline and the DEP-PCFG induced grammar. Significantly, it also out-performs RBRANCH in both precision and recall, and, to our knowledge, it is the first unsupervised system to do so. To facilitate comparison with other recent systems, figure 4(b) gives results where we trained as before but used (all) the sentences from the distributionally different ATIS section of the treebank as a test set. For this experiment, precision and recall were calculated using the EVALB system of measuring precision and recall (as in [6, 17]) – EVALB is a standard for parser evaluation, but complex, and unsuited to evaluating unlabeled constituency. EMILE and ABL are lexical systems described in [17]. The results for CDC-40, from [6], reflect training on much more data (12M words). Our system is superior in terms of both precision and recall (and so F1). These figures are certainly not all that there is to say about an induced grammar; there are a number of issues in how to interpret the results of an unsupervised system when comparing with treebank parses. Errors come in several kinds. First are innocent sins of commission. Treebank trees are very flat; for example, there is no analysis of the inside of many short noun phrases ([two hard drives] rather than [two [hard drives]]). Our system gives a Sequence Example CORRECT FREQUENCY ENTROPY DEP-PCFG CCM DT NN the man 1 2 2 1 1 NNP NNP United States 2 1 – 2 2 CD CD 4 1/2 3 9 – 5 5 JJ NNS daily yields 4 7 3 4 4 DT JJ NN the top rank 5 – – 7 6 DT NNS the people 6 – – – 10 JJ NN plastic furniture 7 3 7 3 3 CD NN 12 percent 8 – – – 9 IN NN on Monday 9 – 9 – – IN DT NN for the moment 10 – – – – NN NNS fire trucks 11 – 6 – 8 NN NN fire truck 22 8 10 – 7 TO VB to go 26 – 1 6 – DT JJ ?the big 78 6 – – – IN DT *of the 90 4 – 10 – PRP VBZ ?he says 95 – – 8 – PRP VBP ?they say 180 – – 9 – NNS VBP ?people are =350 – 4 – – NN VBZ ?value is =532 10 5 – – NN IN *man from =648 5 – – – NNS VBD ?people were =648 – 8 – – Figure 5: Top non-trivial sequences by actual treebank constituent counts, linear frequency, scaled context entropy, and in DEP-PCFG and CCM learned models’ parses. (usually correct) analysis of the insides of such NPs, for which it is penalized on precision (though not recall or crossing brackets). Second are systematic alternate analyses. Our system tends to form modal verb groups and often attaches verbs first to pronoun subjects rather than to objects. As a result, many VPs are systematically incorrect, boosting crossing bracket scores and impacting VP recall. Finally, the treebank’s grammar is sometimes an arbitrary, and even inconsistent standard for an unsupervised learner: alternate analyses may be just as good.8 Notwithstanding this, we believe that the treebank parses have enough truth in them that parsing scores are a useful component of evaluation. Ideally, we would like to inspect the quality of the grammar directly. Unfortunately, the grammar acquired by our system is implicit in the learned feature weights. These are not by themselves particularly interpretable, and not directly comparable to the grammars produced by other systems, except through their functional behavior. Any grammar which parses a corpus will have a distribution over which sequences tend to be analyzed as constituents. These distributions can give a good sense of what structures are and are not being learned. Therefore, to supplement the parsing scores above, we examine these distributions. Figure 5 shows the top scoring constituents by several orderings. These lists do not say very much about how long, complex, recursive constructions are being analyzed by a given system, but grammar induction systems are still at the level where major mistakes manifest themselves in short, frequent sequences. CORRECT ranks sequences by how often they occur as constituents in the treebank parses. DEP-PCFG and CCM are the same, but use counts from the DEP-PCFG and CCM parses. As a baseline, FREQUENCY lists sequences by how often they occur anywhere in the sentence yields. Note that the sequence IN DT (e.g., “of the”) is high on this list, and is a typical error of many early systems. Finally, ENTROPY is the heuristic proposed in [11] which ranks by context entropy. It is better in practice than FREQUENCY, but that isn’t self-evident from this list. Clearly, the lists produced by the CCM system are closer to correct than the others. They look much like a censored version of the FREQUENCY list, where sequences which do not co-exist with higher-ranked ones have been removed (e.g., IN DT often crosses DT NN). This observation may explain a good part of the success of this method. Another explanation for the surprising success of the system is that it exploits a deep fact about language. Most long constituents have some short, frequent equivalent, or proform, which occurs in similar contexts [14]. In the very common case where the proform is a single word, it is guaranteed constituency, which will be transmitted to longer sequences 8For example, transitive sentences are bracketed [subject [verb object]] (The president [executed the law]) while nominalizations are bracketed [[possessive noun] complement] ([The president’s execution] of the law), an arbitrary inconsistency which is unlikely to be learned automatically. via shared contexts (categories like PP which have infrequent proforms are not learned well unless the empty sequence is in the model – interestingly, the empty sequence appears to act as the proform for PPs, possibly due to the highly optional nature of many PPs). 5 Conclusions We have presented an alternate probability model over trees which is based on simple assumptions about the nature of natural language structure. It is driven by the explicit transfer between sequences and their contexts, and exploits both the proform phenomenon and the fact that good constituents must tile in ways that systematically cover the corpus sentences without crossing. The model clearly has limits. Lacking recursive features, it essentially must analyze long, rare constructions using only contexts. However, despite, or perhaps due to its simplicity, our model predicts bracketings very well, producing higher quality structural analyses than previous methods which employ the PCFG model family. Acknowledgements. We thank John Lafferty, Fernando Pereira, Ben Taskar, and Sebastian Thrun for comments and discussion. This paper is based on work supported in part by the National Science Foundation under Grant No. IIS-0085896. References [1] James K. Baker. Trainable grammars for speech recognition. In D. H. Klatt and J. J. Wolf, editors, Speech Communication Papers for the 97th Meeting of the ASA, pages 547–550, 1979. [2] Glenn Carroll and Eugene Charniak. Two experiments on learning probabilistic dependency grammars from corpora. In C. Weir, S. Abney, R. Grishman, and R. Weischedel, editors, Working Notes of the Workshop Statistically-Based NLP Techniques, pages 1–13. AAAI Press, 1992. [3] Eugene Charniak. A maximum-entropy-inspired parser. In NAACL 1, pages 132–139, 2000. [4] Noam Chomsky. Knowledge of Language. Prager, New York, 1986. [5] Noam Chomsky & Morris Halle. The Sound Pattern of English. Harper & Row, NY, 1968. [6] Alexander Clark. Unsupervised induction of stochastic context-free grammars using distributional clustering. In The Fifth Conference on Natural Language Learning, 2001. [7] Michael John Collins. Three generative, lexicalised models for statistical parsing. In ACL 35/EACL 8, pages 16–23, 1997. [8] A.P. Dempster, N.M. Laird, and D.B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. J. Royal Statistical Society Series B, 39:1–38, 1977. [9] Steven Finch and Nick Chater. Distributional bootstrapping: From word class to proto-sentence. In Proceedings of the Sixteenth Annual Conference of the Cognitive Science Society, pages 301– 306, Hillsdale, NJ, 1994. Lawrence Erlbaum. [10] Zellig Harris. Methods in Structural Linguistics. University of Chicago Press, Chicago, 1951. [11] Dan Klein and Christopher D. Manning. Distributional phrase structure induction. In The Fifth Conference on Natural Language Learning, 2001. [12] K. Lari and S. J. Young. The estimation of stochastic context-free grammars using the insideoutside algorithm. Computer Speech and Language, 4:35–56, 1990. [13] Fernando Pereira and Yves Schabes. Inside-outside reestimation from partially bracketed corpora. In ACL 30, pages 128–135, 1992. [14] Andrew Radford. Transformational Grammar. Cambridge University Press, Cambridge, 1988. [15] Hinrich Sch¨utze. Distributional part-of-speech tagging. In EACL 7, pages 141–148, 1995. [16] Andreas Stolcke and Stephen M. Omohundro. Inducing probabilistic grammars by Bayesian model merging. In Grammatical Inference and Applications: Proceedings of the Second International Colloquium on Grammatical Inference. Springer Verlag, 1994. [17] M. van Zaanen and P. Adriaans. Comparing two unsupervised grammar induction systems: Alignment-based learning vs. emile. Technical Report 2001.05, University of Leeds, 2001. [18] J. G. Wolff. Learning syntax and meanings through optimization and distributional analysis. In Y. Levy, I. M. Schlesinger, and M. D. S. Braine, editors, Categories and processes in language acquisition, pages 179–215. Lawrence Erlbaum, Hillsdale, NJ, 1988.
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Estimating Car Insurance Premia: a Case Study in High-Dimensional Data Inference Nicolas Chapados, Yoshua Bengio, Pascal Vincent, Joumana Ghosn, Charles Dugas, Ichiro Takeuchi, Linyan Meng University of Montreal, dept. IRQ, CP 6128, Succ. Centre-Ville, Montreal, Qc, Canada, H3C3J7 {chapadosJbengioy,vincentp,ghosnJdugas,takeuchi,mengl}~iro.umontreal.ca Abstract Estimating insurance premia from data is a difficult regression problem for several reasons: the large number of variables, many of which are .discrete, and the very peculiar shape of the noise distribution, asymmetric with fat tails, with a large majority zeros and a few unreliable and very large values. We compare several machine learning methods for estimating insurance premia, and test them on a large data base of car insurance policies. We find that function approximation methods that do not optimize a squared loss, like Support Vector Machines regression, do not work well in this context. Compared methods include decision trees and generalized linear models. The best results are obtained with a mixture of experts, which better identifies the least and most risky contracts, and allows to reduce the median premium by charging more to the most risky customers. 1 Introduction The main mathematical problem faced by actuaries is that of estimating how much each insurance contract is expected to cost. This conditional expected claim amount is called the pure premium and it is the basis of the gross premium charged to the insured. This expected value is conditionned on information available about the insured and about the contract, which we call input profile here. This regression problem is difficult for several reasons: large number of examples, -large number variables (most of which are discrete and multi-valued), non-stationarity of the distribution, and a conditional distribution of the dependent variable which is very different from those usually encountered in typical applications .of machine learning and function approximation. This distribution has a mass at zero: the vast majority of the insurance contracts do not yield any claim. This distribution is also strongly asymmetric and it has fat tails (on one side only, corresponding to the large claims). In this paper we study and compare several learning algorithms along with methods traditionally used by actuaries for setting insurance premia. The study is performed on a large database of automobile insurance policies. The methods that were tried are the following: the constant (unconditional) predictor as a benchmark, linear regression, generalized linear models (McCullagh and NeIder, 1989), decision tree models (CHAID (Kass, 1980)), support vector machine regression (Vapnik, 1998), multi-layer neural networks, mixtures of neural network experts, and the current premium structure of the insurance company. In a variety of practical applications, we often find data distributions with an asymmetric heavy tail extending out towards more positive values. Modeling data with such an asymmetric heavy-tail distribution is essentially difficult because outliers, which are sampled from the tail of the distribution, have a strong influence on parameter estimation. When the distribution is symmetric (around the mean), the problems caused by outliers can be reduced using robust estimation techniques (Huber, 1982; F.R.Hampel et al., 1986; Rousseeuw and Leroy, 1987) which basically intend to ignore or downweight outliers. Note that these techniques do not work for an asymmetric distribution: most outliers are on the same side of the mean, so downweighting them introduces a strong bias on its estimation: the conditional expectation would be systematically underestimated. There is another statistical difficulty, due to the large number of variables (mostly discrete) and the fact that many interactions exist between them. Thus the traditional actuarial methods based on tabulating average claim amounts for combinations of values are quickly hurt by the curse of dimensionality, unless they make hurtful independence assumptions (Bailey and Simon, 1960). Finally, there is a computational difficulty: we had access to a large database of ~ 8 x 106 examples, and the training effort and numerical stability of some algorithms can be burdensome for such a large number of training examples. This paper is organized as follows: we start by describing the mathematical criteria underlying insurance premia estimation (section 2), followed by a brief review of the learning algorithms that we consider in this study, including our best-performing mixture of positive-output neural networks (section 3). We then highlight our most important experimental results (section 4), and in view of them conclude with an examination of the prospects for applying statistical learning algorithms to insurance modeling (section 5). 2 Mathematical Objectives The first goal of insurance premia modeling is to estimate the expected claim amount for a given insurance contract for a future one-year period (here we consider that the amount is 0 when no claim is filed). Let X E Rm denote the customer and contract input profile, a vector representing all the information known about the customer and the proposed insurance policy before the beginning of the contract. Let A E R+ denote the amount that the customer claims during the contract period; we shall assume that A is non-negative. Our objective is to estimate this claim amount, which is the pure premium Ppure of a given contract x:1 Ppure(X) == E[AIX == x]. (1) The Precision Criterion. In practice, of course, we have no direct access to the quantity (1), which we must estimate. One possible criterion is to seek the most precise estimator, which minimizes the mean-squared error (MSE) over a data set D == {(xl,a£)}r=l. Let P == {p(·;8)} be a function class parametrized by the IThe pure premium is distinguished from the premium actually charged to the customer, which must account for the risk remaining with the insurer, the administrative overhead, desired profit, and other business costs. (3) parameter vector (). The MSE criterion produces the most precise function (on average) within the class, as measured with respect to D: L ()* = argm:n ~ L(P(Xi; (}) - ai)2. (2) i=1 Is it an appropriate criterion and why? First one should note that if PI and P2 are two estimators of E[AIX]' then the MSE criterion is a good indication of how close they are to E[AIX], since by the law of iterated expectations, E[(PI(X) - A)2] - E[(P2(X) - A)2] == E[(PI(X) - E[AIX])2] -E[(P2(X) - E[AIX])2], and of course the expected MSE is minimized when p(X) == E[AIX]. The Fairness Criterion. However, in insurance policy pricing, the precision criterion is not the sole part of the picture; just as important is that the estimated premia do not systematically discriminate against specific segments of the population. We call this objective the fairness criterion. We define the bias of the premia b(P) to be the difference between the average premium and the average incurred amount, in a given population P: 1 b(P) = 1FT L p(Xi) - ai, (xi,ai)EP where IPI denotes the cardinality of the set P, and p(.) is some premia estimation function. A possible fairness criterion would be based on minimizing the norm of the bias over every subpopulation Q of P. From a practical standpoint, such a minimization would be extremely difficult to carry out. Furthermore, the bias over small subpopulations is hard to estimate with statistical significance. We settle instead for an approximation that gives good empirical results. After training a model to minimize the MSE criterion (2), we define a finite number of disjoint subsets (subpopulations) of the test set P, PkC P, Pk n Pj:f;k == 0, and verify that the absolute bias is not significantly different from zero. The subsets Pk can be chosen at convenience; in our experiments, we considered 10 subsets of equal-size delimited by the deciles of the test set premium distribution. In this way, we verify that, for example, for the group of contracts with a premium between the 5th and the 6th decile, the average premium matches the average claim amount. 3 Models Evaluated An important requirement for any model of insurance premia is that it should produce positive premia: the company does not want to charge negative money to its customers! To obtain positive outputs neural networks we have considered using an exponential activation function at the output layer but this created numerical difficulties (when the argument of the exponential is large, the gradient is huge). fustead, we have successfully used the "softplus" activation function (Dugas et al., 2001): softplus(s) == log(1 + e 8 ) where s is the weighted sum of an output neuron, and softplus(s) is the corresponding predicted premium. Note that this function is convex, monotone increasing, and can be considered as a smooth version of the "positive part" function max(O, x). The best model that we obtained is a mixture of experts in which the experts are positive outputs neural networks. The gater network (Jacobs et al., 1991) has softmax outputs to obtain positive w~ights summing to one. X 10-3 Distribution of (claim - prediction) in each prediction quintile 2 1.8 1.6 1.4 1.2 1000 2000 3000 4000 5000 6000 claim - prediction oL-.._..I....=~-L-~-l...-_----L_----l-==:::=:~::=::::::r:::===:±==~ -3000 -2000 -1000 0.8 0.2 0.4 0.6 Proportion of non-zero claims in each prediction quintile 0.25r---r-------,r----------,------,.--------,---, 0.15 0.1 0.05 3 quintile Figure 1: A view of the conditional distribution of the claim amounts in the out-ofsample test set. Top: probability density of (claim amount - conditional expectation) for 5 quintiles of the conditional expectation, excluding zero-claim records. The mode moves left for increasing conditional expectation quintiles. Bottom: proportion of non-zero claim records per quintile of the prediction. The mixture model was compared to other models. The constant model only has intercepts as free parameters. The linear model corresponds to a ridge linear regression (with weight decay chosen with the validation set). Generalized linear models (GLM) estimate the conditional expectation from j(x) == eb+w1x with parameters b and w. Again weight decay is used and tuned on the validation set. There are many variants of GLMs and they are popular for building insurance models, since they provide positive outputs, interpretable parameters, and can be associated to parametric models of the noise. Decision trees are also used by practitioners in the insurance industry, in particular the CHAID-type models (Kass, 1980; Biggs, Ville and Suen, 1991), which use statistical criteria for deciding how to split nodes and when to stop growing the tree. We have compared our models with a CHAID implementation based on (Biggs, Ville and Suen, 1991), adapted for regression purposes using a MANOVA analysis. The threshold parameters were selected based on validation set MSE. Regression Support Vector Machines (SVM) (Vapnik, 1998) were also evaluated 67.1192 67.0851 56.5744 56.5416 Mean-Squared Error .......................................................................................... :;....:-.:--'*..... -*"-----.... ~~--:-~-:-~~..:..:.~:-.:-.. -.. - -~-~--:.-.. _..-.~~--'---~.. : . Test Validation 56.1108 . Training 56.0743 Figure 2: MSE results for eight models. Models have been sorted in ascending order of test results. The training, validation and test curves have been shifted closer together for visualization purposes (the significant differences in MSE between the 3 sets are due to "outliers"). The out-of-sample test performance of the Mixture model is significantly better than any of the other. Validation based model selection is confirmed on test results. CondMean is a constructive greedy version of GLM. but yielded disastrous results for two reasons: (1) SVM regression optimizes an L 1like criterion that finds a solution close to the conditional median, whereas the MSE criterion is minimized for the conditional mean, and because the distribution is highly asymmetric the conditional median is far from the conditional mean; (2) because the output variable is difficult to predict, the required number of support vectors is huge, also yielding poor generalization. Since the median is actually 0 for our data, we tried to train the SVM using only the cases with positive claim amounts, and compared the performance to that obtained with the GLM and the neural network. The SVM is still way off the mark because of the above two reasons. Figure 1 (top) illustrates the fat tails and asymetry of the conditional distribution of the claim amounts. . Finally, we compared the best statistical model with a proprietary table-based and rule-based premium estimation method that was provided to us as the benchmark against which to judge improvements. 4 Experimental Results Data from five kinds of losses were included in the study (Le. a sub-premium was estimated for each type of loss), but we report mostly aggregated results showing the error on the total estimated premium. The input variables contain information about the policy (e.g., the date to deal with inflation, deductibles and options), the car, and the driver (e.g., about past claims, past infractions, etc...). Most variables are subject to discretization and binning. Whenever possible, the bins are chosen such that they contain approximately the same number of observations. For most models except CHAID, the discrete variables are one-hot encoded. The number of input random variables is 39, all discrete except one, but using one-hot encoding this results in an input vector x of length m == 266. An overall data set containing about Table 1: Statistical comparison of the prediction accuracy difference between several individual learning models and the best Mixture model. The p-value is given under the null hypothesis oino difference between Model #1 and the best Mixture model. Note that all differences are statistically significant. Model #1 Model #2 Mean MSE Diff. Std. Error Z p-value Constant Mixture 3.40709e-02 3.32724e-03 10.2400 0 CHAID Mixture 2.35891e-02 2.57762e-03 9.1515 0 GLM Mixture 7.54013e-03 1.15020e-03 6.5555 2.77e-ll Softplus NN Mixture 6.71066e-03 1.09351e-03 6.1368 4.21e-l0 Linear Mixture 5.82350e-03 1.32211e-03 4.4047 5.30e-06 NN Mixture 5.23885e-03 1.41112e-03 3.7125 1.02e-04 Table 2: MSE difference between benchmark and Mixture models across the 5 claim categories (kinds of losses) and the total claim amount. In all cases except category 1, the IvIixture model is statistically significantly (p < 0.05) more precise than the benchmark model. Claim Category MSE Difference 95% Confidence Interval (Kind of Loss) Benchmark minus Mixture Lower Higher Category 1 20669.53 (-4682.83 46021.89 ) Category 2 1305.57 (1032.76 1578.37 ) Category 3 244.34 (6.12 482.55 ) Category 4 1057.51 (623.42 1491.60 ) Category 5 1324.31 (1077.95 1570.67 ) Total claim amount 60187.60 ( 7743.96 112631.24) 8 million examples is randomly permuted and split into a training set, validation set and test set, respectively of size 50%, 25% and 25% of the total. The validation set is used to select among models (includi~g the choice of capacity), and th~ test set is used for final statistical comparisons. Sample-wise paired statistical tests are used to reduce the effect of huge per-sample variability. Figure 1 is an attempt at capturing the shape of the conditional distribution of claim amounts given input profiles, by considering the distributions of claim amounts in different quantiles of the prediction (pure premium), on the test set. The top figure excludes the point mass of zero claims and rather shows the difference between the claim amount and the estimated conditional expectation (obtained with the mixture model). The bottom histogram shows that the fraction of claims increases nicely for the higher predicted pure premia. Table 1 and Figure 2 summarize the comparison between the test MSE of the different tested models. NN is a neural network with linear output activation whereas Softplus NNhas the softplus output activations. The Mixture is the mixture of softplus neural networks. This result identifies the mixture model with softplus neural networks as the best-performing of the tested statistical models. Our conjecture is that the mixture model works better because it is more robust to the effect of "outliers" (large claims). Classical robust regression methods (Rousseeuw and Leroy, 1987) work by discarding or downweighting outliers: they cannot be applied here because the claims distribution is highly asymmetric (the extreme values are always large ones, the claims being all non-negative). Note that the capacity of each model has been tuned on the validation set. Hence, e.g. CHAID could have easily yielded lower training error, but at the price of worse generalization. 1000 500 o -1500 -1000 -500 Difference between premia ($) Rule-Based minus UdeM Mixture -2000 -2500 4 x10 2,...-------,------,-----r-------.-------,-----.---------.----......., OL.-..----L.----L----.L.----..L---~~ -3000 0.5 ~ oc (]) :::l 0(]) u: Mean = -1.5993e-1a Median = 37.5455 1.5 ... Stddev = 154.65 Figure 3: The premia difference distribution is negatively skewed, but has a positive m~dian for a mean of zero. This implies that the benchmark model (current pricing) undercharges risky customers, while overcharging typical customers. Table 2 shows a comparison of this model against the rule-based benchmark. The improvements are shown across the five types of losses. In all cases the mixture improves, and the improvement is significant in four out of the five as well as across the sum of the five. A qualitative analysis of the resulting predicted premia shows that the mixture model has smoother and more spread-out premia than the benchmark. The analysis (figure 3) also reveals that the difference between the mixture premia and the benchmark premia is negatively skewed, with a positive median, i.e., the typical customer will pay less under the new mixture model, but the "bad" (risky) customers will pay much more. To evaluate fairness, as discussed in the previous section, the distribution of premia computed by the best model is analyzed, splitting the contracts in 10 groups according to their premium level. Figure 4 shows that the premia charged are fair for each sub-population. 5 Conclusion This paper illustrates a successful data-mining application in the insurance industry. It shows that a specialized model (the mixture model), that was designed taking into consideration the specific problem posed by the data (outliers, asymmetric distribution, positive outputs), performs significantly better than existing and popular learning algorithms. It also shows that such models can significantly improve over the current practice, allowing to compute premia that are lower for less risky contracts and higher for more risky contracts, thereby reducing the cost of the median contract. Future work should investigate in more detail the role of temporal pon-stationarity, how to optimize fairness (rather than just test for it afterwards), and how to further increase the robustness of the model with respect to large claim amounts. 200 ~ C/} E ~ 0 "'C ~ :5 "~ -200 -5 "§ CD ~ -400 CD ~o -600 Difference with incurred claims (sum of all KOL-groups) ............ 0 ° 0 0 .' .. .. .. . .. I : : :1' \ · . . \ · . . .......: : ·1········· .\. · . \ -B- Mixture Model (normalized premia) -*- Rule-Based Model (normalized premia) 2 4 6 Decile 8 10 Figure 4: We ensure fairness by comparing the average incurred amount and premia within each decile of the premia distribution; both models are generally fair to subpopu1ations. The error bars denote 95% confidence intervals. The comparisqn is for the sum of claim amounts over all 5 kinds of losses (KOL). References Bailey, R. A. and Simon, L. (1960). Two studies in automobile insurance ratemaking. ASTIN Bulletin, 1(4):192-217. Biggs, D., Ville, B., and Suen, E. (1991). A method of choosing multiway partitions for classification and decision trees. Journal of Applied Statistics, 18(1):49-62. Dugas, C., Bengio, Y., Belisle, F., and Nadeau, C. (2001). Incorporating second order functional· knowledge into learning algorithms. In Leen, T., Dietterich, T., and Tresp, V., editors, Advances in Neural Information Processing Systems, volume 13, pages 472-478. F.R.Hampel, E.M.Ronchetti, P.J.Rousseeuw, and W.A.Stahel (1986). Robust Statistics, The Approach based on Influence Functions. John Wiley & Sons. Huber, P. (1982). Robust Statistics. John Wiley & Sons Inc. Jacobs, R. A., Jordan, M. I., Nowlan, S. J., and Hinton, G. E. (1991). Adaptive mixture of local experts. Neural Computation, 3:79-87. Kass, G. (1980). An exploratory technique for investigating large quantities of categorical data. Applied Statistics, 29(2):119-127. McCullagh, P. and NeIder, J. (1989). Generalized Linear Models. Chapman and Hall, London. ' Rousseeuw, P. and Leroy, A. (1987). Robust Regression and Outlier Detection. John Wiley & Sons Inc. Vapnik, V. (1998). Statistical Learning Theory. Wiley, Lecture Notes in Economics and Mathematical Systems, volume 454.
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Classifying Single Trial EEG: Towards Brain Computer Interfacing Benjamin Blankertz1∗, Gabriel Curio2 and Klaus-Robert Müller1,3 1Fraunhofer-FIRST.IDA, Kekuléstr. 7, 12489 Berlin, Germany 2Neurophysics Group, Dept. of Neurology, Klinikum Benjamin Franklin, Freie Universität Berlin, Hindenburgdamm 30, 12203 Berlin, Germany 3University of Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany
"!$#$&% ' ( "')*,+ -/.&! 0, 1"%02435 6 *%-/ 3 7 08 # ) 943 Abstract Driven by the progress in the field of single-trial analysis of EEG, there is a growing interest in brain computer interfaces (BCIs), i.e., systems that enable human subjects to control a computer only by means of their brain signals. In a pseudo-online simulation our BCI detects upcoming finger movements in a natural keyboard typing condition and predicts their laterality. This can be done on average 100–230ms before the respective key is actually pressed, i.e., long before the onset of EMG. Our approach is appealing for its short response time and high classification accuracy (>96%) in a binary decision where no human training is involved. We compare discriminative classifiers like Support Vector Machines (SVMs) and different variants of Fisher Discriminant that possess favorable regularization properties for dealing with high noise cases (inter-trial variablity). 1 Introduction The online analysis of single-trial electroencephalogram (EEG) measurements is a challenge for signal processing and machine learning. Once the high inter-trial variability (see Figure 1) of this complex multivariate signal can be reliably processed, the next logical step is to make use of the brain activities for real-time control of, e.g., a computer. In this work we study a pseudo-online evaluation of single-trial EEGs from voluntary self-paced finger movements and exploit the laterality of the left/right hand signal as one bit of information for later control. Features of our BCI approach are (a) no pre-selection for artifact trials, (b) state-of-the-art learning machines with inbuilt feature selection mechanisms (i.e., sparse Fisher Discriminant Analysis and SVMs) that lead to >96% classification accuracies, (c) non-trained users and (d) short response times. Although our setup was not tuned for speed, the a posteriori determined information transmission rate is 23 bits/min which makes our approach competitive to existing ones (e.g., [1, 2, 3, 4, 5, 6, 7]) that will be discussed in section 2. ∗To whom correspondence should be addressed. Aims and physiological concept of BCI devices. Two key issues to start with when conceiving a BCI are (1) the definition of a behavioral context in which a subject’s brain signals will be monitored and used eventually as surrogate for a bodily, e.g., manual, input of computer commands, and (2) the choice of brain signals which are optimally capable to convey the subject’s intention to the computer. Concerning the behavioral context, typewriting on a computer keyboard is a highly overlearned motor competence. Accordingly, a natural first choice is a BCI-situation which induces the subject to arrive at a particular decision that is coupled to a predefined (learned) motor output. This approach is well known as a two alternative forced choice-reaction task (2AFC) where one out of two stimuli (visual, auditory or somatosensory) has to be detected, categorised and responded to by issuing one out of two alternative motor commands, e.g., pushing a button with either the left or right hand. A task variant without explicit sensory input is the voluntary, endogeneous generation of a ›go‹ command involving the deliberate choice between the two possible motor outputs at a self-paced rate. Here, we chose this latter approach so as to approximate the natural computer input situation of self-paced typewriting. Concerning the selection of brain signals related to such endogeneous motor commands we focussed here on one variant of slow brain potentials which are specifically related to the preparation and execution of a motor command, rather than reflecting merely unspecific modulations of vigilance or attention. Using multi-channel EEG-mapping it has been repeatedly demonstrated that several highly localised brain areas contribute to cerebral motor command processes. Specifically, a negative ›Bereitschaftspotential‹ (BP) precedes the voluntary initiation of the movement. A differential scalp potential distribution can be reliably demonstrated in a majority of experimental subjects with larger BP at lateral scalp positions (C3, C4) positioned over the left or right hemispherical primary motor cortex, respectively, consistenly correlating with the performing (right or left) hand [8, 9]. Because one potential BCI-application is with paralysed patients, one might consider to mimic the ›no-motor-output‹ of these individuals by having healthy experimental subjects to intend a movement but to withhold its execution (motor imagery). While it is true that brain potentials comparable to BP are associated with an imagination of hand movements, which indeed is consistent with the assumption that the primary motor cortex is active with motor imagery, actual motor performance significantly increased these potentials [10]. We therefore chose to instruct the experimental subjects to actually perform the typewriting finger movements, rather than to merely imagine their performance, for two reasons: first, this will increase the BP signal strength optimising the signal-to-noise ratio in BCI-related single trial analyses; and second, we propose that it is important for the subject’s task efficiency not to be engaged in an unnatural condition where, in addition to the preparation of a motor command, a second task, i.e., to ›veto‹ the very same movement, has to be executed. In the following section we will briefly review part of the impressive earlier research towards BCI devices (e.g., [1, 2, 3, 4, 5, 6, 7]) before experimental set-up and classification results are discussed in sections 3 and 4 respectively. Finally a brief conclusion in given. 2 A brief outline of BCI research Birbaumer et al. investigate slow cortical potentials (SCP) and how they can be selfregulated in a feedback scenario. In their thought translation device [2] patients learn to produce cortical negativity or positivity at a central scalp location at will, which is fed back to the user. After some training patients are able to transmit binary decisions in a 4 sec periodicity with accuracy levels up to 85% and therewith control a language support program or an internet browser. Pfurtscheller et al. built a BCI system based on event-related (de-)synchronisation (ERD/ERS, typically of the µ and central β rhythm) for online classification of movement imaginations or preparations into 2–4 classes (e.g., left/right index finger, feet, tongue). Typical preprocessing techniques are adaptive autoregressive parameters, common spatial patterns (after band pass filtering) and band power in subject specific frequency bands. Classification is done by Fisher discriminant analysis, multi-layer neural networks or LVQ variants. In classification of exogeneous movement preparations, rates of 98%, 96% and 75% (for three subjects respectively) are obtained before movement onset1 in a 3 classes task and trials of 8 sec [3]. Only selected, artifact free trials (less that 40%) were used. A tetraplegic patient controls his hand orthosis using the Graz BCI system. Wolpaw et al. study EEG-based cursor control [4], translating the power in subject specific frequency bands, or autoregressive parameters, from two spatially filtered scalp locations over sensorimotor cortex into vertical cursor movement. Users initially gain control by various kinds of motor imagery (the setting favours ›movement‹ vs. ›no movement‹ in contrast to ›left‹ vs. ›right‹), which they report to use less and less as feedback training continues. In cursor control trials of at least 4 sec duration trained subjects reach accuracies of over 90%. Some subjects acquired also considerable control in a 2-d setup. 3 Acquisition and preprocessing of brain signals Experimental setup. The subject sat in a normal chair, relaxed arms resting on the table, fingers in the standard typing position at the computer keyboard. The task was to press with the index and little fingers the corresponding keys in a self-chosen order and timing (›self-paced key typing‹). The experiment consisted of 3 sessions of 6 minutes each, preand postceeded by 60 seconds relaxing phase. All sessions were conducted on the same day with some minutes break inbetween. Typing of a total of 516 keystrokes was done at an average speed of 1 key every 2.1 seconds. Brain activity was measured with 27 Ag/AgCl electrodes at positions of the extended international 10-20 system, 21 mounted over motor and somatosensory cortex, 5 frontal and one occipital, referenced to nasion (sampled at 1000Hz, band-pass filtered 0.05–200Hz). Besides EEG we recorded an electromyogram (EMG) of the musculus flexor digitorum bilaterally (10–200Hz) and a horizontal and vertical electrooculogram (EOG). In an event channel the timing of keystrokes was stored along with the EEG signal. All data were recorded with a NeuroScan device and converted to Matlab format for further analysis. The signals were downsampled to 100 Hz by picking every 10th sample. In a moderate rejection we sorted out only 3 out of 516 trials due to heavy measurement artifacts, while keeping trials that are contaminated by less serious artifacts or eye blinks. Note that 0.6% rejection rate is very low in contrast to most other BCI offline studies. The issue of preprocessing. Preprocessing the data can have a substantial effect on classification in terms of accuracy, effort and suitability of different algorithms. The question to what degree data should be preprocessed prior to classification is a trade-off between the danger of loosing information or overfitting and not having enough training samples for the classifier to generalize from high dimensional, noisy data. We have investigated two options: unprocessed data and preprocessing that was designed to focus on BP related to finger movement: (none) take 200ms of raw data of all relevant channels; (<5Hz) filter the signal low pass at 5 Hz, subsample it at 20 Hz and take 150ms of all relevant channels (see Figure 1); Speaking of classification at a certain time point we strictly mean classification based on EEG signals until that very time point. The following procedure of calculating features of a single trial due to (<5Hz) is easy applicable in an online scenario: Take the last 128 sample points of each channel (to the past relative from the given time point), apply a windowed (w(n) := 1−cos(nπ/128)) FFT, keep only the coefficients corresponding to the pass 1Precisely: before mean EMG onset time, for some trials this is before for others after EMG onset. −500 −400 −300 −200 −100 0 [ms] −20 −10 0 10 20 [µV] average single trial feature Figure 1: Averaged data and two single trials of right finger movements in channel C3. 3 values (marked by circles) of smoothed signals are taken as features in each channel. −260 −240 −220 −200 −180 −160 −140 −120 F3 F1 FZ F2 F4 CA5 CA3 CA1 CAZ CA2 CA4 CA6 C5 C3 C1 CZ C2 C4 C6 CP5 CP3 CP1 CPZ CP2 CP4 CP6 O1 Σ ← ↑ Figure 2: Sparse Fisher Discriminant Analysis selected 68 features (shaded) from 405 input dimensions (27 channels × 15 samples [150 ms]) of raw EEG data. band (bins 2–7, as bin 1 just contains DC information) and transform back. Downsampling to 20 Hz is done by calculating the mean of consecutive 5-tuple of data points. We investigated the alternatives of taking all 27 channels, or only the 21 located over motor and sensorimotor cortex. The 6 frontal and occipital channels are expected not to give strong contributions to the classification task. Hence a comparison shows, whether a classifier is disturbed by low information channels or if it even manages to extract information from them. Figure 1 depicts two single trial EEG signals at scalp location C3 for right finger movements. These two single trials are very well-shaped and were selected for resembling the the grand average over all 241 right finger movements, which is drawn as thick line. Usually the BP of a single trial is much more obscured by non task-related brain activity and noise. The goal of preprocessing is to reveal task-related components to a degree that they can be detected by a classifier. Figure 1 shows also the feature vectors due to preprocessing (<5 Hz) calculated from the depicted raw single trial signals. 4 From response-aligned to online classification We investigate some linear classification methods. Given a linear classifier (w,b) in separating hyperplane formulation (w⊤x+b = 0), the estimated label {1, −1} of an input vector x ∈ N is ˆy = sign(w⊤x+b). If no a priori knowledge on the probability distribution of the data is available, a typical objective is to minimize a combination of empirical risk function and some regularization term that restrains the algorithm from overfitting to the training set {(xk,yk)|k = 1,...,K}. Taking a soft margin loss function [11] yields the empirical risk function ∑K k=1 max(0,1 −yk (w⊤xk + b)). In most approaches of this type there is a hyper-parameter that determines the trade-off between risk and regularization, which has to be chosen by model selection on the training set2. Fisher Discriminant (FD) is a well known classification method, in which a projection vector is determined to maximize the distance between the projected means of the two classes while minimizing the variance of the projected data within each class [13]. In the binary decision case FD is equivalent to a least squares regression to (properly scaled) class labels. Regularized Fisher Discriminant (RFD) can be obtained via a mathematical programming approach [14]: min w,b,ξ 1/2||w||2 2 + C/K||ξ||2 2 subject to yk(w⊤xk +b) = 1−ξk for k = 1,...,K 2As this would be very time consuming in k-fold crossvalidation, we proceed similarly to [12]. filter ch’s FD RFD SFD SVM k-NN <5 Hz mc 3.7±2.6 3.3±2.2 3.3±2.2 3.2±2.5 21.6±4.9 <5 Hz all 3.3±2.5 3.1±2.5 3.4±2.7 3.6±2.5 23.1±5.8 none mc 18.1±4.8 7.0±4.1 6.4±3.4 8.5±4.3 29.6±5.9 none all 29.3±6.1 7.5±3.8 7.0±3.9 9.8±4.4 32.2±6.8 Table 3: Test set error (± std) for classification at 120 ms before keystroke; ›mc‹ refers to the 21 channels over (sensori) motor cortex, ›all‹ refers to all 27 channels. where ||·||2 denotes the ℓ2-norm (||w||2 2 = w⊤w) and C is a model parameter. The constraint yk(w⊤xk +b) = 1−ξk ensures that the class means are projected to the corresponding class labels, i.e., 1 and −1. Minimizing the length of w maximizes the margin between the projected class means relative to the intra class variance. This formalization above gives the opportunity to consider some interesting variants, e.g., Sparse Fisher Discriminant (SFD) uses the ℓ1-norm (||w||1 = Σ|wn|) on the regularizer, i.e., the goal function is 1/N ||w||1 + C/K||ξ||2 2. This choice favours solutions with sparse vectors w, so that this method also yields some feature selection (in input space). When applied to our raw EEG signals SFD selects 68 out of 405 input dimensions that allow for a left vs. right classification with good generalization. The choice coincides in general with what we would expect from neurophysiology, i.e., high loadings for electrodes close to left and right hemisphere motor cortices which increase prior to the keystroke, cf. Figure 2. But here the selection is automatically adapted to subject, electrode placement, etc. Our implementation of RFD and SFD uses the cplex optimizer [15]. Support Vector Machines (SVMs) are well known for their use with kernels [16, 17]. Here we only consider linear SVMs: min w,b,ξ 1/2||w||2 2 + C/K ||ξ||1 subject to yk(w⊤xk +b) 1−ξk, and ξk 0 The choice of regulization keeps a bound on the Vapnik-Chervonenkis dimension small. In an equivalent formulation the objective is to maximize the margin between the two classes (while minimizing the soft margin loss function)3. For comparision we also employed a standard classifier of different type: k-Nearest-Neighbor (k-NN) maps an input vector to that class to which the majority of the k nearest training samples belong. Those neighbors are determined by euclidean distance of the corresponding feature vectors. The value of k chosen by model selection was around 15 for processed and around 25 for unprocessed data. Classification of response-aligned windows. In the first phase we make full use of the information that we have regarding the timing of the keystrokes. For each single trial we calculate a feature vector as described above with respect to a fixed timing relative to the key trigger (›response-aligned‹). Table 3 reports the mean error on test sets in a 10×10fold crossvalidation for classifying in ›left‹ and ›right‹ at 120ms prior to keypress. Figure 4 shows the time course of the classification error. For comparison, the result of EMG-based classification is also displayed. It is more or less at chance level up to 120 ms before the keystroke. After that the error rate decreases rapidly. Based upon this observation we chose t =−120ms for investigating EEG-based classification. From Table 3 we see that FD works well with the preprocssed data, but as dimensionality increases the performance breaks down. k-NN is not successful at all. The reason for the failure is that the variance in the discriminating directions is much smaller that the variance in other directions. So using the euclidean metric is not an appropirate similarity measure for this purpose. All regularized discriminative classifiers attain comparable results. For preprocessed data a very low 3We used the implementation LIBSVM by Chang and Lin, available along with other implementations from
! . −1000 −800 −600 −400 −200 0 [ms] 0 10 20 30 40 50 60 −120 ms→ classification error [%] EMG EEG −200 −100 0 [ms] 0 2 4 6 8 10 12 Figure 4: Comparison of EEG (<5 Hz, mc, SFD) and EMG based classification with respect to the endpoint of the classification interval. The right panel gives a details view: -230 to 50 ms. error rate between 3% and 4% can be reached without a significant difference between the competing methods. For the classification of raw data the error is roughly twice as high. The concept of seeking sparse solution vectors allows SFD to cope very well with the high dimensional raw data. Even though the error is twice as high compared to the the minimum error, this result is very interesting, because it does not rely on preprocessing. So the SFD approach may be highly useful for online situations, when no precursory experiments are available for tuning the preprocessing. The comparison of EEG- and EMG-based classification in Figure 4 demonstrates the rapid response capability of our system: 230ms before the actual keystroke the classification rate exceeds 90%. To assess this result it has to be recalled that movements were performed spontaneously. At −120ms, when the EMG derived classifier is still close to chance, EEG based classification becomes already very stable with less than 3.5% errors. Interpreting the last result in the sense of a 2AFC gives an information transfer rate of 60/2.1B ≈22.9 [bits/min], where B = log2 N + plog2 p+(1−p)log2(1−p/N−1) is the number of bits per selection from N = 2 choices with success probability p = 1−0.035 (under some uniformity assumptions). Classification in sliding windows. The second phase is an important step towards online classification of endogeneous brain signals. We have to refrain from using event timing information (e.g., of keystrokes) in the test set. Accordingly, classification has to be performed in sliding windows and the classifier does not know in what time relation the given signals are to the event—maybe there is even no event. Technically classification could be done as before, as the trained classifiers can be applied to the feature vectors calculated from some arbitrary window. But in practice this is very likely to lead to unreliable results since those classifiers are highly specialized to signals that have a certain time relation to the response. The behavior of the classifier elsewhere is uncertain. The typical way to make classification more robust to time shifted signals is jittered training. In our case we used 4 windows for each trial, ending at -240, -160, -80 and 0 ms relative to the response (i.e., we get four feature vectors from each trial). Movement detection and pseudo-online classification. Detecting upcoming events is a crucial point in online analysis of brain signals in an unforced condition. To accomplish this, we employ a second classifier that distinguishes movement events from the ›rest‹. Figures 5 and 6 display the continuous classifier output w⊤x+b (henceforth called graded) for left/right and movement/rest distinction, respectively. For Figure 5, a classifier was trained as described above and subsequently applied to windows sliding over unseen test samples yielding ›traces‹ of graded classifier outputs. After doing this for several train/test set splits, the borders of the shaded tubes are calculated as 5 and 95 percentile values of −1000 −750 −500 −250 0 250 500 750 [ms] −1.5 −1 −0.5 0 0.5 1 1.5 right left Figure 5: Graded classifier output for left/right distinctions. −1000 −750 −500 −250 0 250 500 750 [ms] −2 −1 0 1 2
median 10, 90 percentile 5−95 perc. tube Figure 6: Graded classifier output for movement detection in endogenous brain signals. those traces, thin lines are at 10 and 90 percent, and the thick line indicates the median. At t =−100ms the median for right events in Figure 5 is approximately 0.9, i.e., applying the classifier to right events from the test set yielded in 50% of the cases an output greater 0.9 (and in 50% an output less than 0.9). The corresponding 10-percentile line is at 0.25 which means that the output to 90% of the right events was greater than 0.25. The second classifier (Figure 6) was trained for class ›movement‹ on all trials with jitters as described above and for class ›rest‹ in multiple windows between the keystrokes. The preprocessing and classification procedure was the same as for left vs. right. The classifier in Figure 5 shows a pronounced separation during the movement (preparation and execution) period. In other regions there is an overlap or even crossover of the classifier outputs of the different classes. From Figure 5 we observe that the left/right classifier alone does not distinguish reliably between ›movement‹ and ›no movement‹ by the magnitude of its output, which explains the need for a movement detector. The elevation for the left class is a little less pronounced (e.g., the median is −1 at t =0 ms compared to 1.2 for right events) which is probably due to the fact that the subject is right-handed. The movement detector in Figure 6 brings up the movement phase while giving (mainly) negative output to the post movement period. This differentiation is not as decisive as desirable, hence further research has to be pursued to improve on this. Nevertheless a pseudo-online BCI run on the recorded data using a combination of the two classifiers gave the very satisfying result of around 10% error rate. Taking this as a 3 classes choice (left, right, none) this corresponds to an information transmission rate of 29 bits/min. 5 Concluding discussion We gave an outline of our BCI system in the experimental context of voluntary self-paced movements. Our approach has the potential for high bit rates, since (1) it works at a high trial frequency, and (2) classification errors are very low. So far we have used untrained individuals, i.e., improvement can come from appropriate training schemes to shape the brain signals. The two-stage process of first a meta classification whether a movement is about to take place and then a decision between left/right finger movement is very natural and an important new feature of the proposed system. Furthermore, we reject only 0.6% of the trials due to artifacts, so our approach seems ideally suited for the true, highly noisy feedback BCI scenario. Finally, the use of state-of-the-art learning machines enables us not only to achieve high decision accuracies, but also, as a by-product of the classification, the few most prominent features that are found match nicely with physiological intuition: the most salient information can be gained between 230–100ms before the movement with a focus on C3/C4 area, i.e., over motor cortices, cf. Figure 2. There are clear perspectives for improvement in this BCI approach: our future research activities will therefore focus on (a) projection techniques like ICA, (b) time-series approaches to capture the (non-linear) dynamics of the multivariate EEG signals, and (c) construction of specially adapted kernel functions (SVM or kernel FD) in the spirit of, e.g., [17] to ultimately obtain a BCI feedback system with an even higher bit rate and accuracy. Acknowledgements. We thank S. Harmeling, M. Kawanabe, J. Kohlmorgen, J. Laub, S. Mika, G. Rätsch, R. Vigário and A. Ziehe for helpful discussions. References [1] J. J. Vidal, “Toward direct brain-computer communication”, Annu. Rev. Biophys., 2: 157–180, 1973. [2] N. Birbaumer, N. Ghanayim, T. Hinterberger, I. Iversen, B. Kotchoubey, A. Kübler, J. Perelmouter, E. Taub, and H. Flor, “A spelling device for the paralysed”, Nature, 398: 297–298, 1999. [3] B. O. Peters, G. Pfurtscheller, and H. Flyvbjerg, “Automatic Differentiation of Multichannel EEG Signals”, IEEE Trans. Biomed. Eng., 48(1): 111–116, 2001. [4] J. R. Wolpaw, D. J. McFarland, and T. M. Vaughan, “Brain-Computer Interface Research at the Wadsworth Center”, IEEE Trans. Rehab. Eng., 8(2): 222–226, 2000. [5] W. D. Penny, S. J. Roberts, E. A. Curran, and M. J. Stokes, “EEG-based cummunication: a pattern recognition approach”, IEEE Trans. Rehab. Eng., 8(2): 214–215, 2000. [6] J. D. Bayliss and D. H. Ballard, “Recognizing Evoked Potentials in a Virtual Environment”, in: S. A. Solla, T. K. Leen, and K.-R. Müller, eds., Advances in Neural Information Processing Systems, vol. 12, 3–9, MIT Press, 2000. [7] S. Makeig, S. Enghoff, T.-P. Jung, and T. J. Sejnowski, “A Natural Basis for Efficient BrainActuated Control”, IEEE Trans. Rehab. Eng., 8(2): 208–211, 2000. [8] W. Lang, O. Zilch, C. Koska, G. Lindinger, and L. Deecke, “Negative cortical DC shifts preceding and accompanying simple and complex sequential movements”, Exp. Brain Res., 74(1): 99–104, 1989. [9] R. Q. Cui, D. Huter, W. Lang, and L. Deecke, “Neuroimage of voluntary movement: topography of the Bereitschaftspotential, a 64-channel DC current source density study”, Neuroimage, 9(1): 124–134, 1999. [10] R. Beisteiner, P. Hollinger, G. Lindinger, W. Lang, and A. Berthoz, “Mental representations of movements. Brain potentials associated with imagination of hand movements”, Electroencephalogr. Clin. Neurophysiol., 96(2): 183–193, 1995. [11] K. P. Bennett and O. L. Mangasarian, “Robust Linear Programming Discrimination of two Linearly Inseparable Sets”, Optimization Methods and Software, 1: 23–34, 1992. [12] G. Rätsch, T. Onoda, and K.-R. Müller, “Soft Margins for AdaBoost”, 42(3): 287–320, 2001. [13] R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification, Wiley & Sons, 2nd edn., 2001. [14] S. Mika, G. Rätsch, and K.-R. Müller, “A mathematical programming approach to the Kernel Fisher algorithm”, in: T. K. Leen, T. G. Dietterich, and V. Tresp, eds., Advances in Neural Information Processing Systems 13, 591–597, MIT Press, 2001. [15] “ILOG Solver, ILOG CPLEX 6.5 Reference Manual”, , 1999. [16] V. Vapnik, The nature of statistical learning theory, Springer Verlag, New York, 1995. [17] K.-R. Müller, S. Mika, G. Rätsch, K. Tsuda, and B. Schölkopf, “An Introduction to KernelBased Learning Algorithms”, IEEE Transactions on Neural Networks, 12(2): 181–201, 2001.
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Kernel Logistic Regression and the Import Vector Machine Ji Zhu Department of Statistics Stanford University Stanford, CA 94305 jzhu@stat.stanford.edu Trevor Hastie Department of Statistics Stanford University Stanford, CA 94305 hastie@stat.stanford.edu Abstract The support vector machine (SVM) is known for its good performance in binary classification, but its extension to multi-class classification is still an on-going research issue. In this paper, we propose a new approach for classification, called the import vector machine (IVM), which is built on kernel logistic regression (KLR). We show that the IVM not only performs as well as the SVM in binary classification, but also can naturally be generalized to the multi-class case. Furthermore, the IVM provides an estimate of the underlying probability. Similar to the “support points” of the SVM, the IVM model uses only a fraction of the training data to index kernel basis functions, typically a much smaller fraction than the SVM. This gives the IVM a computational advantage over the SVM, especially when the size of the training data set is large. 1 Introduction In standard classification problems, we are given a set of training data ,
, , where the output is qualitative and assumes values in a finite set . We wish to find a classfication rule from the training data, so that when given a new input , we can assign a class from to it. Usually it is assumed that the training data are an independently and identically distributed sample from an unknown probability distribution . The support vector machine (SVM) works well in binary classification, i.e. "!$#&%'() , but its appropriate extension to the multi-class case is still an on-going research issue. Another weakness of the SVM is that it only estimates *+-,/.10 2 3$(54687 , while the probability 2 9 is often of interest itself, where 2 9 : ;:<(/= >:? is the conditional probability of a point being in class ( given @:A . In this paper, we propose a new approach, called the import vector machine (IVM), to address the classification problem. We show that the IVM not only performs as well as the SVM in binary classification, but also can naturally be generalized to the multi-class case. Furthermore, the IVM provides an estimate of the probability 2 . Similar to the “support points” of the SVM, the IVM model uses only a fraction of the training data to index the kernel basis functions. We call these training data import points. The computational cost of the SVM is B C$D& , while the computational cost of the IVM is B C
&E8
, where E is the number of import points. Since E does not tend to increase as C increases, the IVM can be faster than the SVM, especially for large training data sets. Empirical results show that the number of import points is usually much less than the number of support points. In section (2), we briefly review some results of the SVM for binary classification and compare it with kernel logistic regression (KLR). In section (3), we propose our IVM algorithm. In section (4), we show some simulation results. In section (5), we generalize the IVM to the multi-class case. 2 Support vector machines and kernel logistic regression The standard SVM produces a non-linear classification boundary in the original input space by constructing a linear boundary in a transformed version of the original input space. The dimension of the transformed space can be very large, even infinite in some cases. This seemingly prohibitive computation is achieved through a positive definite reproducing kernel , which gives the inner product in the transformed space. Many people have noted the relationship between the SVM and regularized function estimation in the reproducing kernel Hilbert spaces (RKHS). An overview can be found in Evgeniou et al. (1999), Hastie et al. (2001) and Wahba (1998). Fitting an SVM is equivalent to minimizing: ( C ( 3$
(1) with : ! ! . is the RKHS generated by the kernel . The classification rule is given by *+-,/.10 7 . By the representer theorem (Kimeldorf et al (1971)), the optimal 9 has the form: : (2) It often happens that a sizeable fraction of the C values of can be zero. This is a consequence of the truncation property of the first part of criterion (1). This seems to be an attractive property, because only the points on the wrong side of the classification boundary, and those on the right side but near the boundary have an influence in determining the position of the boundary, and hence have non-zero ’s. The corresponding 9 ’s are called support points. Notice that (1) has the form ! *&* 2#"&. %$ . The loss function ( 3 & is plotted in Figure 1, along with several traditional loss functions. As we can see, the negative log-likelihood (NLL) of the binomial distribution has a similar shape to that of the SVM. If we replace ( 3 ' in (1) with () (* ",+.-0/ , the NLL of the binomial distribution, the problem becomes a KLR problem. We expect that the fitted function performs similarly to the SVM for binary classfication. There are two immediate advantages of making such a replacement: (a) Besides giving a classification rule, the KLR also offers a natural estimate of the probability 2 9 : "1/ 4 (* "1/ , while the SVM only estimates *+-, .10 2 3 (54687 ; (b) The KLR can naturally be generalized to the multi-class case through kernel multi-logit regression, whereas this is not the case for the SVM. However, because the KLR compromises the hinge loss function of the SVM, it no longer has the “support points” property; in other words, all the ’s in (2) are non-zero. KLR is a well studied problem; see Wahba et al. (1995) and references there; see also Green et al. (1985) and Hastie et al. (1990). yf(x) Loss -3 -2 -1 0 1 2 3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Binomial NLL Squared Error Support Vector Figure 1: Several loss functions,
The computational cost of the KLR is B C D5 ; to save the computational cost, the IVM algorithm will find a sub-model to approximate the full model (2) given by the KLR. The sub-model has the form: 1: (3) where is a subset of the training data # 8
) , and the data in are called import points. The advantage of this sub-model is that the computational cost is reduced, especially for large training data sets, while not jeopardizing the performance in classification. Several other researchers have investigated techniques in selecting the subset . Lin et al. (1998) divide the training data into several clusters, then randomly select a representative from each cluster to make up . Smola et al. (2000) develope a greedy technique to sequentially select E columns of the kernel matrix 0 & 7 , such that the span of these E columns approximates the span of 0 7 well in the Frobenius norm. Williams et al. (2001) propose randomly selecting E points of the training data, then using the Nystrom method to approximate the eigen-decomposition of the kernel matrix 0 & 7 , and expanding the results back up to C dimensions. None of these methods uses the output in selecting the subset (i.e., the procedure only involves ). The IVM algorithm uses both the output / and the input to select the subset , in such a way that the resulting fit approximates the full model well. 3 Import vector machine Following the tradition of logistic regression, we let / ! #5% () for the rest of this paper. For notational simplicity, the constant term in the fitted function is ignored. In the KLR, we want to minimize: : 3 0 3 ( ) ( 7 6
From (2), it can be shown that this is equivalent to the finite dimensional form: : 3 ! #" ($ () ( %
! #" 6 '& (4) where : 8 ; the regressor matrix '" : 0 7 ; and the regularization matrix '& : #" . To find , we set the derivative of with respect to equal to 0, and use the NewtonRaphson method to iteratively solve the score equation. It can be shown that the NewtonRaphson step is a weighted least squares step: : " #" #& + " (5) where is the value of in the th step, : '" + + 3 2 . The weight matrix is : + ,90 2 ( 3 2 7 . As mentioned in section 2, we want to find a subset of #& 8
) , such that the sub-model (3) is a good approximation of the full model (2). Since it is impossible to search for every subset , we use the following greedy forward strategy: 3.1 Basic algorithm ( ( ) Let :
, : #&98
) , : ( . ( 6 ) For each ! , let 9 : Find to minimize : 3 0 3 ( ) ( 7 6
: 3# " ($ () ( " 6 & (6) where the regressor matrix " : 0 7 & , ! #&98
) , ! # ) ; the regularization matrix & : 0 7 & & , ! # ) ; E : = = . ( ) Let : argmin "! Let : # ) , :$# # % ) , : , : ( . ( '& ) Repeat steps ( 6 ) and ( ' ) until converges. We call the points in import points. 3.2 Revised algorithm The above algorithm is computationally feasible, but in step ( 6 ) we need to use the Newton-Raphson method to find iteratively. When the number of import points E becomes large, the Newton-Raphson computation can be expensive. To reduce this computation, we use a further approximation. Instead of iteratively computing until it converges, we can just do a one-step iteration, and use it as an approximation to the converged one. To get a good approximation, we take advantage of the fitted result from the current “optimal” , i.e., the sub-model when = =: E , and use it as the initial value. This one-step update is similar to the score test in generalized linear models (GLM); but the latter does not have a penalty term. The updating formula allows the weighted regression (5) to be computed in B C E time. Hence, we have the revised step ( 6 ) for the basic algorithm: ( 6 ) For each ! , correspondingly augment " with a column, and & with a column and a row. Use the updating formula to find in (5). Compute (6). 3.3 Stopping rule for adding point to In step ( '& ) of the basic algorithm, we need to decide whether has converged. A natural stopping rule is to look at the regularized NLL. Let 8
be the sequence of regularized NLL’s obtained in step ( & ). At each step , we compare with + , where is a pre-chosen small integer, for example : ( . If the ratio +
is less than some pre-chosen small number , for example, : % %%'( , we stop adding new import points to . 3.4 Choosing the regularization paramter So far, we have assumed that the regularization parameter is fixed. In practice, we also need to choose an “optimal” . We can randomly split all the data into a training set and a tuning set, and use the misclassification error on the tuning set as a criterion for choosing . To reduce the computation, we take advantage of the fact that the regularized NLL converges faster for a larger . Thus, instead of running the entire revised algorithm for each , we propose the following procedure, which combines both adding import points to and choosing the optimal : ( ( ) Start with a large regularization parameter . ( 6 ) Let :
, : #&
) , : ( . ( ) Run steps ( 6 ), ( ) and ( & ) of the revised algorithm, until the stopping criterion is satisfied at : # & ) . Along the way, also compute the misclassfication error on the tuning set. ( & ) Decrease to a smaller value. ( ) Repeat steps ( ) and ( & ), starting with :A#& & ) . We choose the optimal as the one that corresponds to the minimum misclassification error on the tuning set. 4 Simulation In this section, we use a simulation to illustrate the IVM method. The data in each class are generated from a mixture of Gaussians (Hastie et al. (2001)). The simulation results are shown in Figure 2. 4.1 Remarks The support points of the SVM are those which are close to the classification boundary or misclassified and usually have large weights [ 2 9 ( 3 2 ]. The import points of the IVM are those that decrease the regularized NLL the most, and can be either close to or far from the classification boundary. This difference is natural, because the SVM is only concerned with the classification *+ , .10 2 9 3 (84687 , while the IVM also focuses on the unknown probability 2 . Though points away from the classification boundary do not contribute to determining the position of the classification boundary, they may contribute to estimating the unknown probability 2 . Figure 3 shows a comparison of the SVM and the IVM. The total computational cost of the SVM is B C$D& , while the computational cost of the IVM method is B C
&E8
, where E is the number of import points. Since E does not •••••••••••••• •• •• ••••••• •• •••• •• •••••••• ••••• •••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• No. of import points added 0 50 100 150 200 100 150 200 250 Regularized NLL for different lambda’s • • • • ••• •• •••••• •• • ••••••• •• •••••••••••••• ••••• •••••••••• •••• •••• •••• •••••••••••• ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• No. of import points added 0 50 100 150 200 0.22 0.24 0.26 0.28 0.30 0.32 0.34 Misclassification rate for different lambda’s • • • •• • • • • •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• No. of import points added 0 50 100 150 200 160 180 200 220 240 Regularized NLL for the optimal lambda Figure 2: Radial kernel is used. . The left and middle panels illustrate how to choose the optimal . ,
, decreases from to . The minimum misclassification rate
is found to correspond to
. The right panel is for the optimal
. The stopping criterion is satisfied when !" # . tend to increase as C increases, the computational cost of the IVM can be smaller than that of the SVM, especially for large training data sets. 5 Multi-class case In this section, we briefly describe a generalization of the IVM to multi-class classification. Suppose there are $ ( classes. We can write the response as an $ -vector , with each component being either 0 or 1, indicating which class the observation is in. Therefore : ( : % &%(' : &%*) $ indicates the response is in the th class, and : %'+%,) $ indicates the response is in the $ A( th class. Using the $ ( th class as the basis, the multi-logit can be written as : ( ) 2 4 2. , : ( ) 2/4 2. , : % . Hence the Bayes classification rule is given by: : argmax 0
1032323240 We use + to index the observations, % to index the classes, i.e. + : ( C , % : ( $ . Then the regularized negative log-likelihood is : 3 0 3 () ( " /65 $ 87977 " /6: 7 6
(7) where : 5
, 1: 8
, and
:
Using the representer theorem (Kimeldorf et al. (1971)), the % th element of , , which minimizes has the form : (8) SVM - with 107 support points +++++++ +++++++ ++++++ ++++++ +++++ +++++ ++++ ++++ ++++ ++++ +++ +++ +++ +++ +++ +++ +++ +++ +++ +++ +++ ++++ ++++ ++++ ++++ +++++ +++++ ++++++ ++++++ +++++++ +++++++ ++++++++ ++++++++ +++++++++ +++++++++ ++++++++++ ++++++++++ +++++++++++ ++++++++++++ ++++++++++++ +++ +++++++++++++ +++++ ++++++++++++++ +++++++ +++++++++++++++ ++++++++ +++++++++++++++ +++++++++ ++++++++++++++++ ++++++++++ +++++++++++++++++ ++++++++++ ++ ++++++++++++++++++ ++++++++++ ++++ +++++++++++++++++++ +++++++++++ ++++ +++++++++++++++++++ +++++++++++ +++++ ++++++++++++++++++++ +++++++++++ ++++++ +++++++++++++++++++++ +++++++++++ +++++++ ++++++++++++++++++++++ ++++++++++++ +++++++ ++++++++++++++++++++++ ++++++++++++ +++++++ +++++++++++++++++++++++ ++++++++++++ +++++++++ ++++++++++++++++++++++++ ++++++++++++ +++++++++ +++++++++++++++++++++++++ ++++++++++++ +++++++++ +++++++++++++++++++++++++ ++++++++++++ +++++++++ ++++++++++++++++++++++++++ +++++++++++++ ++++++++++ +++++++++++++++++++++++++++ +++++++++++++ ++++++++++ +++++++++++++++++++++++++++++ ++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++ +++++++++ ++++++++++++++++++++++++++++++++++++++++++++++ +++++++++ +++++++++++++++++++++++++++++++++++++++++++++++ +++++++++ +++++++++++++++++++++++++++++++++++++++++++++++ +++++++++ +++++++++++++++++++++++++++++++++++++++++++++++ ++++++++ +++++++++++++++++++++++++++++++++++++++++++++++ ++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++ +++++++ +++++++++++++++++++++++++++++++++++++++++++++++++ +++++++ +++++++++++++++++++++++++++++++++++++++++++++++++ +++++++ +++++++++++++++++++++++++++++++++++++++++++++++++ ++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++ +++++ +++++++++++++++++++++++++++++++++++++++++++++++++++ +++++ +++++++++++++++++++++++++++++++++++++++++++++++++++ ++++ +++++++++++++++++++++++++++++++++++++++++++++++++++ ++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++ +++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • Training Error: 0.160 Test Error: 0.218 Bayes Error: 0.210 IVM - with 21 import points ++++++++++++++++ + +++++++++++++++ + ++++++++++++++ + +++++++++++++ + ++++++++++++ + ++++++++++++ + +++++++++++ + ++++++++++ + ++++++++++ + +++++++++ + +++++++++ + +++++++++ + ++++++++ + ++++++++ + ++++++++ + ++++++++ + ++++++++ ++ ++++++++ ++ +++++++ ++ +++++++ ++ +++++++ ++ ++++++++ ++ ++++++++ ++ ++++++++ ++ ++++++++ ++ ++++++++ ++ ++++++++ ++ +++++++++ ++ +++++++++ ++ +++++++++ + ++++++++++ ++++++++++ +++++++++++ +++++++++++ +++++++++++ ++++++++++++ ++++++++++++ +++++++++++++ +++++++++++++ ++++++++++++++ ++ ++++++++++++++ ++++++ +++++++++++++++ ++++++++ ++++++++++++++++ ++++++++ ++ ++++++++++++++++ ++++++++++ +++ +++++++++++++++++ ++++++++++ ++++ ++++++++++++++++++ ++++++++++ ++++ +++++++++++++++++++ +++++++++++ ++++ +++++++++++++++++++ +++++++++++ ++++++ ++++++++++++++++++++ +++++++++++ ++++++ +++++++++++++++++++++ ++++++++++++ ++++++ +++++++++++++++++++++ ++++++++++++ ++++++ ++++++++++++++++++++++ ++++++++++++ ++++++ ++++++++++++++++++++++ ++++++++++++ +++++++ +++++++++++++++++++++++ +++++++++++++ +++++++ ++++++++++++++++++++++++ +++++++++++++ ++++++++ ++++++++++++++++++++++++ ++++++++++++ ++++++++ +++++++++++++++++++++++++ +++++++++++++ ++++++++ +++++++++++++++++++++++++ +++++++++++++ ++++++++ ++++++++++++++++++++++++++ ++++++++++++++ ++++++++ +++++++++++++++++++++++++++ +++++++++++++++ ++++++++ +++++++++++++++++++++++++++++ ++++++++++++++++ ++++++++ ++++++++++++++++++++++++++++++++++++++++++++++ ++++++++ ++++++++++++++++++++++++++++++++++++++++++++++ +++++++++ ++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • Training Error: 0.150 Test Error: 0.219 Bayes Error: 0.210 Figure 3: The solid lines are the classification boundaries; the dotted lines are the Bayes rule boundaries. For the SVM, the dashed lines are the edges of the margin. For the IVM, the dashed lines are the (
and
lines. Hence, (7) becomes : 3 0 " + 3 () ( ( " 0
7 6 & (9) where : : , '" and '& are defined in the same way as in the binary case; and " + is the + th row of " . The multi-class IVM procedure is similar to the binary case, and the computational cost is B $ C
E8
. Figure 4 is a simulation of the multi-class IVM. The data in each class are generated from a mixture of Gaussians (Hastie et al. (2001)). Multi-class IVM - with 32 import points +++++++++++ +++++++++++++++ ++++++++++++++++++ ++++++++++++++++++ +++++++++++++++++++ ++++++++++++++++++ ++++++++++++++++++ +++++++++++++++++ +++++++++++++++++ +++++++++++++++++ ++++++++++++++++++ ++++++++++++++++++++ +++++++++++++++++++++++ ++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ +++++++++++++++++++++++++++ ++++++++++++++++++++++ +++++++++++++ .................................................................................................................. .................................................................................................................. 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( , ( , (
, ! # . 6 Conclusion We have discussed the import vector machine (IVM) method in both binary and multi-class classification. We showed that it not only performs as well as the SVM, but also provides an estimate of the probability 2 9 . The computational cost of the IVM is B C
E8
for the binary case and B $ C
E8
for the multi-class case, where E is the number of import points. Acknowledgments We thank Dylan Small, John Storey, Rob Tibshirani, and Jingming Yan for their helpful comments. Ji Zhu is partially supported by the Stanford Graduate Fellowship. Trevor Hastie is partially supported by grant DMS-9803645 from the National Science Foundation, and grant ROI-CA-72028-01 from the National Institutes of Health. Thanks to Grace Wahba and Chris Williams for pointing out several interesting and important references. We also want to thank the anonymous NIPS referees who helped improve this paper. References [1] Burges, C.J.C. (1998) A tutorial on support vector machines for pattern recognition. In Data Mining and Knowledge Discovery. Kluwer Academic Publishers, Boston. (Volume 2) [2] Evgeniou, T., Pontil, M., & Poggio., T. (1999) Regularization networks and support vector machines. In A.J. Smola, P. Bartlett, B. Sch¨olkopf, and C. Schuurmans, editors, Advances in Large Margin Classifiers. MIT Press. [3] Green, P. & Yandell, B. (1985) Semi-parametric generalized linear models. Proceedings 2nd International GLIM Conference, Lancaster, Lecture notes in Statistics No. 32 44-55 Springer-Verlag, New York. [4] Hastie, T. & Tibshirani, R. (1990) Generalized Additive Models, Chapman and Hall. [5] Hastie, T., Tibshirani, R., & Friedman, J.(2001) The elements of statistical learning. In print. [6] Lin, X., Wahba, G., Xiang, D., Gao, F., Klein, R. & Klein B. (1998), Smoothing spline ANOVA models for large data sets with Bernoulli observations and the randomized GACV. Technical Report 998, Department of Statistics, University of Wisconsin, Madison WI. [7] Kimeldorf, G. & Wahba, G. (1971) Some results on Tchebycheffian spline functions. J. Math. Anal. Applic. 33, 82-95. [8] Smola, A. & Sch¨olkopf, B. (2000) Sparse Greedy Matrix Approximation for Machine Learning. In Proceedings of the Seventeenth International Conference on Machine Learning. Morgan Kaufmann Publishers. [9] Wahba, G. (1998) Support Vector Machine, Reproducing Kernel Hilbert Spaces and the Randomized GACV. Technical Report 984rr, Department of Statistics, University of Wisconsin, Madison WI. [10] Wahba, G., Gu, C., Wang, Y., & Chappell, R. (1995) Soft Classification, a.k.a. Risk Estimation, via Penalized Log Likelihood and Smoothing Spline Analysis of Variance. In D.H. Wolpert, editor, The Mathematics of Generalization. Santa Fe Institute Studies in the Sciences of Complexity. Addison-Wesley Publisher. [11] Williams, C. & Seeger, M (2001) Using the Nystrom Method to Speed Up Kernel Machines. In T. K. Leen, T. G. Diettrich, and V. Tresp, editors, Advances in Neural Information Processing Systems 13. MIT Press.
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A Model of the Phonological Loop: Generalization and Binding Randall C. O'Reilly Department of Psychology University of Colorado Boulder 345 UCB Boulder, CO 80309 oreilly@psych.colorado.edu Rodolfo Soto Department of Psychology University of Colorado Boulder 345 UCB Boulder, CO 80309 Abstract We present a neural network model that shows how the prefrontal cortex, interacting with the basal ganglia, can maintain a sequence of phonological information in activation-based working memory (i.e., the phonological loop). The primary function of this phonological loop may be to transiently encode arbitrary bindings of information necessary for tasks the combinatorial expressive power of language enables very flexible binding of essentially arbitrary pieces of information. Our model takes advantage of the closed-class nature of phonemes, which allows different neural representations of all possible phonemes at each sequential position to be encoded. To make this work, we suggest that the basal ganglia provide a region-specific update signal that allocates phonemes to the appropriate sequential coding slot. To demonstrate that flexible, arbitrary binding of novel sequences can be supported by this mechanism, we show that the model can generalize to novel sequences after moderate amounts of training. 1 Introduction Sequential binding is a version of the binding problem requiring that the identity of an item and its position within a sequence be bound. For example, to encode a phone number (e.g., 492-0054), one must remember not only the digits, but their order within the sequence. It has been suggested that the brain may have developed a specialized system for this form of binding in the domain of phonological sequences, in the form of the phonological loop (Baddeley, 1986; Baddeley, Gathercole, & Papagno, 1998; Burgess & Hitch, 1999). The phonological loop is generally conceived of as a system that can quickly encode a sequence of phonemes and then repeat this sequence back repeatedly. Standard estimates place the capacity of this loop at about 2.5 seconds of "inner speech," and it is widely regarded as depending on the prefrontal cortex (e.g., Paulesu, Frith, & Frackowiak, 1993). We have developed a model of the phonological loop based on our existing framework for understanding how the prefrontal cortex and basal ganglia interact to support activation-based working memory (Frank, Loughry, & O'Reilly, 2001). This model performs binding by using different neural substrates for the different sequential positions of phonemes. This is a viable solution for a small, closed-class set of items like phonemes. However, through the combinatorial power of language, these phonological sequences can represent a huge number of distinct combinations of concepts. Therefore, this basic maintenance mechanism can be leveraged in many different circumstances to bind information needed for immediate use (e.g., in working memory tasks). A good example of this form of transient, phonologically-dependent binding comes from a task studied by Miyake and Soto (in preparation). In this task, participants saw sequentially-presented colored letters one at a time on a computer display, and had to respond to targets of a red X or a green Y, but not to any other color-letter combination (e.g., green X's and red Y's, which were also presented). After an initial series of trials with this set of targets, the targets were switched to be a green X and a red Y. Thus, the task clearly requires binding of color and letter information, and updating of these bindings after the switch condition. Miyake and Soto (in preparation) found that if they simply had participants repeat the word "the" over and over during the task (i.e., articulatory suppression), it interfered significantly with performance. In contrast, performing a similar repeated motor response that did not involve the phonological system (repeated foot tapping) did not interfere (but this task did interfere at the same level as articulatory suppression in a control visual search task, so one cannot argue that the interference was simply a matter of differential task difficulty). Miyake and Soto (in preparation) interpret this pattern of results as showing that the phonological loop supports the binding of stimulus features (e.g., participants repeatedly say to themselves "red X, green y' .. " , which is supported by debriefing reports), and that the use of this phonological system for unrelated information during articulatory suppression leads to the observed performance deficits. This form of phonological binding can be contrasted with other forms of binding that can be used in other situations and subserved by other brain areas besides the prefrontal cortex. O'Reilly, Busby, and Soto (in press) identify two other important binding mechanisms and their neural substrates in addition to the phonological loop mechanism: • Cortical coarse-coded conjunctive binding: This is where each neural unit codes in a graded fashion for a large number of relatively low-order conjunctions, and many such units are used to represent any given input (e.g., Wickelgren, 1969; Mel & Fiser, 2000; O'Reilly & Busby, 2002). This form of binding takes place within the basic representations in the network that are shaped by gradual learning processes and provides a long-lasting (nontransient) form of binding. In short, these kinds of distributed representations avoid the binding problem in the first place by ensuring that relevant conjunctions are encoded, instead of representing different features using entirely separate, localist units (which is what gives rise to binding problems in the first place). However, this form of binding cannot rapidly encode novel bindings required for specific tasks the phonological loop mechanism can thus complement the basic cortical mechanism by providing flexible, transient bindings on an ad-hoc basis. • Hippocampal episodic conjunctive binding: Many theories of hippocampal function converge on the idea that it binds together individual elements of an experience into a unitary representation, which can for example be later recalled from partial cues (see O'Reilly & Rudy, 2001 for a review). These hippocampal conjunctive representations are higher-order and more specific than the lower-order coarse-coded cortical conjunctive representations (i.e., a hippocampal conjunction encodes the combination of many feature elements, while a cortical conjunction encodes relatively few). Thus, the hippocampus can be seen as a specialized system for doing long-term binding of specific episodes, complementing the more generalized conjunctive binding performed by the cortex. Importantly, the hippocampus can also encode these conjunctions rapidly, and therefore it shares some of the same functionality as the phonological loop mechanism (i.e., rapidly encoding arbitrary conjunctions required for tasks). Thus, it is likely that the hippocampus and the prefrontal-mediated working memory system (including the phonological loop) are partially redundant with each other, and work together in many tasks (Cohen & O'Reilly, 1996). 2 Prefrontal Cortex and Basal Ganglia in Working Memory Our model of the phonological loop takes advantage of recent work showing how the prefrontal cortex and basal ganglia can interact to support activation-based working memory (Frank et al., 2001). The critical principles behind this work are as follows: • Prefrontal cortex (PFC) is specialized relative to the posterior cortex for robust and rapidly updatable maintenance of information in an active state (i.e., via persistent firing of neurons). Thus, PFC can quickly update to maintain new information (in this case, the one exposure to a sequence of phonemes), while being able to also protect maintained information from interference from ongoing processing (see O'Reilly, Braver, & Cohen, 1999; Cohen, Braver, & O'Reilly, 1996; Miller & Cohen, 2001 for elaborations and reviews of relevant data). • Robust maintenance and rapid updating are in fundamental conflict, and require a dynamic gating mechanism that can switch between these two modes of operation (O'Reilly et al., 1999; Cohen et al., 1996). • The basal ganglia (BG) can provide this dynamic gating mechanism via modulatory, dis inhibitory connectivity with the PFC. Furthermore, this BG-based gating mechanism provides selectivity, such that separate regions of the PFC can be independently updated or allowed to perform robust maintenance. A possible anatomical substrate for these separably updatable PFC regions are the stripe structures identified by Levitt, Lewis, Yoshioka, and Lund (1993). • Active maintenance in the PFC is implemented via a combination of recurrent excitatory connections and intracellular excitatory ionic conductances. This allows the PFC units to generally reflect the current inputs, except when these units have their intracellular maintenance currents activated, which causes them to reflect previously maintained information. See Frank et al. (2001) for more details on the importance of this mechanism. 3 Phonological Loop Model The above mechanisms motivated our modeling of the phonological loop as follows (see Figure 1). First, separate PFC stripes are used to encode each step in the sequence. Thus, binding of phoneme identity and sequential order occurs in this model by using distinct neural substrates to represent the sequential information. This is entirely feasible because each stripe can represent all of the possible phonemes, given that they represent a closed class of items. Second, the storage of a Figure 1: Phonological loop model. Ten different input symbols are possible at each time step (one unit out of ten activated in the Input layer). A sequence is encoded in one pass by presenting the Input together with the sequential location in the Time input layer for each step in the sequence. The simulated basal ganglia gating mechanism (implemented by fiat in script code) uses the time input to trigger intracellular maintenance currents in the corresponding stripe region of the context (PFC) layer (stripes are shown as the three separate groups of units within the Context layer; individual context units also had an excitatory self-connection for maintenance). Thus, the first stripe must learn to encode the first input, etc. Immediately after encoding, the network is then trained to produce the correct output in response to the time input, without any Input activation (the activation state shown is the network correctly recalling the third item in a sequence). The hidden layer must therefore learn to decode the context representations for this recall phase. Generalization testing involved presenting untrained sequences. new sequence involves the basal ganglia gating mechanism triggering updates of the different PFC stripes in the appropriate order. We assume this can be learned over experience, and we are currently working on developing powerful learning mechanisms for adapting the basal ganglia gating mechanism in this way. This kind of gating control would also likely require some kind of temporal/sequential input that indicates the location within the sequence such information might come from the cerebellum (e.g., Ivry, 1996). In advance of having developed realistic and computationally powerful mechanisms for both the learning and the temporal/sequential control aspects of the model, we simply implemented these by fiat in the simulator. For the temporal signal indicating location within the sequence, we simply activated a different individual time unit for each point in the sequence (the Time input layer in Figure 1). This signal was then used by a simulated gating mechanism (implemented in script code in the simulator) to update the corresponding stripe in prefrontal cortex. Although the resulting model was therefore simplified, it nevertheless still had a challenging learning task to perform. Specifically, the stripe context layers had to learn to encode and maintain the current input value properly, and the Hidden layer had to be able to decode the context layer information as a function of the time input value. The model was implemented using the Leabra algorithm with standard parameters (O'Reilly, 1998; O'Reilly & Munakata, 2000). Phonological Loop Generalization 0.3 .. e ali 0.2 I: .S! "lii .!::! iij 0.1 .. Q) I: Q) Cl 0.0 100 200 300 800 Number of Training Events Figure 2: Generalization results for the phonological loop model as a function of number training patterns. Generalization is over 90% correct with training on less than 20% of the possible input patterns. N = 5. 3.1 Network Training The network was trained as follows. Sequences (of length 3 for our initial work) were presented by sequentially activating an input "phoneme" and a corresponding sequential location input (in the Time input layer) . We only used 10 different phonemes, each of which was encoded locally with a different unit in the Input layer. For example, the network could get Time = 0, Input = 2, then Time = 1, Input = 7, then Time = 2, Input = 3 to encode the sequence 2,7,3. During this encoding phase, the network was trained to activate the current Input on the Output layer, and the simulated gating function simply activated the intracellular maintenance currents for the units in the stripe in the Context (PFC) layer that corresponded to the Time input (i.e., stripe 0 for Time=O, etc). Then, the network was trained to recall this sequence, during which time no Input activation was present. The network received the sequence of Time inputs (0,1,2), and was trained to produce the corresponding Output for that location in the sequence (e.g., 2,7,3). The PFC context layers just maintained their activation states based on the intracellular ion currents activated during encoding (and recurrent activation) once the network has been trained, the active PFC state represents the entire sequence. 3.2 Generalization Results A critical test of the model is to determine whether it can perform systematically with novel sequences only if it demonstrates this capacity can it serve as a mechanism for rapidly binding arbitrary information (such as the task demands studied by Miyake & Soto, in preparation). With 10 input phonemes and sequences of length three, there were 1,000 different sequences possible (we allowed phonemes to repeat). We trained on 100, 200, 300, and 800 of these sequences, and tested generalization on the remaining sequences. The generalization results are shown in Figure 2, which clearly shows that the network learned these sequences in a systematic manner and could transfer its training knowledge to novel sequences. Interestingly, there appears to be a critical transition between 100 and 200 training sequences 100 sequences corresponds to each item within each slot being presented roughly 10 times, which appears to provide sufficient statistical information regarding the independence of individual slots. Figure 3: Hidden unit representations (values are weights into a hidden unit from all other layers). Unit in a) encodes the conjunction of a subset of input/output items at time 2. (b) encodes a different subset of items at time 2. (c) encodes items over times 2 and 3. (d) has no selectivity in the input, but does project to the output and likely participates in recall of items at time step 3. 3.3 Analysis of Representations To understand how the hidden units encode and retrieve information in the maintained context layer in a systematic fashion that supports the good generalization observed, we examined the patterns of learned weights. Some representative examples are shown in Figure 3. Here, we see evidence of coarse-coded representations that encode a subset of items in either one time point in the sequence or a couple of time points. Also we found units that were more clearly associated with retrieval and not encoding. These types of representations are consistent with our other work showing how these kinds of representations can support good generalization (O'Reilly & Busby, 2002). 4 Discussion We have presented a model of sequential encoding of phonemes, based on independently-motivated computational and biological considerations, focused on the neural substrates of the prefrontal cortex and basal ganglia (Frank et al., 2001). Viewed in more abstract, functional terms, however, our model is just another in a long line of computational models of how people might encode sequential order information. There are two classic models: (a) associative chaining, where the activation of a given item triggers the activation of the next item via associative links, and (b) item-position association models where items are associated with their sequential positions and recalled from position cues (e.g., Lee & Estes, 1977). The basic associative chaining model has been decisively ruled out based on error patterns (Henson, Norris, Page, & Baddeley, 1996), but modified versions of it may avoid these problems (e.g., Lewandowsky & Murdock, 1989). Probably the most accomplished current model, Burgess and Hitch (1999), is a version of the itemposition association model with a competitive queuing mechanism where the most active item is output first and is then suppressed to allow other items to be output. Compared to these existing models, our model is unique in not requiring fast associational links to encode items within the sequence. For example, the Burgess and Hitch (1999) model uses rapid weight changes to associate items with a context representation that functions much like the time input in our model. In contrast, items are maintained strictly via persistent activation in our model, and the basalganglia based gating mechanism provides a means of encoding items into separate neural slots that implicitly represent sequential order. Thus, the time inputs act independently on the basal ganglia, which then operates generically on whatever phoneme information is presently activated in the auditory input, obviating the need for specific item-context links. The clear benefit of not requiring associationallinks is that it makes the model much more flexible and capable of generalization to novel sequences as we have demonstrated here (see O'Reilly & Munakata, 2000 for extended discussion of this general issue). Thus, we believe our model is uniquely well suited for explaining the role of the phonological loop in rapid binding of novel task information. Nevertheless, the present implementation of the model has numerous shortcomings and simplifications, and does not begin to approach the work of Burgess and Hitch (1999) in accounting for relevant psychological data. Thus, future work will be focused on remedying these limitations. One important issue that we plan to address is the interplay between the present model based on the prefrontal cortex and the binding that the hippocampus can provide we suspect that the hippocampus will contribute item-position associations and their associated error patterns and other phenomena as discussed in Burgess and Hitch (1999). Acknowledgments This work was supported by ONR grant N00014-00-1-0246 and NSF grant IBN9873492. Rodolfo Soto died tragically at a relatively young age during the preparation of this manuscript this work is dedicated to his memory. 5 References Baddeley, A., Gathercole, S., & Papagno, C. (1998). The phonological loop as a language learning device. Psychological Review, 105, 158. Baddeley, A. D. (1986). Working memory. New York: Oxford University Press. Burgess, N., & Hitch, G. J . (1999). Memory for serial order: A network model of the phonological loop and its timing. Psychological Review, 106, 551- 581. Cohen, J. D., Braver, T. S., & O'Reilly, R. C. (1996). A computational approach to prefrontal cortex, cognitive control, and schizophrenia: Recent developments and current challenges. Philosophical Transactions of the Royal Society (London) B, 351, 1515- 1527. Cohen, J. D., & O'Reilly, R. C. (1996). A preliminary theory of the interactions between prefrontal cortex and hippocampus that contribute to planning and prospective memory. In M. Brandimonte, G. O. Einstein, & M. A. McDaniel (Eds.), Prospective memory: Theory and applications (pp. 267- 296). Mahwah, New Jersey: Erlbaum. Frank, M. J., Loughry, B., & O'Reilly, R. C. (2001). Interactions between the frontal cortex and basal ganglia in working memory: A computational model. Cognitive, Affective, and Behavioral Neuroscience, 1, 137- 160. Henson, R. N. A., Norris, D. G., Page, M. P. A., & Baddeley, A. D. (1996) . Unclaimed memory: Error patterns rule out chaining models of immediate serial recall. Quarterly Journal of Experimental Psychology: Human Experimental Psychology, 49(A), 80- 115. Ivry, R. (1996). The representation of temporal information in perception and motor control. Current Opinion in Neurobiology, 6,851-857. Lee, C. L., & Estes, W. K. (1977). Order and position in primary memory for letter strings. Journal of Verbal Learning and Verbal Behavior, 16, 395- 418. Levitt, J. B., Lewis, D. A., Yoshioka, T., & Lund, J. S. (1993). Topography of pyramidal neuron intrinsic connections in macaque monkey prefrontal cortex (areas 9 & 46). Journal of Comparative Neurology, 338, 360- 376. Lewandowsky, S., & Murdock, B. B. (1989). Memory for serial order. Psychological Review, 96, 25- 57. Mel, B. A., & Fiser, J. (2000). Minimizing binding errors using learned conjunctive features. Neural Computation, 12, 731- 762. Miller, E. K., & Cohen, J. D. (2001). An integrative theory of prefrontal cortex function. Annual Review of Neuroscience, 24, 167- 202. Miyake, A., & Soto, R. (in preparation). The role of the phonological loop in executive control. O'Reilly, R. C. (1998). Six principles for biologically-based computational models of cortical cognition. Trends in Cognitive Sciences, 2(11), 455- 462. O'Reilly, R. C., Braver, T . S., & Cohen, J. D. (1999). A biologically based computational model of working memory. In A. Miyake, & P. Shah (Eds.), Models of working m emory: Mechanisms of active maintenance and executive control. (pp. 375- 411) . New York: Cambridge University Press. O'Reilly, R. C. , & Busby, R. S. (2002). Generalizable relational binding from coarsecoded distributed representations. Advances in Neural Information Processing Systems (NIPS), 2001. O'Reilly, R. C. , Busby, R. S., & Soto, R. (in press). Three forms of binding and their neural substrates: Alternatives to temporal synchrony. In A. Cleeremans (Ed.), The unity of consciousness: Binding, integration, and dissociation. Oxford: Oxford University Press. O'Reilly, R. C., & Munakata, Y. (2000) . Computational explorations in cognitive neuroscience: Understanding the mind by simulating the brain. Cambridge, MA: MIT Press. O'Reilly, R. C. , & Rudy, J. W . (2001). Conjunctive representations in learning and memory: Principles of cortical and hippocampal function. Psychological Review, 108, 311345. Paulesu, E., Frith, C. D., & Frackowiak, R. S. J. (1993). The neural correlates of the verbal component of working memory. Nature, 362,342- 345. 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Thin Junction Trees Francis R. Bach Computer Science Division University of California Berkeley, CA 94720 fbach@cs.berkeley.edu Michael I. Jordan Computer Science and Statistics University of California Berkeley, CA 94720 jordan@cs.berkeley.edu Abstract We present an algorithm that induces a class of models with thin junction trees—models that are characterized by an upper bound on the size of the maximal cliques of their triangulated graph. By ensuring that the junction tree is thin, inference in our models remains tractable throughout the learning process. This allows both an efficient implementation of an iterative scaling parameter estimation algorithm and also ensures that inference can be performed efficiently with the final model. We illustrate the approach with applications in handwritten digit recognition and DNA splice site detection. Introduction Many learning problems in complex domains such as bioinformatics, vision, and information retrieval involve large collections of interdependent variables, none of which has a privileged status as a response variable or class label. In such problems, the goal is generally that of characterizing the principal dependencies in the data, a problem which is often cast within the framework of multivariate density estimation. Simple models are often preferred in this setting, both for their computational tractability and their relative immunity to overfitting. Thus models involving low-order marginal or conditional probabilities— e.g., naive independence models, trees, or Markov models—are in wide use. In problems involving higher-order dependencies, however, such strong assumptions can be a serious liability. A number of methods have been developed for selecting models of higher-order dependencies in data, either within the maximum entropy setting—in which features are selected [9, 16]—and the graphical model setting—in which edges are selected [8]. Simplicity also plays an important role in the design of these algorithms; in particular, greedy methods that add or subtract a single feature or edge at a time are generally employed. The model that results at each step of this process, however, is often not simple, and this is problematic both computationally and statistically in large-scale problems. In the current paper we describe a methodology that can be viewed as a generalization of the Chow-Liu algorithm for constructing tree models [2]. Note that tree models have the property that their junction trees have no more than two nodes in any clique—the treewidth of tree models is one. In our generalization, we allow the treewidth to be a larger, but still controlled, value. We fit data within the space of models having “thin” junction trees. Models with thin junction trees are tractable for exact inference, indeed the complexity of any type of inference (joint, marginal, conditional) is controlled by the upper bound that is imposed on the treewidth. This makes it possible to achieve some of the flexibility that is often viewed as a generic virtue of generative models, but is not always achievable in practice. For example, in the classification setting we are able to classify partially observed data (e.g., occluded digits) in a simple and direct way—we simply marginalize away the unobserved variables, an operation which is tractable in our models. We illustrate this capability in a study of handwritten digit recognition in Section 4.2, where we compare thin junction trees and support vector machines (SVMs), a discriminative technique which does not come equipped with a simple and principled method for handling partially observed data. As we will see, thin junction trees are quite robust to missing data in this domain. There are a number of issues that need to be addressed in our framework. In particular, tree models come equipped with particularly efficient algorithms for parameter estimation and model selection—algorithms which do not generalize readily to non-tree models, including thin junction tree models. It is important to show that efficient algorithms can nonetheless be found to fit such models. We show how this can be achieved in Sections 1, 2 and 3. Empirical results using these algorithms are presented in Section 4. 1 Feature induction We assume an input space with variables and a target probability distribution . Our goal is to find a probability distribution that minimizes the Kullback-Leibler divergence . Consider a vector-valued “feature” or “sufficient statistic”
, where is the dimensionality of the feature space. The feature can also be thought in terms of its components as a set of real-valued features . We focus on exponential family distributions (also known as “Gibbs” or “maximum entropy” distributions) based on these features: ! #"!$ &%' ( ()+* where % ,% ./ is a parameter vector, # is a base-measure (typically uniform), and * is the normalizing constant. (Section 3 considers the closely-related problem of inducing edges rather than features). Each feature is a function of a certain subset of variables, and we let 02143657 8:9:;=<>; ? ?+?@; BA index the subset of variables referred to by feature :1 . Let us consider the undirected graphical model CD 5 ;(E , where the set of edges E is the set of all pairs included in at least one 0 1 . With this definition the 0 1 are the maximal cliques of the graph and, if # is decomposable in this graph, the exponential family distribution with features and reference distribution + is also decomposable in this graph. We assume without loss of generality that the graph is connected. For each possible triangulation of the graph, we can define a junction tree [4], where for all F there exists a maximal clique containing 0G1 . The complexity of exact inference depends on the size of the maximal clique of the triangulated graph. We define the treewidth H of our original graph to be one less than the minimum possible value of this maximal clique size for all possible triangulations. We say that a graphical model has a thin junction tree if its treewidth H is small. Our basic feature induction algorithm is a constrained variant of that proposed by [9]. Given a set of available features, we perform a greedy search to find the set of features that enables the best possible fit to , under the constraint of having a thin junction tree. At each step, candidates are ranked according to the gain in KL divergence, with respect to the empirical distribution, that would be achieved by their addition to the current set of features. Features that would generate a graphical cover with treewidth greater than a given upper bound H are removed from the ranking. The parameter values % are held fixed during each step of the feature ranking process. Once a set of candidate features are chosen, however, we reestimate all of the parameters (using the algorithm to be described in Section 2) and iterate. FEATUREINDUCTION 1. Initialization: , . ; % , a set of available features 2. Repeat steps (a) to (d) until no further progress is made with respect to a model selection criterion (e.g., MDL or cross-validation) (a) Ranking: generate samples from and rank feature candidates according to the KL gain (b) Elimination: remove all candidates that would generate a model with treewidth greater than H (c) Selection: select the best features ; ?+? ?#; and add them to (d) Parameter Estimation: Estimate % using the junction tree implementation of Iterative Scaling (see Section 2) Freezing the parameters during the feature ranking step is suboptimal, but it yields an essential computational efficiency. In particular, as shown by [9], under these conditions we can rank a new feature by solving a polynomial equation whose degree is the number of values can take minus one, and whose coefficients are expectations under of functions of . This equation has only one root and can be solved efficiently by Newton’s method. When the feature is binary the process is even more efficient—the equation is linear and can be solved directly. Consequently, with a single set of samples from , we can rank many features very cheaply. For the feature elimination operation, algorithms exist that determine in time linear in the number of nodes whether a graph has a treewidth smaller than H , and if so output a triangulation in which all cliques are of size less than H [1]. These algorithms are super-exponential in H , however, and thus are applicable only to problems with small treewidths. In practice we have had success using fast heuristic triangulation methods [11] that allow us to guarantee the existence of a junction tree with a maximal clique no larger than H for a given model. (This is a conservative technique that may occasionally throw out models that in fact have small treewidth). A critical bottleneck in the algorithm is the parameter estimation step, and it is important to develop a parameter estimation algorithm that exploits the bounded treewidth property. We now turn to this problem. 2 Iterative Scaling using the junction tree Fitting an exponential family distribution under expectation constraints is a well studied problem; the basic technique is known as Iterative Scaling. A generalization of Iterative Proportional Fitting (IPF), it updates the parameters % sequentially [5]. Algorithms that update the parameters in parallel have also been proposed; in particular the Generalized Iterative Scaling algorithm [6], which imposes the constraint that the features sum to one, and the Improved Iterative Scaling algorithm [9], which removes this constraint. These algorithms have an important advantage in our setting in that, for each set of parameter updates, they only require computations of expectation that can all be estimated with a single set of samples from the current distribution. When the input dimensionality is large, however, we would like to avoid sampling algorithms altogether. To do so we exploit the bounded treewidth of our models. We present a novel algorithm that uses the junction tree and the structure of the problem to speed up parameter estimation. The algorithm generalizes to Gibbs distributions the “effective IPF” algorithm of [10]. When working with a junction tree, a efficient way of performing Iterative Scaling is to update parameters block by block so that each update is performed for a relatively small number of features on a small number of variables. Each block can be fit with any parameter estimation algorithm, in particular Improved Iterative Scaling (IIS). The following algorithm exploits this idea by grouping the features whose supports are in the same clique of the triangulated graph. Thus, parameter estimation is done in spaces of dimensions at most H 9 , and all the needed expectations can be evaluated cheaply. 2.1 Notation Let be our -dimensional feature. Let & denote the maximal cliques of the triangulated graph, with potentials
. We assign each feature :1 to one of the cliques that contains 0G1 . For each clique we denote 1 ;+? ?+?#; 1 as the set of features assigned to . 2.2 Algorithm EFFICIENTITERATIVESCALING 1. Initialization: –Construct a junction tree associated with the subsets 8 0 1 supp 1 A –Assign each 1 to one , such that 0 1 3 (equivalent to determining 1 ; ?+? ?#; 1 for all ) –Set % ,% ; ?+? ? ; % and decompose onto the junction tree –Set 2. Loop until convergence: Repeat step (3) until convergence of the % ’s 3. Loop through all cliques: Repeat steps (a) to (c) for all cliques (a) Define the root of the junction tree to be (b) Collect evidence from the leaves to the root of the junction tree and normalize potential (c) Calculate the maximum likelihood -dimensional exponential family distribution with features and reference distribution , using IIS. Replace
by this distribution and add the resulting parameters (one for each feature in ) to the corresponding % ’s: &% 1 ; ?+? ?+; % 1 . After step (b), the potential is exactly marginalized to , so that performing IIS for the features can be done using instead of the full distribution . Moreover, each pass through all the cliques is equivalent to one pass of Iterative Scaling and therefore this algorithm converges to the maximum likelihood distribution. 3 Edge induction Thus far we have emphasized the exponential family representation. Our algorithm can, however, be adapted readily to the problem of learning the structure of a graphical model. This is achieved by using features that are indicators of subsets of variables, ensuring that there is one such indicator for every combination of values of the variables in a clique. In this case, Iterative Scaling reduces to Iterative Proportional Fitting. We generally employ a further approximation when ranking and selecting edges. In particular, we evaluate an edge only in terms of the two variables associated directly with the edge. The clique formed by the addition of the edge, however, may involve additional higher-order dependencies, which can be parameterized and incorporated in the model. Evaluating edges in this way thus underestimates the potential gain in KL divergence. 0 10 20 30 0 5 10 15 20 Figure 1: (Left) Circular Boltzmann machine of treewidth 4. (Right) Proportion (in ) of edges not in common between the fitted model and the generating model vs the number of available training examples (in thousands). We should not expect to be able to find an exact edge-selection method—recent work by Srebro [15] has shown that the related problem of finding the maximum likelihood graphical model with bounded treewidth is NP-hard. 4 Empirical results 4.1 Small graphs with known generative model In this experiment we generate samples from a known graphical model and fit our model to the data. We consider circular Boltzmann machines of known treewidth equal to 4 as shown in Figure 1. Our networks all have 32 nodes and the weights were selected from a uniform distribution in < 9 9 =< —so that each edge is significant. For an increasing number of training samples, ten replications were performed for each case using our feature induction algorithm with maximum treewidth equal to 4. Figure 1 shows that with enough samples we are able to recover the structure almost exactly (up to ?
of the original edges). 4.2 MNIST digit dataset In this section we study the performance of the thin junction tree method on the MNIST dataset of handwritten digits. While discriminative methods outperform generative methods in this high-dimensional setting [12], generative methods offer capabilities that are not provided by discriminative classifiers; in particular, the ability to deal with large fractions of missing pixels and the ability to to reconstruct images from partial data. It is of interest to see how much performance loss we incur and how much robustness we gain by using a sophisticated generative model for this problem. The MNIST training set is composed of < < 4-bit grayscale pixels that have been resized and cropped to 99 binary images (an example is provided in the leftmost plots in Figure 2). We used thin junction trees as density estimators in the 256-dimensional pixel space by training ten different models, one for each of the ten classes. We used binary features of the form 9 . No vision-based techniques such as de-skewing or virtual examples were used. We utilized ten percent fractions of the training data for crossvalidation and test. Density estimation: The leftmost plot in Figure 3 shows how increasing the maximal allowed treewidth, ranging from 1 (trees) to 15, enables a better fit to data. Classification: We built classifiers from the bank of ten thin junction tree (“TJT”) models using one of the following strategies: (1) take the maximum likelihood among the ten Figure 2: Digit from the MNIST database. From left to right, original digit, cropped and resized digits used in our experiments, 50% of missing values, 75% of missing values, occluded digit. 0 5 10 15 50 55 60 65 70 0 50 100 0 20 40 60 80 100 Figure 3: (Left) Negative log likelihood for the digit 2 vs maximal allowed treewidth. (Right) Error rate as a function of the percentage of erased pixels for the TJT classifier (plain) and a support vector machine (dotted). See text for details. models (TJT-ML), or (2) train a discriminative model using the outputs of the ten models. We used softmax regression (TJT-Softmax) and the support vector machine (TJT-SVM) in the latter case. The classification error rates were as follows: LeNet 0.7, SVM 0.8, Product of experts, 2.0, TJT-SVM 3.8, TJT-Softmax 4.2, TJT-ML 5.3, Chow-Liu 8.5, and Linear classifier 12.0. (See [12] and [13] for further details on the non-TJT models). It is important to emphasize that our models are tractable for full joint inference; indeed, the junction trees have a maximal clique size of 10 in the largest models we used on the ten classes. Thus we can use efficient exact calculations to perform inference. The following two sections demonstrate the utility of this fact. Missing pixels: We ran an experiment in which pixels were chosen uniformly at random and erased, as shown in Figure 2. In our generative model, we treat them as hidden variables that were marginalized out. The rightmost plot in Figure 3 shows the error rate on the testing set as a function of the percentage of unknown pixels, for our models and for a SVM. In the case of the SVM, we used a polynomial kernel of degree four [7] and we tried various heuristics to fill in the value of the non-observed pixels, such as the average of that pixel over the training set or the value of a blank pixel. Best classification performance was achieved with replacing the missing value by the value of a blank pixel. Note that very little performance decrement is seen for our classifier even with up to 50 percent of the pixels missing, while for the SVM, although performance is better for small percentages, performance degrades more rapidly as the percentage of erased digits increases. Reconstruction: We conducted an additional experiment in which the upper halves of images were erased. We ran the junction tree inference algorithm to fill in these missing values, choosing the maximizing value of the conditional probability (max-propagation). Figure 3 shows the results. For each line, from left to right, we show the original digit, the digit after erasure, reconstructions based on the model having the maximum likelihood, and 0 0 2 6 1 1 9 7 2 2 6 5 3 3 5 8 4 4 7 9 5 5 8 3 6 6 0 2 7 7 9 4 8 8 6 3 9 7 9 3 Figure 4: Reconstructions of images whose upper halves have been deleted. See text for details. reconstruction based on the model having the second and third largest values of likelihood. 4.3 SPLICE Dataset The task in this dataset is to classify splice junctions in DNA sequences. Splice junctions can either be an exon/intron (EI) boundary, an intron/exon (IE) boundary, or no boundary. (Introns are the portions of genes that are spliced out during transcription; exons are retained in the mRNA). Each sample is a sequence of 60 DNA bases (where each base can take one of four values, A,G,C, or T). The three different classes are: EI exactly at the middle (between the 30th and the 31st bases), IE exactly at the middle (between the 30th and the 31st bases), no splice junction. The dataset is composed of 3175 training samples. In order to be able to compare to previous experiments using this dataset, performance is assessed by picking 2000 training data points at random and testing on the 1175 others, with 20 replications. We treat classification as a density estimation problem in this case by treating the class variable as another variable. We classify by choosing the value of that maximizes the conditional probability . We tested both feature induction and edge induction; in the former case only binary features that are products of features of the form were tested and induced. MDL was used to pick the number of features or edges. Our feature induction algorithm, with a maximum treewidth equal to 5, gave an error rate of ? , while the edge induction algorithm gave an error rate of ? 9 . This is better than the best reported results in the literature; in particular, neural networks have an error rate of ? and the Chow and Liu algorithm has an error rate of ? [14]. 5 Conclusions We have described a methodology for feature selection, edge selection and parameter estimation that can be viewed as a generalization of the Chow-Liu algorithm. Drawing on the feature selection methods of [9, 16], our method is quite general, building an exponential family model from the general vocabulary of features on overlapping subsets of variables. By maintaining tractability throughout the learning process, however, we build this flexible representation of a multivariate density while retaining many of the desirable aspects of the Chow-Liu algorithm. Our methodology applies equally well to feature or edge selection. In large-scale, sparse domains in which overfitting is of particular concern, however, feature selection may be the preferred approach, in that it provides a finer-grained search in the space of simple models than is allowed by the edge selection approach. Acknowledgements We wish to acknowledge NSF grant IIS-9988642 and ONR MURI N00014-00-1-0637. The results presented here were obtained using Kevin Murphy’s Bayes Net Matlab toolbox and SVMTorch [3]. References [1] H. Bodlaender, A linear-time algorithm for finding tree-decompositions of small treewidth, Siam J. Computing, 25, 105-1317, 1996. [2] C.K. Chow and C.N. Liu, Approximating discrete probability distributions with dependence trees, IEEE Trans. Information Theory, 42, 393-405, 1990. [3] R. Collobert and S. Bengio, SVMTorch: support vector machines for large-scale regression problems, Journal of Machine Learning Research, 1, 143-160, 2001. [4] R.G. Cowell, A.P. Dawid, S.L. Lauritzen, and D.J. Spiegelhalter, Probabilistic Networks and Expert Systems, Springer-Verlag, New York, 1999. [5] I. Csisz´ar, I-divergence geometry of probability distributions and minimization problems, Annals of Probability, 3, 146-158, 1975. [6] J.N. Darroch and D. Ratcliff, Generalized iterative scaling for log-linear models, Ann. Math. Statist., 43, 1470-1480, 1972. [7] D. DeCoste and B. Sch¨olkopf, Training invariant support vector machines, Machine Learning, 46, 1-3, 2002. [8] D. Heckerman, D. Geiger, and D.M. Chickering, Learning Bayesian networks: The combination of knowledge and statistical data, Machine Learning, 20, 197-243, 1995. [9] S. Della Pietra, V. Della Pietra, and J. Lafferty, Inducing features of random fields, IEEE Trans. PAMI, 19, 380-393, 1997. [10] R. Jirousek and S. Preucil, On the effective implementation of the iterative proportional fitting procedure, Computational Statistics and Data Analysis, 19, 177-189, 1995. [11] U. Kjaerulff, Triangulation of graphs—algorithms giving small total state space, Technical Report R90-09, Dept. of Math. and Comp. Sci., Aalborg Univ., Denmark, 1990. [12] Y. Le Cun, http://www.research.att.com/˜yann/exdb/mnist/index.html [13] G. Mayraz and G. Hinton, Recognizing hand-written digits using hierarchical products of experts, Adv. NIPS 13, MIT Press, Cambridge, MA, 2001. [14] M. Meila and M.I. Jordan, Learning with mixtures of trees, Journal of Machine Learning Research, 1, 1-48, 2000. [15] N. Srebro, Maximum likelihood bounded tree-width Markov networks, in UAI 2001. [16] S.C. Zhu, Y.W. Wu, and D. Mumford, Minimax entropy principle and its application to texture modeling, Neural Computation, 9, 1997.
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Linking motor learning to function approximation: Learning in an unlearnable force field Opher Donchin and Reza Shadmehr Dept. of Biomedical Engineering Johns Hopkins University, Baltimore, MD 21205 Email: opher@bme.jhu.edu, reza@bme.jhu.edu Abstract Reaching movements require the brain to generate motor commands that rely on an internal model of the task’s dynamics. Here we consider the errors that subjects make early in their reaching trajectories to various targets as they learn an internal model. Using a framework from function approximation, we argue that the sequence of errors should reflect the process of gradient descent. If so, then the sequence of errors should obey hidden state transitions of a simple dynamical system. Fitting the system to human data, we find a surprisingly good fit accounting for 98% of the variance. This allows us to draw tentative conclusions about the basis elements used by the brain in transforming sensory space to motor commands. To test the robustness of the results, we estimate the shape of the basis elements under two conditions: in a traditional learning paradigm with a consistent force field, and in a random sequence of force fields where learning is not possible. Remarkably, we find that the basis remains invariant. 1 Introduction It appears that in constructing the motor commands to guide the arm toward a target, the brain relies on an internal model (IM) of the dynamics of the task that it learns through practice [1]. The IM is presumably a system that transforms a desired limb trajectory in sensory coordinates to motor commands. The motor commands in turn create the complex activation of muscles necessary to cause action. A major issue in motor control is to infer characteristics of the IM from the actions of subjects. Recently, we took a first step toward mathematically characterizing the IM’s representation in the brain [2]. We analyzed the sequence of errors made by subjects on successive movements as they reached to targets while holding a robotic arm. The robot produced a force field and subjects learned to compensate for the field (presumably by constructing an IM) and eventually produced straight movements within the field. Our analysis sought to draw conclusions about the structure of the IM from the sequence of errors generated by the subjects. For instance, in a velocity-dependent force field (such as the fields we use), the IM must be able to encode velocity in order to anticipate the upcoming force. We hoped that the effect of errors in one direction on subsequent movements in other directions would give information about the width of the elements which the IM used in encoding velocity. For example, if the basis elements were narrow, then movements in a given direction would result in little or no change in performance in neighboring directions. Wide basis elements would mean appropriately larger effects. We hypothesized that an estimate of the width of the basis elements could be calculated by fitting the time sequence of errors to a set of equations representing a dynamical system. The dynamical system assumed that error in a movement resulted from a difference between the IM’s approximation and the actual environment, an assumption that has recently been corroborated [3]. The error in turn changed the IM, affecting subsequent movements: ( y(n) = Dk(n)F (n) −z(n) k(n) z(n+1) l = z(n) l + Bl,k(n)y(n) l = 1, · · · , 8 (1) Here y(n) is the error on the nth movement, made in direction k(n) (8 possible directions); F (n) is the actual force experienced in the movement, and it is scaled by an arm compliance D which is direction dependent; and z(n) k is the current output of the IM in the direction k. The difference between this output and reality results in movement errors. B is a matrix characterizing the effect of errors in one direction on other directions. That is, B can provide the generalization function we sought. By comparing the B produced by a fit to human data to the Bs produced from simulated data (generated using a dynamical simulation of arm movements), we found that the time sequence of the subjects’ errors was similar to that generated by a simulation that represented the IM with gaussian basis elements that encoded velocity with a σ = 0.08 m/sec. But why might this dynamical system be a good model of trial-to-trial behavior in a learning paradigm? Here we demonstrate that, under reasonable assumptions, behavior in accordance with Eq. 1 can be derived within the framework of functional approximation, and that B is closely related to the basis functions in the approximation process. We find that this model gives accurate fits to human data, even when the number of parameters in the model is drastically reduced. Finally, we test the prediction of Eq. 1 that learning involves simple movement-by-movement corrections to the IM, and that these variations depend only on the shape of the basis which the IM uses for representation. Remarkably, when subjects perform movements in a force field that changes randomly from one movement to the next, the pattern of errors predicts a generalization function, and therefore a set of basis elements, indistinguishable from the condition where the force field does not change. That is, “an unlearnable task is learned in exactly the same way as a learnable task.” 2 Approach 2.1 The Learning Process In the current task, subjects grip the handle of a robot and make 10cm reaching movements to targets presented visually. The robot produces a force field F(˙x) proportional and perpendicular to the velocity of the hand, such as F = (0 13; −13 0)· ˙x (with F in Newtons and ˙x in m/s). To simulate the process of learning an IM, we assume that the IM uses scalar valued basis functions that encode velocity g = [g1(˙x), . . . , gn(˙x)]T so that the IM’s expectation of force at a desired velocity is: ˆF(˙x) = Wg(˙x), where W is a 2 × n matrix [4]. To move the hand to a target at direction k, a desired trajectory ˙xk(t) is given as input to the IM, which in turn produces as output ˆF(˙xk) [5, 6]. As a result, forces are experienced F(t) so that a force error can be calculated as ˜F(t) = F(t) −ˆF(˙xk(t)). We adjust W in the direction that minimizes a cost function e which is simply the magnitude of the force error integrated over the entire movement: e = 1 2 Z T 0 ˜F(t)T ˜F(t) dt = 1 2 Z T 0 (F(t) −Wg(t))T (F(t) −Wg(t)) dt Changing W to minimize this value requires that we calculate the gradient of e with respect to the weights and move W in the direction opposite to the gradient: (▽e)Wij = ∂e ∂Wij = − Z T 0 gj(t) ˜Fi(t) dt W (n+1) = W (n) + η Z T t=0 ˜F(n)(t)g(˙xk(n)(t))T dt (2) where W (n) means the W matrix on the nth movement. 2.2 Deriving the Dynamical System Our next step is to represent learning not in terms of weight changes, but in terms of changes in IM output, ˆF. We do this for an arbitrary point in velocity space ˙x0 by multiplying both sides of the Eq. 2 by g(˙x0) with the result that: ˆF(n+1)(˙x0) = ˆF(n)(˙x0) + η Z T t=0 g(˙xk(n))T g(˙x0) ˜F(n) dt (3) Further simplification will require approximation. Because we are considering a case where the actual force, F(˙x), is directly proportional to velocity, it is reasonable to make the approximation that, along a reasonably straight desired trajectory, the force error, ˜F(t), is simply proportional to the velocity, ˜F(˙xk(n)) = ˜F · ˙xk(n). This means that the integral of Eq. 3 is actually of the form ˜F Z T t=0 ˙xk(n)(t)g(˙xk(n))T g(˙x0) dt (4) One more assumption is required to make this tractable. If we approximate the desired trajectory with a triangular function of time, and integrate only over the raising phase of the velocity curve (because the values are the same going up and going down) we can simplify the integral to an integral over speed, drawing out a constant (2K R ˙x= ˙xk(250ms) ˙x=0 G( ˙x, ˙x0) d ˙x). The integral has become a function of the values of ˙xk(n)(250ms) and ˙x0. Calling this function B, Eq. 4 becomes ˆF(n+1)(˙x0) = ˆF(n)(˙x0) + B(˙xk(n), ˙x0)˜F(n) (5) ˙x0 is arbitrary. We restrict our attention to only ˙x0 that equals the peak velocity of the desired trajectory associated with a movement direction l. Since we have only eight different points in velocity space to consider, ˆF can be considered an eightvalued vector, ˆFl rather than a function ˆF(˙x). Similarly, B(˙xl, ˙xk) will become an 8x8 matrix, Bl,k. The simpler notation allows us to write Eq. 5 as ˆF(m+1) l = ˆF(n) l + Bl,k(n) ˜F(n) l = 1, . . . , 8 (6) 6 N 9 N 12 N 3 cm Figure 1: We performed simulations to test the approximation that displacement in arm motion at 250 msec toward a target at 10 cm is proportional to error in the force estimate made by the IM. A system of equations describing a controller, dynamics of a typical human arm, and robot dynamics [7] were simulated for a 500 msec min jerk motion to 8 targets. The simulated robot produced one of 8 force fields scaled to 3 different magnitudes, while the controller remained na¨ıve to the field. The errors in hand motion at 250 msec were fitted to the robot forces using a single compliance matrix. Lighter dashed lines are the displacement predictions of the model, darker solid lines are the actual displacement in the simulations’ movement. One more approximation is to assume that force error ˜F in a given movement will be proportional to position error in that movement when both are evaluated at 250ms. This approximation is justified by the data presented in Fig. 1 which shows that the linear relationship holds for a wide range of movements and force errors. Finally, because the forces are perpendicular to the movement, we will disregard the error parallel to the direction of movement, reducing Eq. 6 to a scalar equation. We are now in a position to write our system of equations in its final form: ( y(n) = Dk(n)(F (n) −ˆF (n) k(n)) ˆF (m+1) l = ˆF (n) l + Bl,k(n) ˜F (n) l = 1, . . . , 8 (7) Note that this is a system of nine equations: a single movement causes a change in all 8 directions for which the IM has an expectation. Let us now introduce a new variable z(n) k(n) ≡Dk(n) ˆF (n) k(n), which represents the error (perpendicular displacement) that would have been experienced during this movement if we had not compensated for the expected field. With this substitution, Eq. 7 reduces to Eq. 1. 2.3 The shape of the generalization function B Our task now is to give subjects a sequence of targets, observe the errors in their movements, and ask whether there are parameters for which the system of Eq. 7 gives a good fit. Given a sequence of N movement directions, forces imposed on each movement, and the resulting errors ({k, F, y}(n), j=1, . . . , N), we search for values of Bl,k, Dk and initial conditions ( ˆF (0) m , m=1, . . . , 8) that minimize the squared difference, summed over the movements, between the y calculated in Eq. 7 and the measured errors. One concern is that, in fitting a model with 80 parameters (64 from the B matrix, 8 from D, and 8 from ˆF (0)), we are likely to be overfitting our data. We address this concern by making the assumption that the B matrix has a special shape: Bl,k = b(̸ ˙xl ˙xk). That is, each entry in the B matrix is determined according to the difference in angle between the two directions represented. This assumption implies that g(˙xk)T g(˙xl) depends only on ̸ ˙xk ˙xl. This reduces the B matrix to 8 parameters, and reduces the number of parameters in the model to 24. 0 100 −20 0 20 Error (mm) σ = 04 m/s 0 100 08 m/s 0 100 12 m/s 0 100 20 m/s 0 100 Subjects −180 −90 0 90 180 −0.5 0 0.5 1 Difference in angle Normalized Simulated Bs 0.04 0.08 0.12 0.20 −180 −90 0 90 180 −0.5 0 0.5 1 Difference in angle Normalized Comparison to subjects 0.08 Subjects Figure 2: We simulated a system of equations representing dynamics of robot, human arm, and adaptive controller for movements to a total of 192 targets spanning 8 directions of movement. The adaptive controller learned by applying gradient descent (η = 0.002) to learn a gaussian basis encoding arm velocity with a σ of 0.04, 0.08, 0.12, or 0.20 m/s. Errors, computed as displacement perpendicular to direction of target were measured at 250 msec and are plotted for one direction of movement (45 deg) (a - d). Simulated data is the solid line and the fit is shown as a dashed line. Circles indicate error on no field trials and triangles indicate error on fielded trials. The data for all 192 targets were then fit to Eq. 7 and the generalization matrix B was estimated (f). Data was also collected from 76 subjects, and fit with the model (e), and it gave a generalization function that is nearly identicals to the generalization function of a controller using gaussians with a width of 0.08 m/s (g). 3 Results We first tested the validity of our approach in an artificial learning system that used a simulation of human arm and robot dynamics to learn an IM of the imposed force field with gaussian basis elements. The result was a sequence of errors to a series of targets. We fit Eq. 7 to the sequence of errors and found an estimate for the generalization function (Fig. 2). As expected, when narrow basis elements are used, the generalization function is narrow. We next fit the same model to data that had been collected from 76 subjects and again found an excellent fit. Plots f and g in Fig. 2 show the generalization function, B, as a function of the angle between ˙xk and ˙xl. The demonstrate that errors in one direction affect movements in other directions both in simulations errors and in the subjects’ errors. The greatest effect of error is in the direction in which the movement was made. The immediately neighboring directions are also significantly affected but the effect drops offwith increasing distance. The generalization function which matched the human data was nearly identical to the one matching data produced by the simulation whose gaussians had σ = 0.08 m/sec. The most interesting aspect of the success we had using the simple system in equation 7 to explain human behavior is that the global learning process is being charac0 200 400 −20 −10 0 10 20 Error (mm) Consistent Field Mvmt Num 200 250 300 350 −20 −10 0 10 20 Fit to Consistent Field Error (mm) Mvmt Num Data Fit 0 200 400 −20 −10 0 10 20 Random Field Mvmt Num 200 250 300 350 −20 −10 0 10 20 Fit to Random Field Mvmt Num −180 −90 0 90 180 −0.5 0 0.5 1 Difference in angle B Matrix Learn 1 Learn 2 Rand 1 Rand 2 Figure 3: Fitting the model in Eq. 7 to a learning situation (a and c, 76 subjects) or a situation where subjects are presented with a random sequence of fields (b and d, 6 subjects) produce nearly identical models. a and b show errors (binned to 5 movements per data point), measured as perpendicular distance from a straight line trajectory at 250ms into the movement. Triangles are field A (F = [0 13; −13 0]· ˙x) movements , wedges are field B (F = [0 −13; 13 0]· ˙x), and filled circles are no field. The data is split into three sets of 192 movements. It can be seen that subjects in the learning paradigm learn to counteract the field, and show after affects. Subjects in the random field do not improve on either field, and do not show after affects. c and d show that the model fit both the learning paradigm and the random field paradigm. The fit is plotted for movements made to 90◦during the first 192 movements following first exposure to the field (movements 193 through 384 in a and b). r2 for the fits is 0.96 and 0.97 respectively. Fits to the last 192 movements in each paradigm gave r2 of 0.96 and 0.98. Finally, in the bottom plot, we compare the generalization function, B, given by each fit. The normalized generalization function is nearly identical for the all four sets. The size of the central peak is 0.21 for both sets of the consistent field and 0.19 and 0.14, respectively, for the two sets of the random field. terized as the accretion of small changes in the state of the controller accumulated over a large number of movements. In order to challenge this surprising aspect of the model, we decided to apply it to data in which human subjects performed movements in fields that varied randomly from trial to trial. In this case, no cumulative learning is possible. The important question is whether the model will still be able to fit the data. If it does fit the data, then the question is whether the parameters of the fit are similar to those derived from the learning paradigm. Fig. 3 is a comparison of fitting a model to a consistent field and a random field. As seen in a and b of the figure, subjects are able to improve their performance through learning in a consistent field but they do not improve in the random field. However, as shown in in c and d, the model is able to fit the performance in both fields. Although the fits of each type of field were performed independently, we can see in e that the B matrixes are nearly identical which indicates that trial-by-trial learning was the same for both types of fields. In the second set of the random paradigm, it seems as though the adjustment of state may slower. This raises the possibility that the process of movement-by-movement adjustment of state is gradually abandoned when it consistently fails to produce improvement. It is likely that in this case subjects come to rely on a feedback driven controller which would be unable to compensate for the errors generated early in the movement but would allow them to more quickly adjust to those errors as information about the field they are moving through is processed. 4 Conclusions We hypothesized that the process of learning an internal model of the arm’s dynamics may be similar to mechanisms of gradient descent in the framework of approximation theory. If so, then errors experienced in a given movement should affect subsequent movements in a meaningful way, and perhaps as simply as those predicted by the dynamical system in Eq. 7. These equations appear to fit both simulations and actual human data exceedingly well, making strong predictions about the shape of the basis with which the IM is apparently learned. Here we find that the shape of the basis remains invariant despite radical changes in pattern of errors, as exhibited when subjects were exposed to a random field as compared to a stationary field. We conclude that even when the task is unlearnable and errors approximate a flat line, the brain is attempting to learn with the same characteristic basis which is used when the task is simple and errors exponentially approach zero. References [1] R. Shadmehr and F. A. Mussa-Ivaldi. Adaptive representation of dynamics during learning of a motor task. J. Neurosci., 14(5 Pt 2):3208–3224, 1994. [2] K. Thoroughman and R. Shadmehr. Learning of action through adaptive combination of motor primitives. Nature, 407(6805):742–747, 2000. [3] R. A. Scheidt, J. B. Dingwell, and F. A. Mussa-Ivaldi. Learning to move amid uncertainty. The Journal of Neurophysiology, 86(2):971–985, 2001. [4] R. M. Sanner and M. Kosha. A mathematical model of the adaptive control of human arm motions. Biol. Cybern., 80(5):369–382, 1999. [5] C. G. Atkeson. Learning arm kinematics and dynamics. Annu. Rev. Neurosci., 12:157– 183, 1989. [6] Y. Uno, M. Kawato, and R. Suzuki. Formation and control of optimal trajectory in human multijoint arm movement. minimum torque-change model. Biol. Cybern., 61(2):89–101, 1989. [7] R. Shadmehr and H. H. Holcomb. Neural correlates of motor memory consolidation. Science, 277(5327):821–825, 1997.
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A Parallel Mixture of SVMs for Very Large Scale Problems Ronan Collobert* Universite de Montreal, DIRG CP 6128, Succ. Centre-Ville Montreal, Quebec, Canada collober©iro.umontreal.ca Samy Bengio IDIAP CP 592, rue du Simp Ion 4 1920 Martigny, Switzerland bengio©idiap.ch Yoshua Bengio Universite de Montreal, DIRG CP 6128, Succ. Centre-Ville Montreal, Quebec, Canada bengioy©iro.umontreal.ca Abstract Support Vector Machines (SVMs) are currently the state-of-the-art models for many classification problems but they suffer from the complexity of their training algorithm which is at least quadratic with respect to the number of examples. Hence, it is hopeless to try to solve real-life problems having more than a few hundreds of thousands examples with SVMs. The present paper proposes a new mixture of SVMs that can be easily implemented in parallel and where each SVM is trained on a small subset of the whole dataset. Experiments on a large benchmark dataset (Forest) as well as a difficult speech database, yielded significant time improvement (time complexity appears empirically to locally grow linearly with the number of examples). In addition, and that is a surprise, a significant improvement in generalization was observed on Forest. 1 Introduction Recently a lot of work has been done around Support Vector Machines [9], mainly due to their impressive generalization performances on classification problems when compared to other algorithms such as artificial neural networks [3, 6]. However, SVMs require to solve a quadratic optimization problem which needs resources that are at least quadratic in the number of training examples, and it is thus hopeless to try solving problems having millions of examples using classical SVMs. In order to overcome this drawback, we propose in this paper to use a mixture of several SVMs, each of them trained only on a part of the dataset. The idea of an SVM mixture is not new, although previous attempts such as Kwok's paper on Support Vector Mixtures [5] did not train the SVMs on part of the dataset but on the whole dataset and hence could not overcome the 'Part of this work has been done while Ronan Collobert was at IDIAP, CP 592, rue du Simplon 4, 1920 Martigny, Switzerland. LUHe CUIHIJ1eJULY vrUUleUI lUI 1i:L1!!,e UaLaOeLO. vve vruvuoe Here a l:i't'fltpte 'fIte~ltuu LU LlalH oUCH a mixture, and we will show that in practice this method is much faster than training only one SVM, and leads to results that are at least as good as one SVM. We conjecture that the training time complexity of the proposed approach with respect to the number of examples is sub-quadratic for large data sets. Moreover this mixture can be easily parallelized, which could improve again significantly the training time. The organization of the paper goes as follows: in the next section, we briefly introduce the SVM model for classification. In section 3 we present our mixture of SVMs, followed in section 4 by some comparisons to related models. In section 5 we show some experimental results, first on a toy dataset, then on two large real-life datasets. A short conclusion then follows. 2 Introduction to Support Vector Machines Support Vector Machines (SVMs) [9] have been applied to many classification problems, generally yielding good performance compared to other algorithms. The decision function is of the form (1) where x E ~d is the d-dimensional input vector of a test example, y E {-I, I} is a class label, Xi is the input vector for the ith training example, Yi is its associated class label, N is the number of training examples, K(x , Xi) is a positive definite kernel function, and 0: = {a1 , ... ,aN} and b are the parameters of the model. Training an SVM consists in finding 0: that minimizes the objective function N 1 N N Q(o:) = - 2..: ai + 22..:2..:aiajYiyjK(Xi, Xj) (2) subject to the constraints and i=l i=l j=l N 2..: aiYi = 0 i=l (3) O:S ai :S C Vi. (4) The kernel K(X,Xi) can have different forms, such as the Radial Basis Function (RBF): K(Xi, Xj) = exp (-llxi(T~ Xj112) (5) with parameter (T. Therefore, to train an SVM, we need to solve a quadratic optimization problem, where the number of parameters is N. This makes the use of SVMs for large datasets difficult: computing K(Xi' Xj) for every training pair would require O(N2) computation, and solving may take up to O(N3). Note however that current state-of-the-art algorithms appear to have training time complexity scaling much closer to O(N2 ) than O(N3) [2]. 3 A New Conditional Mixture of SVMs In this section we introduce a new type of mixture of SVMs. The output of the mixture for an input vector X is computed as follows: f(x) = h (II wm(x)sm(x)) (6) wuen~ lVl 1::; LUe UUUIUel Ul eXvelL::; lU LUe lUIXLUle, ;;m~;,r;) 1::; LUe UULVUL Ul LUe 'fit exvelL given input x, wm(x) is the weight for the mth expert given by a "gater" module taking also x in input, and h is a transfer function which could be for example the hyperbolic tangent for classification tasks. Here each expert is an SVM, and we took a neural network for the gater in our experiments. In the proposed model, the gater is trained to minimize the cost function N C = L [f(xi) - Yi]2 . (7) i=l To train this model, we propose a very simple algorithm: 1. Divide the training set into M random subsets of size near N j M. 2. Train each expert separately over one of these subsets. 3. Keeping the experts fixed, train the gater to minimize (7) on the whole training set. 4. Reconstruct M subsets: for each example (Xi,Yi), • sort the experts in descending order according to the values Wm(Xi), • assign the example to the first expert in the list which has less than (NjM + c) examples*, in order to ensure a balance between the experts. 5. If a termination criterion is not fulfilled (such as a given number of iterations or a validation error going up), goto step 2. Note that step 2 of this algorithm can be easily implemented in parallel as each expert can be trained separately on a different computer. Note also that step 3 can be an approximate minimization (as usually done when training neural networks). 4 Other Mixtures of SVMs The idea of mixture models is quite old and has given rise to very popular algorithms, such as the well-known Mixture of Experts [4] where the cost function is similar to equation (7) but where the gater and the experts are trained, using gradient descent or EM, on the whole dataset (and not subsets) and their parameters are trained simultaneously. Hence such an algorithm is quite demanding in terms of resources when the dataset is large, if training time scales like O(NP) with p > 1. In the more recent Support Vector Mixture model [5], the author shows how to replace the experts (typically neural networks) by SVMs and gives a learning algorithm for this model. Once again the resulting mixture is trained jointly on the whole dataset, and hence does not solve the quadratic barrier when the dataset is large. In another divide-and-conquer approach [7], the authors propose to first divide the training set using an unsupervised algorithm to cluster the data (typically a mixture of Gaussians), then train an expert (such as an SVM) on each subset of the data corresponding to a cluster, and finally recombine the outputs of the experts. Here, the algorithm does indeed train separately the experts on small datasets, like the present algorithm, but there is no notion of a loop reassigning the examples to experts according to the prediction made by the gater of how well each expert performs on each example. Our experiments suggest that this element is essential to the success of the algorithm. Finally, the Bayesian Committee Machine [8] is a technique to partition the data into several subsets, train SVMs on the individual subsets and then use a specific combination scheme based on the covariance of the test data to combine the predictions. This method scales linearly in the 'where c is a small positive constant. In the experiments, c = 1. llUll1ue1 U1 Lld111111!!, UdLd, UUL 1~ 111 1dCL d HUnIjU ·uc~·tVt; ll1eLllUU ~ 1L CdllllUL Uve1dLe Ull d ~U1!!,1e test example. Like in the previous case, this algorithm assigns the examples randomly to the experts (however the Bayesian framework would in principle allow to find better assignments). Regarding our proposed mixture of SVMs, if the number of experts grows with the number of examples, and the number of outer loop iterations is a constant, then the total training time of the experts scales linearly with the number of examples. Indeed, &iven N the total number of examples, choose the number of expert M such that the ratio M is a constant r; Then, if k is the number of outer loop iterations, and if the training time for an SVM with r examples is O(ri3 ) (empirically f3 is slightly above 2), the total training time of the experts is O(kri3 * M) = O(kri3- 1 N), where k, rand f3 are constants, which gives a total training time of O(N). In particular for f3 = 2 that gives O(krN). The actual total training time should however also include k times the training time of the gater, which may potentially grow more rapidly than O(N). However, it did not appear to be the case in our experiments, thus yielding apparent linear training time. Future work will focus on methods to reduce the gater training time and guarantee linear training time per outer loop iteration. 5 Experiments In this section, we present three sets of experiments comparing the new mixture of SVMs to other machine learning algorithms. Note that all the SVMs in these experiments have been trained using SVMTorch [2] . 5.1 A Toy Problem In the first series of experiments, we first tested the mixture on an artificial toy problem for which we generated 10,000 training examples and 10,000 test examples. The problem had two non-linearly separable classes and had two input dimensions. On Figure 1 we show the decision surfaces obtained first by a linear SVM, then by a Gaussian SVM, and finally by the proposed mixture of SVMs. Moreover, in the latter, the gater was a simple linear function and there were two linear SVMs in the mixturet . This artificial problem thus shows clearly that the algorithm seems to work, and is able to combine, even linearly, very simple models in order to produce a non-linear decision surface. 5.2 A Large-Scale Realistic Problem: Forest For a more realistic problem, we did a series of experiments on part of the UCI Forest dataset+. We modified the 7-class classification problem into a binary classification problem where the goal was to separate class 2 from the other 6 classes. Each example was described by 54 input features, each normalized by dividing by the maximum found on the training set. The dataset had more than 500,000 examples and this allowed us to prepare a series of experiments as follows: • We kept a separate test set of 50,000 examples to compare the best mixture of SVMs to other learning algorithms. • We used a validation set of 10,000 examples to select the best mixture of SVMs, varying the number of experts and the number of hidden units in the gater. • We trained our models on different training sets, using from 100,000 to 400,000 examples. • The mixtures had from 10 to 50 expert SVMs with Gaussian kernel and the gater was an MLP with between 25 and 500 hidden units. tNote that the transfer function hO was still a tanhO. tThe Forest dataset is available on the VCI website at the following address: ftp://ftp.ics.uci.edu/pub/rnachine-learning-databases/covtype/covtype.info. (a) Linear SVM (b) Gaussian SVM (c) Mixture of two linear SVMs Figure 1: Comparison of the decision surfaces obtained by (a) a linear SVM, (b) a Gaussian SVM, and (c) a linear mixture of two linear SVMs, on a two-dimensional classification toy problem. Note that since the number of examples was quite large, we selected the internal training parameters such as the (J of the Gaussian kernel of the SVMs or the learning rate of the gater using a held-out portion of the training set. We compared our models to • a single MLP, where the number of hidden units was selected by cross-validation between 25 and 250 units, • a single SVM, where the parameter of the kernel was also selected by cross-validation, • a mixture of SVMs where the gater was replaced by a constant vector, assigning the same weight value to every expert. Table 1 gives the results of a first series of experiments with a fixed training set of 100,000 examples. To select among the variants of the gated SVM mixture we considered performance over the validation set as well as training time. All the SVMs used (J = 1. 7. The selected model had 50 experts and a gater with 150 hidden units. A model with 500 hidden units would have given a performance of 8.1 % over the test set but would have taken 621 minutes on one machine (and 388 minutes on 50 machines). Train Test Time (minutes) Error (%) (1 cpu) (50 cpu) one MLP 17.56 18.15 12 one SVM 16.03 16.76 3231 uniform SVM mixture 19.69 20.31 85 2 gated SVM mixture 5.91 9.28 237 73 Table 1: Comparison of performance between an MLP (100 hidden units), a single SVM, a uniform SVM mixture where the gater always output the same value for each expert, and finally a mixture of SVMs as proposed in this paper. As it can be seen, the gated SVM outperformed all models in terms of training and test error. Note that the training error of the single SVM is high because its hyper-parameters were selected to minimize error on the validation set (other values could yield to much lower training error but larger test error). It was also much faster, even on one machine, than the SVM and since the mixture could easily be parallelized (each expert can be trained separately), we also reported Lue LIUIe IL LUUK LU LldUI UU ClV UldCUIUei:>. .1U d UIi:>L dLLeUIVL LU UUUeli:>LdUU LUei:>e lei:>UILi:>, uue can at least say that the power of the model does not lie only in the MLP gater, since a single MLP was pretty bad, it is neither only because we used SVMs, since a single SVM was not as good as the gated mixture, and it was not only because we divided the problem into many sub-problems since the uniform mixture also performed badly. It seems to be a combination of all these elements. We also did a series of experiments in order to see the influence of the number of hidden units of the gater as well as the number of experts in the mixture. Figure 2 shows the validation error of different mixtures of SVMs, where the number of hidden units varied from 25 to 500 and the number of experts varied from 10 to 50. There is a clear performance improvement when the number of hidden units is increased, while the improvement with additional experts exists but is not as strong. Note however that the training time increases also rapidly with the number of hidden units while it slightly decreases with the number of experts if one uses one computer per expert. 2!'50 Validation error as a function of the number of hidden units of the gater and the number of experts 100 150200 250 Number of hidden units of the gater 500 10 50 Figure 2: Comparison of the validation error of different mixtures of SVMs with various number of hidden units and experts. In order to find how the algorithm scaled with respect to the number of examples, we then compared the same mixture of experts (50 experts, 150 hidden units in the gater) on different training set sizes. Table 3 shows the validation error of the mixture of SVMs trained on training sets of sizes from 100,000 to 400,000. It seems that, at least in this range and for this particular dataset, the mixture of SVMs scales linearly with respect to the number of examples, and not quadratically as a classical SVM. It is interesting to see for instance that the mixture of SVMs was able to solve a problem of 400,000 examples in less than 7 hours (on 50 computers) while it would have taken more than one month to solve the same problem with a single SVM. Finally, figure 4 shows the evolution of the training and validation errors of a mixture of 50 SVMs gated by an MLP with 150 hidden units, during 5 iterations of the algorithm. This should convince that the loop of the algorithm is essential in order to obtain good performance. It is also clear that the empirical convergence of the outer loop is extremely rapid. 5.3 Verification on Another Large-Scale Problem In order to verify that the results obtained on Forest were replicable on other large-scale problems, we tested the SVM mixture on a speech task. We used the Numbers95 dataset [1] and 450 ,----~--~-~--~-~-_ 400 350 _300 c: E -;250 E i= 200 150 100 1~ 2 2~ 3 3~ 4 Number of train examples x 105 Figure 3: Comparison of the training time of the same mixture of SVMs (50 experts, 150 hidden units in the gater) trained on different training set sizes, from 100,000 to 400,000. 14 13 12 11 ~10 g 9 w 8 7 6 Error as a function of the number of training iterations 1 Train error Validation Error ~L---~2~--~3---~4---~5 Number of training iterations Figure 4: Comparison of the training and validation errors of the mixture of SVMs as a function of the number of training iterations. turned it into a binary classification problem where the task was to separate silence frames from non-silence frames. The total number of frames was around 540,000 frames. The training set contained 100,000 randomly chosen frames out of the first 400,000 frames. The disjoint validation set contained 10,000 randomly chosen frames out of the first 400,000 frames also. Finally, the test set contained 50,000 randomly chosen frames out of the last 140,000 frames. Note that the validation set was used here to select the number of experts in the mixture, the number of hidden units in the gater, and a. Each frame was parameterized using standard methods used in speech recognition (j-rasta coefficients, with first and second temporal derivatives) and was thus described by 45 coefficients, but we used in fact an input window of three frames, yielding 135 input features per examples. Table 2 shows a comparison between a single SVM and a mixture of SVMs on this dataset. The number of experts in the mixture was set to 50, the number of hidden units of the gater was set to 50, and the a of the SVMs was set to 3.0. As it can be seen, the mixture of SVMs was again many times faster than the single SVM (even on 1 cpu only) but yielded similar generalization performance. Train Test Time (minutes) Error (%) (1 cpu) (50 cpu) one SVM 0.98 7.57 6787 gated SVM mixture 4.41 7.32 851 65 Table 2: Comparison of performance between a single SVM and a mixture of SVMs on the speech dataset. 6 Conclusion In this paper we have presented a new algorithm to train a mixture of SVMs that gave very good results compared to classical SVMs either in terms of training time or generalization performance on two large scale difficult databases. Moreover, the algorithm appears to scale linearly with the number of examples, at least between 100,000 and 400,000 examples. .1 uebe lebUILb dle eXLleIuelY e UCUUli:t!!,l1l!!, dllU bu!!,!!,ebL LUi:tL Lue plupUbeu lueLuuu CUUIU dllUW training SVM-like models for very large multi-million data sets in a reasonable time. If training of the neural network gater with stochastic gradient takes time that grows much less than quadratically, as we conjecture it to be the case for very large data sets (to reach a "good enough" solution), then the whole method is clearly sub-quadratic in training time with respect to the number of training examples. Future work will address several questions: how to guarantee linear training time for the gater as well as for the experts? can better results be obtained by tuning the hyper-parameters of each expert separately? Does the approach work well for other types of experts? Acknowledgments RC would like to thank the Swiss NSF for financial support (project FN2100-061234.00). YB would like to thank the NSERC funding agency and NCM2 network for support. References [1] RA. Cole, M. Noel, T. Lander, and T. Durham. New telephone speech corpora at CSLU. Proceedings of the European Conference on Speech Communication and Technology, EUROSPEECH, 1:821- 824, 1995. [2] R Collobert and S. Bengio. SVMTorch: Support vector machines for large-scale regression problems. Journal of Machine Learning Research, 1:143- 160, 200l. [3] C. Cortes and V. Vapnik. Support vector networks. Machine Learning, 20:273- 297, 1995. [4] Robert A. Jacobs, Michael I. Jordan, Steven J. Nowlan, and Geoffrey E. Hinton. Adaptive mixtures of local experts. Neural Computation, 3(1):79- 87, 1991. [5] J. T. Kwok. Support vector mixture for classification and regression problems. In Proceedings of the International Conference on Pattern Recognition (ICPR), pages 255-258, Brisbane, Queensland, Australia, 1998. [6] E. Osuna, R Freund, and F. Girosi. Training support vector machines: an application to face detection. In IEEE conference on Computer Vision and Pattern Recognition, pages 130- 136, San Juan, Puerto Rico, 1997. [7] A. Rida, A. Labbi, and C. Pellegrini. Local experts combination trough density decomposition. In International Workshop on AI and Statistics (Uncertainty'99). Morgan Kaufmann, 1999. [8] V. Tresp. A bayesian committee machine. Neural Computation, 12:2719-2741,2000. [9] V. N. Vapnik. The nature of statistical learning theory. Springer, second edition, 1995.
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Switch Packet Arbitration via Queue-Learning Timothy X Brown Electrical and Computer Engineering Interdisciplinary Telecommunications University of Colorado Boulder, CO 80309-0530 timxb@colorado.edu Abstract In packet switches, packets queue at switch inputs and contend for outputs. The contention arbitration policy directly affects switch performance. The best policy depends on the current state of the switch and current traffic patterns. This problem is hard because the state space, possible transitions, and set of actions all grow exponentially with the size of the switch. We present a reinforcement learning formulation of the problem that decomposes the value function into many small independent value functions and enables an efficient action selection. 1 Introduction Reinforcement learning (RL) has been applied to resource allocation problems in telecommunications. e.g., channel allocation in wireless systems, network routing, and admission control in telecommunication networks [1, 3, 7, 11]. These have demonstrated reinforcement learning can find good policies that significantly increase the application reward within the dynamics of the telecommunications problems. However, a key issue is how to scale these problems when the state space grows quickly with problem size. This paper focuses on packet arbitration for data packet switches. Packet switches are unlike telephone circuit switches in that packet transmissions are uncoordinated and clusters of traffic can simultaneously contend for switch resources. A packet arbitrator decides the order packets are sent through the switch in order to minimize packet queueing delays and the switch resources needed. Switch performance depends on the arbitration policy and the pattern of traffic entering the switch. A number of packet arbitration strategies have been developed for switches. Many have fixed policies for sending packets that do not depend on the actual patterns of traffic in the network [10]. Under the worse case traffic, these arbitrators can perform quite poorly [8]. Theoretical work has shown consideration of future packet arrivals can have significant impact on the switch performance but is computationally intractable (NP-Hard) to use [4]. As we will show, a dynamic arbitration policy is difficult since the state space, possible transitions, and set of actions all grow exponentially with the size of the switch. In this paper, we consider the problem of finding an arbitration policy that dynamically and efficiently adapts to traffic conditions. We present queue-learning, a formulation that effectively decomposes the problem into many small RL sub-problems. The independent 0.3 0.3 0.3 0.6 0 0.3 0 0.6 0.3 Switch Arrivals 1 3 2 1 2 1 1 2 3 3 x 3 Out Queues 2 1 1 1 0 0 0 1 0 (a) (b) (c) Figure 1: The packet arbitration model. (a) In each time slot, packet sources generate packets on average at input for output . (b) Packets arrive at an input-queued switch and are stored in queues. The number label on each packet indicates to which output the packet is destined. (c) The corresponding queue states, where
indicates the number of packets waiting at input destined for output . RL problems are coupled via an efficient algorithm that trades off actions in the different sub-problems. Results show significant performance improvements. 2 Problem Description The problem is comprised of traffic sources generating traffic at each of inputs to a packet data switch as shown in Figure 1. Time is divided into discrete time slots and in each time slot each source generates 0 or 1 packets. Each packet that arrives at the input is labeled with which of the outputs the packet is headed. In every time slot, the switch takes packets from inputs and delivers them at their intended output. We describe the specific models for each used in this paper and then state the packet arbitration problem. 2.1 The Traffic Sources At input , a traffic source generates a packet destined for output with probability at the beginning of each time slot. If is the load on input and is the load on output , then for stability we require !#" and $ % !&" . The matrix (' *) only represents long term average loads between input and output . We treat the case where packet arrivals are uncorrelated over time and between sources so that in each time slot, a packet arrives at input with probability and given that we have an arrival, it is destined for output with probability ,+* . Let the set of packet arrivals be - . 2.2 The Switch The switch alternates between accepting newly arriving packets and sending packets in every time slot. At the start of the time slot the switch sends packets waiting in the input queues and delivers them to the correct output where they are sent on. Let . /',0 ) represent the set of packets sent where 0 if a packet is sent from input to output and 0 ! otherwise. The packets it can send are limited by the input and output constraints: the switch can send at most one packet per input and can deliver at most one packet to each output. After sending packets, the new arrivals are added at the input and the switch moves to the next time slot. Other switches are possible, but this is the simplest and a common architecture in high-speed switches. 2.3 The Input Queues Because the traffic sources are un-coordinated, it is possible for multiple packets to arrive in one time slot at different inputs, but destined for the same output. Because of the output constraint, only one such packet may be sent and the others buffered in queues, one queue per input. Thus packet queueing is unavoidable and the goal is to limit the delays due to queueing. The queues are random access which means packets can be sent in any order from a queue. For the purposes of this paper, all packets waiting at an input and destined for the same output are considered equivalent. Let '
*) be a matrix where
is the number of packets waiting at input for output as shown in Figure 1c. 2.4 Packet Arbitration The packet arbitration problem is: Given the state of the input queues, , choose a set of packets to send, . , so at most one packet is sent from each input and at most one packet is delivered to each output. We want a packet arbitration policy that minimizes the expected packet wait time. When . is sent the remaining packets must wait at least one more time slot before they can be sent. Let be the total number of packets in all the input queues, let be the number of new arrivals, and let . be the number of packets sent. Thus, the total wait of all packets is increased by the number of packets that remain: . . By Little’s theorem, the expected wait time is proportional to the expected number of packets waiting in each time slot [10]). Thus, we want a policy that minimizes the expected value of . . The complexity of this problem is high. Given an input and output switch. The input and output constraints are met with equality if . is a subset of a permutation matrix (zeros everywhere except that every row has at most one one and every column has one one). This implies there are as many as possible . to choose from. In each time slot at each input, a packet can arrive for one of outputs or not at all. This implies as many as
possible transitions after each send. If each
ranges from 0 to packets, then the number of states in the system is . A minimal representation would only indicate whether each sub-queue is empty or not, resulting in states. Thus, every aspect of the problem grows exponentially in the size of the switch. Traditionally switching solves these problems by not considering the possible next arrivals, and using a search algorithm with time-complexity polynomial in that considers only the current state . For instance the problem can be formulated as a so-called matching problem and polynomial algorithms exist that will send the largest . possible [2, 6, 8]. While maximizing the packets sent in every time slot may seem like a solution, the problem is more interesting than this. In general, many possible . will maximize the number of packets that are sent. Which one can we send now so that we will be in the best possible state for future time slots? Some heuristics can guide this choice, but these are insensitive to the traffic pattern [9]. Further, it can be shown that to minimize the total wait it may be necessary to send less than the maximum number of packets in the current time slot [4]. So, we look to a solution that efficiently finds policies that minimize the total wait by adapting to the current traffic pattern. The problem is especially amenable to RL for two reasons. (1) Packet rates are fast, up to millions of packets per second so that many training examples are available. (2) Occasional bad decisions are not catastrophic. They only increase packet delays somewhat, and so it is possible to freely learn in an online system. The next section describes our solution. 3 Queue-Learning Solution At any given time slot, , the system is in a particular state, . New packets, , arrive and the packet arbitrator can choose to send any valid . . The cost,
. is the
. packets that remain. The task of the learner is to determine a packet arbitration policy that minimizes the total average cost. We use the Tauberian approximation, that is, we assume the discount factor is close enough to 1 so that the discounted reward policy is equivalent to the average reward policy [5]. Since minimizing the expected value of this cost is equivalent to minimizing the expected wait time, this formulation provides an exact match between RL and the problem task. As shown already every aspect of this problem scales badly. The solution to this problem is three fold. First we use online learning and afterstates [12] to eliminate the need to average over the
possible next states. Second, we show how the value function can yield a set of inputs into a polynomial algorithm for choosing actions. Third, we decompose the value function so the effective number of states is much smaller than . We describe each in turn. 3.1 Afterstates RL methods solve MDP problems by learning good approximations to the optimal value function, . A single time slot consists of two stages: new arrivals are added to the queues and then packets are sent (see Figure 2). The value function could be computed after either of these stages. We compute it after packets are sent since we can use the notion of afterstates to choose the action. Since the packet sending process is deterministic, we know the state following the send action. In this case, the Bellman equation is:
. !
#" %$'& where (
is the set of actions available in the current state after arrival event - ,
. . is the effective immediate cost, is the discount factor, and '*) ) is the expectation over possible events and the resulting next state is " . We learn an approximation to using TD(0) learning. At time-step on a transition from state to ,+- on action . after event , we update an estimate to via
/.
. !
,+
%$ where . is the learning step size. With afterstates, the action (which set of packets to send) depends on both the current state and the event. The best action is the one that results in the lowest value function in the next state (which is known deterministically given , , and . ). In this way, afterstated eliminates the need to average over a large number of non-zero transitions to find the best action. 3.2 Choosing the Action We compare every action with the action of not sending any packets. The best action, is the set of packets meeting the input and output constraints that will reduce the value function the most compared to not sending any packets. Each input-output pair
has an associated queue at the input,
. Packets in
contend with other packets at input and other packets destined for output . If we send a packet from
, then no packet at the same input or output will be sent. In other words, packets at Queue State Packet Arrivals Queue After Arrivals
Packets Sent . Next State ,+2 1 1 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 2 2 1 1 0 0 0 2 0 0 0 1 1 0 0 0 1 0 2 2 0 0 0 0 0 1 0 Stochastic Step Deterministic Step
Decision
,+ Figure 2: Timing of packet arrivals and sends relative to decisions and the value function.
interact primarily with packets in the same row and column. Packets in other rows and columns only have an indirect effect on the value of sending a packet from
. This suggests the following approximation. Let
be the number of packets in every subqueue after arrivals - in state and before the decision. Let
be the reduction in the value function if one packet is sent from subqueue
(
! if the subqueue is empty). We can reformulate the best action as: . argmax
0 subject to the constraints: 0 ' ! ) " 0 " 0 " This problem can be solved as a linear program and is also known as the weighted matching or the assignment problem which has a polynomial time solution [13]. In this way, we reduce the search over the
possible actions to a polynomial time solution. 3.3 Decomposing the Value Function The interaction between queues in the same row or the same column is captured primarily by the input and output constraints. This suggests a further simplifying approximation with the following decomposition. We compute a separate value function for each
, denoted
. In principle, this can depend on the entire state , but can be reduced to consider only elements of the state relevant to
. Every
estimates its associated value function
based on the packets at input and packets destined for output . Many forms of
could be considered, but we consider a linear approximation. Let
be the total number of packets waiting at input . Let
$ be the total number of packets waiting for output . With these variables we define a linear approximation with parameters
:
$
(1) It follows the value of sending a packet (compared to not sending a packet) from
is
*
*
*
$
This is computed for each
and used in the weighted matching of Section 3.2 to compute which packets to send. Learning for this problem is standard TD(0) for linear approximations [12]. The combination of decomposition and linear value function approximation reduces the problem to estimating
parameters. No explicit exploration is used since from the perspective of , enough stochasticity already exists in the packet arrival and send processes. To assist the switch early in the learning, the switch sends the packets from a maximum matching in each time slot (instead of the packets selected by queue learning). This initial assist period during the training was found to bring the switch into a good operating regime from which it could learn a better policy. In summary, we simplify the exponential computation for this problem by decomposing the state into substates. Each substate computes the value of sending a packet versus not sending a packet, and a polynomial algorithm computes the action that maximizes the total value across substates subject to the input and output constraints. 4 Implementation Issues A typical high speed link rate is at OC-3 rates (155Mbps). In ATM at this rate, the packet rate is 366k time slots/s or less than 30 sec for ! time slots. For learning, the number of floating point operations per time slot is approximately where is the number of parameters in the linear approximation. At the above packet rate, for an switch, this translates into 650 MFLOPS which is within existing highend microprocessor capacity. For computation of the packets to send, the cost is approximately to compute the weights. To compute the maximum weight matching an
algorithm exists [13]. New optical transport technologies are pushing data rates one and two orders of magnitude greater than OC-3 rates. In this case, if computing is limited then the queue-learning can learn on a subsample of time slots. To compute the packets to send, the decomposition has a natural parallel implementation that can divide it among processors. Massively parallel neural networks can also be used to compute the maximum weighted matching [2, 9]. 5 Simulation Results We applied our procedure to
switches under different loads. The parameters used in the experiment are shown in Table 1. In each experiment, the queue-learning was trained for an initial period, and then the mean wait time, is measured over a test period. We compared performance to two alternatives. One alternative sends the largest number of packets in every time slot. If multiple sets are equally large it chooses randomly between them. We simulate this arbitrator and measure the mean packet wait time, . The best possible switch is a so-called output-queued switch [10]. Such a switch is difficult to build at high-speeds, but we can compute its mean packet wait time, , via simulation. The results are specified in normalized form as
+
$ . Thus if our queue-learning solution is no better than a max send arbitrator, the gain will be 0 and if we achieve the performance of the output-queued switch, the gain will be 1. We experimented on five different traffic loads. - is a uniform load of ! packets per input per time slot with each packet uniformly destined for one of the outputs. Similarly, is a uniform load of ! ! . The uniform load is a common baseline scenario for evaluating switches. and are random matrices where the sum of loads per row and column are 0.6 and 0.9 (as in - and ) but the distribution is not uniform. This is generated by summing permutation matrices and than scaling the entries to yield the desired row and column sums Table 1: RL parameters. Parameter Value Discount, 0.99 Learn Rate, . + Assist Period ! time slots Train Period ! time slots Test Period ! time slots Table 2: Simulation Results. Switch Loading Normalized Wait Reduction ( ) - (uniform 0.6 load) 10% (uniform 0.9 load) 50% (random 0.6 load) 14% (random 0.9 load) 70% (truncated 0.9 load) 84% (e.g. Figure 1a). The random load is a more realistic in that loads tend to vary among the different input/output pairs. is , except that all for the last + outputs is set to zero. This simulates the more typical case of traffic being concentrated on a few outputs. We emphasize that a different policy is learned for each of these loads. The different loads suggest the kinds of improvements that we might expect if queue-learning is implemented. The results for the five loads are given in Table 2. 6 Conclusion This paper showed that queue learning is able to learn a policy that significantly reduces the wait times of packets in a high-speed switch. It uses a novel decomposition of the value function combined with efficient computation of the action to overcome the problems a traditional RL approach would have with the large number of states, actions, and transitions. This is able to gain 10% to 84% of the possible reductions in wait times. The largest gains are when the network is more heavily loaded and delays are largest. The gains are also largest when the switch load is least uniform which is what is most likely to be encountered in practice. Traditional thinking in switching is that input-queued switches are much worse than the optimal output-queuedswitches and improving performance would require increasing switching speeds (the electronic switching is already the slowest part of the otherwise optical networking), or using information of future arrivals (which may not exists and in any case is NP-Hard to use optimally). The queue-learning approach is able to use its estimates of the future impact of its packet send decisions in a consistent framework that is able to bridge the majority of the gap between current input queueing and optimal output queueing. Acknowledgment This work was supported by CAREER Award: NCR-9624791. References [1] Boyan, J.A., Littman, M.L., “Packet routing in dynamically changing networks: a reinforcement learning approach,” in Cowan, J.D., et al., ed. Advances in NIPS 6, Morgan Kauffman, SF, 1994. pp. 671–678. [2] Brown, T.X, Lui, K.H., “Neural Network Design of a Banyan Network Controller,” IEEE JSAC, v. 8, n. 8, pp. 1428–1438, Oct., 1990. [3] Brown, T.X, Tong, H., Singh, S., “Optimizing admission control while ensuring quality of service in multimedia networks via reinforcement learning,” in Advances NIPS 11, ed. M. Kearns et al., MIT Press, 1999. [4] Brown, T.X, Gabow, H.N., “Future Information in Input Queueing,” submitted to Computer Networks, April 2001. [5] Gabor, Z., Kalmar, Z., Szepesvari, C., “Multi-criteria Reinforcement Learning,” International Conference on Machine Learning, Madison, WI, July, 1998. [6] J. Hopcroft and R. Karp, “An algorithm for maximum matchings in bipartite graphs”, SIAM J. Computing 2, 4, 1973, pp 225-231. [7] Marbach, P., Mihatsch, M., Tsitsiklis, J.N., “Call admission control and routing in integrated service networks using neuro-dynamic programming,” IEEE J. Selected Areas in Comm., v. 18, n. 2, pp. 197–208, Feb. 2000. [8] McKeown, N., Anantharam, V., Walrand, J., “Achieving 100% Throughput in an Input-Queued Switch,” Proc. of IEEE INFOCOM ’96, San Francisco, March 1996. [9] Park, Y.-K., Lee, G., “NN Based ATM Scheduling with Queue Length Based Priority Scheme,” IEEE J. Selected Areas in Comm., v. 15, n. 2 pp. 261–270, Feb. 1997. [10] Pattavina, A., Switching Theory: Architecture and Performance in Broadband ATM Networks, John Wiley and Sons, New York, 1998. [11] Singh, S.P., Bertsekas, D.P., “Reinforcement learning for dynamic channel allocation in cellular telephone systems,” in Advances in NIPS 9, ed. Mozer, M., et al., MIT Press, 1997. pp. 974–980. [12] Sutton, R.S., Barto, A.G., Reinforcement Learning: an Introduction, MIT Press, 1998. [13] Tarjan, R.E., Data Structures and Network Algorithms, Soc. for Industrial and Applied Mathematics, Philidelphia, 1983.
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Multi Dimensional ICA to Separate Correlated Sources Roland Vollgraf, Klaus Obermayer Department of Electrical Engineering and Computer Science Technical University of Berlin Germany { vro, oby} @cs.tu-berlin.de Abstract We present a new method for the blind separation of sources, which do not fulfill the independence assumption. In contrast to standard methods we consider groups of neighboring samples ("patches") within the observed mixtures. First we extract independent features from the observed patches. It turns out that the average dependencies between these features in different sources is in general lower than the dependencies between the amplitudes of different sources. We show that it might be the case that most of the dependencies is carried by only a small number of features. Is this case - provided these features can be identified by some heuristic - we project all patches into the subspace which is orthogonal to the subspace spanned by the "correlated" features. Standard ICA is then performed on the elements of the transformed patches (for which the independence assumption holds) and robustly yields a good estimate of the mixing matrix. 1 Introduction ICA as a method for blind source separation has been proven very useful in a wide range of statistical data analysis. A strong criterion, that allows to detect and separate linearly mixed source signals from the observed mixtures, is the independence of the source signals amplitude distribution. Many contrast functions rely on this assumption, e.g. in the way, that they estimate the Kullback-Leibler distance to a (non-Gaussian) factorizing multivariate distribution [1, 2, 3]. Others consider higher order moments of the source estimates [4, 5]. Naturally these algorithms fail when the independence assumption does not hold. In such situations it can be very useful to consider temporal/spatial statistical properties of the source signals as well. This has been done in form of suitable linear filtering [6] to achieve a sparse and independent representation of the signals. In [7] the author suggests to model the sources as a stochastic process and to do the ICA on the innovations rather than on the signals them self. In this work we extend the ICA to multidimensional channels of neighboring realizations. The used data model is explained in detail in the following section. In section 3 it will be shown, that there are optimal features, that carry lower dependencies between the sources and can be used for source separation. A heuristic is introduced, that allows to discard those features, that carry most of the dependencies. This leads to the Two-Step algorithm described in section 4. Our method requires (i) sources which exhibit correlations between neighboring pixels (e.g. continuous sources like images or sound signals), and (ii) sources from which sparse and almost independent features can be extracted. In section 5 we show separation results and benchmarks for linearly mixed passport photographs. The method is fast and provides good separation results even for sources, whose correlation coefficient is as large as 0.9. 2 Sources and observations Let us consider a set of N source signals Si(r), i = 1, ... , N of length L, where r is a discrete sample index. The sample index could be of arbitrary dimension, but we assume that it belongs to some metric space so that neighborhood relations can be defined. The sample index might be a scalar for sources which are time series and a two-dimensional vector for sources which are images1 . The sources are linearly combined by an unknown mixing matrix A of full rank to produce a set of N observations Xi(r), N Xi(r) = l: AijSj(r) , (1) j=l and we assume that the mixing process is stationary, i.e. that the mixing matrix A is independent of r. In the following we refer to the vectors S(r) = (Sl (r), ... ,SN(r))T and X(r) = (X1(r), ... , XN(r))T as a source and an observation stack. The goal is to find an appropriate demixing matrix W which - when applied to the observations X(r) - recovers good estimates S(r), S(r) = WX(r) ~ S(r) (2) of the original source signals (up to a permutation and scaling of the sources). Since the mixing matrix A is not known its inverse W has to be detected blindly, i.e. only properties of the sources which are detectable in the mixtures can be exploited. For a large class of ICA algorithms one assumes that the sources are non-Gaussian and independent, i.e. that the random vector S which is sampled by L realizations S: {S(rd, 1= I, ... ,L} (3) has a factorizing and non-Gaussian joint probability distribution2 . In situations, however, where the independence assumption does not hold, it can be helpful to take into account spatial dependencies, which can be very prominent for natural signals, and have been subject for a number of blind source separation algorithms [8, 9, 6]. Let us now consider patches si(r), s(r) = (4) 1 In the following we will mostly consider images, hence we will refer to the abovementioned neighborhood relations as spatial relations. 2In the following, symbols without sample index will refer to the random variable rather than to the particular realization. of M « L neighboring source samples. si(r) could be a sequence of M adjacent samples of an audio signal or a rectangular patch of M pixels in an image. Instead of L realizations of a random N-vector S (cf. eq. (3)) we now obtain a little less than L realizations of a random N x M matrix s, s: {s(r)}. (5) Because of the stationarity of the mixing process we obtain x = As and s = Wx, (6) where x is an N x M matrix of neighboring observations and where the matrices A and W operate on every column vector of sand x. 3 Optimal spatial features Let us now consider a set of sources which are not statistically independent, i.e. for which N p(S) = p(Slk"'" SNk) :j:. IIp(sik) for all k = 1 ... M. (7) i=1 Our goal is to find in a first step a linear transformation 0 E IRMxM which when applied to every patch - yields transformed sources u = sOT for which the independence assumption, p(Ulk, ... ,UNk) = rr~1p(Uik) does hold for all k = 1 .. . M, at least approximately. When 0 is applied to the observations x, v = xOT, we obtain a modified source separation problem (8) where the demixing matrix W can be estimated from the transformed observations v in a second step using standard ICA. Eq. (7) is tantamount to positive transinformation of the source amplitudes. (9) where DKL is the Kullback-Leibler distance. As all elements of the patches are equally distributed, this quantity is the same for all k. Clearly, the dependencies, that are carried by single elements of the patches, are also present between whole patches, i.e. J(S1 , S2,"', SN) > O. However, since neighboring samples are correlated, it holds M J(S1 ,S2, "' ,SN) < LJ(Slk,S2k"",SNk) . k=1 (10) Only if the sources where spatially white and s would consist of independent column vectors, this would hold with equality. When 0 is applied to the source patches, the trans-information between patches is not changed, provided 0 is a non-singular transformation. Neither information is introduced nor discarded by this transformation and it holds (11) For the optimal 0 now the column vectors of u = sOT shall be independent. From (10) and (11) it follows that M M I(u1 ,u2, " ',uN) = 2::I(ulk,u2k"",uNk) < 2::I(slk,s2k"",sNk) (12) k=1 k=1 The column vectors of u are in general not equally distributed anymore, however the average trans-information has decreased to the level of information carried between the patches. In the experiments we shall see that this can be sufficiently small to reliably estimate the de-mixing matrix W. So it remains to estimate a matrix 0 that provides a matrix u with independent columns. We approach this by estimating 0 so that it provides row vectors of u that have independent elements, i.e. P(Ui) = IT;!1 P(Uik) for all i. With that and under the assumption that all sources may come from the same distribution and that there are no "cross dependencies" in u (i.e. p( Uik) is independent from p( Ujl) for k :j:. l), the independence is guaranteed also for whole column vectors of u. Thus, standard leA can be applied to patches of sources which yields 0 as the de-mixing matrix. For real world applications however, 0 has to be estimated from the observations xOT = v. It holds the relation v = Au, i.e. A only interchanges rows. So column vectors of u are independent to each other if, and only if columns of v are independent3 . Thus, 0 can be computed from x as well. According to Eq. (12) the trans-information of the elements of columns of u has decreased in average, but not necessarily uniformly. One can expect some columns to have more independent elements than others. Thus, it may be advantageous to detect these columns rsp. the corresponding rows of 0 and discard them prior to the second leA step. Each source patch Si can be considered as linear combination of independent components, that are given by the columns of 0- 1 , where the elements of Ui are the coefficients. In the result of the leA, the coefficients have normalized variance. Therefore, those components, that have large Euklidian norm, occur as features with high entropy in the source patches. At the same time it is clear that, if there are features, that are responsible for the source dependencies, these features have to be present with large entropy, otherwise the source dependencies would have been low. Accordingly we propose a heuristic that discards the rows of 0 with the smallest Euklidian norm prior to the second leA step. How many rows have to be discarded and if this type of heuristic is applicable at all, depends of the statistical nature of the sources. In section 5 we show that for the test data this heuristic is well applicable and almost all dependencies are contained in one feature. 4 The Two-Step algorithm The considerations of the previous section give rise to a Two-Step algorithm. In the first step the transformation 0 has to be estimated. Standard leA [1, 2, 5] is performed on M -dimensional patches, which are chosen with equal probability from all of the observed mixtures and at random positions. The positions may overlap but don't overlap the boundaries of the signals. The resulting "demixing matrix" 0 is applied to the patches of observations, generating a matrix v(r) = x(r )OT, the columns of which are candidates for the input for the second leA. A number of M D columns that belong to rows of 0 with small norm are discarded as they very likely represent features , that carry dependencies between the sources. M D is chosen as a model parameter or it can be determined empirically, given the data at hand (for instance by detecting a major jump in the 3We assume non-Gaussian distributions for u and v. increase of the row norm of n). For the remaining columns it is not obvious which one represents the most sparse and independent feature. So any of them with equal probability now serve as input sample for the second ICA, which estimates the demixing matrix W. When the number N of sources is large, the first ICA may fail to extract the independent source features, because, according to the central limit theorem, the distribution of their coefficients in the mixtures may be close to a Gaussian distribution. In such a situation we recommend to apply the abovementioned two steps repeatedly. The source estimates Wx(r) are used as input for the first ICA to achieve a better n, which in turn allows to better estimate W. Figure 1: Results of standard and multidimensional ICA performed on a set of 8 correlated passport images. Top row: source images; Second row: linearly mixed sources; Third row: separation results using kurtosis optimization (FastICA Matlab package); Bottom row: separation results using multidimensional ICA (For explanation see text). 5 Numerical experiments We applied our method to a linear mixture of 8 passport photographs which are shown in Fig. 1, top row. The images were mixed (d. Fig. 1, second row) using a matrix whose elements were chosen randomly from a normal distribution with mean zero and variance one. The mixing matrix had a condition number of 80. The correlation coefficients of the source images were between 0.4 and 0.9 so that standard ICA methods failed to recover the sources: Fig. 1, 3rd row, shows the results of a kurtosis optimization using the FastICA Matlab package4 . Fig. 1, bottom row, shows the result of the Two-Step multidimensional ICA described in section 4. For better comparison images were inverted manually to appear positive. In the first step n was estimated using FastICA on 105 patches, 6 x 6 pixels in size, which were taken with equal probability from random positions from all mixtures. The result of the first ICA is displayed in Fig. 2. The top row shows the row vectors of n sorted by the logarithm of their norm. The second row shows the features (the corresponding columns of n- 1 ) which are extracted by n. In the dia4http://www.cis.hut.fi/projects/ica/fastica/ gram below the stars indicate the logarithm of the row norm, log V'Lt!1 0%1' and the squares indicate the mutual information J(Ulk,U7k) between the k-th features in sources 1 and 7 5, calculated using a histogram estimator. It is quite prominent that (i) a small norm of a column vector corresponds to a strongly correlated feature, and (ii) there is only one feature which carries most of the dependencies between the sources. Thus, the first column of v was discarded. The second ICA was applied to any of the remaining components, chosen randomly and with equal probability. A comparison between Figs. 1, top and bottom rows, shows that all sources were successfully recovered. Figure 2: Result of an ICA (kurtosis optimization) performed on patches of observations (cf. Fig. 1, 2nd row), 6 x 6 pixels in size. Top row: Row vectors of the demixing matrix O. Second row: Corresponding column vectors of 0- 1 . Vectors are sorted by increasing norm of the row vectors; dark and bright pixels indicate positive and negative values. Bottom diagram: Logarithm of the norm of row vectors (stars) and mutual information J(Ulk' U7k) (squares) between the coefficients of the corresponding features in the source images 1 and 7. In the next experiment we examined the influence of selecting columns of v prior to the second ICA. In Fig. 3 we show the reconstruction error (cf. appendix A), that could be achieved with the second ICA when only a single column of v served as input. From the previous experiment we have seen, that only the first component has considerable dependencies. As expected, only the first column yields poor reconstruction error. Fig. 4 shows the reconstruction error vs. M D when the M D smallest norm rows of 0 (rsp. columns of v) are discarded. We see, that for all values a good reconstruction is achieved (re < 0.6). Even if no row is discarded the result is only slightly worse than for one or two discarded rows. The dependencies of the first component are "averaged" by the vast majority of components, that carry no dependencies, in this case. The conspicuous large variance of the error for larger numbers M D might be due to convergence instabilities or close to Gaussian distributed columns of u. In either case it gives rise to discard as few components as possible. To evaluate the influence of the patch size M, the Two-Step algorithm was applied to 9 different mixtures of the sources shown in Fig. 1, top row, and using patch sizes between M = 2 x 2 and M = 6 x 6. Table 1 shows the mean and standard deviation of the achieved reconstruction error. The mixing matrix A was randomly chosen from a normal distribution with mean zero and variance one. FastICA was used for both steps, where 5.105 sample patches were used to extract the optimal features and 2.5.104 samples were used to estimate W. The smallest row of 0 was always discarded. The algorithm shows a quite robust performance, and even for patch sizes of 2 x 2 pixels a fairly good separation result is achieved 5Images no. 1 and 7 were chosen exemplarily as the two most strongly correlated sources. Jl ·~. ==1 !C.,,". : .. ::. :':,. ::!·;::=I 1 6 11 16 21 26 31 36 0 5 10 large row norm small row norm Figure 3: Every single row of 0 used to generate input for the second leA. Only the first (smallest norm) row causes bad reconstruction error for the second leA step. patch size M J-lre (Jre 2x2 0.4361 0.0383 3x3 0.2322 0.0433 4x4 0.1667 0.0263 5x5 0.1408 0.0270 6x6 0.1270 0.0460 Figure 4: M D rows with smallest norm discarded. All values of M D provide good reconstruction error in the second step. Note the slidely worse result for MD=O! Table 1: Separation result of the TwoStep algorithm performed on a set of 8 correlated passport images (d. Fig. 1, top row). The table shows the average reconstruction error J-lre and its standard deviation (Jre calculated from 9 different mixtures. (Note, for comparison, that the reconstruction error of the separation in Fig. 1, bottom row, was 0.2). 6 Summary and outlook We extended the source separation model to multidimensional channels (image patches). There are two linear transformations to be considered, one operating inside the channels (0) and one operating between the different channels (W). The two transformations are estimated in two adjacent leA steps. There are mainly two advantages, that can be taken from the first transformation: (i) By arranging independence among the columns of the transformed patches, the average transinformation between different channels is decreased. (ii) A suitable heuristic can be applied to discard those columns of the transformed patches, that carry most of the dependencies between different channels. A heuristic, that identifies the dependence carrying components by a small norm of the corresponding rows of 0 has been introduced. It shows, that for the image data only one component carries most of the dependencies. Due this fact, the described method works well also when all components are taken into account . In future work, we are going to establish a Maximum Likelihood model for both transformations. We expect a performance gain due to the mutual improvement of the estimates of W and 0 during the iterations. It remains to examine what the model has to be in case some rows of 0 are discarded. In this case the transformations don't preserve the dimensionality of the observation patches. A Reconstruction error The reconstruction error re is a measure for the success of a source separation. It compares the estimated de-mixing matrix W with the inverse of the original mixing matrix A with respect to the indeterminacies: scalings and permutations. It is always nonnegative and equals zero if, and only if P = W A is a nonsingular permutation matrix. This is the case when for every row of P exactly one element is different from zero and the rows of P are orthogonal, i.e. ppT is a diagonal matrix. The reconstruction error is the sum of measures for both aspects N N N N N N re 2LlogL P 7j - Llog LPij + Llog L P 7j -log detppT i=1 j=1 i=1 j=1 i=1 j=1 N N N N 3 L log L P 7j - L log L pij - log det ppT . (13) i=1 j=1 i=1 j=1 Acknowledgment: This work was funded by the German Science Foundation (grant no. DFG SE 931/1-1 and DFG OB 102/3-1 ) and Wellcome Trust 061113/Z/00. References [1] Anthony J. Bell and Terrence J . Sejnowski, "An information-maximization approach to blind separation and blind deconvolution," Neural Computation, vol. 7, no. 6, pp. 1129-1159, 1995. [2] S. Amari, A. Cichocki, and H. H. Yang, "A new learning algorithm for blind signal separation," in Advances in Neural Information Processing Systems, D. S. Touretzky, M. C. Mozer, and M. E. Hasselmo, Eds., 1995, vol. 8. [3] J . F. Cardoso, "Infomax and maximum likelihood for blind source separation," IEEE Signal Processing Lett., 1997. [4] Jean-Franc;ois Cardoso, Sandip Bose, and Benjamin Friedlander, "On optimal source separation based on second and fourth order cumulants," in Proc. IEEE Workshop on SSAP, Co rfou, Greece, 1996. [5] A. Hyvarinen and E. Oja, "A fast fixed point algorithm for independent component analysis.," Neural Comput., vol. 9, pp. 1483- 1492,1997. [6] M. Zibulevski and B. A. Pearlmutter, "Blind source separation by sparse decomposition in a signal dictionary," Neural Computation, vol. 12, no. 3, pp. 863- 882, April 200l. [7] A. Hyvi:irinen, "Independent component analysis for time-dependent stochastic processes," in Proc. Int. Conf. on Artificial Neural Networks (ICANN'98), 1998, pp. 541-546. [8] 1. Molgedey and H. G. Schuster, "Separation of a mixture of independent signals using time delayed correlations," Phys. Rev. Lett., vol. 72, pp. 36343637, 1994. [9] H. Attias and C. E. Schreiner, "Blind source separation and deconvolution: The dynamic component analysis algorithm," Neural Comput., vol. 10, pp. 1373- 1424, 1998. [10] Anthony J. Bell and Terrence J. Sejnowski, "The 'independent components' of natural scenes are edge filters," Vision Res., vol. 37, pp. 3327- 3338, 1997.
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On the Generalization Ability of On-line Learning Algorithms Nicol`o Cesa-Bianchi DTI, University of Milan via Bramante 65 26013 Crema, Italy cesa-bianchi@dti.unimi.it Alex Conconi DTI, University of Milan via Bramante 65 26013 Crema, Italy conconi@dti.unimi.it Claudio Gentile DSI, University of Milan via Comelico 39 20135 Milano, Italy gentile@dsi.unimi.it Abstract In this paper we show that on-line algorithms for classification and regression can be naturally used to obtain hypotheses with good datadependent tail bounds on their risk. Our results are proven without requiring complicated concentration-of-measure arguments and they hold for arbitrary on-line learning algorithms. Furthermore, when applied to concrete on-line algorithms, our results yield tail bounds that in many cases are comparable or better than the best known bounds. 1 Introduction One of the main contributions of the recent statistical theories for regression and classification problems [21, 19] is the derivation of functionals of certain empirical quantities (such as the sample error or the sample margin) that provide uniform risk bounds for all the hypotheses in a certain class. This approach has some known weak points. First, obtaining tight uniform risk bounds in terms of meaningful empirical quantities is generally a difficult task. Second, searching for the hypothesis minimizing a given empirical functional is often computationally expensive and, furthermore, the minimizing algorithm is seldom incremental (if new data is added to the training set then the algorithm needs be run again from scratch). On-line learning algorithms, such as the Perceptron algorithm [17], the Winnow algorithm [14], and their many variants [16, 6, 13, 10, 2, 9], are general methods for solving classification and regression problems that can be used in a fully incremental fashion. That is, they need (in most cases) a short time to process each new training example and adjust their current hypothesis. While the behavior of these algorithms is well understood in the so-called mistake bound model [14], where no assumptions are made on the way the training sequence is generated, there are fewer results concerning how to use these algorithms to obtain hypotheses with small statistical risk. Littlestone [15] proposed a method for obtaining small risk hypotheses from a run of an arbitrary on-line algorithm by using a cross validation set to test each one of the hypotheses generated during the run. This method does not require any convergenceproperty of the online algorithm and provides risk tail bounds that are sharper than those obtainable choosing, for instance, the hypothesis in the run that survived the longest. Helmbold, Warmuth, and others [11, 6, 8] showed that, without using any cross-validation sets, one can obtain expected risk bounds (as opposed to the more informative tail bounds) for a hypothesis randomly drawn among those generated during the run. In this paper we prove, via refinements and extensions of the previous analyses, that online algorithms naturally lead to good data-dependent tail bounds without employing the complicated concentration-of-measure machinery needed by other frameworks [19]. In particular we show how to obtain, from an arbitrary on-line algorithm, hypotheses whose risk is close to with high probability (Theorems 2 and 3), where is the amount of training data and is a data-dependent quantity measuring the cumulative loss of the online algorithm on the actual training data. When applied to concrete algorithms, the loss bound translates into a function of meaningful data-dependent quantities. For classification problems, the mistake bound for the -norm Perceptron algorithm yields a tail risk bound in terms of the empirical distribution of the margins — see (4). For regression problems, the square loss bound for ridge regression yields a tail risk bound in terms of the eigenvalues of the Gram matrix — see (5). 2 Preliminaries and notation Let be arbitrary sets and
. An example is a pair , where is an instance belonging to and is the label associated with . Random variables will be denoted in upper case and their realizations will be in lower case. We let be the pair of random variables , where and take values in and , respectively. Throughout the paper, we assume that data are generated i.i.d. according to an unknown probability distribution over . All probabilities and expectations will be understood with respect to this underlying distribution. We use the short-hand "! to denote the vectorvalued random variable # %$&(')'('*+ ! . A hypothesis , is any (measurable) mapping from instances - to predictions ,./0 1 , where 1 is a given decision space. The risk of , is defined by 2(3(#,45 7698;:<#, =>? @BA , where :DC 1 E GFIH is a nonnegative loss function. Unless otherwise specified, we will assume that : takes values in 8 JKLA for some known J MNKOM7P . The on-line algorithms we investigate are defined within a well-known mathematical model, which is a generalization of a learning model introduced by Littlestone [14] and Angluin [1]. Let a training sequence Q ! RS$T U$V>(')'('W( ! ! X#YZ 0 ! be fixed. In this learning model, an on-line algorithm processes the examples in Q ! one at a time in trials, generating a sequence of hypotheses ,\[]?, $ )'(')'*+, ! . At the beginning of the ^ -th trial, the algorithm receives the instance 4_ and uses its current hypothesis ,/_ `*$ to compute a prediction ,4_ `*$T/_Ba 1 for the label ]_ associated with /_ . Then, the true value of the label b_ is disclosed and the algorithm suffers a loss :<#,/_`W$&/_#V <_B , measuring how bad is the prediction ,S_`W$&/_c for the label _ . Before the next trial begins, the algorithm generates a new hypothesis , _ which may or may not be equal to ,/_ `*$ . We measure the algorithm’s performance on Q ! by its cumulative loss Q ! 5 ! d _fe.$ :<g, _`W$ _ > _ >' In our analysis, we will write h and ij[<(')'('Wi ! when we want to stress the fact that the cumulative loss and the hypotheses of the on-line algorithm are functions of the random sample ! . In particular, throughout the paper i [ will denote the (deterministic) initial hypothesis of an arbitrary on-line algorithm and, for each kml7^nl , io_ will be a random variable denoting the ^ -th hypothesis of the on-line algorithm and such that the value of ij_B 5$&)')'('*+ ! does not change upon changes in the values of p_fq.$(')'('W+ ! . Our goal is to relate the risk of the hypotheses produced by an on-line algorithm running on an i.i.d. sequence r! to the cumulative loss hs# r!t of the algorithm on that sequence. The cumulative loss hsB !t will be our key empirical (data-dependent) quantity. Via our analysis we will obtain bounds of the form 2)3 gi [ )'(')'*i ! hsB n!t k
k l where gi [<(')'('Wi ! is a specific function of the sequence of hypotheses i9[])')'('Wi ! produced by the algorithm, and is a suitable positive constant. We will see that for specific on-line algorithms the ratio hs# r!t & can be further bounded in terms of meaningful empirical quantities. Our method centers on the following simple concentration lemma about bounded losses. Lemma 1 Let : be an arbitrary bounded loss satisfying J l:Zl K . Let an arbitrary on-line algorithm output (not necessarily distinct) hypotheses i [ (')'('Wi ! when it is run on n! . Then for any JjM l k we have k ! d _fe.$ 2)3 i _ `*$ h K k l ' Proof. For each ^ Gk<(')')'* , set /_ `*$ 2)3 ij_`W$) :Lij_ `*$< 9_#V /_# . We have k ! d _fe.$ _ `*$ k ! d _fe.$ 2)3 i _ `*$ h ' Furthermore, KOl _ `*$ lNK , since : takes values in 8 J?K A . Also, 698\_ `*$ _`W$+AS 72)3g,4_`W$) =6 8 : g,\_ `*$< 9_gV /_# _`W$?Ab+ J where _ `*$ denotes the -algebra generated by %$(')')'*> _ `*$ . A direct application of the Hoeffding-Azuma inequality [3] to the bounded random variables S[b)'(')'*! ! `W$ proves the lemma. " 3 Concentration for convex losses In this section we investigate the risk of the average hypothesis ,$#&%(' k ! d _ e.$ , _`W$ where ,\[<?, $ (')'('*?, ! are the hypotheses generated by some on-line algorithm run on training examples.1 The average hypothesis generates valid predictions whenever the decision space 1 is convex. Theorem 2 Let 1 be convex and : C 1 F 8 J\K A be convex in the first argument. Let an arbitrary on-line algorithm for : output (not necessarily distinct) hypotheses i [ )')'('Wi ! when the algorithm is run on r! . Then for any J M M k the following holds 2)3( i
h ) K
* k l ' 1Notice that the last hypothesis +-, is not used in this average. Proof. Since : is convex in the first argument, by Jensen’s inequality we have : , /V l $ ! ! _ e $ :Lg,4_`W$& />G' Taking expectation with respect to ? @ yields 2(3 ,4 l $ ! ! _ e $ 2)3 g, _`W$ . Using the last inequality along with Lemma 1 yields the thesis. " This theorem, which can be viewed as the tail bound version of the expected bound in [11], implies that the risk of the average hypothesis is close to Q !c & for “most” samples Q ! . On the other hand, note that it is unlikely that ! _fe.$ 2(3 gi _`W$ & concentrates around 6L8 h A , at least without taking strong assumptions on the underlying on-line algorithm. An application of Theorem 2 will be shown is Section 5. Here we just note that by applying this theorem to the Weighted Majority algorithm [16], we can prove a version of [5, Theorem 4] for the absolute loss without resorting to sophisticated concentration inequalities (details in the full paper). 4 Penalized risk estimation for general losses If the loss function : is nonconvex (such as the 0-1 loss) then the risk of the average hypothesis cannot be bounded in the way shown in the previous section. However, the risk of the best hypothesis, among those generated by the on-line algorithm, cannot be higher than the average risk of the same hypotheses. Hence, Lemma 1 immediately tells us that, under no conditions on the loss function other than boundedness, for most samples Q ! at least one of the hypotheses generated has risk close to Q !t . In this section we give a technique (Lemma 4) that, using a penalized risk estimate, finds with high probability such a hypothesis. The argument used is a refinement of Littlestone’s method [15]. Unlike Littlestone’s, our technique does not require a cross validation set. Therefore we are able to obtain bounds on the risk whose main term is Q !c & , where is the size of the whole set of examples available to the learning algorithm (i.e., training set plus validation set in Littlestone’s paper). Similar observations are made in [4], though the analysis there does actually refer only to randomized hypotheses with 0-1 loss (namely, to absolute loss). Let us define the penalized risk estimate of hypothesis ,S_ by _ ^ =^t5 where ^ is the length of the suffix Q _fq.$ (')')' Q ! of the training sequence that the on-line algorithm had not seen yet when ,/_ was generated, _ is the cumulative loss of ,4_ on that suffix, and /5 k k ' Our algorithm chooses the hypothesis ,o , _ , where ^
<3 [ _ ! `W$ _ ^ ^t ' For the sake of simplicity, we will restrict to losses : with range 8 J(kVA . However, it should be clear that losses taking values in arbitrary bounded real interval can be handled using techniques similar to those shown in Section 3. We prove the following result. Theorem 3 Let an arbitrary on-line algorithm output (not necessarily distinct) hypotheses i [<(')')'i ! when it is run on ! . Then, for any JjM l k , the hypothesis i chosen using the penalized risk estimate based on satisfies 2)3 i h k * k l ' The proof of this theorem is based on the two following technical lemmas. Lemma 4 Let an arbitrary on-line algorithm output (not necessarily distinct) hypotheses i [<(')')'i ! when it is run on ! . Then for any J M M k the following holds: 2)3 i [ _ ! `*$ 2)3 i _ ^t l ' Proof. Let
<3 [ _ ! `*$ 2(3( ij_B ^t S' Let further i i , h h , and set for brevity _ hZ_ ^ r h ' For any fixed J we have 2)3 i= 2(3( i l ! `W$ d _fe [ _ =^t%l S2)3 ij_B 2(3( i <S' (1) Now, if _ =^t%l ( holds then either _ lO2)3 ij_# =^t or 2)3 i ( ) or 2(3 gi _ 2)3 i %M hold. Hence for any fixed ^ we can write _ ^t%l )*2)3 i _ 2)3 i ) l _ l 2(3(gi _ ^tS2(3( i _ 2(3(gi < 2(3( i *2)3 ij_g 2)3 i < 2)3(gij_#2)3 i )%M )S\2)3 ij_# O2)3 i ( < l _ l 2(3(gi _ ^t 2)3 i ( ) (2) 2)3(gi _ 2)3 i %M S\2)3 i _ O2)3 i <9' (3) Probability (3) is zero if . Hence, plugging (2) into (1) we can write 2(3 i O2)3(gi ( ( l ! `*$ d _fe [ _ l 2(3 gi _B ^t 2(3(gi l k ! `*$ d _fe [ _ O2)3(gi _ =^t l k k where in the last two inequalities we applied Chernoff-Hoeffding bounds. " Lemma 5 Let an arbitrary on-line algorithm output (not necessarily distinct) hypotheses i [ (')')'i ! when it is run on ! . Then for any J M M k the following holds: [ _ ! `*$ 2)3 ij_B ^t h k k k l ' Proof. We have [ _ ! `W$ 2)3g,4_g =^t %l k ! `*$ d _fe [ g2)3 g,4_# =^t k ! `*$ d _fe [ 2(3 #, _ ! `W$ d _ e [ k ^t * k M k ! `*$ d _fe [ 2(3 #,\_# ! `W$ d _ e [ k =^ k l k ! `*$ d _fe [ 2(3 #,\_# k k where the last inequality follows from ! _fe.$ k ^ l . Therefore [ _ ! `*$ 2(3( ij_B ^t h ) k k k l k ! `W$ d _fe [ 2)3 ij_B h
k l by Lemma 1 (with K Gk ). " Proof (of Theorem 3). The proof follows by combining Lemma 4 and Lemma 5, and by overapproximating the square root terms therein. " 5 Applications For the sake of concreteness we now sketch two generalization bounds which can be obtained through a direct application of our techniques. The -norm Perceptron algorithm [10, 9] is a linear threshold algorithm which keeps in the ^ -th trial a weight vector _ `*$ H . On instance _
N=H C lGk , the algorithm predicts by ,/_`W$T _g5 sign S_ `*$V _B% k] k , where *L $ and . If the algorithm’s prediction is wrong (i.e., if , _ `*$ _ 7 _ ) then the algorithm performs the weight update S_ 4_ `*$ <_ _ . Notice that yields the classical Perceptron algorithm [17]. On the other hand, ! " # gets an algorithm which performs like a multiplicative algorithm, such as the Normalized Winnow algorithm [10]. Applying Theorem 3 to the bound on the number h of mistakes for the -norm Perceptron algorithm shown in [9], we immediately obtain that, with probability at least k with respect to the draw of the training sample r! , the risk 2(3( i of the penalized estimator i is at most k %$'& (r> ! Ok ) k )* Ok $'& +(r> ! k * k (4) for any ) YJ and for any ( such that * ( `*$ lRk . The margin-based quantity $ & (r Q !tL ! _ e $
J\)k
<_( _ ) is called soft margin in [20] and accounts for the distribution of margin values achieved by the examples in Q ! with respect to hyperplane ( . Traditional data-dependent bounds using uniform convergence methods (e.g., [19]) are typically expressed in terms of the sample margin (^aC5 _ ( _ l ) - & , i.e., in terms of the fraction of training points whose margin is at most ) . The ratio $ & +(" Q ! occurring in (4) has a similar flavor, though the two ratios are, in general, incomparable. We remark that bound (4) does not have the extra log factors appearing in the analyses based on uniform convergence. Furthermore, it is significantly better than the bound in [20] whenever $ & is constant, which typically occurs when the data sequence is not linearly separable. As a second application, we consider the ridge regression algorithm [12] for square loss. Assume H and 8 p 0A . This algorithm computes at the beginning of the ^ -th trial the vector _ `*$ which minimizes * _ `*$ e.$ $ , where NJ . On instance _ the algorithm predicts with ,/_ `*$< _gp
E_`W$ _g , where is the “clipping” function r L% if , r L if Nl and L if Xl Nl . The losses $ _ , _ `*$ _ are thus bounded by . We can apply Theorem 2 to the bound on the cumulative loss h for ridge regression (see [22, 2]) and obtain that, with probability at least k with respect to the draw of the training sample ! , the risk 2)3 i of the average hypothesis estimator i is at most k ( hs+(r+ ! ! d _fe.$ _ _ # k (5) for any ( H , where hs(r+ !c ! _ e $ $ _ ( _ , denotes the determinant of matrix , is the # -dimensional identity matrix and is the transpose of .2 Let us denote by ! the matrix whose columns are the data vectors _ , ^ Gk<(')')'* . Then simple linear algebra shows that ! _fe.$ j_T _ # * ! ! # _fe.$ tk _ / where the _ ’s are the eigenvalues of ! ! . The nonzero eigenvalues of ! ! are the same as the nonzero eigenvalues of the Gram matrix ! ! . Risk bounds in terms of the eigenvalues of the Gram matrix were also derived in [23]; we defer to the full paper a comparison between these results and ours. Finally, our bound applies also to kernel ridge regression [18] by replacing the eigenvalues of ! ! with the eigenvalues of the kernel Gram matrix _ , k l ^+ l , where is the kernel being considered. References [1] Angluin, D. Queries and concept learning, Machine Learning, 2(4), 319-342, 1988. [2] Azoury, K., and Warmuth, M. K. Relative loss bounds for on-line density estimation with the exponential family of distributions, Machine Learning, 43:211–246, 2001. [3] K. Azuma. Weighted sum of certain dependend random variables. Tohoku Mathematical Journal, 68, 357–367, 1967. 2Using a slightly different linear regression algorithm, Forster and Warmuth [7] have proven a sharper bound on the expected relative loss. In particular, they have exhibited an algorithm computing hypothesis ! ,#" such that in expectation (over , ) the relative risk $&%' ")(+*-,/.102354768 9 ;:=< >@?BA "#C is bounded by D AFEHGHI . [4] A. Blum, A. Kalai, and J. Langford. Beating the hold-out: bounds for k-fold and progressive cross-validation. In 12th COLT, pages 203–208, 1999. [5] S. Boucheron, G. Lugosi, and P. Massart. A sharp concentration inequality with applications. Random Structures and Algorithms, 16, 277–292, 2000. [6] N. Cesa-Bianchi, Y. Freund, D. Haussler, D. P. Helmbold, R. E. Schapire, and M. K. Warmuth. How to use expert advice. Journal of the ACM, 44(3), 427–485, 1997. [7] J. Forster, and M. K. Warmuth. Relative expected instantaneous loss bounds. 13th COLT, 90–99, 2000. [8] Y. Freund and R. Schapire. Large margin classification using the perceptron algorithm. Machine Learning, 37(3), 277–296, 1999. [9] C. Gentile The robustness of the -norm algorithms. Manuscript, 2001. An extended abstract (co-authored with N. Littlestone) appeared in 12th COLT, 1–11, 1999. [10] A. J. Grove, N. Littlestone, and D. Schuurmans. General convergence results for linear discriminant updates, Machine Learning, 43(3), 173–210, 2001. [11] D. Helmbold and M. K. Warmuth. On weak learning. JCSS, 50(3), 551–573, June 1995. [12] A. Hoerl, and R. Kennard, Ridge regression: biased estimation for nonorthogonal problems. Technometrics, 12, 55–67, 1970. [13] J. Kivinen and M. K. Warmuth. Additive versus exponentiated gradient updates for linear prediction. Information and Computation, 132(1), 1–64, 1997. [14] N. Littlestone. Learning quickly when irrelevant attributes abound: A new linearthreshold algorithm. Machine Learning, 2, 285–318, 1988. [15] N. Littlestone. From on-line to batch learning. In 2nd COLT, 269–284, 1989. [16] N. Littlestone and M. K. Warmuth. The weighted majority algorithm. Information and Computation, 108(2), 212–261, 1994. [17] F. Rosenblatt. Principles of neurodynamics: Perceptrons and the theory of brain mechanisms. Spartan Books, Washington, D.C., 1962. [18] C. Saunders, A. Gammerman, and V. Vovk. Ridge Regression Learning Algorithm in Dual Variables, In 15th ICML, 1998. [19] J. Shawe-Taylor, P. Bartlett, R. Williamson, and M. Anthony, Structural Risk Minimization over Data-dependent Hierarchies. IEEE Trans. IT, 44, 1926–1940, 1998. [20] J. Shawe-Taylor and N. Cristianini, On the generalization of soft margin algorithms, 2000. NeuroCOLT2 Tech. Rep. 2000-082, http://www.neurocolt.org. [21] V.N. Vapnik, Statistical learning theory. J. Wiley and Sons, NY, 1998. [22] V. Vovk, Competitive on-line linear regression. In NIPS*10, 1998. Also: Tech. Rep. Department of Computer Science, Royal Holloway, University of London, CSD-TR97-13, 1997. [23] R. C. Williamson, J. Shawe-Taylor, B. Sch¨olkopf and A. J. Smola, Sample Based Generalization Bounds, IEEE Trans. IT, to appear.
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Effective size of receptive fields of inferior temporal visual cortex neurons in natural scenes Thomas P. Trappenberg Dalhousie University Faculty of Computer Science 5060 University Avenue, Halifax B3H 1W5, Canada tt@cs.dal.ca Edmund T. Rolls and Simon M. Stringer University of Oxford, Centre for Computational Neuroscience, Department of Experimental Psychology, South Parks Road, Oxford OX1 3UD, UK edmund.rolls,simon.stringer@psy.ox.ac.uk Abstract Inferior temporal cortex (IT) neurons have large receptive fields when a single effective object stimulus is shown against a blank background, but have much smaller receptive fields when the object is placed in a natural scene. Thus, translation invariant object recognition is reduced in natural scenes, and this may help object selection. We describe a model which accounts for this by competition within an attractor in which the neurons are tuned to different objects in the scene, and the fovea has a higher cortical magnification factor than the peripheral visual field. Furthermore, we show that top-down object bias can increase the receptive field size, facilitating object search in complex visual scenes, and providing a model of object-based attention. The model leads to the prediction that introduction of a second object into a scene with blank background will reduce the receptive field size to values that depend on the closeness of the second object to the target stimulus. We suggest that mechanisms of this type enable the output of IT to be primarily about one object, so that the areas that receive from IT can select the object as a potential target for action. 1 Introduction Neurons in the macaque inferior temporal visual cortex (IT) that respond to objects or faces have large receptive fields when a single object or image is shown on an otherwise blank screen [1, 2, 3]. The responsiveness of the neurons to their effective stimuli independent of their position on the retina over many degrees is termed translation invariance. Translation invariant object recognition is an important property of visual processing, for it potentially enables the neurons that receive information from the inferior temporal visual cortex to perform memory operations to determine whether for example the object has been seen before or is associated with reward independently of where the image was on the retina. This allows correct generalization over position, so that what is learned when an object is shown at one position on the retina generalizes correctly to other positions [4]. If more than one object is present on the screen, then there is evidence that the neuron responds more to the object at the fovea than in the parafoveal region [5, 6]. More recently, it has been shown that if an object is presented in a natural background (cluttered scene), and the monkey is searching for the object in order to touch it to obtain a reward, then the receptive fields are smaller than when the monkey performs the same task with the object against a blank background [7]. We define the size of a receptive field as twice the distance from the fovea (the centre of the receptive field) to locations at which the response decreases to half maximal. An analysis of IT neurons that responded to the target stimulus showed that the average size of the receptive fields shrinks from approximately 56 degrees in a blank background to approximately 12 degrees with a complex scene [8]. The responses of an IT cell with a large receptive field are illustrated in Figure 1A. There the average firing rates of the cell to an effective stimulus that the monkey had to touch on a touch-screen to receive reward is shown as a function of the angular distance of the object from the fovea. The solid line represents the results from experiments with the object placed in a blank background. This demonstrates the large receptive fields of IT cells that have often been reported in the literature [3]. In contrast, when the object is placed in a natural scene (cluttered background), the size of the receptive field is markedly smaller (dashed line). 2 The model We formalized our understanding of how the dependence of the receptive field size on various conditions could be implemented in the ventral visual processing pathway by developing a neural network model with the components sufficient to produce the above effects. The model utilizes an attractor network representing the inferior temporal visual cortex, and a neural input layer with several retinotopically organized modules representing the visual scene in an earlier visual cortical area such as V4 (see Figure 1B). Each independent module within ‘V4’ represents a small part of the visual field and receives input from earlier visual areas represented by an input vector for each possible location which is unique for each object. Each module was 6 deg in width, matching the size of the objects presented to the network. For the simulations we chose binary random input vectors representing objects with components set to ones and the remaining
components set to zeros. is the number of nodes in each module and is the sparseness of the representation. The structure labeled ‘IT’ represents areas of visual association cortex such as the inferior temporal visual cortex and cortex in the anterior part of the superior temporal sulcus in which neurons provide distributed representations of faces and objects [9, 3]. The activity ! " of nodes in this structure are governed by leaky integrator dynamics with time constant # # $ ! " $
% "'&)(+*, % *
/. 10 * ! "'&2(435 76 3 0 3 % "'&28 9 :;'< =9> (1) The firing rate 0 of the ? th node is determined by a sigmoidal function from the activation @A as 0 ! " CBD " E&FHGDIKJL
+M @A % "N
POQ1R! , where the parameters M and O represent the gain and the offset, respectively. The constant . represents the strength of the activity-dependent global inhibition simulating the effects of inhibitory interneurons. The external ‘top-down’ input vector < =9S> produces object-selective inputs, which are used as the attentional drive when a visual search task is simulated. The strength of this IT V4 Visual Input Object bias A. B. C. 0 10 20 30 40 50 60 50 60 70 80 90 100 110 120 130 Distance of gaze from target object Average firing rate blank
background natural
background 0 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 Eccentricity Weight factor from cortical
magnification
factor Gaussian
Figure 1: A) Average activity of a macaque inferior temporal cortex neuron as a function of the distance of the object from the fovea recorded in a visual search task when the object was in a blank or a cluttered natural background. B) Outline of the model used in this study with an attractor network labelled ‘IT’ that receives topgraphical organised inputs from an input neural layer labeled ‘V4’. Objects close to the fovea produce stronger inputs to reflect the higher magnification factor of the visual representation close to the fovea. The attractor network also receives top-down object-based inputs, to incorporate object-based attention in a visual search task. C) The modulation factor used to weight inputs to IT from V4 shown as a function of their distance from the fovea. The values on the solid line are derived from cortical magnification factors, and were used in the simulations, whereas the dotted line corresponds to a shifted Gaussian function. object bias is modulated by the value of 8 9SA:; . The recognition functionality of this structure is modeled as an attractor neural network (ANN) with trained memories indexed by representing particular objects. The memories are formed through Hebbian learning on sparse patterns, * 8 (
PQ *
PN (2) where 8 A (set to 1 in the simulations below) is a normalization constant that depends on the learning rate, S is the sparseness of the training pattern in IT, and are the components of the pattern used to train the network. The weights A 76 * between the V4 nodes and IT nodes are trained by Hebbian learning of the form 6 * 8 A 76 8S(
*
P (3) to produce object representations in IT based on inputs in V4. The normalizing modulation factor 8 76 8 allows the gain of inputs to be modulated as a function of their distance from the fovea, and depends on the module 8 to which the presynaptic node belongs. The weight values between V4 and IT support translation invariant object recognition of a single object in the visual field if the normalization factor is the same for each module and the model is trained with the objects placed at every possible location in the visual field. The translation invariance of the weight vectors between each V4 module and the IT nodes is however explicitly modulated in our model by the module-dependent modulation factor 8 6 8S as indicated in Figure 1B by the width of the lines connecting V4 with IT. The strength of the foveal module is strongest, and the strength decreases for modules representing increasing eccentricity. The form of this modulation factor was derived from the parameterization of the cortical magnification factors given by [10],1 and is illustrated in Figure 1C as a solid line. Similar results to the ones presented here can be achieved with different forms of the modulation factor such as a shifted Gaussian as illustrated by the dashed line in Figure 1C. 3 Results To study the ability of the model to recognize trained objects at various locations relative to the fovea we tested the network with distorted versions of the objects, and measured the ‘correlation’ between the target object and the final state of the attractor network. The correlation was estimated from the normalized dot product between the target object vector and the state of the IT network after a fixed amount of time sufficient for the network to settle into a stable state. The objects were always presented on backgrounds with some noise (introduced by flipping 2% of the bits in the scene) because the input to IT will inevitably be noisy under normal conditions of operation. All results shown in the following represent averages over 10 runs and over all patterns on which the network was trained. 3.1 Receptive fields are large in scenes with blank backgrounds In the first experiments we placed only one object in the visual scene with different eccentricities relative to the fovea. The results of this simulation are shown in Figure 2A with the line labeled ‘blank background’. The value of the object bias 8 9 :; was set to 0 in these simulations. Good object retrieval (indicated by large correlations) was found even when the object was far from the fovea, indicating large IT receptive fields with a blank background. The reason that any drop is seen in performance as a function of eccentricity is because flipping 2% of the bits in the V4 modules introduces some noise into the recall process. The results demonstrate that the attractor dynamics can support translation invariant object recognition even though the weight vectors between V4 and IT are not translation invariant but are explicitly modulated by the modulation factor 8 76 derived from the cortical magnification factor. 3.2 The receptive field size is reduced in scenes with complex background In a second experiment we placed individual objects at all possible locations in the visual scene representing natural (cluttered) visual scenes. The resulting correlations between the target pattern and asymptotic IT state are shown in Figure 2A with the line labeled ’natural background’. Many objects in the visual scene are now competing for recognition by the attractor network, while the objects around the foveal position are enhanced through the modulation factor derived by the cortical magnification factor. This results in a much smaller size of the receptive field of IT neurons when measured with objects in natural 1This parameterization is based on V1 data. However, it was shown that similar forms of the magnification factor hold also in V4 [11] 0 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 Eccentricity Correlation A. B. blank
background natural background Without object bias With object bias 0 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 Eccentricity Correlation blank
background natural
background Figure 2: Correlations as measured by the normalized dot product between the object vector used to train IT and the state of the IT network after settling into a stable state with a single object in the visual scene (blank background) or with other trained objects at all possible locations in the visual scene (natural background). There is no object bias included in the results shown in graph A, whereas an object bias is included in the results shown in B with 8 9SA:; in the experiments with a natural background and 8 9SA:; in the experiments with a blank background. backgrounds. 3.3 Object-based attention increases the receptive field size, facilitating object search in complex visual scenes In addition to this major effect of the background on the size of the receptive field, which parallels and we suggest may account for the physiological findings outlined in the introduction, there is also a dependence of the size of the receptive fields on the level of object bias provided to the IT network. Examples are shown in Figure 2B where we used an object bias. The object bias biasses the IT network towards the expected object with a strength determined by the value of 8 9SA:; , and has the effect of increasing the size of the receptive fields in both blank and natural backgrounds (see Figure 2B and compare to Figure 2A). This models the effect found neurophysiologically [8].2 3.4 A second object in a blank background reduces the receptive field size depending on the distance between the second object and the fovea In the last set of experiments we placed two objects in an otherwise blank background. The IT network was biased towards one of the objects designated as the target object (in for example a visual search task), which was placed on one side of the fovea at different eccentricities from the fovea. The second object, a distractor object, was placed on the opposite side of the fovea at a fixed distance of degrees from the fovea. Results for different values of are shown in 3A. The results indicate that the size of the receptive field (for the target object) decreases with decreasing distance of the distractor object from the fovea. The size of the receptive fields (the width at half maximal response) is shown 2The larger values of in the experiments with a natural background compared to the experiments in a blank background reflects the fact that more attention may be needed to find objects in natural cluttered environments. in 3B. The size starts to increase linearly with increasing distance of the distractor object from the fovea until the influence of the distractor on the size of the receptive field levels off and approaches the value expected for the situation with one object in a visual scene and a blank background. A. B. 0 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 Eccentricity Correlation d=6 d=12 d=18 d=24 5 10 15 20 25 30 35 40 10 20 30 40 50 60 70 d, distance of distractor from fovea Size of receptive field Figure 3: A) Correlations between the target object and the final state of the IT network in experiments with two objects in a visual scene with a blank background. The different curves correspond to different distances of the distractor object from the fovea. The eccentricity refers to the distance between the target object and the fovea. B) The size of the receptive field for the target as a function of the distance of the distractor object from the fovea. 4 Discussion When single objects are shown in a scene with a blank background, the attractor network helps neurons to respond to an object with large eccentricities of this object relative to the fovea of the agent. When the object is presented in a natural scene, other neurons in the inferior temporal cortex become activated by the other effective stimuli present in the visual field, and these forward inputs decrease the response of the network to the target stimulus by a competitive process. The results found fit well with the neurophysiological data, in that IT operates with almost complete translation invariance when there is only one object in the scene, and reduces the receptive field size of its neurons when the object is presented in a cluttered environment. The model here provides an explanation of the real IT neuronal responses in natural scenes and makes several predictions that can be explored experimentally. The model is compatible with the models developed by Gustavo Deco and colleagues (see, for example, [12, 13]) while specific simplifications and addition have been made to explore the variations in the size of receptive fields in IT. The model accounts for the larger receptive field sizes from the fovea of IT neurons in natural backgrounds if the target is the object being selected compared to when it is not selected [8]. The model accounts for this by an effect of top-down bias which simply biasses the neurons towards particular objects compensating for their decreasing inputs produced by the decreasing magnification factor modulation with increasing distance from the fovea. Such object based attention signals could originate in the prefrontal cortex and could provide the object bias for the inferotemporal cortex [14]. We proposed that the effective variation of the size of the receptive field in the inferior temporal visual cortex enables the brain areas that receive from this area (including the orbitofrontal cortex, amygdala, and hippocampal system) to read out the information correctly from the inferior temporal visual cortex about individual objects, because the neurons are responding effectively to the object close to the fovea, and respond very much less to objects away from the fovea.3 This enables, for example, the correct reward association of an object to be determined by pattern association in the orbitofrontal cortex or amygdala, because they receive information essentially about the object at the fovea. Without this shrinkage in the receptive field size, the areas that receive from the inferior temporal visual cortex would respond to essentially all objects in a visual scene, and would therefore provide an undecipherable babel of information about all objects present in the visual scene. It appears that part of the solution to this potential binding problem that is used by the brain is to limit the size of the receptive fields of inferior temporal cortex neurons when natural environments are being viewed. The suggestion is that by providing an output about what is at the fovea in complex scenes, the inferior temporal visual cortex enables the correct reward association to be looked up in succeeding brain regions, and then for the object to be selected for action if appropriate. Part of the hypothesis here is that the coordinates of the object in the visual scene being selected for action are provided by the position in space to which the gaze is directed [7]. Acknowledgments This research was supported by the Medical Research Council, grant PG9826105, and by the MRC Interdisciplinary Research Centre for Cognitive Neuroscience. References [1] C. G. Gross, R. Desimone, T. D. Albright, and Schwartz E. L. Inferior temporal cortex and pattern recognition. Experimental Brain Research, 11:179–201, 1985. [2] M. J. Tovee, E. T. Rolls, and P. Azzopardi. Translation invariance and the responses of neurons in the temporal visual cortical areas of primates. Journal of Neurophysiology, 72:1049–1060, 1994. [3] E. T. Rolls. Functions of the primate temporal lobe cortical visual areas in invariant visual object and face recognition. Neuron, 27:205– 218, 2000. [4] E. T. Rolls and A. Treves. Neural Networks and Brain Function. Oxford University Press, Oxford, 1998. [5] T. Sato. Interactions of visual stimuli in the receptive fields of inferior temporal neurons in macaque. Experimental Brain Research, 77:23–30, 1989. [6] E. T. Rolls and M. J. Tovee. The responses of single neurons in the temporal visual cortical areas of the macaque when more than one stimulus is present in the visual field. Experimental Brain Research, 103:409–420, 1995. [7] E. T. Rolls, B. Webb, and M. C. A. Booth. Responses of inferior temporal cortex neurons to objects in natural scenes. Society for Neuroscience Abstracts, 26:1331, 2000. [8] E. T. Rolls, F. Zheng, and N. Aggelopoulos. Responses of inferior temporal cortex neurons to objects in natural scenes. Society for Neuroscience Abstracts, 27, 2001. [9] M. C. A. Booth and E. T. Rolls. View-invariant representations of familiar objects by neurons in the inferior temporal visual cortex. Cerebral Cortex, 8:510–523, 1998. 3Note that it is possible that a “spotlight of attention” [15] can be moved away from the fovea, but at least during normal visual search tasks, the neurons are sensitive to the object at which the monkey is looking, that is which is on the fovea, as shown by [8]. Thus, spatial modulation of the responsiveness of neurons at the V4 level can be influenced by location-specific attentional modulations originating, for example, in the posterior parietal cortex, which may be involved in directing visual spatial attention [15]. [10] B.W. Dow, A.Z. Snyder, R.G. Vautin, and R. Bauer. Magnification factor and receptive field size in foveal striate cortex of the monkey. Exp. Brain. Res., 44:213:228, 1981. [11] R. Gattass, A.P.B. Sousa, and E. Covey. Cortical visual areas of the macaque: Possible substrates for pattern recognition mechanisms. Exp. Brain. Res., Supplement 11, 1985. [12] G. Deco and J. Zihl. Top-down selective visual attention: A neurodynamical approach. Visual Cognition, 8:119–140, 2001. [13] E. T. Rolls and G. Deco. Computational neuroscience of vision. Oxford University Press, Oxford, 2002. [14] A. Renart, N. Parga, and E. T. Rolls. A recurrent model of the interaction between the prefrontal cortex and inferior temporal cortex in delay memory tasks. In S.A. Solla, T.K. Leen, and K.-R. Mueller, editors, Advances in Neural Information Processing Systems. MIT Press, Cambridge Mass, 2000. in press. [15] R. Desimone and J. Duncan. Neural mechanisms of selective visual attention. Annual Review of Neuroscience, 18:193–222, 1995.
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A kernel method for multi-labelled classification Andr´e Elisseeff and Jason Weston BIOwulf Technologies, 305 Broadway, New York, NY 10007 andre,jason @barhilltechnologies.com Abstract This article presents a Support Vector Machine (SVM) like learning system to handle multi-label problems. Such problems are usually decomposed into many two-class problems but the expressive power of such a system can be weak [5, 7]. We explore a new direct approach. It is based on a large margin ranking system that shares a lot of common properties with SVMs. We tested it on a Yeast gene functional classification problem with positive results. 1 Introduction Many problems in Text Mining or Bioinformatics are multi-labelled. That is, each point in a learning set is associated to a set of labels. Consider for instance the classification task of determining the subjects of a document, or of relating one protein to its many effects on a cell. In either case, the learning task would be to output a set of labels whose size is not known in advance: one document can for instance be about food, meat and finance, although another one would concern only food and fat. Two-class and multi-class classification or ordinal regression problems can all be cast into multi-label ones. This makes the latter quite attractive but at the same time it gives a warning: their generality hides their difficulty to solve them. The number of publications is not going to contradict this statement: we are aware of only a few works about the subject [4, 5, 7] and they all concern text mining applications. In Schapire and Singer’s work about Boostexter, one of the only general purpose multilabel ranking systems [7], they observe that overfitting occurs on learning sets of relatively small size ( ). They conclude that controlling the complexity of the overall learning system is an important research goal. The aim of the current paper is to provide a way of controlling this complexity while having a small empirical error. For that purpose, we consider only architectures based on linear models and follow the same reasoning as for the definition of Support Vector Machines [1]. Defining a cost function (section 2) and margin for multi-label models, we focus our attention mainly on an approach based on a ranking method combined with a predictor of the size of the sets (section 3 and 4). Sections 5 and 6 present experiments on a toy problem and on a real dataset. 2 Cost functions Let
be a d-dimensional input space. We consider as an output space the space formed by all the sets of integer between 1 and identified here as the labels of the learning problem. Such an output space contains elements and one output corresponds to one set of labels. The learning problem we are interested in is to find from a learning set
, drawn identically and independently from an unknown distribution , a function such that the following generalization error is as low as possible: !#"%$ &(')+*-, . / 021 (1) The function . is a real-valued loss and can take different forms depending on how is computed. Here, we consider only linear models. Given vectors 3 4 3 and bias 5 4 5 , we follow two schemes: With the binary approach: sign 6 3 7+8 5 96 3 7+8 5 , where the sign function applies component-wise. The value of is a binary vector from which the set of labels can be retrieved easily by stating that label : is in the set iff sign 6 3<; 748 5 ; (= . For example this can be achieved by using a SVM for each binary problem and applying the latter rule [4]. With the ranking approach: assume that > , the size of the label set for the input , is known. We define: ?@; 6 3A; 78 5 ; and consider that a label : is in the label set of iff ?9; is among the largest > elements ? #B ? . The algorithm Boostexter [7] is an example of such a system. The ranking approach is analyzed more precisely in section 3. We consider the same loss functions as in [7] for any multi-label system built from real functions BB . It includes the so-called Hamming Loss defined as C0D / 0 E FG HE where F stands for the symmetric difference of sets. When E HE a multi-label system is in fact a multi-class one and the Hamming Loss is I times the loss of the usual classification loss. We also consider the one-error: 1-err / 0 J if argmax ; K; ML
otherwise which is exactly the same as the classification error for multi-class problems (it ignores the rankings apart from the highest ranked one and so does not address the quality of the other labels). Other losses concern only ranking systems (a system that specifies a ranking but no set size predictor > ). Let us denote by N the complementary set of in #B . We define the Ranking Loss [7] to be: D / 0 E HE@O O N PO O O O QR TSU(LV N s.t. ?9W YX ?4Z @O O (2) It represents the average fraction of pairs that are not correctly ordered. For ranking systems, this loss is natural and is related to the precision which is a common error measure in Information Retrieval: precision + [ E HE/\ ;9] & E 9^_L s.t. ?` Ma ?9; E E ^_L #B s.t. ?` Ma ? ; E from which a loss can be directly deduced. All these loss functions have been discussed in [7]. Good systems should have a high precision and a low Hamming or Ranking Loss. We do not consider the one-error to be a good loss for multi-label systems but we retain it because it was measured in [7]. For multi-label linear models, we need to define a way of minimizing the empirical error measured by the appropriate loss and at the same time to control the complexity of the resulting model. A direct method would be to use the binary approach and thus take the benefit of good two-class systems. However, as it has been raised in [5, 7], the binary approach does not take into account the correlation between labels and therefore does not capture the structure of some learning problems. We propose here to instead focus on the ranking approach. This will be done by introducing notions of margin and regularization as has been done for the two-class case in the definition of SVMs. 3 Ranking based system Our goal is to define a linear model that minimizes the Ranking Loss while having a large margin. For systems that rank the values of 6 3<; 7 8 5 ; , the decision boundaries for are defined by the hyperplanes whose equations are 6 3 ; 3 ` 7 8 5 ; 5 ` , where : belongs to the label sets of and ^ does not. So, the margin of + [ can be expressed as: ;9] & $ `] & 6 3Y; 3M` 7+8 5 ; 5 ` 3 ; 3 ` It represents the signed I distance of to the decision boundary. Considering that all the data in the learning set are well ranked, we can normalize the parameters 3 ; such that: 6 3 ; 3 ` 7+8 5 ; 5 ` a with equality for some L , and : ^TYL N . Maximizing the margin on the whole learning set can then be done via the following problem:
$ Z $ $ !#"%$ &(' ] ;9] & $ `] & ! (3) subject to: 6 3 ; 3M` W 7+8 5 ; 5 ` a : ^T(LV W N W (4) In the case where the problem is not ill-conditioned (two labels are always co-occurring), the objective function can be replaced by:
" # ; $ ` " $
; $ ` 3 ; 3M` I . In order to get a simpler optimization procedure we approximate this maximum by the sum and, after some calculations (see [3] for details), we obtain: $ Z $ $ \ ;% 3A; I (5) subject to: 6 3 ; 3M` W 7+8 5 ; 5 ` a : ^T(LV W N W (6) To generalize this problem in the case where the learning set can not be ranked exactly we follow the same reasoning as for the binary case: the ultimate goal would be to maximize the margin and at the same time to minimize the Ranking Loss. The latter can be expressed quite directly by extending the constraints of the previous problems. Indeed, if we have 6 3A; 3M` W 7 8 5 ; 5 ` a '& WB;` for : ^T LH W( N W , then the Ranking Loss on the learning set is: ( \ W E W EBE N W E \ ; $ ` ] ! &*),+ &*)' 8 & W ;R` where - is the Heaviside function. As for SVMs we approximate the functions 8 & WB;R` by only & WB;R` and this gives the final quadratic optimization problem: $ Z $ $ \ ;% 3 ; I 8/. \ W E W E#E N W E \ ! ; $ ` ' ] &*),+ &*) & WB;R` (7) subject to: 6 3A; 3M` W 7+8 5 ; 5 ` a 0& WB;` - : ^ (LV W N W (8) & W ;R` a (9) In the case where the label sets W all have a size of we find the same optimization problem as has been derived for multi-class Support Vector Machines [8]. For this reason, we call the solution of this problem a ranking Support Vector Machine (Rank-SVM). Another common property with SVM is the possibility to use kernels rather than linear dot products. This can be achieved by computing the dual of the former optimization problem. We refer the reader to [3] for the dual formluation and to [2] and references therein for more information about kernels and SVMs. Solving a constrained quadratic problem like those we just introduced requires an amount of memory that is quadratic in terms of the learning set size and it is generally solved in ( computational steps where we have put into the the number of labels. Such a complexity is too high to apply these methods in many real datasets. To circumvent this limitation, we propose to use a linearization method in conjunction with a predictor-corrector logarithmic barrier procedure. Details are described in [3] with all the calculations relative to the implementation. The memory cost of the method then becomes ( " where "
W E W E is the maximum number of labels. In many applications is much larger than " . The time cost of each iteration is ( I . 4 Set size prediction So far we have only developed ranking systems. To obtain a complete multi-label system we need to design a set size predictor > . A natural way of doing this is to look for inspiration from the binary approach. The latter can indeed be interpreted as a ranking system whose ranks are derived from the real values #B . The predictor of the set size is then quite simple: > E ; M= E is the number of ; that are greater than . The function > is computed from a threshold value that differentiates labels in the target set from others. For the ranking system introduced in the previous section we generalize this idea by designing a function > E %; M=4 E . The remaining problem now is to choose 4 which is done by solving a learning problem. The training data are composed by the W #B W given by the ranking system, and by the target values defined by: 4 W argmin E : L s.t. K; W (X E8 O O : L N s.t. K; W (a
+O O When the minimum is not unique and the optimal values are a segment, we choose the middle of this segment. We refer to this method of predicting the set size as the threshold based method. In the following, we have used linear least squares, and we applied it not only to Rank-SVM but also to Boostexter in order to transform these algorithms from ranking methods to multi-label ones. Note that we could have followed a much simpler scheme to build the function > . A naive method would be to consider the set size prediction as a regression problem on the original training data with the targets E W E W $ $ and to use any regression learning system. This however does not provide a satisfactory solution mainly because it does not take into account how the ranking is performed. In particular, when there are some errors in the ranking, it does not learn how to compensate these errors although the threshold based approach tries to learn the best threshold with respect to these errors. 5 Toy problem As previously noticed the binary approach is not appropriate for problems where correlation between labels exist. To illustrate this point consider figure 2. There are only three labels. One of them (label ) is present for all points in the learning set. The binary approach leads to a system that will fail to separate, for instance, points with label from points of label sets not containing , that is, on points of label and . We see then that the expressible power of a binary system can be quite low when simple configurations occur. If we consider the ranking approach, one can imagine the following solution: 3 , 5 , 3 I 5 I is the hyperplane separating class 2 from class 3, and 3 5 3 I 5 I . By taking the number of labels at point to be > 6 3 7 8 5 where 3 and 5 , we have a simple multi-label system that separates all the regions exactly. Figure 2: Three labels and three regions in the input space. The upper left region is labelled with . The bottom right region is partitioned into two sub-regions with labels or . 1 1,2 1,3 To make this point more concrete we sampled points uniformly on , 1 I and solved all optimization problems with . . On the learning set the Hamming Loss for the binary approach was although for the direct approach it was as expected. 6 Experiments on real data Yeast Saccharomyces cerevisiae Metabolism Energy Cell Growth, Cell Division, DNA synthesis Protein Synthesis Transposable elements Viral and Plasmid proteins Cellular Organization Ionic Homeostasis Cell. Rescue, Defense, Cell Death and Aging Cell. communication, Signal Transduction Cellular Biogenesis Cell. Transport, Transport Mechanisms Transport Facilitation Protein Destination Transcription YAL041W Figure 3: First level of the hierarchy of the gene functional classes. There are 14 groups. One gene, for instance the gene YAL041W can belong to different groups (shaded in grey on the figure). The Yeast dataset is formed by micro-array expression data and phylogenetic profiles with 1500 genes in the learning set and 917 in the test set. The input dimension is . Each gene is associated with a set of functional classes whose maximum size can be potentially more than . This dataset has already been analyzed with a two-class approach [6] and is known to be difficult. In order to make it easier, we used the known structure of the functional classes. The whole set of classes is indeed structured in a tree whose leaves are the functional categories (see http://mips.gsf.de/proj/yeast/catalogues/funcat/ for more details). Given a gene, knowing which edge to take from one level to another leads directly to a leaf and thus to a functional class. Here we try to predict which edge to take from the root to the first level of the tree (see figure 3). Since one gene can have many functional classes this is a multi-label problem: one gene is associated to different edges. We then have and the average number of labels for all genes in the learning set is . We assessed the quality of our method from two perspectives. First as a ranking system with the Ranking Loss and the precision. In that case, for the binary approach, the real outputs of the two-class SVMs were used as ranking values. Second, the methods were compared as multi-label systems using the Hamming Loss. We computed the latter for the binary approach used in conjunction with SVMs, for the Rank-SVM and for Boostexter. To measure the Hamming Loss with Boostexter we used a threshold based > function in combination with the ranking given by the algorithm. Rank-SVM Binary-SVM degree 2 3 4 5 2 3 4 5 Precision
Ranking Loss
! "
! "#
! #$ %& ! ! ' Hamming Loss (
#
"
) * % % ) one-error
#
$$
# % ) % Figure 4: Polynomials of degree 2-5. Loss functions for the rank-SVM and the binary approach based on two-class SVMs. Considering the size of the problem, two values different from less than are not significantly different. Bold values represent superior performance comparing classifiers with the same kernel. For rank-SVMs and for two-class SVMs in the binary approach we choose polynomial kernels of degrees two to nine (experiments on two-class problems using the Yeast data in [6] already showed that polynomial kernels were appropriate for this task). Boostexter was used with the standard stump weak learner and was stopped after 1000 iterations. Results are reported in tables 4, 5 and 6. Rank-SVM Binary-SVM degree 6 7 8 9 6 7 8 9 Precision
$ #
% % Ranking Loss
! ## + %
! + % %!* !* %+ ) !& Hamming Loss
! "" + %"( !& ! ) % !&
! "$
! "$ one-error
#
$ % %&' Figure 5: Polynomials of degree 6-9. Loss functions for the rank-SVM and the binary approach based on two-class SVMs. Considering the size of the problem, two values different from less than are not significantly different. Bold values represent superior performance comparing classifiers with the same kernel. Boostexter (1000 iterations) Precision % * Ranking Loss % ' Hamming Loss % ) one-error % ) Figure 6: Loss functions for Boostexter. Note that these results are worse than with the binary approach or with rank-SVM. Note that Boostexter performs quite poorly on this dataset compared to SVM-based approaches. This may be due to the simple decision function realized by Boostexter. One of the main advantages of the SVM-based approaches is the ability to incorporate priori knowledge into the kernel and control complexity via the kernel and regularization. We believe this may also be possible with Boostexter but we are not aware of any work in this area. To compare the binary and the rank-SVM we put in bold the best results for each kernel. For all kernels and for almost all losses, the combination ranking based SVM approach is better than the binary one. In terms of the Ranking Loss, the difference is significantly in favor of the rank-SVM. It is consistent with the fact that this system tends to minimize this particular loss function. It is worth noticing that when the kernel becomes more and more complex the difference between rank-SVM and the binary method disappears. 7 Discussion and conclusion In this paper we have defined a whole system to deal with multi-label problems. The main contribution is the definition of a ranking based SVM that extends the use of the latter to many problems in the area of Bioinformatics and Text Mining. We have seen on complex, real data that rank-SVMs lead to better performance than Boostexter and the binary approach. On its own this could be interpreted as a sufficient argument to motivate the use of such a system. However, we can also extend the rank-SVM system to perform feature selection on ranking problems [3] . This application can be very useful in the field of bioinformatics as one is often interested in interpretability of a multilabel decision rule. For example one could be interested in a small set of genes which is discriminative in a multi-condition physical disorder. We have presented only first experiments using multi-labelled systems applied to Bioinformatics. Our future work is to conduct more investigations in this area. References [1] B. Boser, I. Guyon, and V. Vapnik. A training algorithm for optimal margin classifiers. In Fifth Annual Workshop on Computational Learning Theory, pages 144–152, Pittsburgh, 1992. ACM. [2] N. Cristianini and J. Shawe-Taylor. Introduction to Support Vector Machines. Cambridge University Press, 2000. [3] Andr´e Elisseeff and Jason Weston. Kernel methods for multi-labelled classification and categorical regression problems. Technical report, BIOwulf Technologies, 2001. http://www.bhtlabs.com/public/. [4] T. Joachims. Text categorization with support vector machines: learning with many relevant features. In Claire N´edellec and C´eline Rouveirol, editors, Proceedings of ECML-98, 10th European Conference on Machine Learning, number 1398, pages 137–142, Chemnitz, DE, 1998. Springer Verlag, Heidelberg, DE. [5] A. McCallum. Multi-label text classification with a mixture model trained by em. AAAI’99 Workshop on Text Learning., 1999. [6] P. Pavlidis, J. Weston, J. Cai, and W.N. Grundy. Combining microarray expression data and phylogenetic profiles to learn functional categories using support vector machines. In RECOMB, pages 242–248, 2001. [7] R.E. Schapire and Y. Singer. Boostexter: A boosting-based system for text categorization. Machine Learning, 39(2/3):135–168, 2000. [8] J. Weston and C. Watkins. Multi-class support vector machines. Technical Report 98-04, Royal Holloway, University of London, 1998.
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Learning Lateral Interactions for Feature Binding and Sensory Segmentation Heiko Wersing HONDA R&D Europe GmbH Carl-Legien-Str.30, 63073 Offenbach/Main, Germany heiko.wersing@hre-ftr.f.rd.honda.co.jp Abstract We present a new approach to the supervised learning of lateral interactions for the competitive layer model (CLM) dynamic feature binding architecture. The method is based on consistency conditions, which were recently shown to characterize the attractor states of this linear threshold recurrent network. For a given set of training examples the learning problem is formulated as a convex quadratic optimization problem in the lateral interaction weights. An efficient dimension reduction of the learning problem can be achieved by using a linear superposition of basis interactions. We show the successful application of the method to a medical image segmentation problem of fluorescence microscope cell images. 1 Introduction Feature binding has been proposed to provide elegant solution strategies to the segmentation problem in perception [11, 12, 14]. A lot of feature binding models have thus tried to reproduce groping mechanisms like the Gestalt laws of visual perception, e.g. connectedness and good continuation, using temporal synchronization [12] or spatial coactivation [9, 14] for binding. Quite generally in these models, grouping is based on lateral interactions between feature-representing neurons, which characterize the degree of compatibility between features. Currently in most of the approaches this lateral interaction scheme is chosen heuristically, since the experimental data on the corresponding connection patterns in the visual cortex is insufficient. Nevertheless, in more complex feature spaces this heuristic approach becomes infeasible, raising the question for more systematic learning methods for lateral interactions. Mozer et al. [4] suggested supervised learning for a dynamic feature binding model of complex-valued directional units, where the connections to hidden units guiding the grouping dynamics were adapted by recurrent backpropagation learning. The application was limited to synthetic rectangle patterns. Hofmann et al. [2] considered unsupervised texture segmentation by a pairwise clustering approach on feature vectors derived from Gabor filter banks at different frequencies and orientations. In their model the pairwise feature compatibilities are determined by a divergence measure of the local feature distributions which was shown to achieve good segmentation results for a range of image types. The problem of segmentation can also be phrased as a labeling problem, where relaxation labeling algorithms have been used as a popular tool in a wide range of computer vision applications. Pelillo & Refice [7] suggested a supervised learning method for the compatibility coefficients of relaxation labeling algorithms, based on minimizing the distance between a target labeling vector and the output after iterating a fixed number of relaxation steps. The main problem are multiple local minima arising in this highly nonlinear optimization problem. Recent results have shown that linear threshold (LT) networks provide interesting architectures for combining properties of digital selection and analogue context-sensitive amplification [1, 13] with efficient hardware implementation options [1]. Xie et al. [16] demonstrated how these properties can be used to learn winner-take-all competition between groups of neurons in an LT network with lateral inhibition. The CLM binding model is implemented by a large-scale topographically organized LT network, and it was shown that this leads to consistency conditions characterizing its binding states [14]. In this contribution we show how these conditions can be used to formulate a learning approach for the CLM as a quadratic optimization problem. In Section 2 we briefly introduce the competitive layer binding model. Our learning approach is elaborated in Section 3. In Section 4 we show application results of the approach to a cell segmentation problem and give a discussion in the final Section 5. 2 The CLM architecture The CLM [9, 14] consists of a set of layers of feature-selective neurons (see Fig. 1). The activity of a neuron at position in layer is denoted by , and a column denotes the set of the neuron activities ,
, sharing a common position . With each column a particular “feature” is associated, which is described by a set of parameters like e.g. local edge elements characterized by position and orientation
. A binding between two features, represented by columns and
, is expressed by simultaneous activities ! and "# ! that share a common layer $ . All neurons in a column are equally driven by an external input %& , which represents the significance of the detection of feature by a preprocessing step. The afferent input %& is fed to the activities & with a connection weight ' ( . Within each layer the activities are coupled via lateral connections ) " which characterize the degree of compatibility between features and and which is a symmetric function of the feature parameters, thus ) " *) "+ . The purpose of the layered arrangement in the CLM is to enforce an assignment of the input features to the layers by the dynamics, using the contextual information stored in the lateral interactions. The unique assignment to a single layer is realized by a columnar Winner-Take-All (WTA) circuit, which uses mutual symmetric inhibitory interactions with absolute strength ' , between neural activities - and ./ that share a common column . Due to the WTA coupling, for a stable equilibrium state of the CLM only a neuron from one layer can be active within each column [14]. The number of layers does not predetermine the number of active groups, since for sufficiently many layers only those are active that carry a salient group. The combination of afferent inputs and lateral and vertical interactions is combined into the standard linear threshold additive activity dynamics 0 .1324&6587:9
';<%.=2?> / &@/.5A> " ) " ."BDCE (1) where 7F&G!HJIK - . For ' large compared to the lateral weights ) " , the single active neuron in a column reproduces its afferent input, ML % . As was shown [14], the stable states of (1) satisfy the consistency conditions > " ) / " "N/ O > " ) " "P for all RQTS U$ GVW (2) which express the assignment of a feature to the layer $ =V with highest lateral support. xrL xr’L xr2 xr1 rh r’ h xr’1 xr’2 lateral interaction vertical WTA interaction input layer 1 layer 2 layer L Figure 1: The competitive layer model architecture (see text for description). 3 Learning of CLM Lateral Interactions Formulation of the Learning Problem. The overall task of the learning algorithm is to adapt the lateral interactions, given by the interaction coefficients ) " , such that the CLM architecture performs appropriate segmentation on the labeled training data and also generalizes to new test data. We assume that the training data consists of a set of labeled training patterns ,
, where each pattern consists of a subset
of different features with their corresponding labels $ . For each labeled training pattern a target labeling vector is constructed by choosing
/ for all Q,S U$ V (3) for the labeled columns, assuming % . Columns for features which are not contained in the training pattern are filled with zeroes according to for all S . In the following indices run over all possible features, e.g. all edges of different orientations at different image positions, while V run over the subset of features realized in a particular pattern, e.g. only one oriented edge at each image position. The assignment vectors =
form the basis of the learning approach since they represent the target activity distribution, which we want to obtain after iterating the CLM with appropriately adjusted lateral interactions. In the following the abbreviation $ for $ V is used to keep the notation readable. The goal of the learning process is to make the training patterns consistent, which is in accordance with (2) expressed by the inequalities > " ) / " " / OT> " ) " " for all # @Q!S $ (4) These 2A "! inequalities define the learning problem that we want to solve in the following. Let us develop a more compact notation. We can rewrite (4) as > #$% " & / #$% " ) # % " O! for all # @QTS U$ G (5) where & @/ # ' " )( +* ( / # + " / 2,( # + "P "- . The form of the inequalities can be simplified by introducing multiindices . and / which correspond to /10 32E , .40 5# RQE and )5670 ) # % " , &98 6 0 & / #$% " . The index . runs over all 2 ! consistency relations defined for the labeled columns of the assignment vectors. The vectors : 8 with components &;8 6 are called consistency vectors and represent the consistency constraints for the lateral interaction. The index / runs over all entries in the lateral interaction matrix. The vector ) @) ) )
with components contains the corresponding matrix entries. The inequalities (4) can then be written in the form > 6 & 8 6 )56 O for all .F (6) This illustrates the nature of the learning problem. The problem is to find a weight vector which leads to a lateral interaction matrix, such that the consistency vectors lie in the opposite half space of the weight state space. Since the conditions (6) determine the attractivity of the training patterns, it is customary to introduce a positive margin to achieve greater robustness. This gives the target inequalities > 6 & 8 6 )5665 O! for all .F (7) which we want to solve in for given training data. If the system of inequalities admits a solution for it is called compatible. If there is no satisfying all constraints, the system is called incompatible. Superposition of Basis Interactions. If the number of features is large, the number of parameters in the complete interaction matrix ) # % " may be too large to be robustly estimated from a limited number of training examples. To achieve generalization from the training data, it is necessary to reduce the number of parameters which have to be adapted during learning. This is also useful to incorporate a priori knowledge into the interaction. An example is to choose basis functions which incorporate invariances such as translation and rotation invariance, or which satisfy the constraint that the interaction is equal in all layers. A simple but powerful approach is to choose a set of fixed basis interactions with compatibilities # % " J
W , with an interaction ) # ' " obtained by linear superposition ) # % " *> # % " > 6 (8) with weight coefficients
W . Now the learning problem of solving the inequalities (7) can be recast in the new free parameters . After inserting (8) into (7) we obtain the transformed problem > 6 & 8 6 > 6 51 >
8 5 OA for all .F (9) where 8 1! 6 &8 6 6 is the component of the consistency vector : 8 in the basis interaction . The basis interactions can thus be used to reduce the dimensionality of the learning problem. To avoid any redundancy, the basis interactions should be linearly independent. Although the functions are here denoted “basis” functions, they need neither be orthogonal nor span the whole space of interactions . Quadratic Consistency Optimization. The generic case in any real world application is that the majority of training vectors contains relevant information, while single spurious vectors may be present due to noise or other disturbing factors. Consequently, in most applications the equations (7) or (9) will be incompatible and can only be satisfied approximately. This will be especially the case, if a low-dimensional embedding is used for the basis function templates as described above. We therefore suggest to adapt the interactions by minimizing the following convex cost function QCO > 8 9 > 6 & 8 6 ) 6 5 C (10) A similar minimization approach was suggested for the imprinting of attractors for the Brain-State-in-a-Box (BSB) model [8], and a recent study has shown that the approach is competitive with other methods for designing BSB associative memories [6]. For a fixed positive margin A , the cost function (10) is minimized by making the inner products of the weight vector and the consistency vectors negative. The global minimum with QCO is attained if the inner products are all equal to 2 , which can be interpreted such that all consistency inequalities are fulfilled in an equal manner. Although this additional regularizing constraint is hard to justify on theoretical grounds, the later application shows that it works quite well for the application examples considered. If we insert the expansion of in the basis of function templates we obtain according to (8) QCO > 8 9 > 8 5 C (11) which results in a -dimensional convex quadratic minimization problem in the parameters. The coefficients 8 , which give the components of the training patterns in the basis interactions, are given by 8 ! 6 & 8 6 6 ! " "N/ @/ " 2 ! " " " . The quadratic optimization problem is then given by minimizing QCO > 5 > 5 (12) where ! 8 8 8 and ! 8 8 . If the coefficients are unconstrained, then the minimum of (12) can be obtained by solving the linear system of equations ! 5 for all . 4 Application to Cell Segmentation The automatic detection and segmentation of individual cells in fluorescence micrographs is a key technology for high-throughput analysis of immune cell surface proteins [5]. The strong shape variability of cells in tissue, however, poses a strong challenge to any automatic recognition approach. Figure 2a shows corresponding fluorescence microscopy images from a tissue section containing lymphocyte cells (courtesy W. Schubert). In the bottom row corresponding image patches are displayed, where individual cell regions were manually labeled to obtain training data for the learning process. For each of the image patches, a training vector consists of a list of labeled edge features parameterized by E
, where E is the position in the image and E is a unit local edge orientation vector computed from the intensity gradient. For a pixel image this amounts to a set of labeled edge features. Since the figure-groundseparating mechanism as implemented by the CLM [14] is also used for this cell segmentation application, features which are not labeled as part of a cell obtain the corresponding background label, given by . Each training pattern contains one additional free layer, to enable the learning algorithm to generalize over the number of layers. The lateral interaction to be adapted is decomposed into the following weighted basis components: i) A constant negative interaction between all features, which facilitates group separation, ii) a self-coupling interaction in the background layer which determines the attractivity of the background for figure-ground segmentation, and iii) an angular interaction with limited range, which is in itself decomposed into templates, capturing the interaction for a particular combination of the relative angles between two edges. This angular decomposition is done using a discretization of the space of orientations, turning the unit-vector representation into an angular orientation variable . To achieve rotation invariance of the interaction, it is only dependent on the edge orientations relative 2 3 4 5 6 7 8 2 3 4 5 7 6 8 9 a) Manually labelled training patterns b) Grouping results after learning Figure 2: a) Original images and manually labeled training patterns from a fluorescence micrograph. b) Test patterns and resulting CLM segmentation with learned lateral interaction. Grayscale represents different layer activations, where a total of 20 layers plus one background layer (black) was used. to their mutual position difference vector 2 . The angles and are discretized by partitioning the interval into 8 subintervals. For each combination of the two discretized edge orientations there is an interaction template generated, which is only responding in this combined orientation interval. Thus the angular templates do not overlap in the combined space, i.e. if # W for a particular , then # P for all S . Since the interaction must be symmetric under feature exchange, this does not result in different combinations, but only 36 independent templates. Apart form the discretization, the interaction represents the most arbitrary angular-dependent interaction within the local neighborhood, which is symmetric under feature exchange. We use two sets of angular templates for O and O O respectively, where is the maximal local interaction range. With the abovementioned two components, the resulting optimization problem is 36+36+2=74dimensional. Figure 3 compares the optimized interaction field to earlier heuristic lateral interactions for contour grouping. See [15] for a more detailed discussion. The performance of the learning approach was investigated by choosing a small number of the manually labeled patterns as training patterns. For all the training examples we used, the resulting inequalities (9) were in fact incompatible, rendering a direct solution of (9) infeasible. After training was completed by minimizing (12), a new image patch was selected as a test pattern and the CLM grouping was performed with the lateral interaction learned before, using the dynamical model as described in [14]. The quadratic consistency optimization was performed as described in the previous section, exploring the free margin parameter . For a set of two training patterns as shown in Fig. (2)a with a total of 1600 features each, a learning sweep takes about 4 minutes on a standard desktop computer. Typical segmentation results obtained with the quadratic consistency optimization approach are shown in Figure 2b, where the margin was given by * . The grouping results were not very sensitive to in a range of O O
. The grouping results show a good segmentation performance where most of the salient cells are detected as single groups. There are some spurious groups where a dark image region forms an additional group and some smaller cells are rejected into the background layer. Apart from these minor errors, the optimization has achieved an adequate balancing of the different lateral interaction components for this segmentation task. n n n2 1 2 d p p 2 1 n n 1 2 a) Plotting scheme b) Edge parameters d) Learned interaction field c) Standard continuity interaction field Figure 3: Comparison between heuristic continuity grouping interaction field and a learned lateral interaction field for cell segmentation. The interaction depends on the difference vector and two unit vectors
, shown in b), encoding directed orientation. a) explains the interaction visualizations c) and d) by showing a magnification of the plot c) of the interaction field of a single horizontal edge pointing to the left. The plots are generated by computing the interaction of the central directed edge with directed edges of all directions (like a cylindrical plot) at a spatial grid. Black edges share excitatory, white edges share inhibitory interaction with the central edge and length codes for interaction strength. The cocircular continuity field in c) depends on position and orientation but is not direction-selective. It supports pairs of edges which are cocircular, i.e. lie tangentially to a common circle and has been recently used for contour segmentation [3, 14]. The learned lateral interaction field is shown in d). It is direction-selective and supports pairs of edges which “turn right”. The strong local support is balanced by similarly strong long-range inhibition. 5 Discussion The presented results show that appropriate lateral interactions can be obtained for the CLM binding architecture from the quadratic consistency optimization approach. The only a priori conditions which were used for the template design were the properties of locality, symmetry, and translation as well as rotation invariance. This supervised learning approach has clear advantages over the manual tuning of complex feature interactions in complex feature spaces with many parameters. We consider this as an important step towards practical applicability of the feature binding concept. The presented quadratic consistency optimization method is based on choosing equal margins for all consistency inequalities. There exist other approaches to large margin classification, like support vector machines [10], where more sophisticated methods were suggested for appropriate margin determination. The application of similar methods to the supervised learning of CLM interactions provides an interesting field for future work. Acknowledgments: This work was supported by DFG grant GK-231 and carried out at the Faculty of Technology, University of Bielefeld. The author thanks Helge Ritter and Tim Nattkemper for discussions and Walter Schubert for providing the cell image data. References [1] R. Hahnloser, R. Sarpeshkar, M. A. Mahowald, R. J. Douglas, and H. S. Seung. Digital selection and analogue amplification coexist in a cortex-inspired silicon circuit. Nature, 405:947–951, 2000. [2] T. Hofmann, J. Puzicha, and J. Buhmann. Unsupervised texture segmentation in a deterministic annealing framework. IEEE Trans. Pattern Analysis and Machine Intelligence, 20(8):803–818, 1998. [3] Z. Li. A neural model of contour integration in the primary visual cortex. Neural Computation, 10:903–940, 1998. [4] M. Mozer, R. S. Zemel, M. Behrmann, and C. K. I. Williams. Learning to segment images using dynamic feature binding. Neural Computation, 4(5):650–665, 1992. [5] T. W. Nattkemper, H. Ritter, and W. Schubert. A neural classificator enabling high-throughput topological analysis of lymphocytes in tissue sections. IEEE Trans. Inf. Techn. in Biomed., 5(2):138–149, 2001. [6] J. Park, H. Cho, and D. Park. On the design of BSB associative memories using semidefinite programming. Neural Computation, 11:1985–1994, 1999. [7] M. Pelillo and M Refice. Learning compatibility coefficients for relaxation labeling processes. IEEE Trans. Pattern Analysis and Machine Intelligence, 16(9):933–945, 1994. [8] Renzo Perfetti. A synthesis procedure for Brain-State-in-a-Box neural networks. IEEE Transactions on Neural Networks, 6(5):1071–1080, September 1995. [9] H. Ritter. A spatial approach to feature linking. In Proc. International Neural Network Conference Paris Vol.2, pages 898–901, 1990. [10] V. Vapnik. The nature of statistical learning theory. Springer, New York, 1995. [11] C. von der Malsburg. The what and why of binding: The modeler’s perspective. Neuron, 24:95–104, 1999. [12] D. Wang and D. Terman. Image segmentation based on oscillatory correlation. Neural Computation, 9(4):805–836, 1997. [13] H. Wersing, W.-J. Beyn, and H. Ritter. Dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions. Neural Computation, 13(8):1811– 1825, 2001. [14] H. Wersing, J. J. Steil, and H. Ritter. A competitive layer model for feature binding and sensory segmentation. Neural Computation, 13(2):357–387, 2001. [15] Heiko Wersing. Spatial Feature Binding and Learning in Competitive Neural Layer Architectures. PhD thesis, University of Bielefeld, 2000. Published by Cuvillier, Goettingen. [16] X. Xie, R. Hahnloser, and H.S. Seung. Learning winner-take-all competition between groups of neurons in lateral inhibition networks. In Advances in Neural Information Processing Systems, volume 13. The MIT Press, 2001.
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Very loopy belief propagation for unwrapping phase images Brendan J . Freyl, Ralf Koetter2, Nemanja Petrovic1,2 1 Probabilistic and Statistical Inference Group, University of Toronto http://www.psi.toronto.edu 2 Electrical and Computer Engineering, University of Illinois at Urbana Abstract Since the discovery that the best error-correcting decoding algorithm can be viewed as belief propagation in a cycle-bound graph, researchers have been trying to determine under what circumstances "loopy belief propagation" is effective for probabilistic inference. Despite several theoretical advances in our understanding of loopy belief propagation, to our knowledge, the only problem that has been solved using loopy belief propagation is error-correcting decoding on Gaussian channels. We propose a new representation for the two-dimensional phase unwrapping problem, and we show that loopy belief propagation produces results that are superior to existing techniques. This is an important result, since many imaging techniques, including magnetic resonance imaging and interferometric synthetic aperture radar, produce phase-wrapped images. Interestingly, the graph that we use has a very large number of very short cycles, supporting evidence that a large minimum cycle length is not needed for excellent results using belief propagation. 1 Introduction Phase unwrapping is an easily stated, fundamental problem in image processing (Ghiglia and Pritt 1998). Each real-valued observation on a 1- or 2-dimensional grid is measured modulus a known wavelength, which we take to be 1 without loss of generality. Fig. Ib shows the wrapped, I-dimensional waveform obtained from the original waveform shown in Fig. la. Every time the original waveform goes above 1 or below 0, it is wrapped to 0 or 1, respectively. The goal of phase unwrapping is to infer the original, unwrapped curve from the wrapped measurements, using using knowledge about which signals are more probable a priori. In two dimensions, exact phase unwrapping is exponentially more difficult than 1dimensional phase unwrapping and has been shown to be NP-hard in general (Chen and Zebker 2000). Fig. lc shows the wrapped output of a magnetic resonance imaging device, courtesy of Z.-P. Liang. Notice the "fringe lines" - boundaries across which wrappings have occurred. Fig. Id shows the wrapped terrain height measurements from an interferometric synthetic aperture radar, courtesy of Sandia National Laboratories, New Mexico. (a) (b) (d) Figure 1: (a) A waveform measured on a 1-dimensional grid. (b) The phase-wrapped version ofthe waveform in (a), where the wavelength is 1. (c) A wrapped intensity ma p from a magnetic resonance imaging device, measured on a 2-dimensional grid (courtesy of Z.-P. Liang). (d) A wrapped topographic map measured on a 2-dimensional grid (courtesy of Sandia National Laboratories, New Mexico) . A sensible goal in phase unwrapping is to infer the gradient field of the original surface. The surface can then be reconstructed by integration. Equivalently, the goal is to infer the number of relative wrappings, or integer "shifts", between every pair of neighboring measurements. Positive shifts correspond to an increase in the number of wrappings in the direction of the x or y coordinate, whereas negative shifts correspond to a decrease in the number of wrappings in the direction of the x or y coordinate. After arbitrarily assigning an absolute number of wrappings to one point, the absolute number of wrappings at any other point can be determined by summing the shifts along a path connecting the two points. To account for direction, when taking a step against the direction of the coordinate, the shift should be subtracted. When neighboring measurements are more likely closer together than farther apart a priori, I-dimensional waveforms can be unwrapped optimally in time that is linear in the waveform length. For every pair of neighboring measurements, the shift that makes the unwrapped values as close together as possible is chosen. For example, the shift between 0.4 and 0.5 would be 0, whereas the shift between 0.9 and 0.0 would be -1. For 2-dimensional surfaces and images, there are many possible I-dimensional paths between any two points. These paths should be examined in combination, since the sum of the shifts along every such path should be equal. Viewing the shifts as state variables, the cut-set between any two points is exponential in the size of the grid, making exact inference for general priors NP-hard (Chen and Zebker 2000). The two leading fully-automated techniques for phase unwrapping are the least squares method and the branch cut technique (Ghiglia and Pritt 1998). (Some other techniques perform better in some circumstances, but need additional information or require hand-tweaking.) The least squares method begins by making a greedy guess at the gradient between every pair of neighboring points. The resulting vector field is not the gradient field of a surface, since in a valid gradient field, the sum of the gradients around every closed loop must be zero (that is, the curl must be 0). For example, the 2 x 2 loop of measurements 0.0, 0.3, 0.6, 0.9 will lead to gradients of 0.3,0.3,0.3, 0.1 around the loop, which do not sum to O. The least squares method proceeds by projecting the vector field onto the linear subspace of gradient fields. The result is integrated to produce the surface. The branch cut technique also begins with greedy decisions for the gradients and then identifies untrustworthy regions of the image whose gradients should not be used during integration. As shown in our results section, both of these techniques are suboptimal. Previously, we attempted to use a relaxed mean field technique to solve this problem (Achan, Frey and Koetter 2001). Here, we take a new approach that works better and is motivated by the impressive results of belief propagation in cycle-bound graphs for error-correcting decoding (Wiberg, Loeliger and Koetter 1995; MacKay and Neal 1995; Frey and Kschischang 1996; Kschischang and Frey 1998; McEliece, MacKay and Cheng 1998). In contrast to other work (Ghiglia and Pritt 1998; Chen and Zebker 2000; Koetter et al. 2001), we introduce a new framework for quantitative evaluation, which impressively places belief propagation much closer to the theoretical limit than other leading methods. It is well-known that belief propagation (a.k.a. the sum-product algorithm, probability propagation) is exact in graphs that are trees (Pearl 1988), but it has been discovered only recently that it can produce excellent results in graphs with many cycles. Impressive results have been obtained using loopy belief propagation for super-resolution (Freeman and Pasztor 1999) and for infering layered representations of scenes (Frey 2000). However, despite several theoretical advances in our understanding of loopy belief propagation (c.f. (Weiss and Freeman 2001)) and proposals for modifications to the algorithm (c.f. (Yedidia, Freeman and Weiss 2001)), to our knowledge, the only problem that has been solved by loopy belief propagation is error-correcting decoding on Gaussian channels. We conjecture that although phase unwrapping is generally NP-hard, there exists a near-optimal phase unwrapping algorithm for Gaussian process priors. Further, we believe that algorithm to be loopy belief propagation. 2 Loopy Belief Propagation for Phase Unwrapping As described above, the goal is to infer the number of relative wrappings, or integer "shifts" , between every pair of neighboring measurements. Denote the x-direction shift at (x,y) by a(x, y) and the y-direction shift at (x , y) by b(x, y), as shown in Fig.2a. If the sum of the shifts around every short loop of 4 shifts (e.g., a(x,y) + b(x + l,y) - a(x, y + 1) - b(x,y) in Fig. 2a) is zero, then perturbing a path will not change the sum of the shifts along the path. So, a valid set of shifts S = {a(x,y) ,b(x, y) : x = 1, ... ,N -1;y = 1, .. . , M -I} in an N x M image must satisfy the constraint a(x,y) + b(x + l,y) - a(x,y + 1) - b(x,y) = 0, (1) for x = 1, ... , N -1, Y = 1, ... , M -1. Since a(x, y) +b(x+ 1, y) -a(x, y+ 1) -b(x, y) is a measure of curl at (x, y), we refer to (1) as a "zero-curl constraint", reflecting the fact that the curl of a gradient field is O. In this way, phase unwrapping is formulated as the problem of inferring the most probable set of shifts subject to satisfying all zero-curl constraints. We assume that given the set of shifts, the unwrapped surface is described by a loworder Gaussian process. The joint distribution over the shifts S = {a(x, y), b(x, y) : x = 1, ... , N - 1; Y = 1, ... , M - I} and the wrapped measurements <I> = {¢(x, y) : (a) (b) x-direction shifts (' a's) Vi' t.: (x, y + l ) X a(x,y+ l ) X (x+ l ,y+ l ) <fl -7 'E .<::: ~-r1 X X X X X [f X X X X X <fl b(x, y ) I I b(x+ l ,y) c 0 t5 ~ '6 >. X -7 X (x,y) a(x,y) (x+ l ,y) (d) ~ it2 t a(x, y) it l t X X X X X X X X ) X X X X X X ) X X X X X X X X X X Figure 2: (a) Positive x-direction shifts (arrows labeled a) and positive y-direction shifts (arrows labeled b) between neighboring measurements in a 2 X 2 patch of points (marked by X 's), (b) A graphical model that describes the zero-curl constraints (black discs) between neighboring shift variables (white discs), 3-element probability vectors (J-L's) on the relative shifts between neighboring variables (-1, 0, or +1) are propagated across the network: (c) Constraint-to-shift vectors are computed from incoming shift-to-constraint vectors; (d) Shiftto-constraint vectors are computed from incoming constraint-to-shift vectors; (d) Estimates of the marginal probabilities of the shifts given the data are computed by combining incoming constra i nt-to-sh ift vectors, 0 :::; r/J(x, y) < 1, x = 1, .. . , N; y = 1, . . . , M } can be expressed in the form N - l M - l P(S, <I» ex: II II 5(a(x,y) +b(x +1,y) - a(x,y +1 ) -b(x,y)) x=l y=l N-l M N M-l . II II e-(c/>(x+l,y)-c/>(x,y)-a(x,y))2/2u2 II II e-(C/>(x,y+1)-c/>(x,y)-b(x,y))2/2u2 . (2) x= l y=l x= l y=l The zero-curl constraints are enforced by 5 (.), which evaluates to 1 if its argument is o and evaluates to 0 otherwise. We assume the slope of the surface is limited so that the unknown shifts take on the values -1, 0 and 1. a 2 is the variance between two neighboring measurements in the unwrapped image, but we find that in practice it can be estimated directly from the wrapped image. Phase unwrapping consists of making inferences about the a's and b's in the above probability model. For example, the marginal probability that the x-direction shift at (x,y) is k given an observed wrapped image <I>, is P (a(x,y) = kl<I» ex: L P(S, <I» . (3) S:a(x ,y)=k For an N x M grid, the above sum has roughly 32N M terms and so exact inference is intractable. The factorization of the joint distribution in (2) can be described by a graphical model, as shown in Fig. 2b. In this graph, each white disc sits on the border between two measurements (marked by x's), and corresponds to either an x-direction shift (a's ) or a y-direction shift (b's) . Each black disc corresponds to a zero-curl constraint (5(·) in (2)), and is connected to the 4 shifts that it constrains to sum to O. Probability propagation computes messages (3-vectors denoted by J-L) which are passed in both directions on every edge in the network. The elements of each 3vector correspond to the allowed values of the neighboring shift, -1, 0 and 1. Each of these 3-vectors can be thought of as a probability distribution over the 3 possible values that the shift can take on. Each element in a constraint-to-shift message summarizes the evidence from the other 3 shifts involved in the constraint, and is computed by averaging the allowed configurations of evidence from the other 3 shifts in the constraint. For example, if ILl ' IL2 and IL3 are 3-vectors entering a constraint as shown in Fig. 2c, the outgoing 3-vector, IL4' is computed using 1 1 f.t4i = L L L J(k + l- i - j)f.tljf.t2kf.t31, (4) j=-l k=-ll=-l and then normalized for numerical stability. The other 3 messages produced at the constraint are computed in a similar fashion. Shift-to-constraint messages are computed by weighting incoming constraint-to-shift messages with the likelihood for the shift. For example, if ILl is a 3-vector entering an x-direction shift as shown in Fig. 2d, the outgoing 3-vector, IL2 is computed using (5) and then normalized. Messages produced by y-direction shifts are computed in a similar fashion. At any step in the message-passing process, the messages on the edges connected to a shift variable can be combined to produce an approximation to the marginal probability for that shift, given the observations. For example, if ILl and IL2 are the 3-vectors entering an x-direction shift as shown in Fig. 2e, the approximation is P(a(x,y) = il<I» = (f.tlif.t2i)/(Lf.tljf.t2j). j (6) Given a wrapped image, the variance a 2 is estimated directly from the wrapped image, the probability vectors are initialized to uniform distributions, and then probability vectors are propagated across the graph in an iterative fashion. Different message-passing schedules are possible, ranging from fully parallel, to a "forwardbackward-up-down" -type schedule, in which messages are passed across the network to the right, then to the left, then up and then down. For an N x M grid, each iteration takes O(N M) scalar computations. After probability propagation converges (or, after a fixed number of iterations), estimates of the marginal probabilities of the shifts given the data are computed, and the most probable value of each shift variable is selected. The resulting configuration of the shifts can then be integrated to obtain the unwrapped surface. If some zero-curl constraints remain violated, a robust integration technique, such as least squares integration (Ghiglia and Pritt 1998), should be used. 3 Experimental results Generally, belief propagation in cycle-bound graphs is not guaranteed to converge. Even if it does converge, the approximate marginals may not be close to the true marginals. So, the algorithm must be verified by experiments. On surfaces drawn from Gaussian process priors, we find that the belief propagation algorithm produces significantly lower reconstruction errors than the least squares method and the branch cut technique. Here, we focus on the performances of the algorithms for real data recorded from a synthetic aperture radar device (Fig. 1d). Since our algorithm assumes the surface is Gaussian given the shifts, a valid concern is that it will not perform well when the Gaussian process prior is incorrect. (a) (b) en c Q) '6 CO ..... 0) -o ..... 10- 1 o ..... ..... Q) "0 Minimum wavelength ~ required for error-free ::J unwrapping using algs g that infer relative ~ § 10-2 shifts of -1 , 0 and + 1 Q) ~ 6 8 10 12 14 16 18 20 22 24 26 Wavelength, "Figure 3: (a) After 10 iterations of belief propagation using the phase-wrapped surface from Fig. Id, hard decisions were made for the shift variables and the resulting shifts were integrated to produce this unwrapped surface. (b) Reconstruction error versus wrapping wavelength for our technique, the least squares method and the branch cuts technique. Fig. 3a shows the surface that is obtained by setting (/2 to the mean squared difference between neighboring wrapped values, applying 10 iterations of belief propagation, making hard decisions for the integer shifts, and integrating the resulting gradients. Since this is real data, we do not know the "ground truth" . However, compared to the least squares method, our algorithm preserves more detail. The branch cut technique is not able to unwrap the entire surface. To obtain quantitative results on reconstruction error, we use the surface produced by the least squares method as the "ground truth" . To determine the effect of wrapping wavelength on algorithm performance, we rewrap this surface using different wavelengths. For each wavelength, we compute the reconstruction error for belief propagation, least squares and branch cuts. Note that by using least squares to obtain the ground truth, we may be biasing our results in favor of least squares. Fig. 3b shows the logarithm of the mean squared error in the gradient field of the reconstructed surface as a function of the wrapping wavelength, >., on a log-scale. (The plot for the mean squared error in the surface heights looks similar.) As >. -+ 0, unwrapping becomes impossible and as >. -+ 00, unwrapping becomes trivial (since no wrappings occur), so algorithms have waterfall-shaped curves. The belief propagation algorithm clearly obtains significantly lower reconstruction errors. Viewed another way, belief propagation can tolerate much lower wrapping wavelengths for a given reconstruction error. Also, it turns out that for this surface, it is impossible for an algorithm that infers relative shifts of -1,0 and 1 to obtain a reconstruction error of 0, unless A ::::: 12.97. Belief propagation obtains a zero-error wavelength that is significantly closer to this limit than the least squares method and the branch cuts technique. 4 Conclusions Phase unwrapping is a fundamental problem in image processing and although it has been shown to be NP-hard for general priors (Chen and Zebker 2000), we conjecture there exists a near-optimal phase unwrapping algorithm for Gaussian process priors. Further, we believe that algorithm to be loopy belief propagation. Our experimental results show that loopy belief propagation obtains significantly lower reconstruction errors compared to the least squares method and the branch cuts technique (Ghiglia and Pritt 1998), and performs close to the theoretical limit for techniques that infer relative wrappings of -1, 0 and + 1. The belief propagation algorithm runs in about the same time as the other techniques. References Achan, K., Frey, B. J. , and Koetter, R. 2001. A factorized variational technique for phase unwrapping in Markov random fields. In Uncertainty in Artificial Intelligence 2001. Seattle, Washington. Chen, C. W. and Zebker, H. A. 2000. Network approaches to two-dimensional phase unwrapping: intractability and two new algorithms. Journal of the Optical Society of America A, 17(3):401- 414. Freeman, W. and Pasztor, E. 1999. Learning low-level vision. In Proceedings of the International Conference on Computer Vision, pages 1182- 1189. Frey, B. J. 2000. Filling in scenes by propagating probabilities through layers and into appearance models. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. Frey, B. J. and Kschischang, F. R. 1996. Probability propagation and iterative decoding. In Proceedings of the 34th Allerton Conference on Communication, Control and Computing 1996. Ghiglia, D. C. and Pritt, M. D. 1998. Two-Dimensional Phase Unwrapping. Theory, Algorithms and Software. John Wiley & Sons. Koetter, R., Frey, B. J ., Petrovic, N., and Munson, Jr., D. C. 2001. Unwrapping phase images by propagating probabilities across graphs. In Proceedings of the International Conference on Acoustics, Speech and Signal Processing. IEEE Press. Kschischang, F. R. and Frey, B. J. 1998. Iterative decoding of compound codes by probability propagation in graphical models. IEEE Journal on Selected Areas in Communications, 16(2):219- 230. MacKay, D. J . C. and Neal, R. M. 1995. Good codes based on very sparse matrices. In Boyd, C., editor, Cryptograph,!! and Coding. 5th IMA Conference, number 1025 in Lecture Notes in Computer SCience, pages 100- 111. Springer, Berlin Germany. McEliece, R. J. , MacKay, D. J . C., and Cheng, J. F. 1998. Turbo-decoding as an instance of Pearl's 'belief propagation' algorithm. IEEE Journal on Selected Areas in Communications, 16. Pearl, J. 1988. Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann, San Mateo CA. Weiss, Y. and Freeman, W. 2001. On the optimaility of solutions of the max-product belief propagation algorithm in artbitrary graphs. IEEE Transactions on Information Theory, Special Issue on Codes on Graphs and Iterative Algorithms, 47(2):736- 744. Wiberg, N., Loeliger, H.-A., and Koetter, R. 1995. Codes and iterative decoding on general graphs. European Transactions on Telecommunications, 6:513- 525. Yedidia, J., Freeman, W . T., and Weiss, Y. 2001. Generalized belief propagation. In Advances in Neural Information Processing Systems 13. 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Modularity in the motor system: decomposition of muscle patterns as combinations of time-varying synergies Andrea d’Avella and Matthew C. Tresch Department of Brain and Cognitive Sciences Massachusetts Institute of Technology, E25-526 Cambridge, MA 02139 davel, mtresch @ai.mit.edu Abstract The question of whether the nervous system produces movement through the combination of a few discrete elements has long been central to the study of motor control. Muscle synergies, i.e. coordinated patterns of muscle activity, have been proposed as possible building blocks. Here we propose a model based on combinations of muscle synergies with a specific amplitude and temporal structure. Time-varying synergies provide a realistic basis for the decomposition of the complex patterns observed in natural behaviors. To extract time-varying synergies from simultaneous recording of EMG activity we developed an algorithm which extends existing non-negative matrix factorization techniques. 1 Introduction In order to produce movement, every vertebrate has to coordinate the large number of degrees of freedom in the musculoskeletal apparatus. How this coordination is accomplished by the central nervous system is a long standing question in the study of motor control. According to one common proposal, this task might be simplified by a modular organization of the neural systems controlling movement [1, 2, 3, 4]. In this scheme, specific output modules would control different but overlapping sets of degrees of freedom, thereby decreasing the number of variables controlled by the nervous system. By activating different output modules simultaneously but independently, the system may achieve the flexibility necessary to control a variety of behaviors. Several studies have sought evidence for such a modular controller by examining the patterns of muscle activity during movement, in particular looking for the presence of muscle synergies. A muscle synergy is a functional unit coordinating the activity of a number of muscles. The simplest model for such a unit would be the synchronous activation of a set of muscles with a specific activity balance, i.e. a vector in the muscle activity space. Using techniques such as the correlation between pairs of muscles, these studies have generally failed to provide strong evidence in support of such units. However, using a new analysis that allows for simultaneous combinations of more than one synergy, our group has recently provided evidence in support of this basic hypothesis of the neural control of movement. We used a non-negative matrix factorization algorithm to examine the composition of muscle activation patterns in spinalized frogs [5, 6]. This algorithm, similarly to that developed independently by others [7], extracts a small number of non-negative1 factors which can be combined to reconstruct a set of high-dimensional data. However, this analysis assumed that the muscle synergies consisted of a set of muscles which were activated synchronously. In examinations of complex behaviors produced by intact animals, it became clear that muscles within a putative synergy were often activated asynchronously. In these cases, although the temporal delay between muscles was nonzero, the dispersion around this delay was very small. These observations suggested that the basic units of motor production might involve not only a fixed coordination of relative muscle activation amplitudes, but also a coordination of relative muscle activation timings. We therefore have developed a new algorithm to factorize muscle activation patterns produced during movement into combinations of such time-varying muscle synergies. 2 Combinations of time-varying muscle synergies We model the output of the neural controller as a linear combination of muscle patterns with a specific time course in the activity of each muscle. In discrete time, we can represent each pattern, or time-varying synergy, as a sequence of vectors in muscle activity space. The data set which we consider here consists of episodes of a given behavior, e.g. a set of jumps in different directions and distances, or a set of walking or swimming cycles. In a particular episode , each synergy is scaled by an amplitude coefficient and timeshifted by a delay . The sequence of muscle activity for that episode is then given by:
(1) Fig. 1 illustrates the model with an example of the construction of a muscle pattern by combinations of three synergies. Compared to the model based on combinations of synchronous muscle synergies this model has more parameters describing each synergy ( vs. , with muscles and maximum number of time steps in a synergy) but less overall parameters. In fact, with synchronous synergies there is a combination coefficient for each time step and each synergy, whereas with time-varying synergies there are only two parameters ( and ) for each episode and each synergy. 3 Iterative minimization of the reconstruction error For a given set of episodes, we search for the set of non-negative time-varying synergies ! " " , $# %&(')'*' +-,.0/ , of maximum duration time steps and the set of coefficients 1-%& and that minimize the reconstruction error 243 2 3 2 3 65 7 8 :9
;
< 9 3 1The non-negativity constraint arises naturally in the context of motor control from the fact that firing rates of motoneurons, and consequently muscle activities, cannot be negative. While it is conceivable that a negative contribution on a motoneuronal pool from one factor would always be cancelled by a larger positive contribution from other factors, we chose a model based on non-negative factors to ensure that each factor could be independently activated. 5 4 3 2 1 Synergy1 5 4 3 2 1 Synergy2 5 4 3 2 1 Synergy3 10 20 30 40 50 60 70 80 90 100 5 4 3 2 1 Time Muscles C1 C2 C3 T1 T2 T3 Figure 1: An example of construction of a muscle pattern by the combinations of three time-varying synergies. In this example, each time-varying synergy (left) is constituted by a sequence of 50 activation levels in 5 muscles chosen as samples from Gaussian functions with different centers, widths, and amplitudes. To construct the muscle pattern (top right, shaded area), the activity levels of each synergy are first scaled by an amplitude coefficient ( , represented in the bottom right by the height of an horizontal bar) and shifted in time by a delay ( , represented by the position of the same bar). Then, at each time step, the scaled and shifted components (top right, broken lines) are summed together. with % for -% or 1- . After initializing synergies and coefficients to random positive values, we minimize the error by iterating the following steps: 1. For each episode, given the synergies and the scaling coefficients , find the delays using a nested matching procedure based on the cross-correlation of the synergies with the data (see 3.1 below). 2. For each episode, given the synergies and the delays , update the scaling coefficients by gradient descent
7 2 3 Here and below, we enforce non-negativity by setting to zero any negative value. 3. Given delays and scaling coefficients, update the synergy elements by gradient descent 243 3.1 Matching the synergy delays To find the best delay of each synergy in each episode we use the following procedure: i. Compute the sum of the scalar products between the s-th data episode and the i-th synergy time-shifted by 5 < (2) or scalar product cross-correlation at delay , for all possible delays. ii. Select the synergy and the delay with highest cross-correlation. iii. Subtract from the data the selected synergy (after scaling and time-shifting by the selected delay). iv. Repeat the procedure for the remaining synergies. 4 Results We tested the algorithm on simulated data in order to evaluate its performance and then applied it to EMG recordings from 13 hindlimb muscles of intact bullfrogs during several episodes of natural behaviors [8]. 4.1 Simulated data We first tested whether the algorithm could reconstruct known synergies and coefficients from a dataset generated by those same synergies and coefficients. We used two different types of simulated synergies. The first type was generated using a Gaussian function of different center, width, and amplitude for each muscle. The second type consisted of synergies generated by uniformly distributed random activities. For each type, we generated sets of three synergies involving five muscles with a duration of 15 time steps. Using these synergies, 50 episodes of duration 30 time steps were generated by scaling each synergy with random coefficients and shifting it by random delays . In figure 2 the results of a run with Gaussian synergies are shown. Using as a convergence criterion a change in 3 of less than , % for 20 iterations, after 474 iterations the solution had 3 % ' . Generating and reconstructed synergy activations are shown side by side on the left, in gray scale. Scatter plots of generating vs. reconstructed scaling coefficients and temporal delays are shown in the center and on the right respectively. Both synergies and coefficients were accurately reconstructed by the algorithm. In table 1, a summary of the results from 10 runs with Gaussian and random synergies is presented. We used the maximum of the scalar product cross-correlation between two normalized synergies (see eq. 2) to characterize their similarity. We compared two sets of synergies by matching the pairs in each set with the highest similarity and computing the mean similarity (
) between these pairs. All the synergy sets that we reconstructed ( ) had a high similarity with the generating set ( ). We also compared the generating and reconstructed scaling coefficients . using their correlation coefficient , and delays by counting the number of delay coefficients that were reconstructed correctly after compensating for possible lags in the synergies ( 8 ). The match in scaling coefficients and delays was in general very good. Only in a few runs with Gaussian synergies were the data correctly reconstructed (high 3 ) but with synergies slightly different from the generating ones (as indicated by the lower
) and consequently not perfectly matching coefficients (lower "! and # 8 $ 8 ! ). Synergy1 1 2 3 4 5 0 2 0 2 0 16 0 16 Synergy2 1 2 3 4 5 0 2 0 2 0 16 0 16 Synergy3 Wgen 5 10 15 1 2 3 4 5 Wrec 5 10 15 0 1 0 1 Cgen vs. Crec 0 16 0 16 Tgen vs. Trec Figure 2: An example of reconstruction of known synergies and coefficients from simulated data. The first column ( ) shows three time-varying synergies, generated from Gaussian functions, as three matrices each representing, in gray scale, the activity of 5 muscles (rows) over 15 time steps (columns). The second column ( ) shows the three synergies reconstructed by the algorithm: they accurately match the generating synergies (except for a temporal shift compensated by an opposite shift in the reconstructed delays). The third and fourth columns show scatter plots of generating vs. reconstructed scaling coefficients and delays in 50 simulated episodes. Both sets of coefficients are accurately reconstructed in almost all episodes. 4.2 Time-varying muscle synergies in frog’s muscle patterns We then applied the algorithm to EMG recordings of a large set ( , , , ) of hindlimb kicks, a defensive reflex that frogs use to remove noxious stimuli from the foot. Each kick consists of a fast extension followed by a slower flexion to return the leg to a crouched posture. The trajectory of the foot varies with the location of the stimulation on the skin and, as a consequence, the set of kicks spans a wide range of the workspace of the frog. Correspondingly, across different episodes the muscle activity patterns in the 13 muscles that we recorded showed considerable amplitude and timing variations that we sought to explain by combinations of time-varying synergies. After rectifying and integrating the EMGs over 10 ms intervals, we performed the optimization procedure with sets of synergies, with '*')' . We chose the maximum duration of each synergy to be 20 time steps, i.e. 200 ms, a duration larger than the duration of a typical muscle burst observed in this behavior. We repeated the procedure 10 times for each . Gaussian synergies 8 3
! 8 $ 8 ! max 561 0.9989 0.9996 0.9983 0.9467 median 451 0.9952 0.9990 0.9974 0.9233 min 297 0.9874 0.8338 0.2591 0.3133 Random synergies 8 3
! 8 $ 8 ! max 555 0.9999 1.0000 0.9996 0.9867 median 395 0.9998 1.0000 0.9990 0.9800 min 208 0.9998 1.0000 0.9984 0.9733 Table 1: Comparison between generated and reconstructed synergies and coefficients for 10 runs with Gaussian and random synergies. See text for explanation. In figure 3 the result of the extraction of four synergies with the highest 3 is shown. The convergence criterion of a change in 3 smaller than , % for 20 iterations was reached after 100 iterations with a final 3 % ' . The synergies extracted in the other nine runs were in general very similar to this set, as indicated by a mean similarity (
) ranging from % ' to % ' (median % ' ) and a correlation between scaling coefficients ranging from % ' to % ' (median % ' ). In the case with the lowest similarity, only one synergy in the set shown in figure 3 was not properly matched. The four synergies captured the basic features of the muscle patterns observed during different kicks. The first synergy, recruiting all the major knee extensor muscles (VI, RA, and VE), is highly activated in laterally directed kicks, as seen in the first kick shown in figure 3, which involved a large knee extension. The second synergy, recruiting two large hip extensor muscles (RI and SM) and an ankle extensor muscle (GA), is highly activated in caudally and medially directed kicks, i.e. kicks involving hip extension. The third synergy involves a specific temporal sequencing of several muscles: BI and VE first, followed by RI, SM, and GA, and then by AD and VI at the end. The fourth synergy has long activation profiles in many flexor muscles, i.e. those involved in the return phase of the kick, with a specific temporal pattern (long activation of IP; BI and SA before TA). When this set of EMGs was reconstructed using different numbers of muscle synergies, we found that the synergies identified using N synergies were generally preserved in the synergies identified using N+1 synergies. For instance, the first two synergies shown in figure 3 were seen in all sets of synergies, from to . Therefore, increasing the number of synergies allowed the data to be reconstructed more accurately (as seen by a higher 3 ) but without a complete reorganization of the synergies. 5 Discussion The algorithm that we introduced here represents a new analytical tool for the investigation of the organization of the motor system. This algorithm is an extension of previous nonnegative matrix factorization procedures, providing a means of capturing structure in a set of data not only in the amplitude domain but also in the temporal domain. Such temporal structure is a natural description of motor systems where many behaviors are characterized by a particular temporal organization. The analysis applied to behaviors produced by the frog, as described here, was able to capture significant physiologically relevant characteristics in the patterns of muscle activations. The motor system is not unique, however, in having structure in both amplitude and temporal domains and the techniques used here could easily be extended to other systems. Synergy1 RI AD SM ST IP VI RA GA TA PE BI SA VE Synergy2 RI AD SM ST IP VI RA GA TA PE BI SA VE Synergy3 RI AD SM ST IP VI RA GA TA PE BI SA VE Synergy4 5 10 15 20 RI AD SM ST IP VI RA GA TA PE BI SA VE 10 20 30 VE SA BI PE TA GA RA VI IP ST SM AD RI Time Muscles 5 10 15 20 25 Time C1 C2 C3 C4 T1 T2 T3 T4 Figure 3: Reconstruction of rectified and integrated (10 ms) EMGs for two kicks by timevarying synergies. Left: four extracted synergies constituted by activity levels (in gray scale) for 20 time steps in 13 muscles: rectus internus major (RI), adductor magnus (AD), semimembranosus (SM), ventral head of semitendinosus (ST), ilio-psoas (IP), vastus internus (VI), rectus anterior (RA), gastrocnemius (GA), tibialis anterior (TA), peroneous (PE), biceps (BI), sartorius (SA), and vastus externus (VE) [8]. Top right: the observed EMGs (thin line and shaded area) and their reconstruction (thick line) by combinations of the four synergies, scaled in amplitude ( ) and shifted in time ( ). Our model can be naturally extended to include temporal scaling of the synergies, i.e. allowing different durations of a synergy in different episodes. Work is in progress to implement an algorithm similar to the one presented here to extract time-varying and timescalable synergies. We will also address the issue of how to identify time-varying muscle synergies from continuous recordings of EMG patterns, without any manual segmentation into different episodes. A possibility that we are investigating is to extend the approach based on a sparse and overcomplete basis used by Lewicki and Sejnowski [9]. Finally, future work will aim to the development of a probabilistic model to address the issue of the dimensionality of the synergy set in terms of Bayesian model selection [10]. Acknowledgments We thank Zoubin Ghahramani, Emanuel Todorov, Emilio Bizzi, Sebastian Seung, Simon Overduin, and Maura Mezzetti for useful discussions and comments. References [1] E. Bizzi, P. Saltiel, and M. Tresch. Modular organization of motor behavior. Z Naturforsch [C], 53(7-8):510–7, 1998. [2] F. A. Mussa-Ivaldi. Modular features of motor control and learning. Curr Opin Neurobiol, 9(6):713–7, 1999. [3] W. J. Kargo and S. F. Giszter. Rapid correction of aimed movements by summation of forcefield primitives. J Neurosci, 20(1):409–26, 2000. [4] Z. Ghahramani and D. M. Wolpert. Modular decomposition in visuomotor learning. Nature, 386(6623):392–5, 1997. [5] M. C. Tresch, P. Saltiel, and E. Bizzi. The construction of movement by the spinal cord. Nature Neuroscience, 2(2):162–7, 1999. [6] P. Saltiel, K. Wyler-Duda, A. d’Avella, M. C. Tresch, and E. Bizzi. Muscle synergies encoded within the spinal cord: evidence from focal intraspinal nmda iontophoresis in the frog. Journal of Neurophysiology, 85(2):605–19, 2001. [7] D. D. Lee and H. S. Seung. Learning the parts of objects by non-negative matrix factorization. Nature, 401(6755):788–91, 1999. [8] A. d’Avella. Modular control of natural motor behavior. PhD thesis, MIT, 2000. [9] M. S. Lewicki and T. J. Sejnowski. Coding time-varying signals using sparse, shift-invariant representations. In M. S. Kearns, S. A. Solla, and D. A. Cohn, editors, Advances in Neural Information Processing Systems 11. MIT Press, 1999. [10] L. Wasserman. Bayesian model selection and model averaging. Journal of Mathematical Psychology, 44:92–107, 2000.
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The Method of Quantum Clustering David Horn and Assaf Gottlieb School of Physics and Astronomy Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv 69978, Israel Abstract We propose a novel clustering method that is an extension of ideas inherent to scale-space clustering and support-vector clustering. Like the latter, it associates every data point with a vector in Hilbert space, and like the former it puts emphasis on their total sum, that is equal to the scalespace probability function. The novelty of our approach is the study of an operator in Hilbert space, represented by the Schr¨odinger equation of which the probability function is a solution. This Schr¨odinger equation contains a potential function that can be derived analytically from the probability function. We associate minima of the potential with cluster centers. The method has one variable parameter, the scale of its Gaussian kernel. We demonstrate its applicability on known data sets. By limiting the evaluation of the Schr¨odinger potential to the locations of data points, we can apply this method to problems in high dimensions. 1 Introduction Methods of data clustering are usually based on geometric or probabilistic considerations [1, 2, 3]. The problem of unsupervised learning of clusters based on locations of points in data-space, is in general ill defined. Hence intuition based on other fields of study may be useful in formulating new heuristic procedures. The example of [4] shows how intuition derived from statistical mechanics leads to successful results. Here we propose a model based on tools that are borrowed from quantum mechanics. We start out with the scale-space algorithm of [5] that uses a Parzen-window estimator of the probability distribution based on the data. Using a Gaussian kernel, one generates from the data points in a Euclidean space of dimension a probability distribution given by, up to an overall normalization, the expression
(1) where
are the data points. It seems quite natural [5] to associate maxima of this function with cluster centers. The same kind of Gaussian kernel was the basis of another method, Support Vector Clustering (SVC) [6], associating the data-points
with vectors in an abstract Hilbert space. Here we will also consider a Hilbert space, but, in contradistinction with kernel methods where the Hilbert space is implicit, here we work with a Schr¨odinger equation that serves as the basic framework of the Hilbert space. Our method was introduced in [7] and is further expanded in this presentation. Its main emphasis is on the Schr¨odinger potential, whose minima will determine the cluster centers. This potential is part of the Schr¨odinger equation that is a solution of. 2 The Schr¨odinger Potential We define[7] the Schr¨odinger equation
(2) for which is a solution, or eigenstate.1 The simplest case is that of a single Gaussian, when represents a single point at . Then it turns out that . This quadratic function, whose center lies at , is known as the harmonic potential in quantum mechanics (see, e.g., [8]). Its eigenvalue is the lowest possible eigenvalue of , hence the Gaussian function is said to describe the ground state of . Conventionally, in quantum mechanics, one is given and one searches for solutions, or eigenfunctions, . Here, we have already , as determined by the data points, we ask therefore for the whose solution is the given . This can be easily obtained through
(3) is still left undefined. For this purpose we require to be positive definite, i.e. min =0. This sets the value of "!$# (4) and determines uniquely. Using Eq. 3 it is easy to prove that %"& (' (5) 3 2D Examples 3.1 Crab Data To show the power of our new method we discuss the crab data set taken from Ripley’s book [9]. This data set is defined over a five-dimensional parameter space. When analyzed in terms of the 2nd and 3rd principal components of the correlation matrix one observes a nice separation of the 200 instances into their four classes. We start therefore with this problem as our first test case. In Fig. 1 we show the data as well as the Parzen probability distribution using the width parameter *) . It is quite obvious that this width is not small enough to deduce the correct clustering according to the approach of [5]. Nonetheless, the potential displayed in Fig. 2 shows the required four minima for the same width parameter. Thus we conclude that the necessary information is already available. One needs, however, the quantum clustering approach, to bring it out. 1 + (the Hamiltonian) and , (potential energy) are conventional quantum mechanical operators, rescaled so that + depends on one parameter, - . . is a (rescaled) energy eigenvalue in quantum mechanics. −2 −1 0 1 2 −3 −2 −1 0 1 2 PC3 PC2 Figure 1: A plot of Roberts’ probability distribution for Ripley’s crab data [9] as defined over the 2nd and 3rd principal components of the correlation matrix. Using a Gaussian width of ) we observe only one maximum. Different symbols label the four classes of data. −2 −1 0 1 2 −3 −2 −1 0 1 2 0 0.2 0.4 0.6 0.8 1 1.2 PC3 PC2 V/E Figure 2: A plot of the Schr¨odinger potential for the same problem as Fig. 1. Here we clearly see the required four minima. The potential is plotted in units of . Note in Fig. 2 that the potential grows quadratically outside the domain over which the data are located. This is a general property of Eq. 3. sets the relevant scale over which one may look for structure of the potential. If the width is decreased more structure is to be expected. Thus, for , two more minima appear, as seen in Fig. 3. Nonetheless, they lie high and contain only a few data points. The major minima are the same as in Fig. 2. 3.2 Iris Data Our second example consists of the iris data set [10], which is a standard benchmark obtainable from the UCI repository [11]. Here we use the first two principal components to define the two dimensions in which we apply our method. Fig. 4, which shows the case for % , provides an almost perfect separation of the 150 instances into the three classes into which they should belong. 4 Application of Quantum Clustering The examples displayed in the previous section show that, if the spatial representation of the data allows for meaningful clustering using geometric information, quantum clustering (QC) will do the job. There remain, however, several technical questions to be answered: What is the preferred choice of ? How can QC be applied in high dimensions? How does one choose the appropriate space, or metric, in which to perform the analysis? We will confront these issues in this section. 4.1 Varying In the crabs-data we find that as is decreased to , the previous minima of get deeper and two new minima are formed. However the latter are insignificant, in the sense that they lie at high values (of order ), as shown in Fig. 3. Thus, if we classify data-points to clusters according to their topographic location on the surface of , roughly the same clustering assignment is expected for as for . By the way, the wave function acquires only one additional maximum at . As is being further decreased, more and more maxima are expected in and an ever increasing number of minima (limited by ) in . The one parameter of our problem, , signifies the distance that we probe. Accordingly we expect to find clusters relevant to proximity information of the same order of magnitude. One may therefore vary continuously and look for stability of cluster solutions, or limit oneself to relatively high values of and decide to stop the search once a few clusters are being uncovered. 4.2 Higher Dimensions In the iris problem we obtained excellent clustering results using the first two principal components, whereas in the crabs problem, clustering that depicts correctly the classification necessitates components 2 and 3. However, once this is realized, it does not harm to add the 1st component. This requires working in a 3-dimensional space, spanned by the three leading PCs. Calculating on a fine computational grid becomes a heavy task in high dimensions. To cut down complexity, we propose using the analytic expression of Eq. 3 and evaluating the potential on data points only. This should be good enough to give a close estimate of where the minima lie, and it reduces the complexity to irrespective of dimension. In the gradient-descent algorithm described below, we will require further computations, also restricted to well defined locations in space. −2 −1 0 1 2 −3 −2 −1 0 1 2 0 0.2 0.4 0.6 0.8 1 1.2 PC2 PC3 V/E Figure 3: The potential for the crab data with displays two additional, but insignificant, minima. The four deep minima are roughly at the same locations as in Fig. 2. 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 PC1 PC2 Figure 4: Quantum clustering of the iris data for % in a space spanned by the first two principal components. Different symbols represent the three classes. Equipotential lines are drawn at When restricted to the locations of data points, we evaluate on a discrete set of points
. We can then express in terms of the distance matrix
as
(6) with chosen appropriately so that min
=0. All problems that we have used as examples were such that data were given in some space, and we have exercised our freedom to define a metric, using the PCA approach, as the basis for distance calculations. The previous analysis tells us that QC can also be applied to data for which only the distance information is known. 4.3 Principal Component Metrics The QC algorithm starts from distance information. The question how the distances are calculated is another - very important - piece of the clustering procedure. The PCA approach defines a metric that is intrinsic to the data, determined by their second order statistics. But even then, several possibilities exist, leading to non-equivalent results. Principal component decomposition can be applied both to the correlation matrix and to the covariance matrix. Moreover, whitening normalization may be applied. The PCA approach that we have used is based on a whitened correlation matrix. This turns out to lead to the good separation of crab-data in PC2-PC3 and of iris-data in PC1-PC2. Since our aim was to convince the reader that once a good metric is found, QC conveys the correct information, we have used the best preprocessing before testing QC. 5 The Gradient Descent Algorithm After discovering the cluster centers we are faced with the problem of allocating the data points to the different clusters. We propose using a gradient descent algorithm for this purpose. Defining
%
we define the process
(7) letting the points
reach an asymptotic fixed value coinciding with a cluster center. More sophisticated minimum search algorithms, as given in chapter 10 of [12], may be used for faster convergence. To demonstrate the results of this algorithm, as well as the application of QC to higher dimensions, we analyze the iris data in 4 dimensions. We use the original data space with only one modification: all axes are normalized to lie within a unified range of variation. The results are displayed in Fig. 5. Shown here are different windows for the four different axes, within which we display the values of the points after descending the potential surface and reaching its minima, whose values are shown in the fifth window. These results are very satisfactory, having only 5 misclassifications. Applying QC to data space without normalization of the different axes, leads to misclassifications of the order of 15 instances, similar to the clustering quality of [4]. 6 Discussion In the literature of image analysis one often looks for the curve on which the Laplacian of the Gaussian filter of an image vanishes[13]. This is known as zero-crossing and serves as 0 50 100 150 0.5 1 1.5 dim 1 0 50 100 150 1 dim 2 0 50 100 150 0 1 2 dim 3 0 50 100 150 0 1 2 dim 4 20 40 60 80 100 120 140 0 0.1 0.2 serial number V/E Figure 5: The fixed points of the four-dimensional iris problem following the gradientdescent algorithm. The results show almost perfect clustering into the three families of 50 instances each for % . a measure of segmentation of the image. Its analogue in the scale-space approach is where % . Clearly each such contour can also be viewed as surrounding maxima of the probability function, and therefore representing some kind of cluster boundary, although different from the conventional one [5]. It is known that the number of such boundaries [13] is a non-decreasing function of . Note that such contours can be read off Fig. 4. Comparison with Eq. 3 tells us that they are the contours on the periphery of this figure. Clearly they surround the data but do not give a satisfactory indication of where the clusters are. Cluster cores are better defined by % curves in this figure. One may therefore speculate that equipotential levels of may serve as alternatives to % curves in future applications to image analysis. Image analysis is a 2-dimensional problem, in which differential operations have to be formulated and followed on a fine grid. Clustering is a problem that may occur in any number of dimensions. It is therefore important to develop a tool that can deal with it accordingly. Since the Schr¨odinger potential, the function that plays the major role in our analysis, has minima that lie in the neighborhood of data points, we find that it suffices to evaluate it at these points. This enables us to deal with clustering in high dimensional spaces. The results, such as the iris problem of Fig. 5, are very promising. They show that the basic idea, as well as the gradient-descent algorithm of data allocation to clusters, work well. Quantum clustering does not presume any particular shape or any specific number of clusters. It can be used in conjunction with other clustering methods. Thus one may start with SVC to define outliers which will be excluded from the construction of the QC potential. This would be one example where not all points are given the same weight in the construction of the Parzen probability distribution. It may seem strange to see the Schr¨odinger equation in the context of machine learning. Its usefulness here is due to the fact that the two different terms of Eq. 2 have opposite effects on the wave-function. The potential represents the attractive force that tries to concentrate the distribution around its minima. The Laplacian has the opposite effect of spreading the wave-function. In a clustering analysis we implicitly assume that two such effects exist. QC models them with the Schr¨odinger equation. Its success proves that this equation can serve as the basic tool of a clustering method. References [1] A.K. Jain and R.C. Dubes. Algorithms for clustering data. Prentice Hall, Englewood Cliffs, NJ, 1988. [2] K. Fukunaga. Introduction to Statistical Pattern Recognition. Academic Press, San Diego, CA, 1990. [3] R.O. Duda, P.E. Hart and D.G. Stork. Pattern Classification. Wiley-Interscience, 2nd ed., 2001. [4] M. Blat, S. Wiseman and E. Domany. Super-paramagnetic clustering of data. Phys. Rev. Letters 76:3251-3255, 1996. [5] S.J. Roberts. Non-parametric unsupervised cluster analysis. Pattern Recognition, 30(2):261–272, 1997. [6] A. Ben-Hur, D. Horn, H.T. Siegelmann, and V. Vapnik. A Support Vector Method for Clustering. in Advances in Neural Information Processing Systems 13: Proceedings of the 2000 Conference Todd K. Leen, Thomas G. Dietterich and Volker Tresp eds., MIT Press 2001, pp. 367–373. [7] David Horn and Assaf Gottlieb. Algorithm for Data Clustering in Pattern Recognition Problems Based on Quantum Mechanics. Phys. Rev. Lett. 88 (2002) 018702. [8] S. Gasiorowicz. Quantum Physics. Wiley 1996. [9] B. D. Ripley Pattern Recognition and Neural Networks. Cambridge University Press, Cambridge UK, 1996. [10] R.A. Fisher. The use of multiple measurements in taxonomic problems. Annual Eugenics, 7:179–188, 1936. [11] C.L. Blake and C.J. Merz. UCI repository of machine learning databases, 1998. [12] W. H. Press, S. A. Teuklosky, W. T. Vetterling and B. P. Flannery. Numerical Recipes - The Art of Scientific Computing 2nd ed. Cambridge Univ. Press, 1992. [13] A. L. Yuille and T. A. Poggio. Scaling theorems for zero crossings. IEEE Trans. Pattern Analysis and Machine Intelligence PAMI-8, 15-25, 1986.
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Generating velocity tuning by asymmetric recurrent connections Xiaohui Xie and Martin A. Giese Dept. of Brain and Cognitive Sciences and CBCL Massachusetts Institute of Technology Cambridge, MA 02139 Dept. for Cognitive Neurology, University Clinic T¨ubingen Max-Planck-Institute for Biological Cybernetics 72076 T¨ubingen, Germany E-mail: xhxie|giese @mit.edu Abstract Asymmetric lateral connections are one possible mechanism that can account for the direction selectivity of cortical neurons. We present a mathematical analysis for a class of these models. Contrasting with earlier theoretical work that has relied on methods from linear systems theory, we study the network’s nonlinear dynamic properties that arise when the threshold nonlinearity of the neurons is taken into account. We show that such networks have stimulus-locked traveling pulse solutions that are appropriate for modeling the responses of direction selective cortical neurons. In addition, our analysis shows that outside a certain regime of stimulus speeds the stability of this solutions breaks down giving rise to another class of solutions that are characterized by specific spatiotemporal periodicity. This predicts that if direction selectivity in the cortex is mainly achieved by asymmetric lateral connections lurching activity waves might be observable in ensembles of direction selective cortical neurons within appropriate regimes of the stimulus speed. 1 Introduction Classical models for the direction selectivity in the primary visual cortex have assumed feed-forward mechanisms, like multiplication or gating of afferent thalamo-cortical inputs (e.g. [1, 2, 3]), or linear spatio-temporal filtering followed by a nonlinear operation (e.g. [4, 5]). The existence of strong lateral connectivity has motivated modeling studies, which have shown that the properties of direction selective cortical neurons can also be accurately reproduced by recurrent neural network models with asymmetric lateral excitatory or inhibitory connections [6, 7]. Since these biophysically detailed models are not accessible for mathematical analysis, more simplified models appropriate for a mathematical analysis have been proposed. Such analysis was based on methods from linear systems theory by neglecting the nonlinear properties of the neurons [6, 8, 9]. The nonlinear dynamic phenomena resulting from the interplay between the recurrent connectivity and the nonlinear threshold characteristics of the neurons have not been tractable in this theoretical framework. In this paper we present a mathematical analysis that takes the nonlinear behavior of the individual neurons into account. We present the result of the analysis of such networks for two types of threshold nonlinearities, for which closed-form analytical solutions of the network dynamics can be derived. We show that such nonlinear networks have a class of form-stable solutions, in the following signified as stimulus-locked traveling pulses, which are suitable for modeling the activity of direction selective neurons. Contrary to networks with linear neurons, the stability of the traveling pulse solutions in the nonlinear network can break down giving raise to another class of solutions (lurching activity waves) that is characterized by spatio-temporal periodicity. Our mathematical analysis and simulations showed that recurrent models with biologically realistic degrees of direction selectivity typically also show transitions between traveling pulse and lurching solutions. 2 Basic model Dynamic neural fields have been proposed to model the average behavior of a large ensembles of neurons [10, 11, 12]. The scalar neural activity distribution characterizes the average activity at time of an ensemble of functionally similar neurons that code for the position , where can be any abstract stimulus parameter. By the continuous approximation of biophysically discrete neuronal dynamics it is in some cases possible to treat the nonlinear neural dynamics analytically. The field dynamics of neural activation variable is described by:
"! # %$ (1) This dynamics is essentially a leaky integrator with a total input on the right hand side, which includes a feedfoward input term ! and a feedback term that integrates the recurrent contributions from other laterally connected neurons. The interaction kernel &' characterizes the average synaptic connection strength between the neurons coding position and the neurons coding position . is the activation function of the neurons. This function is nonlinear and monotonically increasing, and introduces the nonlinearity that makes it difficult to analyze the network dynamics. With a moving stimulus at constant velocity ( , it is often convenient to transform the static coordinate to the moving frame by changing variable ) *+ ( . Under the new frame, the stimulus is stationary. Let , )
- ( . . The dynamics for , reads , ) ( , ) ) , )
/0 ) ) , ) 1 2 ) "! ) .$ (2) A stationary solution in the moving frame has to satisfy the following equation: ( ,43 ) ) , 3 )
) ) , 3 ) 2 ) "! ) .$ (3) ,43 ) corresponds to a traveling pulse solution with velocity ( in the original static coordinate. Therefore the traveling pulse solution driven by the moving stimulus can be found by solving Eq. (3), and the stability of the traveling pulse can be studied by perturbing the stationary solution in Eq. (2). The neural field dynamics Eq. (2) is a nonlinear integro-differential equation. In most cases an analytic treatment of such equations is impossible. In this paper, we consider two biologically inspired special cases, which can be analytically solved. For this purpose we consider only one-dimensional neural fields and assume that the nonlinear activation function is either a step function or a linear threshold function. 3 Step activation function We first consider step activation function where when and zero otherwise. This form of activation function approximates activities of neurons, which, by saturation, are either active or inactive. For the one-dimensional case, we assume that only a single stationary excited regime with ( , 3 ) )exists that is located between the points ) 3 ) 3 . Only neurons inside this regime contribute to the integral, and accordingly Eq. (3) can be simplified following [11]. The spatial shape , 3 ) of the stationary solution obeys the ordinary differential equation ( , 3 ) ) , 3 )
) ) 3
) ) 3 ! ) . (4) where
. The solution of the above equation can be found by treating the boundaries ) 3 and ) 3 as fixed parameters, and solving Eq. (4). To facilitate notation we define an integral operator with parameter as / "! $# &%(' *),+.-/10 (# (5) where 2 for 43 and 2 otherwise. Using this operator we define the two functions 5
6 (( $7 2 ( and 8
! (9 7 ( %$ The solution of Eq. (4) can be written with these functions in the form , 3 )
5 ) ) 3 5 ) ) 3 8 ) %$ (6) For the boundary points , 3 ) 3 , 3 ) 3 4 must be satisfied, leading to the transcendent equation system 5 5 ) 3 ) 3 8 ) 3 (7) 5 5 ) 3 ) 3 8 ) 3 . (8) from which ) 3 and ) 3 can be determined. 3.1 Stability of the traveling pulse solution The stability of the traveling pulse solution can be analyzed by perturbing the stationary solution in the moving coordinate system. Let : , ) be a small perturbation of , 3 ) . The linearized perturbation dynamics reads
: , ( : , ) : , )
) ) 3 : ) ) ) 3 : ) (9) where : )*; ( < = 1> ) are the perturbations of the boundary points of the exited regime from the stationary values of ) 3 ; with , ) 3 ; : ) ; 4 . However, : ) ; is not independent of : , ) , and the dependence can be found by noting that , ) ; , ) 3 ; : ) ; , ) 3 ; , ) 3 ; ) : ) ; : ) ; $ Since , ) 3 ; : , ) 3 ; to the first order we have : ) ; : , ) 3 ; 7@? 3 ; , where ? 3 ; , 3 ) ; 7 ) . Substituting this back into the perturbed dynamics, we have
: , ( : , ) : , )
) ) 3 ? 3 : , ) 3 ) ) 3 ? 3 : , ) 3 %$ Substitute solution of the form : , ) % ) into the above dynamics. After some calculation, the eigenvalue equation for reads ? 3 ? 3 ) 3 ) 3 ) 3 ) 3 . (10) where function is defined as
6 ( 7 7 2 ( $ From the transcendent Eq. (10), can be found. The traveling pulse solution is asymptotically stable only if the real parts of all eigenvalues are negative. 3.2 Simulation results of step activation function model We use the following function
; #2 ; as an example interaction kernel, numerically simulate the dynamics and compare the simulation results with the above mathematical analysis. The stimulus used is a moving bar with constant width and amplitude. The results are shown in the left (a-e) panels of Fig. (1). Panel (a) shows the speed tuning curve plotted as the dependence of the peak activity of the traveling pulse as function of the stimulus velocity ( . The solid lines indicate the results from the numerical simulation and the dotted lines represent results from the analytical solution. Panel (b) shows the maximum real part of the eigenvalues obtained from Eq. (10). For small and large stimulus velocities maximum of the real parts of becomes positive indicating a loss of stability of the form-stable solution. To verify this result we calculated the variability of the peak activity over time in simulation. Panel (c) shows the average variability as function of the stimulus velocity. At the velocities for which the eigenvalues indicate a loss of stability the variability of the amplitudes suddenly increases, consistent with our interpretation as a loss of the form stability of the solution. An interesting observation is illustrated in panels (d) and (e) that show a color-coded plot of the space-time evolution of the activity. Panel (e) shows the propagation of the form-stable traveling pulse. Panel (d) shows the solution that arises when stability is lost. This solution is characterized by a spatio-temporal periodicity that is defined in the moving coordinate system by , # "! 4 , # , where and ! are constants that depend on the network dynamics. Solutions of similar type have been described before in spiking networks [13]. 4 Linear threshold activation function In this case, the activation function is taken to be #$ &%(' " . Cortical neurons typically operate far below the saturation level. The linear threshold activation function is thus more suitable to capture the properties of real neurons while still permitting a relatively simple theoretical analysis. We consider a ring network with periodic boundary conditions. The dynamics is given by # ) # #) +* -, ) , ) >. )/) # ) 1 "! ) 10 $ $ (11) This network can be shown equivalent to the the standard one in Eq. (1) by changing variables and transforming stimulus. We chose this form because it simplifies the mathematical analysis of ring networks. Again, we consider a moving stimulus with velocity ( and analyze the network in the moving frame. 4.1 General solutions and stability analysis Because the activation function has linear threshold characteristics, inside the excited regime for which the total input ( #) ) is positive the system is linear. One approach to solve this dynamics is therefore to find the solutions to the differential equation assuming the boundaries of the excited regime are given. The conditions at the boundaries lead to a set of self-consistent equations for the solutions to satisfy, from which the boundaries can be determined. By denoting activities in moving coordinates as ) ( .
# ) , the dynamics can be written as: ) ( ) ) #) *1 , ) , #)4) ) >. ) ) 8 ) 0 $ $ Supposing the excited regime is ) ) % ) , we solve the dynamics by Fourier transforming the above equation in the spatial domain . . . Let #
' , ) , ) &% ;
>.# ) ) / , ) , ) &% ; >. ) ) % ; ' ) >. ) ) ! 8 #) &% ; >. ) ) $ where 4 $ $ $ is the frequency. The stationary solution in moving coordinates can then be written as 3 < (! ) " (12) where matrix is defined as the diagonal matrix $# '&% ! . The components of the vector are #' , and those of " are ! . The above solution has to satisfy two boundary conditions, from which ) and ) can be determined. Stability of this traveling pulse solution can be analyzed by linear perturbation. Note that perturbed boundaries points do not contribute to the linearized perturbed dynamics since : ) ; 3 #) ; < 1> ,where 3 ) is the total input at the stationary solution of the moving frame on right hand side of Eq. (11). Therefore, the linearized perturbation dynamics can be fully characterized by the perturbed Fourier modes with fixed boundaries. Hence, the stability of the traveling pulse solution is determined by the eigenvalues of matrix ( 4) < ( . If the largest real part of eigenvalues of ( is negative, then the stimulus locking traveling pulse is stable. 4.2 Simplified linear threshold network The general solution introduced above requires the solution of an equation system. In practice, the Fourier series have to be truncated in order to obtain a finite number of Fourier components at the expense of an approximation error. Next we consider a special simple model for which an exact solution can be found that contains only two Fourier components for the interaction kernel and the input ! . For this model a closed form solution and stability analysis is presented, that at the same time provides insight in some rather general properties of linear threshold networks. The interaction kernel and feedforward input are assumed to have the following form: ) +* * -,.
/ ) 10 ! )
-,./ #) (13) This network was used by Hansel and Sompolinsky as model of cortical orientation selectivity [14]. However different from their network, we consider here an asymmetric interaction kernel ) and a form-constant moving stimulus ! ) ( . Since the interaction kernel and input ! only involve first two Fourier components, the dynamics can be fully determined in terms of its order parameters defined by
, ) , ) >. # ) , ) , ) >. # ) % ; ' ) (14) where phase variable is to restrict to being real. In terms of these two order parameters plus the phase variable, the stimulus-locked traveling pulse solution and its stability conditions can be expressed analytically. Due to space limitation, the detailed derivations are omitted here. We show the theoretical results in right five panels of Fig. (1) and compare them with numerical simulations. Similar to the results of step function model, panel (A) shows the speed tuning curve plotted as values of order parameters and as function of different stimulus velocities ( . Panel (B) shows the largest real part of the eigenvalues of a stability matrix that can be obtained by linearizing the order parameter dynamics around the stationary solution. Panel (C) shows the average variations as function of the stimulus velocity. The space-time evolution of the form-stable traveling pulse is shown in panel (E); the form-unstable lurching wave is shown in panel (D). Thus we found that lurching wave solution type arises very robustly for both types of threshold functions when the network achieved substantial direction selective behavior. 5 Conclusion We have presented different methods for an analysis of the nonlinear dynamics of simple recurrent neural models for the direction selectivity of cortical neurons. Compared to earlier works, we have taken into account the essentially nonlinear effects that are introduced by the nonlinear threshold characteristics of the cortical neurons. The key result of our work is that such networks have a class of form-stable traveling pulse solutions that behave similar as the solutions of linear spatio-temporal filtering models within a certain regime of stimulus speeds. By the essential nonlinearity of the network, however, bifurcations can arise for which the traveling pulse solutions become unstable. We observed that in this case a new class of spatio-temporally periodic solutions (”lurching activity waves”) arises. Since we found this solution type very frequently for networks with substantial direction selectivity our analysis predicts that such ”lurching behavior” might be observable in visual cortex areas if, in fact, the direction selectivity is essentially based on asymmetric lateral connectivity. Acknowledgments We acknowledge helpful discussions with H.S. Seung and T. Poggio. References [1] C Koch and T Poggio. The synaptic veto mechanism: does it underlie direction and orientation selectivity in the visual cortex. In D Rose and V G Dobson, editors, Models of the Visual Cortex, pages 15–34. John Wiley, 1989. [2] J.P. van Santen and G. Sperling. Elaborated reichardt detectors. J Opt Soc Am A, 256:300–21, 1985. [3] W. Reichardt. A principle for the evaluation of sensory information by the central nervous system, 1961. [4] E. H. Adelson and J. R. Bergen. Spatiotemporal energy models for the perception of motion. J Opt Soc Am A, 256:284–99, 1985. a b c d e
SPACE TIME TIME SPACE
A B C D E -40 -30 -20 10 0 10 0 1 2 3 Peak Activity theory simulation -40 -30 -20 10 0 10 -2 0 2 Real(λ) -40 -30 -20 10 0 10 0 0.1 0.2 Peak variance Velocity -50 0 50 100 0 0.01 0.02 0.03 Activity -50 0 50 100 -0.5 0 0.5 Real Eig -50 0 50 100 0 1 2 3 x 10 -3 Variation Velocity r 0 r 1 Figure 1: Traveling pulse solution and its stability in two classes of models. In the left side shown is the step activation function model, while the linear threshold model is drawn in the right. Panel (a) and (A) show the velocity tuning curves of the traveling pulse in terms of its peak activity in (a) or order parameters in (A). The solid lines indicate the results from calculation, and the dotted lines represents the results from simulaion. Panel (b) and (B) plot the largest real parts of eigenvalues of a stability matrix obtained from perturbed linear dynamics around the stationary solution. Outside certain range of stimulus velocities the largest real part of the eigenvalues become positive indicating a loss of stability of the form-stable solution. Panel (c) and (C) plots the average variations of peak activity, and order parameters (blue curve) and (green curve) respectively, over time during simulation. A nonzero variance signifies a loss of stability for traveling pulse solutions, which is consistent with eigenvalue analysis in Panel (b) and (B). A color coded plot of spatialtemporal evolution of the activity # is shown in panels (d) and (e), and # # in (D) and (E). Panel (e) and (E) show the propagation of the form-stable peak over time; panel (d) and (D) show the lurching activity wave that arises when stability is lost. The interaction kernel used in step function model is
2 ;
; with ; $ > ; $ and . The stimulus is a moving bar with width * and amplitude > . Parameters used in linear threshold model are * $ , * $ , $ and $ . [5] A. B. Watson and A. J. Ahumada. Model of human visual-motion sensing. J Opt Soc Am A, 256:322–41, 1985. [6] H. Suarez, C. Koch, and R. Douglas. Modeling direction selectivity of simple cells in striate visual cortex within the framework of the canonical microcircuit. J Neurosci, 15:6700–19, 1995. [7] R. Maex and G. A. Orban. Model circuit of spiking neurons generating directional selectivity in simple cells. J Neurophysiol, 75:1515–45, 1996. [8] P. Mineiro and D. Zipser. Analysis of direction selectivity arising from recurrent cortical interactions. Neural Comput, 10:353–71, 1998. [9] S. P. Sabatini and F. Solari. An architectural hypothesis for direction selectivity in the visual cortex: the role of spatially asymmetric intracortical inhibition. Biol Cybern, 80:171–83, 1999. [10] HR Wilson and JD Cowan. A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik, 13(2):55–80, 1973. [11] S Amari. Dynamics of pattern formation in lateral-inhibition type neural fields. Biol Cybern, 27(2):77–87, 1977. [12] E. Salinas and L.F. Abbott. A model of multiplicative neural responses in parietal cortex. Proc. Natl. Acad. Sci. USA, 93:11956–11961, 1996. [13] D. Golomb and G. B. Ermentrout. Effects of delay on the type and velocity of travelling pulses in neuronal networks with spatially decaying connectivity. Network, 11:221–46, 2000. [14] David Hansel and Haim Sompolinsky. Modeling feature selectivity in local cortical circuits. In C. Koch and I. Segev, editors, Methods in Neuronal Modeling, chapter 13, pages 499–567. MIT Press, Cambridge, Massachusetts, 1998.
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Incremental Learning and Selective Sampling via Parametric Optimization Framework for SVM Shai Fine IBM T. J. Watson Research Center fshai@us.ibm.com Katya Scheinberg IBM T. J. Watson Research Center katyas@us.ibm.com Abstract We propose a framework based on a parametric quadratic programming (QP) technique to solve the support vector machine (SVM) training problem. This framework, can be specialized to obtain two SVM optimization methods. The first solves the fixed bias problem, while the second starts with an optimal solution for a fixed bias problem and adjusts the bias until the optimal value is found. The later method can be applied in conjunction with any other existing technique which obtains a fixed bias solution. Moreover, the second method can also be used independently to solve the complete SVM training problem. A combination of these two methods is more flexible than each individual method and, among other things, produces an incremental algorithm which exactly solve the 1-Norm Soft Margin SVM optimization problem. Applying Selective Sampling techniques may further boost convergence. 1 Introduction SVM training is a convex optimization problem which scales with the training set size rather than the input dimension. While this is usually considered to be a desired quality, in large scale problems it may cause training to be impractical. The common way to handle massive data applications is to turn to active set methods, which gradually build the set of active constraints by feeding a generic optimizer with small scale sub-problems. Active set methods guarantee to converge to the global solution, however, convergence may be very slow, it may require too many passes over the data set, and at each iteration there's an implicit computational overhead of the actual active set selection. By using some heuristics and caching mechanisms, one can, in practice, reduce this load significantly. Another common practice is to modify the SVM optimization problem such that it wont handle the bias term directly. Instead, the bias is either fixed in advance! (e.g. [6]) or added as another dimension to the feature space (e.g. [4]). The advantage is that the resulting dual optimization problem does not contain the linear constraint, in which case one can suggest a procedure which updates only IThroughout this sequel we will refer to such solution as the fixed bias solution. one Lagrange multiplier at a time. Thus, an incremental approach, which efficiently updates an existing solution given a new training point, can be devised. Though widely used, the solution resulting from this practice has inferior generalization performances and the number of SY tends to be much higher [4]. To the best of our knowledge, the only incremental algorithm suggested so far to exactly solve the 1-Norm Soft Margin2 optimization problem, have been described by Cauwenberghs and Poggio at [3]. This algorithm, handles Adiabatic increments by solving a system of linear equations resulted from a parametric transcription of the KKT conditions. This approach is somewhat close to the one independently developed here and we offer a more thorough comparison in the discussion section. In this paper3 we introduce two new methods derived from parametric QP techniques. The two methods are based on the same framework, which we call Parametric Optimization for Kernel methods (POKER), and are essentially the same methodology applied to somewhat different problems. The first method solves the fixed bias problem, while the second one starts with an optimal solution for a fixed bias problem and adjusts the bias until the optimal value is found. Each of these methods can be used independently to solve the SYM training problem. The most interesting application, however, is alternating between the two methods to obtain a unique incremental algorithm. We will show how by using this approach we can adjust the optimal solution as more data becomes available, and by applying Selective Sampling techniques we may further boost convergence rate. Both our methods converge after a finite number of iterations. In principle, this number may be exponential in the training set size, n. However, since parametric QP methods are based on the well-known Simplex method for linear programming, a similar behavior is expected: Though in theory the Simplex method is known to have exponential complexity, in practice it hardly ever displays exponential behavior. The per-iteration complexity is expected to be O(nl), where l is the number of active points at that iteration, with the exception of some rare cases in which the complexity is expected to be bounded by O(nl2). 2 Parametric QP for SVM Any optimal solution to the 1-Norm Soft Margin SYM optimization problem must satisfy the Karush-Kuhn-Tucker (KKT) necessary and sufficient conditions: 1 exiSi = 0, i = 1, ... ,n 2 (c exi)~i = 0, i = 1, . . . ,n 3 4 5 T Y ex = 0, -Qex + by + S ~ = -e, ° ~ ex ~ c, S :::: 0, ~:::: 0. (1) 2 A different incremental approach stems from a geometric interpretation of the primal problem: Keerthi et al. [7] were the first to suggest a nearest point batch algorithm and Kowalczyk [8] provided the on-line version. They handled the inseparable with the well-known transformation W ~ (W, .;c~) and b ~ b, which establish the equivalence between the Hard Margin and the 2-Norm Soft Margin optimization problems. Although the i-Norm and the 2-Norm have been shown to yield equivalent generalization properties, it is often observed (cf. [7]) that the former method results in a smaller number of SV. It is obvious by the above transformation that the i-Norm Soft Margin is the most general SVM optimization problem. 3The detailed statements of the algorithms and the supporting lemmas were omitted due to space limitation, and can be found at [5]. where a E Rn is the vector of Lagrange multipliers, b is the bias (scalar) and sand ~ are the n-dimensional vectors of slack and surplus variables, respectively. Y is a vector oflabels, ±1. Q is the label encoded kernel matrix, i.e. Qij = YiyjK(Xi,Xj), e is the vector of all 1 's of length n and c is the penalty associated with errors. If we assume that the value of the bias is fixed to some predefined value b, then condition 3 disappears from the system (1) and condition 4 becomes -Qa + S ~ = -e - by (2) Consider the following modified parametric system of KKT conditions aiSi = 0, i = 1, ... ,n (c ai)~i = 0, i = 1, ... ,n -Qa + S ~ = p + u( -e - yb - p) , o ::::: a ::::: c, S ~ 0, ~ ~ 0, (3) for some vector p. It is easy to find p, a S and ~ satisfying (3) for u = O. For example, one may pick a = 0, S = e, ~ = 0 and p = -Qa + s. For u = 1 the systems (3) reduces to the fixed bias system. Our fixed bias method starts at a solution to (3) for u = 0 and by increasing u while updating a, s and ~ so that they satisfy (3), obtains the optimal solution for u = 1. Similarly we can obtain solution to (1) by starting at a fixed bias solution and update b, while maintaining a, s and ~ feasible for (2), until the optimal value for b is reached. The optimal value of the bias is recognized when the corresponding solution satisfy (1), namely aT y = O. Both these methods are based on the same framework of adjusting a scalar parameter in the right hand side of a KKT system. In the next section we will present the method for adjusting the bias (adjusting u in (3) is very similar, save for a few technical differences). An advantage of this special case is that it solves the original problem and can, in principal, be applied "from scratch" . 3 Correcting a "Fixed Bias" Solution Let (a(b), s(b), ~(b)) be a fixed bias solution for a given b. The algorithm that we present here is based on increasing (or decreasing) b monotonically, until the optimal b* is found, while updating and maintaining (a(b),s(b),~(b)). Let us introduce some notation. For a given b and and a fixed bias solution, (a(b), s(b), ~(b)), we partition the index set I = {I, ... , n} into three sets 10 (b), Ie(b) and Is(b) in the following way: Vi E Io(b) si(b) > 0 and ai(b) = 0, Vi E Ie(b) ~i(b) > 0 and ai(b) = c and Vi E Is(b) si(b) = ~i(b) = 0 and 0::::: ai(b) ::::: c. It is easy to see that Io(b)Ule(b)UIs(b) = I and Io(b)nle(b) = Ie(b)nIs(b) = Io(b)nIs(b) = 0. We will call the partition (Io(b),Ie(b),Is(b)) - the optimal partition for a given b. We will refer to Is as the active set. Based on partition (Io,Ie,Is) we define Qss (Qes Qse Qee, Qos, Qoo) as the submatrix of Q whose columns are the columns of Q indexed by the set Is (Ie, Is, Ie, 10 , 10 ) and whose rows are the rows of Q indexed by Is (Is, Ie, Ie, Is , 10). We also define Ys (Ye, Yo) and as (ae , ao) and the subvectors of Y and a whose entries are indexed by Is (Ie, 10). Byes (ee) we denote a vector of all ones of the appropriate size. Assume that we are given an initial guess4 bO < b*. To initiate the algorithm we 4Whether bO < b* can be determined by evaluating -y T a(bO): if -y T a(bO) > 0 then bO < b*, otherwise bO > b*, in which case the algorithm is essentially the same, save for obvious changes. assume that we know the optimal partition (Ioo'!eo,Iso) = (Io(bO),!c(bO),!s(bO)) that corresponds to aO = a(bO). We know that Vi E 10 ai = 0 and Vi E Ie ai = c. We also know that -Qia + Yib = -1, Vi E Is (here Qi is the i-th row of Q). We can write the set of active constraints as (4) If Qss is nonsingular (the nondegenerate case), then as depends linearly on scalar b. Similarly, we can express So and ~e as linear functions of b. If Q ss is singular (the degenerate case), then, the set of all possible solutions as changes linearly with b as long as the partition remains optimal. In either case, if 0 < as < c, So > 0 and ~e > 0 then sufficiently small changes in b preserve these constraints. At each iteration b can increase until one of the four types of inequality constraints becomes active. Then, the optimal partition is updated, new linear expressions of the active variables through b are computed, and the algorithm iterates. We terminate when T Y a < 0, that is b > b*. The final iteration gives us the correct optimal active set and optimal partition; from that we can easily compute b* and a*. A geometric interpretation of the algorithmic steps suggest that we are trying to move the separating hyperplane by increasing its bias and at the same time adjusting its orientation so it stays optimal for the current bias. At each iteration we move the hyperplane until either a support vector is dropped from the support set, a support vector becomes violated, a violated point becomes a support vector or an inactive point joins the support vector set. The algorithm is guaranteed to terminate after finitely many iterations. At each iteration the algorithm covers an interval that corresponds to an optimal partition. The same partition cannot correspond to two different intervals and the number of partitions is finite, hence so is the number of iterations (d. [1, 9]). Per-iteration complexity depends on whether an iteration is degenerate or not. A nondegenerate iteration takes O(niIs I) + O(IIs 13 ) arithmetic operations, while a degenerate iteration should in theory take 0(n21Is 12) operations, but in practice it only takes5 0(nIIsI2). Note that the degeneracy occurs when the active support vectors are linearly dependent. The larger is the rank of the kernel matrix the less likely is such a situation. The storage requirement of the algorithm is O(n) + 0(IIsI2). 4 Incremental Algorithm Incremental and on-line algorithms are aimed at training problems for which the data becomes available in the course of training. Such an algorithm, when given an optimal solution for a training set of size n, and additional m training points, has to efficiently find the optimal solution to the extended n + m training set. Assume we have an optimal solution (a, b, s,~) for a given data set X of size n. For each new point that is added, we take the following actions: a new Lagrange multiplier a n+l = 0 is added to the set of multipliers, then the distance to the margin is evaluated for this point. If the point is not violated, that is if Sn+l = W T xn+l_yn+1b_1 > 0, then the new positive slack Sn+l is added to the set of slack variables. If the point is violated then sn+1 = 1 is added to the set of slack variables. (Notice, that at this point the condition w T x n+1 + yn+1b + sn+1 = -1 is violated.) A surplus variable ~n+l = 0 is also added to the set of surplus variables. The optimal partition is adjusted accordingly. The process is repeated for all the points that have to be added at the given step. If no violated points were encountered, 5This assumes solving such a problem by an interior point method o 1 2 3 4 Given dataset <X,y>, asolution(oo , bo , so,~o) , and new points <x,y>~t~ S b t 0 1 b n+i ( n+i )T et p = - e y, On+i = <,n+i = , Sn+i = y + x w, i = 1, ... , m n+i T If Sn+i ::::: 0, Set pn+i := -(x ) w + 1, Sn+i = 1 Else pn+i := -1 - byn+i X := XU {xn+l , ... , xn+m}, y := (yl , ... , yn, yn+l , ... , yn+m) If p #- - e - by Call POKERfixedbias(X, y , 0 , b, s, ~ , p) Call POKERadjustbias (X, y, 0, b, s, ~) 5 If there are more data points go to O. Figure 1: Outline of the incremental algorithm (AltPOKER) then no further action is necessary. The current solution is optimal and the bias is unchanged. If at least one point is violated, then the new set (Q, b, s,~) is not feasible for the KKT system (1) with the extended data set. However, it is easy to find p such that (Q, b, s, ~) is optimal for (3). Thus we can first apply the fixed bias algorithm to find a new solution and then apply the adjustable bias algorithm to find the optimal solution to the new extended problem (see Figure 1). In theory adding even one point may force the algorithm to work as hard as if it were solving the problem "from scratch". But in practice it virtually never happens. In our experiments, just a few iterations of the fixed bias and adjustable bias algorithms were sufficient to find the solution to the extended problem. Overall, the computational complexity ofthe incremental algorithm is expected to be O(n2 ) . 5 Experiments Convergence in Batch Mode: The most straight-forward way to activate POKER in a batch mode is to construct the trivial partition6 and then apply the adjustable bias algorithm to get the optimal solution. We term this method SelfInit POKER. Note that the initial value of the bias is most likely far away from the global solution, and as such, the results presented here should be regarded as a lower bound. We examined performances on a moderate size problem, the Abalone data set from the VCI Repository [2]. We fed the training algorithm with increasing subsets up to the whole set (of size 4177). The gender encoding (male/female/infant) was mapped into {(I,O,O),(O,I,O),(O,O,I)}. Then, the data was scaled to lie in the [-1,1] interval. We demonstrate convergence for polynomial kernel with increasing degree, which in this setting corresponds to level of difficulty. However naive our implementation is, one can observe (see Figure 2) a linear convergence rate in the batch mode. Convergence in Incremental Mode: AltPOKER is the incremental algorithm described in section 4. We examined the performance on the" diabetes" problem 7 that have been used by Cauwenberghs and Poggio in [3] to test the performance of their algorithm. We demonstrate convergence for the RBF kernel with increasing penalty ("C"). Figure 3 demonstrates the advantage of the more flexible approach 6Fixing the bias term to be large enough (positive or negative) and the Lagrange multipliers to 0 or C based on their class (negative/positive) membership. 7 available at http://bach. ece.jhu. edu/pub/ gert/svm/incremental Selflnil POKER: No. ollleralions VS. Problem Size 16000 _ ,near erne 2000 _ POJyKemel:(<>:.y>+1r • POJyKemel:(<>:.y>+1r POJy Kemel:(<>:.y>+1t POJyKemel:(<>:.y>+1r ProblernSize AUPOKER: No. ol lleralions VS. Chunk Size 2500' Chunk Size C:O.l C"l C,,10 C,,25 C,,50 C,,75 C"l00 Figure 2: SelfInit POKER - Convergence Figure 3: AltPOKER - Convergence in in Batch mode Incremental mode which allows various increment sizes: using increments of only one point resulted in a performance of a similar scale as that of Cauwenberghs and Poggio, but with the increase of the chunk sizes we observe rapid improvement in the convergence rate. Selective Sampling: We can use the incremental algorithm even in case when all the data is available in advance to improve the overall efficiency. If one can select a good representative small subset of the data set, then one can use it for training, hoping that the majority of the data points are classified correctly using the initial sampled data8 . We applied selective sampling as a preprocess in incremental mode: At each meta-iteration, we ranked the points according to a predefined selection criterion, and then picked just the top ones for the increment. The following selection criteria have been used in our experiments: CIs2W picks the closest point to the current hyperplane. This approach is inspired by active learning schemes which strive to halve the version space. However, the notion of a version space is more complex when the problem is inseparable. Thus, it is reasonable to adapt a greedy approach which selects the point that will cause the larger change in the value of the objective function. While solving the optimization problem for all possible increments is impracticable, it may still worthwhile to approximate the potential change: MaxSlk picks the most violating point. This corresponds to an upper bound estimate of the change in the objective, since the value of the slack (times c) is an upper bound to the feasibility gap. dObj perform only few iterations of the adjustable bias algorithm and examine the change in the objective value. This is similar to Strong Branching technique which is used in branch and bound methods for integer programming. Here it provides a lower bound estimate to the change in the objective value. Although performing only few iterations is much cheaper than converging to the optimal solution, this technique is still more demanding then previous selection methods. Hence we first ranked the points using CIs2W (MaxSlk) and then applied dObj only to the top few. Table 1 presents the application of the above mentioned criteria to three different problems. The results clearly shows that advantage of using the information obtained by dObj estimate. 8This is different from a full-fledged Active Learning scheme in which the data is not labeled, but rather queried at selected points. Selection a· I Is I Ie I 10 a· I Is I Ie I 10 II a· I Is I Ie I 10 Criteria 400 I 4 I 11 I 9985 8 I 73 I 1 I 277 II 40 I 20 I 313 I 243 No Selection 234 871 3078 MaxSlk 112 303 3860 MaxSlk+dObj 92 269 3184 ClsW 128 433 2576 ClsW+dObj 116 407 2218 Table 1: The impact of Selective Sampling on the No. of iterations of AltPOKER: Synthetic data (10Kx2), "ionosophere" [2] and "diabetes" (columns ordered resp.) 6 Conclusions and Discussion We propose a new finitely convergent method that can be applied in both batch and incremental modes to solve the 1-Norm Soft Margin SVM problem. Assuming that the number of support vectors is small compared to the size of the data, the method is expected to perform O(n2 ) arithmetic operations, where n is the problem size. Applying Selective Sampling techniques may further boost convergence and reduce computation load. Our method is independently developed, but somewhat similar to that in [3]. Our method, however, is more general - it can be applied to solve fixed bias problems as well as obtain optimal bias from a given fixed bias solution; It is not restricted to increments of size one, but rather can handle increments of arbitrary size; And, it can be used to get an estimate of the drop in the value of the objective function, which is a useful selective sampling criterion. Finally, it is possible to extend this method to produce a true on-line algorithm, by assuming certain properties of the data. This re-introduces some very important applications of the on-line technology, such as active learning, and various forms of adaptation. Pursuing this direction with a special emphasis on massive data applications (e.g. speech related applications), is left for further study. References [1] A. B. Berkelaar, B. Jansen, K. Roos, and T. Terlaky. Sensitivity analysis in (degenerate) quadratic programming. Technical Report 96-26, Delft University, 1996. [2] C. L. Blake and C. J Merz. UCI repository of machine learning databases, 1998. [3] G. Cauwenberghs and T . Poggio. Incremental and decremental support vector machine learning. In Adv. in Neural Information Processing Systems 13, pages 409- 415, 2001. [4] N. Cristianini and J. Shawe-Taylor. An Introductin to Support Vector Macines and Other Kernel-Based Learning Methods. Cambridge University Press, 2000. [5] S. Fine and K. Scheinberg. Poker: Parametric optimization framework for kernel methods. Technical report, IBM T. J. Watson Research Center, 2001. Submitted. [6] T. T. Friess, N. Cristianini, and C. Campbell. The kernel-adaraton algorithm: A fast simple learning procedure for SVM. In Pmc. of 15th ICML, pages 188- 196, 1998. [7] S. S. Keerthi, S. K. Shevade, C. Bhattacharyya, and K. R. K. Murthy. A fast iterative nearest point algorithm for SVM classifier design. IEEE Trnas. NN, 11:124- 36, 2000. [8] A. Kowalczyk. Maximal margin perceptron. In Advances in Large Margin Classifiers, pages 75-113. MIT Press, 2000. [9] R. T. Rockafellar. Conjugate Duality and Optimization. SIAM, Philadelphia, 1974.
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Bayesian Predictive Profiles with Applications to Retail Transaction Data Igor V. Cadez Information and Computer Science University of California Irvine, CA 92697-3425, U.S.A. icadez@ics.uci.edu Padhraic Smyth Information and Computer Science University of California Irvine, CA 92697-3425, U.S.A. smyth@ics.uci.edu Abstract Massive transaction data sets are recorded in a routine manner in telecommunications, retail commerce, and Web site management. In this paper we address the problem of inferring predictive individual profiles from such historical transaction data. We describe a generative mixture model for count data and use an an approximate Bayesian estimation framework that effectively combines an individual’s specific history with more general population patterns. We use a large real-world retail transaction data set to illustrate how these profiles consistently outperform non-mixture and non-Bayesian techniques in predicting customer behavior in out-of-sample data. 1 Introduction Transaction data sets consist of records of pairs of individuals and events, e.g., items purchased (market basket data), telephone calls made (call records), or Web pages visited (from Web logs). Of significant practical interest in many applications is the ability to derive individual-specific (or personalized) models for each individual from the historical transaction data, e.g., for exploratory analysis, adaptive personalization, and forecasting. In this paper we propose a generative model based on mixture models and Bayesian estimation for learning predictive profiles. The mixture model is used to address the heterogeneity problem: different individuals purchase combinations of products on different visits to the store. The Bayesian estimation framework is used to address the fact that we have different amounts of data for different individuals. For an individual with very few transactions (e.g., only one) we can “shrink” our predictive profile for that individual towards a general population profile. On the other hand, for an individual with many transactions, their predictive model can be more individualized. Our goal is an accurate and computationally efficient modeling framework that smoothly adapts a profile to each individual based on both their own historical data as well as general population patterns. Due to space limitations only selected results are presented here; for a complete description of the methodology and experiments see Cadez et al. (2001). The idea of using mixture models as a flexible approach for modeling discrete and categorical data has been known for many years, e.g., in the social sciences for latent class analysis (Lazarsfeld and Henry, 1968). Traditionally these methods were only applied to relatively small low-dimensional data sets. More recently there has been a resurgence of interest in mixtures of multinomials and mixtures of conditionally independent Bernoulli models for modeling high-dimensional document-term data in text analysis (e.g., McCallum, 1999; Hoffman, 1999). The work of Heckerman et al. (2000) on probabilistic model-based collaborative filtering is also similar in spirit to the approach described in this paper except that we focus on explicitly extracting individual-level profiles rather than global models (i.e., we have explicit models for each individual in our framework). Our work can be viewed as being an extension of this broad family of probabilistic modeling ideas to the specific case of transaction data, where we deal directly with the problem of making inferences about specific individuals and handling multiple transactions per individual. Other approaches have also been proposed in the data mining literature for clustering and exploratory analysis of transaction data, but typically in a non-probabilistic framework (e.g., Agrawal, Imielinski, and Swami, 1993; Strehl and Ghosh, 2000; Lawrence et al., 2001). The lack of a clear probabilistic semantics (e.g., for association rule techniques) can make it difficult for these models to fully leverage the data for individual-level forecasting. 2 Mixture-Basis Models for Profiles We have an observed data set D = {D1, . . . , DN}, where Di is the observed data on the ith customer, 1 ≤i ≤N. Each individual data set Di consists of one or more transactions for that customer , i.e., Di = {yi1, . . . , yij, . . . , yini}, where yij is the jth transaction for customer i and ni is the total number of transactions observed for customer i. The jth transaction for individual i, yij, consists of a description of the set of products (or a “market basket”) that was purchased at a specific time by customer i (and yi will be used to denote an arbitrary transaction from individual i). For the purposes of the experiments described in this paper, each individual transaction yij is represented as a vector of d counts yij = (nij1, . . . nijc, . . . , nijC), where nijc indicates how many items of type c are in transaction yij, 1 ≤c ≤C. We define a predictive profile as a probabilistic model p(yi), i.e., a probability distribution on the items that individual i will purchase during a store-visit. We propose a simple generative mixture model for an individual’s purchasing behavior, namely that a randomly selected transaction yi from individual i is generated by one of K components in a K-component mixture model. The kth mixture component, 1 ≤k ≤K is a specific model for generating the counts and we can think of each of the K models as “basis functions” describing prototype transactions. For example, in a clothing store, one might have a mixture component that acts as a prototype for suit-buying behavior, where the expected counts for items such as suits, ties, shirts, etc., given this component, would be relatively higher than for the other items. There are several modeling choices for the component transaction models for generating item counts. In this paper we choose a particularly simple memoryless multinomial model that operates as follows. Conditioned on nij (the total number of items in the basket) each of the individual items is selected in a memoryless fashion by nij draws from a multinomial distribution Pk = (θk1, . . . , θkC) on the C possible items, θkj = 1. 0 10 20 30 40 50 0 0.2 0.4 0.6 COMPONENT 1 Probability 0 10 20 30 40 50 0 0.2 0.4 0.6 COMPONENT 2 0 10 20 30 40 50 0 0.2 0.4 0.6 COMPONENT 3 Probability 0 10 20 30 40 50 0 0.2 0.4 0.6 COMPONENT 4 0 10 20 30 40 50 0 0.2 0.4 0.6 COMPONENT 5 Department Probability 0 10 20 30 40 50 0 0.2 0.4 0.6 COMPONENT 6 Department Figure 1: An example of 6 “basis” mixture components fit to retail transaction data. Figure 1 shows an example of K = 6 such basis mixture components that have been learned from a large retail transaction data (more details on learning will be discussed below). Each window shows a different set of component probabilities Pk, each modeling a different type of transaction. The components show a striking bimodal pattern in that the multinomial models appear to involve departments that are either above or below department 25, but there is very little probability mass that crosses over. In fact the models are capturing the fact that departments numbered lower than 25 correspond to men’s clothing and those above 25 correspond to women’s clothing, and that baskets tend to be “tuned” to one set or the other. 2.1 Individual-Specific Weights We further assume that for each individual i there exists a set of K weights, and in the general case these weights are individual-specific, denoted by αi = (αi1, . . . , αiK), where P k αik = 1. Weight αik represents the probability that when individual i enters the store their transactions will be generated by component k. Or, in other words, the αik’s govern individual i’s propensity to engage in “shopping behavior” k (again, there are numerous possible generalizations, such as making the αik’s have dependence over time, that we will not discuss here). The αik’s are in effect the profile coefficients for individual i, relative to the K component models. This idea of individual-specific weights (or profiles) is a key component of our proposed approach. The mixture component models Pk are fixed and shared across all individuals, providing a mechanism for borrowing of strength across individual data. The individual weights are in principle allowed to freely vary for each individual within a K-dimensional simplex. In effect the K weights can be thought as basis coefficients that represent the location of individual i within the space spanned by the K basis functions (the component Pk multinomials). This approach is quite similar in spirit to the recent probabilistic PCA work of Hofmann (1999) on mixture models for text documents, where he proposes a general mixture model framework that represents documents as existing within a K-dimensional simplex of multinomial component models. The model for each individual is an individual-specific mixture model, where the 0 5 10 15 20 25 30 35 40 45 50 0 2 4 6 8 Number of items TRAINING PURCHASES 0 5 10 15 20 25 30 35 40 45 50 0 2 4 6 8 Number of items Department TEST PURCHASES Figure 2: Histograms indicating which products a particular individual purchased, from both the training data and the test data. 0 5 10 15 20 25 30 35 40 45 50 0 0.05 0.1 0.15 0.2 Probability PROFILE FROM GLOBAL WEIGHTS 0 5 10 15 20 25 30 35 40 45 50 0 0.05 0.1 0.15 0.2 Probability SMOOTHED HISTOGRAM PROFILE (MAP) 0 5 10 15 20 25 30 35 40 45 50 0 0.05 0.1 0.15 0.2 Probability PROFILE FROM INDIVIDUAL WEIGHTS Figure 3: Inferred “effective” profiles from global weights, smoothed histograms, and individual-specific weights for the individual whose data was shown in Figure 2. weights are specific to individual i: p(yij) = K X k=1 αikp(yij|k) = K X k=1 αik C Y c=1 θnijc kc . where θkc is the probability that the cth item is purchased given component k and nijc is the number of items of category c purchased by individual i, during transaction ij. As an example of the application of these ideas, in Figure 2 the training data and test data for a particular individual are displayed. Note that there is some predictability from training to test data, although the test data contains (for example) a purchase in department 14 (which was not seen in the training data). Figure 3 plots the effective profiles1 for this particular individual as estimated by three different schemes in our modeling approach: (1) global weights that result in everyone 1We call these “effective profiles” since the predictive model under the mixture assumpbeing assigned the same “generic” profile, i.e., αik = αk, (2) a maximum a posteriori (MAP) technique that smooths each individual’s training histogram with a population-based histogram, and (3) individual weights estimated in a Bayesian fashion that are “tuned” to the individual’s specific behavior. (More details on each of these methods are provided later in the paper; a complete description can be found in Cadez et al. (2001)). One can see in Figure 3 that the global weight profile reflects broad population-based purchasing patterns and is not representative of this individual. The smoothed histogram is somewhat better, but the smoothing parameter has “blurred” the individual’s focus on departments below 25. The individual-weight profile appears to be a better representation of this individual’s behavior and indeed it does provide the best predictive score (of the 3 methods) on the test data in Figure 2. Note that the individual-weights profile in Figure 3 “borrows strength” from the purchases of other similar customers, i.e., it allows for small but non-zero probabilities of the individual making purchases in departments (such as 6 through 9) if he or she has not purchased there in the past. This particular individual’s weights, the αik’s, are (0.00, 0.47, 0.38, 0.00, 0.00.0.15) corresponding to the six component models shown in Figure 1. The most weight is placed on components 2, 3 and 6, which agrees with our intuition given the individual’s training data. 2.2 Learning the Model Parameters The unknown parameters in our model consist of both the parameters of the K multinomials, θkc, 1 ≤k ≤K, 1 ≤c ≤C, and the vectors of individual-specific profile weights αi, 1 ≤i ≤N. We investigate two different approaches to learning individual-specific weights: • Mixture-Based Maximum Likelihood (ML) Weights: We treat the weights αi and component parameters θ as unknown and use expectationmaximization (EM) to learn both simultaneously. Of course we expect this model to overfit given the number of parameters being estimated but we include it nonetheless as a baseline. • Mixture-Based Empirical Bayes (EB) Weights: We first use EM to learn a mixture of K transaction models (ignoring individuals). We then use a second EM algorithm in weight-space to estimate individualspecific weights αi for each individual. The second EM phase uses a fixed empirically-determined prior (a Dirichlet) for the weights. In effect, we are learning how best to represent each individual within the K-dimensional simplex of basis components. The empirical prior uses the marginal weights (α’s) from the first run for the mean of the Dirichlet, and an equivalent sample size of n = 10 transactions is used in the results reported in the paper. In effect, this can be viewed as an approximation to either a fully Bayesian hierarchical estimation or an empirical Bayesian approach (see Cadez et al. (2001) for more detailed discussion). We did not pursue the fully Bayesian or empirical Bayesian approaches for computational reasons since the necessary integrals cannot be evaluated in closed form for this model and numerical methods (such as Markov Chain Monte Carlo) would be impractical given the data sizes involved. We also compare two other approaches for profiling for comparison: (1) Global Mixture Weights: instead of individualized weights we set each individual’s tion is not a multinomial that can be plotted as a bar chart: however, we can approximate it and we are plotting one such approximation here 0 10 20 30 40 50 60 70 80 90 100 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 Number of Mixture Components [k] Negative LogP Score [bits/token] Individualized MAP weights Mixtures: Individualized ML weights Mixtures: Global mixture weights Mixtures: Individualized EB weights Figure 4: Plot of the negative log probability scores per item (predictive entropy) on out-of-sample transactions, for various weight models as a function of the number of mixture components K. weight vector to the marginal weights (αik = αk), and (2) Individualized MAP weights: a non-mixture approach where we use an empirically-determined Dirichlet prior directly on the multinomials, and where the equivalent sample size of this prior was “tuned” on the test set to give optimal performance. This provides an (optimistic) baseline of using multinomial profiles directly, without use of any mixture models. 3 Experimental Results To evaluate our approach we used a real-world transaction data set. The data consists of transactions collected at a chain of retail stores over a two-year period. We analyze the transactions here at the store department level (50 categories of items). We separate the data into two time periods (all transactions are timestamped), with approximately 70% of the data being in the first time period (the training data) and the remainder in the test period data. We train our mixture and weight models on the first period and evaluate our models in terms of their ability to predict transactions that occur in the subsequent out-of-sample test period. The training data contains data on 4339 individuals, 58,866 transactions, and 164,000 items purchased. The test data consists of 4040 individuals, 25,292 transactions, and 69,103 items purchased. Not all individuals in the test data set appear in the training data set (and vice-versa): individuals in the test data set with no training data are assigned a global population model for scoring purposes. To evaluate the predictive power of each model, we calculate the log-probability (“logp scores”) of the transactions as predicted by each model. Higher logp scores mean that the model assigned higher probability to events that actually occurred. Note that the mean negative logp score over a set of transactions, divided by the total number of items, can be interpreted as a predictive entropy term in bits. The lower this entropy term, the less uncertainty in our predictions (bounded below by zero of course, corresponding to zero uncertainty). Figure 4 compares the out-of-sample predictive entropy scores as a function of the −400 −350 −300 −250 −200 −150 −100 −50 0 −400 −350 −300 −250 −200 −150 −100 −50 0 logP, global weights logP, individual weights −100 −95 −90 −85 −80 −75 −70 −65 −60 −55 −50 −100 −95 −90 −85 −80 −75 −70 −65 −60 −55 −50 logP, global weights logP, individual weights Figure 5: Scatter plots of the log probability scores for each individual on out-ofsample transactions, plotting log probability scores for individual weights versus log probability scores for the global weights model. Left: all data, Right: close up. number of mixture components K for the mixture-based ML weights, the mixturebased Global weights (where all individuals are assigned the same marginal mixture weights), the mixture-based Empirical Bayes weights, and the non-mixture MAP histogram method (as a baseline). The mixture-based approaches generally outperform the non-mixture MAP histogram approach (solid line). The ML-based mixture weights start to overfit after about 6 mixture components (as expected). The Global mixture weights and individualized mixture weights improve up to about K = 50 components and then show some evidence of overfitting. The mixture-based individual weights method is systematically the best predictor, providing a 15% decrease in predictive entropy compared to the MAP histogram method, and a roughly 3% decrease compared to non-individualized global mixture weights. Figure 5 shows a more detailed comparison of the difference between individual mixtures and the Global profiles, on a subset of individuals. We can see that the Global profiles are systematically worse than the individual weights model (i.e., most points are above the bisecting line). For individuals with the lowest likelihood (lower left of the left plot) the individual weight model is consistently better: typically lower weight total likelihood individuals are those with more transactions and items. In Cadez et al. (2001) we report more detailed results on both this data set and a second retail data set involving 15 million items and 300,000 individuals. On both data sets the individual-level models were found to be consistently more accurate out-of-sample compared to both non-mixture and non-Bayesian approaches. We also found (empirically) that the time taken for EM to converge is roughly linear as both a function of number of components and the number of transactions (plots are omitted due to lack of space), allowing for example fitting of models with 100 mixture components to approximately 2 million baskets in a few hours. 4 Conclusions In this paper we investigated the use of mixture models and approximate Bayesian estimation for automatically inferring individual-level profiles from transaction data records. On a real-world retail data set the proposed framework consistently outperformed alternative approaches in terms of accuracy of predictions on future unseen customer behavior. Acknowledgements The research described in this paper was supported in part by NSF award IRI9703120. The work of Igor Cadez was supported by a Microsoft Graduate Research Fellowship. References Agrawal, R., Imielenski, T., and Swami, A. (1993) Mining association rules between sets of items in large databases, Proceedings of the ACM SIGMOD Conference on Management of Data (SIGMOD’98), New York: ACM Press, pp. 207–216. Cadez, I. V., Smyth, P., Ip, E., Mannila, H. (2001) Predictive profiles for transaction data using finite mixture models, Technical Report UCI-ICS-01-67, Information and Computer Science, University of California, Irvine (available online at www.datalab.uci.edu. Heckerman, D., Chickering, D. M., Meek, C., Rounthwaite, R., and Kadie, C. (2000) Dependency networks for inference, collaborative filtering, and data visualization. Journal of Machine Learning Research, 1, pp. 49–75. Hoffmann, T. (1999) Probabilistic latent sematic indexing, Proceedings of the ACM SIGIR Conference 1999, New York: ACM Press, 50–57. Lawrence, R.D., Almasi, G.S., Kotlyar, V., Viveros, M.S., Duri, S.S. (2001) Personalization of supermarket product recommendations, Data Mining and Knowledge Discovery, 5 (1/2). Lazarsfeld, P. F. and Henry, N. W. (1968) Latent Structure Analysis, New York: Houghton Mifflin. McCallum, A. (1999) Multi-label text classification with a mixture model trained by EM, in AAAI’99 Workshop on Text Learning. Strehl, A. and J. Ghosh (2000) Value-based customer grouping from large retail datasets, Proc. SPIE Conf. on Data Mining and Knowledge Discovery, SPIE Proc. Vol. 4057, Orlando, pp 33–42.
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A Rational Analysis of Cognitive Control in a Speeded Discrimination Task Michael C. Mozer , Michael D. Colagrosso , David E. Huber Department of Computer Science Department of Psychology Institute of Cognitive Science University of Colorado Boulder, CO 80309 mozer,colagrom,dhuber @colorado.edu Abstract We are interested in the mechanisms by which individuals monitor and adjust their performance of simple cognitive tasks. We model a speeded discrimination task in which individuals are asked to classify a sequence of stimuli (Jones & Braver, 2001). Response conflict arises when one stimulus class is infrequent relative to another, resulting in more errors and slower reaction times for the infrequent class. How do control processes modulate behavior based on the relative class frequencies? We explain performance from a rational perspective that casts the goal of individuals as minimizing a cost that depends both on error rate and reaction time. With two additional assumptions of rationality—that class prior probabilities are accurately estimated and that inference is optimal subject to limitations on rate of information transmission—we obtain a good fit to overall RT and error data, as well as trial-by-trial variations in performance. Consider the following scenario: While driving, you approach an intersection at which the traffic light has already turned yellow, signaling that it is about to turn red. You also notice that a car is approaching you rapidly from behind, with no indication of slowing. Should you stop or speed through the intersection? The decision is difficult due to the presence of two conflicting signals. Such response conflict can be produced in a psychological laboratory as well. For example, Stroop (1935) asked individuals to name the color of ink on which a word is printed. When the words are color names incongruous with the ink color— e.g., “blue” printed in red—reaction times are slower and error rates are higher. We are interested in the control mechanisms underlying performance of high-conflict tasks. Conflict requires individuals to monitor and adjust their behavior, possibly responding more slowly if errors are too frequent. In this paper, we model a speeded discrimination paradigm in which individuals are asked to classify a sequence of stimuli (Jones & Braver, 2001). The stimuli are letters of the alphabet, A–Z, presented in rapid succession. In a choice task, individuals are asked to press one response key if the letter is an X or another response key for any letter other than X (as a shorthand, we will refer to non-X stimuli as Y). In a go/no-go task, individuals are asked to press a response key when X is presented and to make no response otherwise. We address both tasks because they elicit slightly different decision-making behavior. In both tasks, Jones and Braver (2001) manipulated the relative frequency of the X and Y stimuli; the ratio of presentation frequency was either 17:83, 50:50, or 83:17. Response conflict arises when the two stimulus classes are unbalanced in frequency, resulting in more errors and slower reaction times. For example, when X’s are frequent but Y is presented, individuals are predisposed toward producing the X response, and this predisposition must be overcome by the perceptual evidence from the Y. Jones and Braver (2001) also performed an fMRI study of this task and found that anterior cingulate cortex (ACC) becomes activated in situations involving response conflict. Specifically, when one stimulus occurs infrequently relative to the other, event-related fMRI response in the ACC is greater for the low frequency stimulus. Jones and Braver also extended a neural network model of Botvinick, Braver, Barch, Carter, and Cohen (2001) to account for human performance in the two discrimination tasks. The heart of the model is a mechanism that monitors conflict—the posited role of the ACC—and adjusts response biases accordingly. In this paper, we develop a parsimonious alternative account of the role of the ACC and of how control processes modulate behavior when response conflict arises. 1 A RATIONAL ANALYSIS Our account is based on a rational analysis of human cognition, which views cognitive processes as being optimized with respect to certain task-related goals, and being adaptive to the structure of the environment (Anderson, 1990). We make three assumptions of rationality: (1) perceptual inference is optimal but is subject to rate limitations on information transmission, (2) response class prior probabilities are accurately estimated, and (3) the goal of individuals is to minimize a cost that depends both on error rate and reaction time. The heart of our account is an existing probabilistic model that explains a variety of facilitation effects that arise from long-term repetition priming (Colagrosso, in preparation; Mozer, Colagrosso, & Huber, 2000), and more broadly, that addresses changes in the nature of information transmission in neocortex due to experience. We give a brief overview of this model; the details are not essential for the present work. The model posits that neocortex can be characterized by a collection of informationprocessing pathways, and any act of cognition involves coordination among pathways. To model a simple discrimination task, we might suppose a perceptual pathway to map the visual input to a semantic representation, and a response pathway to map the semantic representation to a response. The choice and go/no-go tasks described earlier share a perceptual pathway, but require different response pathways. The model is framed in terms of probability theory: pathway inputs and outputs are random variables and microinference in a pathway is carried out by Bayesian belief revision. To elaborate, consider a pathway whose input at time is a discrete random variable, denoted , which can assume values
corresponding to alternative input states. Similarly, the output of the pathway at time is a discrete random variable, denoted , which can assume values
. For example, the input to the perceptual pathway in the discrimination task is one of visual patterns corresponding to the letters of the alphabet, and the output is one of letter identities. (This model is highly abstract: the visual patterns are enumerated, but the actual pixel patterns are not explicitly represented in the model. Nonetheless, the similarity structure among inputs can be captured, but we skip a discussion of this issue because it is irrelevant for the current work.) To present a particular input alternative, ! , to the model for " time steps, we clamp #$! for %&'(" . The model computes a probability distribution over given , i.e., P #)*+( . Y X(1) (0) ... Y Y Y X X X(T) (T) (3) (2) (2) (1) Y(3) 0 0.2 0.4 0.6 0.8 1 10 20 30 40 50 Probability of responding Reaction time Figure 1: (left panel) basic pathway architecture—a hidden Markov model; (right panel) time course of inference in a pathway A pathway is modeled as a dynamic Bayes network; the minimal version of the model used in the present simulations is simply a hidden Markov model, where the are observations and the are inferred state (see Figure 1, left panel). (In typical usage, an HMM is presented with a sequence of distinct inputs, whereas we maintain the same input for many successive time steps. Further, in typical usage, an HMM transitions through a sequence of distinct hidden states, whereas we attempt to converge with increasing confidence on a single state. Thus, our model captures the time course of information processing for a single event.) To compute P %)*+( , three probability distributions must be specified: (1) P ) * , which characterizes how the pathway output evolves over time, (2) P ) , which characterizes the strength of association between inputs and outputs, and (3) P , the prior distribution over outputs in the absence of any information about the input. The particular values hypothesized for these three distributions embody the knowledge of the model—like weights in a neural networks—and give rise to predictions from the model. To give a sense of how the Mozer et al. (2000) model operates, the right panel of Figure 1 depicts the time course of inference in a single pathway which has 26 input and output alternatives, with one-to-one associations. The solid line in the Figure shows, as a function of time , P )( * , i.e., the probability that a given input will produce its target output. Due to limited association strengths, perceptual evidence must accumulate over many iterations in order for the target to be produced with high probability. The densely dashed line shows the same target probability when the target prior is increased, and the sparsely dashed line shows the target probability when the association strength to the target is increased. Increasing either the prior or the association strength causes the speed-accuracy curve to shift to the left. In our previous work, we proposed a mechanism by which priors and association strengths are altered through experience. 1.1 Model Details The simulations we report in this paper utilize two pathways in cascade. A perceptual pathway maps visual patterns (26 alternatives) to a letter-identity representation (26 alternatives), and a response pathway maps the letter identity to a response. For the choice task, the response pathway has two outputs, corresponding to the two response keys; for the go/no-go task, the response pathway also has two outputs, which are interpreted as “go” and “no go.” The interconnection between the pathways is achieved by copying the output of the perceptual pathway, , to the input of the response pathway, , at each time. The free parameters of the model are mostly task and experience related. Nonetheless, in the current simulations we used the same parameter values as Mozer et al. (2000), with one exception: Because the speeded perceptual discrimination task studied here is quite unlike the tasks studied by Mozer et al., we allowed ourselves to vary the association-strength parameter in the response pathway. This parameter has only a quantitative, not qualitative, influence on predictions of the model. In our simulations, we also use the priming mechanism proposed by Mozer et al. (2000), which we briefly describe. The priors for a pathway are internally represented in a nonnormalized form: the nonnormalized prior for alternative ! is , and the normalized prior is P !( . On each trial, the priming mechanism increases the nonnormalized prior of alternative ! in proportion to its asymptotic activity at final time " , and and all priors undergo exponential decay:
P " ! ) , where
is the strength of priming, and is the decay rate. (The Mozer et al. model also performs priming in the association strengths by a similar rule, which is included in the present simulation although it has a negligible effect on the results here.) This priming mechanism yields priors on average that match the presentation probabilities in the task, e.g., .17 and .83 for the two responses in the 17:83 condition of the Jones and Braver experiment. Consequently, when we report results for overall error rate and reaction time in a condition, we make the assumption of rationality that the model’s priors correspond to the true priors of the environment. Although the model yields the same result when the priming mechanism is used on a trial-by-trial basis to adjust the priors, the explicit assumption of rationality avoids any confusion about the factors responsible for the model’s performance. We use the priming mechanism on a trial-by-trial basis to account for performance conditional on recent trial history, as explained later. 1.2 Control Processes and the Speed-Accuracy Trade Off The response pathway of the model produces a speed-accuracy performance function much like that in Figure 1b. This function characterizes the operation of the pathway, but it does not address the control issue of when in time to initiate a response. A control mechanism might simply choose a threshold in accuracy or in reaction time, but we hypothesize a more general, rational approach in which a response cost is computed, and control mechanisms initiate a response at the point in time when a minimum in cost is attained. When stimulus S is presented and the correct response is R, we posit a cost of responding at time following stimulus onset: %) S R P R ) S (1) This cost involves two terms—the error rate and the reaction time—which are summed, with a weighting factor, , that determines the relative importance of the two terms. We assume that is dependent on task instructions: if individuals are told to make no errors, should be small to emphasize the error rate; if individuals are told to respond quickly and not concern themselves with occasional errors, should be large to emphasize the reaction time. The cost ') S R cannot be computed without knowing the correct response R. Nonetheless, the control mechanism could still compute an expected cost over the alternative responses based on the model’s current estimate of the likelihood of each: E #) S R "!$# P ' !') S %) S !( (2) It is this expected cost that is minimized to determine the appropriate point in time at which to respond. We index by the response R because it is not sensible to assign a time cost to a “no go” response, where no response is produced. Consequently, &%(')(% ; for the “go” response and for the two responses in the choice task, we searched for the parameter that best fit the data, yielding +* . 17:83 50:50 83:17 0 0.2 0.4 0.6 0.8 1 10 20 30 40 50 Probability of responding Reaction time R — R 0 0.2 0.4 0.6 0.8 1 10 20 30 40 50 Probability of responding Reaction time R — R 0 0.2 0.4 0.6 0.8 1 10 20 30 40 50 Probability of responding Reaction time R — R 0.4 0.6 0.8 1 1.2 1.4 10 20 30 40 50 Expected cost Reaction time 0.4 0.6 0.8 1 1.2 1.4 10 20 30 40 50 Expected cost Reaction time 0.4 0.6 0.8 1 1.2 1.4 10 20 30 40 50 Expected cost Reaction time Figure 2: (upper row) Output of response pathway when stimulus S, associated with response R, is presented, and relative frequency of R and the alternative response, R, is 17:83, 50:50, and 83.17. (lower row) Expected cost of responding (Eqn. 2). 2 RESULTS Figure 2 illustrates the model’s performance on the choice task when presented with a stimulus, S, associated with a response, R, and the relative frequency of R and the alternative response, R, is 17:83, 50:50, or 83:17 (left, center, and right columns, respectively). Each graph in the top row plots the probability of R and R against time. Although R wins out asymptotically in all three conditions, it must overcome the effect of its low prior in the 17:83 condition. Each graph in the bottom row plots the expected cost, , over time. In the early part of the cost function, error rate dominates the cost, and in the late part, reaction time dominates. In fact, at long times, the error rate is essentially 0, and the cost grows linearly with reaction time. Our rational analysis suggests that a response should be initiated at the global minimum—indicated by asterisks in the figure—implying that both the reaction time and error rate will decrease as the response prior is increased. Figure 3 presents human and simulation data for the choice task. The data consist of mean reaction time and accuracy for the two target responses, # and , for the three conditions corresponding to different # : presentation ratios. Figure 4 presents human and simulation data for the go/no-go task. Note that reaction times are shown only for the “go” trials, because no response is required for the “no go” trials. For both tasks, the model provides an extremely good fit to the human data. The qualities of the model giving rise to the fit can be inferred by inspection of Figure 2—namely, accuracy is higher and reaction times are faster when a response is expected. Figure 5 reveals how the recent history of experimental trials influences reaction time and error rate in the choice task. The trial context along the x-axis is coded as # , where specifies that trial ! required the same (“S”) or different (“D”) response as trial !( . For example, if the current trial required response X, and the four trials leading up to the current trial were—in forward temporal order—Y, Y, Y, and X, the current trial’s context would be coded as “SSDS.” The correlation coefficient between human and simulation data is .960 for reaction time and .953 for error rate. The model fits the human data extremely well. The simple priming mechanism proposed previously by Mozer et al. (2000), which aims to adapt the model’s priors rapidly to the statistics of the environment, is responsible: On a coarse time scale, the mechanism produces priors in the model that match priors in the environment. On a fine time scale, changes to and decay of the priors results in a strong effect of recent trial history, consistent with the human data: The graphs in Figure 5 show that the fastest and most accurate trials Human Data Simulation 320 340 360 380 400 420 17:83 50:50 83:17 Reaction time R1:R2 frequency R1 R2 26 28 30 17:83 50:50 83:17 Reaction time R1:R2 frequency R1 R2 0.8 0.9 1 17:83 50:50 83:17 Accuracy R1:R2 frequency R1 R2 0.8 0.9 1 17:83 50:50 83:17 Accuracy R1:R2 frequency R1 R2 Figure 3: Human data (left column) and simulation results (right column) for the choice task. Human data from Jones and Braver (2001). The upper and lower rows show mean reaction time and accuracy, respectively, for the two responses ( # and ) in the three conditions corresponding to different # : frequencies. Human Data Simulation 300 320 340 360 380 400 420 17:83 50:50 83:17 Reaction time go:no-go frequency go 24 26 28 30 17:83 50:50 83:17 Reaction time go:no-go frequency go 0.8 0.9 1 17:83 50:50 83:17 Accuracy go:no-go frequency go no-go 0.8 0.9 1 17:83 50:50 83:17 Accuracy go:no-go frequency go no-go Figure 4: Human data (left column) and simulation results (right column) for the go/nogo task. Human data from Jones and Braver (2001). The upper and lower rows show mean reaction time and accuracy, respectively, for the two responses (go and no-go) in the three conditions corresponding to different go:no-go presentation frequencies. are clearly those in which the previous two trials required the same response as the current trial (the leftmost four contexts in each graph). The fit to the data is all the more impressive given that Mozer et al. priming mechanism was used to model perceptual priming, and here the same mechanism is used to model response priming. 3 DISCUSSION We introduced a probabilistic model based on three principles of rationality: (1) perceptual inference is optimal but is subject to rate limitations on information transmission, (2) response class prior probabilities are accurately estimated, and (3) the goal of individuals is to minimize a cost that depends both on error rate and reaction time. The model provides a parsimonious account of the detailed pattern of human data from two speeded discrimination tasks. The heart of the model was proposed previously by Mozer, Colagrosso, and Huber (2000), and in the present work we fit experimental data with only two free parameters, one relating to the rate of information flow, and the other specifying the relative cost of speed and errors. The simplicity and elegance of the model arises from having adopted the rational perspective, which imposes strong constraints on the model and removes arbitrary choices and degrees of freedom that are often present in psychological models. Jones and Braver (2001) proposed a neural network model to address response conflict in a speeded discrimination task. Their model produces an excellent fit to the data too, but -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 SSSS DSSS SDSS DDSS SSDS DSDS SDDS DDDS SSSD DSSD SDSD DDSD SSDD DSDD SDDD DDDD Reaction Time Z-score Sequence of stimuli Human Data Simulation 0 0.05 0.1 0.15 0.2 0.25 SSSS DSSS SDSS DDSS SSDS DSDS SDDS DDDS SSSD DSSD SDSD DDSD SSDD DSDD SDDD DDDD Error Rate Sequence of stimuli Human Data Simulation Figure 5: Reaction time (left curve) and accuracy (right curve) data for humans (solid line) and model (dashed line), contingent on the recent history of experimental trials. involves significantly more machinery, free parameters, and ad hoc assumptions. In brief, their model is an associative net mapping activity from stimulus units to response units. When response units # and both receive significant activation, noise in the system can push the inappropriate response unit over threshold. When this conflict situation is detected, a control mechanism acts to lower the baseline activity of response units, requiring them to build up more evidence before responding and thereby reducing the likelihood of noise determining the response. Their model includes a priming mechanism to facilitate repetition of responses, much as we have in our model. However, their model also includes a secondary priming mechanism to facilitate alternation of responses, which our model does not require. Both models address additional data; for example, a variant of their model predicts a neurophysiological marker of conflict called error-related negativity (Yeung, Botvinick, & Cohen, 2000). Jones and Braver posit that the role of the ACC is conflict detection. Our account makes an alternative proposal—that ACC activity reflects the expected cost of decision making. Both hypotheses are consistent with the fMRI data indicating that the ACC produces a greater response for a low frequency stimulus. We are presently considering further experiments to distinguish these contrasting hypotheses. Acknowledgments We thank Andrew Jones and Todd Braver for generously providing their raw data and for helpful discussions leading to this research. This research was supported by Grant 97–18 from the McDonnell-Pew Program in Cognitive Neuroscience, NSF award IBN–9873492, and NIH/IFOPAL R01 MH61549–01A1. References Botvinick, M. M., Braver, T. S., Barch, D. M., Carter, C. S., & Cohen, J. D. (2001). Evaluating the demand for control: anterior cingulate cortex and conflict monitoring. Submitted for publication. Colagrosso, M. (in preparation). A Bayesian cognitive architecture for analyzing information transmission in neocortex. Ph.D. Dissertation in preparation. Jones, A. D., & Braver, T. S. (2001). Sequential modulations in control: Conflict monitoring and the anterior cingulate cortex. Submitted for publication. Mozer, M. C., Colagrosso, M. D., & Huber, D. H. (2000). A Bayesian Cognitive Architecture for Interpreting Long-Term Priming Phenomena. Presentation at the 41st Annual Meeting of the Psychonomic Society, New Orleans, LA, November 2000. Yeung, N., Botvinick, M. M., & Cohen, J. D. (2000). The neural basis of error detection: Conflict monitoring and the error-related negativity. Submitted for publication.
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Analysis of Sparse Bayesian Learning Anita C. Fanl Michael E. Tipping Microsoft Research St George House, 1 Guildhall St Cambridge CB2 3NH, U.K. Abstract The recent introduction of the 'relevance vector machine' has effectively demonstrated how sparsity may be obtained in generalised linear models within a Bayesian framework. Using a particular form of Gaussian parameter prior, 'learning' is the maximisation, with respect to hyperparameters, of the marginal likelihood of the data. This paper studies the properties of that objective function, and demonstrates that conditioned on an individual hyperparameter, the marginal likelihood has a unique maximum which is computable in closed form. It is further shown that if a derived 'sparsity criterion' is satisfied, this maximum is exactly equivalent to 'pruning' the corresponding parameter from the model. 1 Introduction We consider the approximation, from a training sample, of real-valued functions, a task variously referred to as prediction, regression, interpolation or function approximation. Given a set of data {xn' tn};;=l the 'target' samples tn = f(xn) + En are conventionally considered to be realisations of a deterministic function f, potentially corrupted by some additive noise process. This function f will be modelled by a linearly-weighted sum of M fixed basis functions {4>m (X)}~= l: M f(x) = L wm¢>m(x), (1) m=l and the objective is to infer values of the parameters/weights {Wm}~=l such that f is a 'good' approximation of f. While accuracy in function approximation is generally universally valued, there has been significant recent interest [2, 9, 3, 5]) in the notion of sparsity, a consequence of learning algorithms which set significant numbers of the parameters Wm to zero. A methodology which effectively combines both these measures of merit is that of 'sparse Bayesian learning', briefly reviewed in Section 2, and which was the basis for the recent introduction of the relevance vector machine (RVM) and related models [6, 1, 7]. This model exhibits some very compelling properties, in particular a dramatic degree of sparseness even in the case of highly overcomplete basis sets (M »N). The sparse Bayesian learning algorithm essentially involves the maximisation of a marginalised likelihood function with respect to hyperparameters in the model prior. In the RVM, this was achieved through re-estimation equations, the behaviour of which was not fully understood. In this paper we present further relevant theoretical analysis of the properties of the marginal likelihood which gives a much fuller picture of the nature of the model and its associated learning procedure. This is detailed in Section 3, and we close with a summary of our findings and discussion of their implications in Section 4 (and which, to avoid repetition here, the reader may wish to preview at this point). 2 Sparse Bayesian Learning We now very briefly review the methodology of sparse Bayesian learning, more comprehensively described elsewhere [6]. To simplify and generalise the exposition, we omit to notate any functional dependence on the inputs x and combine quantities defined over the training set and basis set within N- and M-vectors respectively. Using this representation, we first write the generative model as: t = f + €, (2) where t = (t1"'" tN )T, f = (11, ... , fN)T and € = (E1"'" EN)T. The approximator is then written as: f = <I>w, (3) where <I> = [«Pl'" «PM] is a general N x M design matrix with column vectors «Pm and w = (W1, ... ,WM)T. Recall that in the context of (1), <I>nm = ¢m(xn) and f = {f(x1), .. .j(XN)P. The sparse Bayesian framework assumes an independent zero-mean Gaussian noise model, with variance u 2 , giving a multivariate Gaussian likelihood of the target vector t: p(tlw, ( 2) = (27r)-N/2 U-N exp { _lit ~:"2 } . (4) The prior over the parameters is mean-zero Gaussian: M ( a W2) p(wlo:) = (27r)-M/21I a~,e exp m2 m , (5) where the key to the model sparsity is the use of M independent hyperparameters 0: = (a1 " '" aM)T, one per weight (or basis vector), which moderate the strength of the prior. Given 0:, the posterior parameter distribution is Gaussian and given via Bayes' rule as p(wlt, 0:) = N(wIIL,~) with ~ = (A + u- 2<I>T<I» - 1 IL = u - 2~<I>Tt, (6) and A defined as diag(a1, ... ,aM). Sparse Bayesian learning can then be formulated as a type-II maximum likelihood procedure, in that objective is to maximise the marginal likelihood, or equivalently, its logarithm £(0:) with respect to the hyperparameters 0:: £(0:) = logp(tlo:, ( 2) = log i: p(tlw, ( 2) p(wlo:) dw, 1 = -"2 [Nlog27r + log ICI + t T C-1t] , with C = u2I + <I> A - l<I>T. (7) (8) Once most-probable values aMP have been found1 , in practice they can be plugged into (6) to give a posterior mean (most probabletpoint estimate for the parameters J.tMP and from that a mean final approximator: fMP = ()J.tMp· Empirically, the local maximisation of the marginal likelihood (8) with respect to a has been seen to work highly effectively [6, 1, 7]. Accurate predictors may be realised, which are typically highly sparse as a result of the maximising values of many hyperparameters being infinite. From (6) this leads to a parameter posterior infinitely peaked at zero for many weights Wm with the consequence that J.tMP correspondingly comprises very few non-zero elements. However, the learning procedure in [6] relied upon heuristic re-estimation equations for the hyperparameters, the behaviour of which was not well characterised. Also, little was known regarding the properties of (8), the validity of the local maximisation thereof and importantly, and perhaps most interestingly, the conditions under which a-values would become infinite. We now give, through a judicious re-writing of (8), a more detailed analysis of the sparse Bayesian learning procedure. 3 Properties of the Marginal Likelihood £(0:) 3.1 A convenient re-writing We re-write C from (8) in a convenient form to analyse the dependence on a single hyperparameter ai: C = (]"21 + 2..: am¢m¢~' = (]"21 + 2..: a~1¢m¢~ + a-;1¢i¢T, m m # i = C_i + a-;1¢i¢T, (9) where we have defined C-i = (]"21+ Lm#i a;r/¢m¢~ as the covariance matrix with the influence of basis vector ¢i removed, equivalent also to ai = 00. Using established matrix determinant and inverse identities, (9) allows us to write the terms of interest in £( a) as: which gives £(a) = -~ [Nlog(2n) + log IC-il + tTC=;t (¢TC-1t)2 -logai + log(ai + ¢TC=;¢i) . • T-' -1 ], a. + ¢i C_i ¢i 1 (¢TC-1t)2 = £(a-i) + -2 [logai -log(ai + ¢TC=;¢i) + • T-' -1 ] , ai + ¢i C_i ¢i (10) (11) = £( a-i) + £( ai), (12) where £(a-i) is the log marginal likelihood with ai (and thus Wi and ¢i) removed from the model and we have now isolated the terms in ai in the function £(ai). IThe most-probable noise variance (]"~p can also be directly and successfully estimated from the data [6], but for clarity in this paper, we assume without prejudice to our results that its value is fixed. 3.2 First derivatives of £(0:) Previous results. In [6], based on earlier results from [4], the gradient of the marginal likelihood was computed as: (13) with fJi the i-th element of JL and ~ ii the i-th diagonal element of~. This then leads to re-estimation updates for O::i in terms of fJi and ~ii where, disadvantageously, these latter terms are themselves functions of O::i. A new, simplified, expression. In fact, by instead differentiating (12) directly, (13) can be seen to be equivalent to: (14) where advantageously, O::i now occurs only explicitly since C- i is independent of O::i. For convenience, we combine terms and re-write (14) as: o£(o:) _ o::;lS;- (Qr - Si) OO::i 2( O::i + Si)2 (15) where, for simplification of this and forthcoming expressions, we have defined: (16) The term Qi can be interpreted as a 'quality' factor: a measure of how well c/>i increases £(0:) by helping to explain the data, while Si is a 'sparsity' factor which measures how much the inclusion of c/>i serves to decrease £(0:) through 'inflating' C (i. e. adding to the normalising factor). 3.3 Stationary points of £(0:) Equating (15) to zero indicates that stationary points of the marginal likelihood occur both at O::i = +00 (note that, being an inverse variance, O::i must be positive) and for: S2 . _ t O::t - Q; _ Si' (17) subject to Qr > Si as a consequence again of O::i > o. Since the right-hand-side of (17) is independent of O::i, we may find the stationary points of £(O::i) analytically without iterative re-estimation. To find the nature of those stationary points, we consider the second derivatives. 3.4 Second derivatives of £(0:) 3.4.1 With respect to O::i Differentiating (15) a second time with respect to O::i gives: -0::;2S;(O::i + Si)2 - 2(O::i + Si) [o::;lS;- (Qr - Si)] 2(O::i + Si)4 and we now consider (18) for both finite- and infinite-O::i stationary points. (18) Finite 0::. In this case, for stationary points given by (17), we note that the second term in the numerator in (18) is zero, giving: (19) We see that (19) is always negative, and therefore £(O::i) has a maximum, which must be unique, for Q; - Si > ° and O::i given by (17). Infinite 0::. For this case, (18) and indeed, all further derivatives, are uninformatively zero at O::i = 00 , but from (15) we can see that as O::i --+ 00, the sign of the gradient is given by the sign of - (Q; - Si). If Q; - Si > 0, then the gradient at O::i = 00 is negative so as O::i decreases £(O::i) must increase to its unique maximum given by (17). It follows that O::i = 00 is thus a minimum. Conversely, if Q; - Si < 0, O::i = 00 is the unique maximum of £(O::i) . If Q; - Si = 0, then this maximum and that given by (17) coincide. We now have a full characterisation of the marginal likelihood as a function of a single hyperparameter, which is illustrated in Figure 1. u. 10° I 10' Figure 1: Example plots of £(ai) against a i (on a log scale) for Q; > Si (left), showing the single maximum at finite ai, and Q; < Si (right), showing the maximum at a i = 00. 3.4.2 With respect to O::j, j i:- i To obtain the off-diagonal terms of the second derivative (Hessian) matrix, it is convenient to manipulate (15) to express it in terms of C. From (11) we see that and (20) Utilising these identities in (15) gives: (21) We now write: (22) where 6ij is the Kronecker 'delta' function, allowing us to separate out the additional (diagonal) term that appears only when i = j. Writing, similarly to (9) earlier, C = C_j + ajl¢j¢j, substituting into (21) and differentiating with respect to aj gives: while we have (24) If all hyperparameters ai are individually set to their maximising values, i. e. a = aMP such that alI8£(a)/8ai = 0, then even if all 82£(a)/8a; are negative, there may still be a non-axial direction in which the likelihood could be increasing. We now rule out this possibility by showing that the Hessian is negative semi-definite. First, we note from (21) that if 8£(a)/8ai = 0, 'V;i = 0. Then, if v is a generic nonzero direction vector: < (25) where we use the Cauchy-Schwarz inequality. If the gradient vanishes, then for all i = 1, ... , M either ai = 00, or from (21), ¢rC-1¢i = (¢rC-1t)2. It follows directly from (25) that the Hessian is negative semi-definite, with (25) only zero where v is orthogonal to all finite a values. 4 Summary Sparse Bayesian learning proposes the iterative maximisation of the marginal likelihood function £(a) with respect to the hyperparameters a. Our analysis has shown the following: 1. As a function of an individual hyperparameter ai, £( a) has a unique maximum computable in closed-form. (This maximum is, of course, dependent on the values of all other hyperparameters.) II. If the criterion Qr - Si (defined in Section 3.2) is negative, this maximum occurs at O:i = 00, equivalent to the removal of basis function i from the model. III. The point where all individual marginal likelihood functions £(O:i) are maximised is a joint maximum (not necessarily unique) over all O:i. These results imply the following consequences. • From I, we see that if we update, in any arbitrary order, the O:i parameters using (17), we are guaranteed to increase the marginal likelihood at each step, unless already at a maximum. Furthermore, we would expect these updates to be more efficient than those given in [6], which individually only increase, not maximise, £ (O:i) . • Result III indicates that sequential optimisation of individual O:i cannot lead to a stationary point from which a joint maximisation over all 0: may have escaped. (i.e. the stationary point is not a saddle point.) • The result II confirms the qualitative argument and empirical observation that many O:i -+ 00 as a result of the optimisation procedure in [6]. The inevitable implication of finite numerical precision prevented the genuine sparsity of the model being verified in those earlier simulations. • We conclude by noting that the maximising hyperparameter solution (17) remains valid if O:i is already infinite. This means that basis functions not even in the model can be assessed and their corresponding hyperparameters updated if desired. So as well as the facility to increase £(0:) through the 'pruning' of basis functions if Qr - Si ::::: 0, new basis functions can be introduced if O:i = 00 but Qr - Si > O. This has highly desirable computational consequences which we are exploiting to obtain a powerful 'constructive' approximation algorithm [8]. References [1] C. M. Bishop and M. E. Tipping. Variational relevance vector machines. In C. Boutilier and M. Goldszmidt, editors, Proceedings of the 16th Conference on Uncertainty in Artificial Intelligence, pages 46- 53. Morgan Kaufmann, 2000. [2] S. Chen, D. L. Donoho, and M. A. Saunders. Atomic decomposition by basis pursuit. Technical Report 479, Department of Statistics, Stanford University, 1995. [3] Y. Grandvalet. Least absolute shrinkage is equivalent to quadratic penalisation. In L. Niklasson, M. Boden, and T. Ziemske, editors, Proceedings of the Eighth International Conference on Artificial Neural Networks (ICANN98), pages 201- 206. Springer, 1998. [4] D. J. C. MacKay. Bayesian interpolation. Neural Computation, 4(3):415- 447, 1992. [5] A. J. Smola, B. Scholkopf, and G. Ratsch. Linear programs for automatic accuracy control in regression. In Proceedings of the Ninth International Conference on Artificial Neural Networks, pages 575- 580, 1999. [6] M. E. Tipping. The Relevance Vector Machine. In S. A. Solla, T . K. Leen, and K.-R. Muller, editors, Advances in Neural Information Processing Systems 12, pages 652- 658. MIT Press, 2000. [7] M. E. Tipping. Sparse kernel principal component analysis. In Advances in Neural Information Processing Systems 13. MIT Press, 200l. [8] M. E. Tipping and A. C. Faul. Bayesian pursuit. Submitted to NIPS*Ol. [9] V. N. Vapnik. Statistical Learning Theory. Wiley, 1998.
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Learning spike-based correlations and conditional probabilities in silicon Aaron P. Shon David Hsu Chris Diorio Department of Computer Science and Engineering University of Washington Seattle, WA 98195-2350 USA {aaron, hsud, diorio}@cs.washington.edu Abstract We have designed and fabricated a VLSI synapse that can learn a conditional probability or correlation between spike-based inputs and feedback signals. The synapse is low power, compact, provides nonvolatile weight storage, and can perform simultaneous multiplication and adaptation. We can calibrate arrays of synapses to ensure uniform adaptation characteristics. Finally, adaptation in our synapse does not necessarily depend on the signals used for computation. Consequently, our synapse can implement learning rules that correlate past and present synaptic activity. We provide analysis and experimental chip results demonstrating the operation in learning and calibration mode, and show how to use our synapse to implement various learning rules in silicon. 1 Introduction Computation with conditional probabilities and correlations underlies many models of neurally inspired information processing. For example, in the sequence-learning neural network models proposed by Levy [1], synapses store the log conditional probability that a presynaptic spike occurred given that the postsynaptic neuron spiked sometime later. Boltzmann machine synapses learn the difference between the correlations of pairs of neurons in the sleep and wake phase [2]. In most neural models, computation and adaptation occurs at the synaptic level. Hence, a silicon synapse that can learn conditional probabilities or correlations between pre- and post-synaptic signals can be a key part of many silicon neural-learning architectures. We have designed and implemented a silicon synapse, in a 0.35µm CMOS process, that learns a synaptic weight that corresponds to the conditional probability or correlation between binary input and feedback signals. This circuit utilizes floating-gate transistors to provide both nonvolatile storage and weight adaptation mechanisms [3]. In addition, the circuit is compact, low power, and provides simultaneous adaptation and computation. Our circuit improves upon previous implementations of floating-gate based learning synapses [3,4,5] in several ways. First, our synapse appears to be the first spike-based floating-gate synapse that implements a general learning principle, rather than a particular learning rule [4,5]. We demon strate that our synapse can learn either the conditional probability or the correlation between input and feedback signals. Consequently, we can implement a wide range of synaptic learning networks with our circuit. Second, unlike the general correlational learning synapse proposed by Hasler et. al. [3], our synapse can implement learning rules that correlate pre- and postsynaptic activity that occur at different times. Learning algorithms that employ time-separated correlations include both temporal difference learning [6] and recently postulated temporally asymmetric Hebbian learning [7]. Hasler’s correlational floating-gate synapse can only perform updates based on the present input and feedback signals, and is therefore unsuitable for learning rules that correlate signals that occur at different times. Because signals that control adaptation and computation in our synapse are separate, our circuit can implement these time-dependent learning rules. Finally, we can calibrate our synapses to remove mismatch between the adaptation mechanisms of individual synapses. Mismatch between the same adaptation mechanisms on different floating-gate transistors limits the accuracy of learning rules based on these devices. This problem has been noted in previous circuits that use floating-gate adaptation [4,8]. In our circuit, different synapses can learn widely divergent weights from the same inputs because of component mismatch. We provide a calibration mechanism that enables identical adaptation across multiple synapses despite device mismatch. To our knowledge, this circuit is the first instance of a floating-gate learning circuit that includes this feature. This paper is organized as follows. First, we provide a brief introduction to floating-gate transistors. Next, we provide a description and analysis of our synapse, demonstrating that it can learn the conditional probability or correlation between a pair of binary signals. We then describe the calibration circuitry and show its effectiveness in compensating for adaptation mismatches. Finally, we discuss how this synapse can be used for silicon implementations of various learning networks. 2 Floating-gate transistors Because our circuit relies on floating-gate transistors to achieve adaptation, we begin by briefly discussing these devices. A floating-gate transistor (e.g. transistor M3 of Fig.1(a)) comprises a MOSFET whose gate is isolated on all sides by SiO2. A control gate capacitively couples signals to the floating gate. Charge stored on the floating gate implements a nonvolatile analog weight; the transistor’s output current varies with both the floating-gate voltage and the control-gate voltage. We use Fowler-Nordheim tunneling [9] to increase the floating-gate charge, and impact-ionized hot-electron injection (IHEI) [10] to decrease the floating-gate charge. We tunnel by placing a high voltage on a tunneling implant, denoted by the arrow in Fig.1(a). We inject by imposing more than about 3V across the drain and source of transistor M3. The circuit allows simultaneous adaptation and computation, because neither tunneling nor IHEI interfere with circuit operation. Over a wide range of tunneling voltages Vtun, we can approximate the magnitude of the tunneling current Itun as [4]: ( ) 0 exp / tun tun tun fg I I V V Vχ = − (1) where Vtun is the tunneling-implant voltage, Vfg is the floating-gate voltage, and Itun0 and Vχ are fit constants. Over a wide range of transistor drain and source voltages, we can approximate the magnitude of the injection current Iinj as [4]: ( ) 1 / 0 exp ( )/ t U V inj inj s s d I I I V V V γ γ − = − (2) where Vs and Vd are the drain and source voltages, Iinj0 is a pre-exponential current, Vγ is a constant that depends on the VLSI process, and Ut is the thermal voltage kT/q. 3 The silicon synapse We show our silicon synapse in Fig.1. The synapse stores an analog weight W, multiplies W by a binary input Xin, and adapts W to either a conditional probability P(Xcor|Y) or a correlation P(XcorY). Xin is analogous to a presynaptic input, while Y is analogous to a postsynaptic signal or error feedback. Xcor is a presynaptic adaptation signal, and typically has some relationship with Xin. We can implement different learning rules by altering the relationship between Xcor and Xin. For some examples, see section 4. We now describe the circuit in more detail. The drain current of floating-gate transistor M4 represents the weight value W. Because the control gate of M4 is fixed, W depends solely on the charge on floating-gate capacitor C1. We can switch the drain current on or off using transistor M7; this switching action corresponds to a multiplication of the weight value W by a binary input signal, Xin. We choose values for the drain voltage of the M4 to prevent injection. A second floating-gate transistor M3, whose gate is also connected to C1, controls adaptation by injection and tunneling. Simultaneously high input signals Xcor and Y cause injection, increasing the weight. A high Vtun causes tunneling, decreasing the weight. We either choose to correlate a high Vtun with signal Y or provide a fixed high Vtun throughout the adaptation process. The choice determines whether the circuit learns a conditional probability or a correlation, respectively. Because the drain current sourced by M4 provides is the weight W, we can express W in terms of M4’s floating-gate voltage, Vfg. Vfg includes the effects of both the fixed controlgate voltage and the variable floating-gate charge. The expression differs depending on whether the readout transistor is operating in the subthreshold or above-threshold regime. We provide both expressions below: 2 0 2 2 0 exp( /(1 ) ) below threshold above threshold (1 ) fg t fg I V U W V V κ κ κ β κ − + = − + (3) Here V0 is a constant that depends on the threshold voltage and on Vdd, Ut is the thermal voltage kT/q, κ is the floating-gate-to-channel coupling coefficient, and I0 is a fixed bias current. Eq. 3 shows that W depends solely on Vfg, (all the other factors are constants). These equations differ slightly from standard equations for the source current through a transistor due to source degeneration caused by M4. This degeneration smoothes the nonlinear relationship between Vfg and Is; its addition to the circuit is optional. 3.1 Weight adaptation Because W depends on Vfg, we can control W by tunneling or injecting transistor M3. In this section, we show that these mechanisms enable our circuit to learn the correlation or conditional probability between inputs Xcor (which we will refer to as X) and Y. Our analysis assumes that these statistics are fixed over some period during which adaptation occurs. The change in floating-gate voltage, and hence the weight, discussed below should therefore be interpreted in terms of the expected weight change due to the statistics of the inputs. We discuss learning of conditional probabilities; a slight change in the tunneling signal, described previously, allows us to learn correlations instead. We first derive the injection equation for the floating-gate voltage in terms of the joint probability P(X,Y) by considering the relationship between the input signals and Is, Vs, and Vd of M3. We assume that transistor M1 is in saturation, constraining Is at M3 to be constant. Presentation of a joint binary event (X,Y) closes nFET switches M5 and M6, pulling the drain voltage Vd of M3 to 0V and causing injection. Therefore the probability that Vd is low enough to cause injection is the probability of the joint event Pr(X,Y). By Eq.2, the amount of the injection is also dependent on M3’s source voltage Vs. Because M3 is constrained to a fixed channel current, a drop in the floating-gate voltage, ∆Vfg, causes a drop in Vs of magnitude κ∆Vfg. Substituting these expressions into Eq.2 results in a floating-gate voltage update of: 0 fg ( / ) Pr( , )exp( V / ) fg inj inj dV dt I X Y Vγ κ = − (4) where Iinj0 also includes the constant source current. Eq.4 shows that the floating-gate voltage update due to injection is a function of the probability of the joint event (X,Y). Next we analyze the effects of tunneling on the floating-gate voltage. The origin of the tunneling signal determines whether the synapse is learning a conditional probability or a correlation. If the circuit is learning a conditional probability, occurrence of the conditioning event Y gates a corresponding high-voltage (~9V) signal onto the tunneling implant. Consequently, we can express the change in floating-gate voltage due to tunneling in terms of the probability of Y, and the floating-gate voltage. 0 ( / ) Pr( )exp( / ) fg tun tun fg dV dt I Y V Vχ = − (5) Eq.5 shows that the floating-gate voltage update due to tunneling is a function of the probability of the event Y. 0 0.2 0.4 0.6 0.8 1 2 2.5 3 3.5 o chip data − fit Weq (µA) Pr(X|Y) (c) Xin Vtun M3 M4 M7 W Xcor Y M5 M6 Vb M1 M2 C1 synaptic output (a). Pr(X|Y) Weq (nA) 0 0.2 0.4 0.6 0.8 1 20 40 60 80 o chip data − fit: P(X|Y)0.78 (b) Fig. 1. (a) Synapse schematic. (b) Plot of equilibrium weight in the subthreshold regime versus the conditional probability P(X|Y), showing both experimental chip data and a fit from Eq.7 (c). Plot of equilibrium weight versus conditional probability in the above-threshold regime, again showing chip data and a fit from Eq.7. 3.2 Weight equilibrium To demonstrate that our circuit learns P(X|Y), we show that the equilibrium weight of the synapse is solely a function of P(X|Y). The equilibrium weight of the synapse is the weight value where the expected weight change over time equals zero. This weight value corresponds to the floating-gate voltage where injection and tunneling currents are equal. To find this voltage, we equate Eq’s. 4 and 5 and solve: ( ) 0 0 1 log Pr( | ) log / 1/ inj eq fg tun y x I V X Y I V V κ − = + + (6) To derive the equilibrium weight, we substitute Eq.6 into Eq.3 and solve: ( ) 0 0 0 2 0 0 0 2 2 Pr( | ) below threshold log log Pr( | ) above threshold where and . (1 ) ( / 1/ ) (1 )( / 1/ ) inj tun eq inj tun t I I X Y I W I V X Y I U V V V V α γ χ γ χ β η κ κ α η κ κ κ κ = + + = = + + + + (7) Consequently, the equilibrium weight is a function of the conditional probability below threshold and a function of the log-squared conditional probability above threshold. Note that the equilibrium weight is stable because of negative feedback in the tunneling and injection processes. Therefore, the weight will always converge to the equilibrium value shown in Eq.7. Figs. 1(b) and (c) show the equilibrium weight versus the conditional P(X|Y) for both sub- and above-threshold circuits, along with fits to Eq.7. Note that both the sub- and above-threshold relationship between P(X|Y) and the equilibrium weight enables us to compute the probability of a vector of synaptic inputs X given a post-synaptic response Y. In both cases, we can apply the outputs currents of an array of synapses through diodes, and then add the resulting voltages via a capacitive voltage divider, resulting in a voltage that is a linear function of log P(X|Y). 3.3 Calibration circuitry Mismatch between injection and tunneling in different floating-gate transistors can greatly reduce the ability of our synapses to learn meaningful values. Experimental data from floating-gate transistors fabricated in a 0.35µm process show that injection varies by as much as 2:1 across a chip, and tunneling by up to 1.2:1. The effect of this mismatch on our synapses causes the weight equilibrium of different synapses to differ by a multiplicative gain. Fig.2 (b) shows the equilibrium weights of an array of six synapses exposed to identical input signals. The variation of the synaptic weights is of the same order of magnitude as the weights themselves, making large arrays of synapses all but useless for implementing many learning algorithms. We alleviate this problem by calibrating our synapses to equalize the pre-exponential tunneling and injection constants. Because the dependence of the equilibrium weight on these constants is determined by the ratio of Iinj0/Itun0, our calibration process changes Iinj to equalize the ratio of injection to tunneling across all synapses. We choose to calibrate injection because we can easily change Iinj0 by altering the drain current through M1. Our calibration procedure is a self-convergent memory write [11], that causes the equilibrium weight of every synapse to equal the current Ical. Calibration requires many operat ing cycles, where, during each cycle, we first increase the equilibrium weight of the synapse, and second, we let the synapse adapt to the new equilibrium weight. We create the calibrated synapse by modifying our original synapse according to Fig. 2(a). We convert M1 into a floating-gate transistor, whose floating-gate charge thereby sets M3’s channel current, providing control of Iinj0 of Eq.7. Transistor M8 modifies M1’s gate charge by means of injection when M9’s gate is low and Vcal is low. M9’s gate is only low when the equilibrium weight W is less than Ical. During calibration, injection and tunneling on M3 are continuously active. We apply a pulse train to Vcal; during each pulse period, Vcal is predominately high. When Vcal is high, the synapse adapts towards its equilibrium weight. When Vcal pulses low, M8 injects, increasing the synapse’s equilibrium weight W. We repeat this process until the equilibrium weight W matches Ical, causing M9’s gate voltage to rise, disabling Vcal and with it injection. To ensure that a precalibrated synapse has an equilibrium weight below Ical, we use tunneling to erase all bias transistors prior to calibration. Fig.2(c) shows the equilibrium weights of six synapses after calibration. The data show that calibration can reduce the effect of mismatched adaptation on the synapse’s learned weight to a small fraction of the weight itself. Because M1 is a floating-gate transistor, its parasitic gate-drain capacitance causes a mild dependence between M1’s drain voltage and source current. Consequently, M3’s floatinggate voltage now affects its source current (through M1’s drain voltage), and we can model M3 as a source-degenerated pFET [3]. The new expression for the injection current in M3 is: Verase Vb Vcal Ical M1 M3 M8 M2 M4 M5 M6 M7 M9 Vtun synaptic output (a) 0 0.2 0.4 0.6 0.8 1 20 40 60 80 P(X|Y) Weq (nA) (b) Fig. 2. (a) Schematic of calibrated synapse with signals used during the calibration procedure. (b) Equilibrium weights for array of synapses shown in Fig.1a. (c) Equilibrium weights for array of calibrated synapses after calibration. 0.2 0.4 0.6 0.8 1 0 20 40 60 80 P(X|Y) Weq (nA) (c) 1 0 fg ( / ) Pr( , )exp V fg inj inj t k dV dt I X Y V U γ κ κ = − − (8) where k1 is close to zero. The new expression for injection slightly changes the α and η terms of the weight equilibrium in Eq.7, although the qualitative relationship between the weight equilibrium and the conditional probability remains the same. 4 Implementing silicon synaptic learning rules In this section we discuss how to implement a variety of learning rules from the computational-neurobiology and neural-network literature with our synapse circuit. We can use our circuit to implement a Hebbian learning rule. Simultaneously activating both M5 and M6 is analogous to heterosynaptic LTP based on synchronized pre- and postsynaptic signals, and activating tunneling with the postsynaptic Y is analogous to homosynaptic LTD. In our synapse, we tie Xin and Xcor together and correlate Vtun with Y. Our synapse is also capable of emulating a Boltzmann weight-update rule [2]. This weight-update rule derives from the difference between correlations among neurons when the network receives external input, and when the network operates in a free running phase (denoted as clamped and unclamped phases respectively). With weight decay, a Boltzmann synapse learns the difference between correlations in the clamped and unclamped phase. We can create a Boltzmann synapse from a pair of our circuits, in which the effective weight is the difference between the weights of the two synapses. To implement a weight update, we update one silicon synapse based on pre- and postsynaptic signals in the clamped phase, and update the other synapse in the unclamped phase. We do this by sending Xin to Xcor of one synapse in the clamped phase, and sending Xin to Xcor of the other synapse in the negative phase. Vtun remains constant throughout adaptation. Finally, we consider implementing a temporally asymmetric Hebbian learning rule [7] using our synapse. In temporally asymmetric Hebbian learning, a synapse exhibits LTP or LTD if the presynaptic input occurs before or after the postsynaptic response, respectively. We implement an asymmetric learning synapse using two of our circuits, where the synaptic weight is the difference in the weights of the two circuit. We show the circuit in Fig. 3. Each neuron sends two signals: a neuronal output, and an adaptation time window that is active for some time afterwards. Therefore, the combined synapse receives two presynaptic signals and two postsynaptic signals. The relative timing of a postsynaptic response, Y, with the presynaptic input, X, determines whether the synapse undergoes Presynaptic neuron Postsynaptic neuron X Activation window Synapse Injection Injection W+ W− Y Fig. 3. A method for achieving spike-time dependent plasticity in silicon. LTP or LTD. If Y occurs before X, Y’s time window correlates with X, causing injection on the negative synapse, decreasing the weight. If Y occurs after X, Y correlates with X’s time window, causing injection on the positive synapse, increasing the weight. Hence, our circuit can use the relative timing between presynaptic and postsynaptic activity to implement learning. 5 Conclusion We have described a silicon synapse that implements a wide range of spike-based learning rules, and that does not suffer from device mismatch. We have also described how we can implement various silicon-learning networks using this synapse. In addition, although we have only analyzed the learning properties of the synapse for binary signals, we can instead use pulse-coded analog signals. One possible avenue for future work is to analyze the implications of different pulse-coded schemes on the circuit’s adaptive behavior. Acknowledgements This work was supported by the National Science Foundation and by the Office of Naval Research. Aaron Shon was also supported by a NDSEG fellowship. We thank Anhai Doan and the anonymous reviewers for helpful comments. References [1] W.B.Levy, “A computational approach to hippocampal function,” in R.D. Hawkins and G.H. Bower (eds.), Computational Models of Learning in Simple Neural Systems, The Psychology of Learning and Motivation vol.23, pp.243-305, San Diego, CA: Academic Press, 1989. [2] D.H.Ackley, G.Hinton, and T.Sejnowski, “A learning algorithm for Boltzmann machines,” Cognitive Science vol.9, pp.147-169, 1985. [3 ] P.Hasler, B.A.Minch, J.Dugger, and C.Diorio, “Adaptive circuits and synapses using pFET floating-gate devices, ” in G.Cauwenberghs and M.Bayoumi (eds.) Learning in Silicon, pp.33-65, Kluwer Academic, 1999. [4] P.Hafliger, A spike-based learning rule and its implementation in analog hardware, Ph.D. thesis, ETH Zurich, 1999. [5] C.Diorio, P.Hasler, B.A.Minch, and C.Mead, “A floating-gate MOS learning array with locally computer weight updates,” IEEE Transactions on Electron Devices vol.44(12), pp.2281-2289, 1997. [6] R.Sutton, “Learning to predict by the methods of temporal difference,” Machine Learning, vol.3, pp.9-44, 1988. [7] H.Markram, J.Lübke, M.Frotscher, and B.Sakmann, “Regulation of synaptic efficacy by coincidence of postsynaptic APs and EPSPs,” Science vol.275, pp.213-215, 1997. [8] A.Pesavento, T.Horiuchi, C.Diorio, and C.Koch, “Adaptation of current signals with floating-gate circuits,” in Proceedings of the 7th International Conference on Microelectronics for Neural, Fuzzy, and Bio-Inspired Systems (Microneuro99), pp.128-134, 1999. [9] M.Lenzlinger and E.H.Snow. “Fowler-Nordheim tunneling into thermally grown SiO2,” Journal of Applied Physics vol.40(1), pp.278--283, 1969. [10] E.Takeda, C.Yang, and A.Miura-Hamada, Hot Carrier Effects in MOS Devices, San Diego, CA: Academic Press, 1995. [11] C.Diorio, “A p-channel MOS synapse transistor with self-convergent memory writes,” IEEE Journal of Solid-State Circuits vol.36(5), pp.816-822, 2001.
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A Natural Policy Gradient Sham Kakade Gatsby Computational Neuroscience Unit 17 Queen Square, London, UK WC1N 3AR http://www.gatsby.ucl.ac.uk sham@gatsby.ucl.ac.uk Abstract We provide a natural gradient method that represents the steepest descent direction based on the underlying structure of the parameter space. Although gradient methods cannot make large changes in the values of the parameters, we show that the natural gradient is moving toward choosing a greedy optimal action rather than just a better action. These greedy optimal actions are those that would be chosen under one improvement step of policy iteration with approximate, compatible value functions, as defined by Sutton et al. [9]. We then show drastic performance improvements in simple MDPs and in the more challenging MDP of Tetris. 1 Introduction There has been a growing interest in direct policy-gradient methods for approximate planning in large Markov decision problems (MDPs). Such methods seek to find a good policy 7r among some restricted class of policies, by following the gradient of the future reward. Unfortunately, the standard gradient descent rule is noncovariant. Crudely speaking, the rule !:l.()i = oJ] f / a()i is dimensionally inconsistent since the left hand side has units of ()i and the right hand side has units of l/()i (and all ()i do not necessarily have the same dimensions). In this paper, we present a covariant gradient by defining a metric based on the underlying structure of the policy. We make the connection to policy iteration by showing that the natural gradient is moving toward choosing a greedy optimal action. We then analyze the performance of the natural gradient in both simple and complicated MDPs. Consistent with Amari's findings [1], our work suggests that the plateau phenomenon might not be as severe using this method. 2 A Natural Gradient A finite MDP is a tuple (S, So, A, R, P) where: S is finite set of states, So is a start state, A is a finite set of actions, R is a reward function R : S x A --+ [0, Rmax], and P is the transition model. The agent's decision making procedure is characterized by a stochastic policy 7r(a; s) , which is the probability of taking action a in state s (a semi-colon is used to distinguish the random variables from the parameters of the distribution). We make the assumption that every policy 7r is ergodic, ie has a well-defined stationary distribution p7f. Under this assumption, the average reward (or undiscounted reward) is 1]( 7r) == 2:: s,a p7f (s )7r(a; S )R(s, a), the state-action value is Q7f(S, a) == E7f{2:::oR(st,at) -1](7r)lso = s,ao = a} and the value function is J7f(s) == E7f(a' ;s) {Q7f(s, a')}, where and St and at are the state and action at time t. We consider the more difficult case where the goal of the agent is to find a policy that maximizes the average reward over some restricted class of smoothly parameterized policies, fr = {7rO : 8 E ~m}, where tro represents the policy 7r(a; S, 8). The exact gradient of the average reward (see [8, 9]) is: \11](8) = Lp7f(s)\17r(a;s, 8)Q7f(s,a) (1) s,a where we abuse notation by using 1](8) instead of 1](7ro). The steepest descent direction of 1](8) is defined as the vector d8 that minimizes 1](8 + d8) under the constraint that the squared length Id812 is held to a small constant. This squared length is defined with respect to some positive-definite matrix G(8), ie Id812 == 2::ij Gij (8)d8id8j = d8T G(8)d8 (using vector notation). The steepest descent direction is then given by G- 1\11](8) [1]. Standard gradient descent follows the direction \11](8) which is the steepest descent under the assumption that G(8) is the identity matrix, I. However, this as hoc choice of a metric is not necessarily appropriate. As suggested by Amari [1], it is better to define a metric based not on the choice of coordinates but rather on the manifold (ie the surface) that these coordinates parameterize. This metric defines the natural gradient. Though we slightly abuse notation by writing 1](8), the average reward is technically a function on the set of distributions {7rO : 8 E ~m}. To each state s, there corresponds a probability manifold, where the distribution 7r(a; S, 8) is a point on this manifold with coordinates 8. The Fisher information matrix of this distribution 7r(a; s,8) is F (8) = E [81og7r(a; s,8) o log 7r(a; s,8)] s 7f(a;s,O) 08i 08j ' (2) and it is clearly positive definite. As shown by Amari (see [1]), the Fisher information matrix, up to a scale, is an invariant metric on the space of the parameters of probability distributions. It is invariant in the sense that it defines the same distance between two points regardless of the choice of coordinates (ie the parameterization) used, unlike G = I. Since the average reward is defined on a set of these distributions, the straightforward choice we make for the metric is: (3) where the expectation is with respect to the stationary distribution of 7ro. Notice that although each Fs is independent of the parameters of the MDP's transition model, the weighting by the stationary distribution introduces dependence on these parameters. Intuitively, Fs (8) measures distance on a probability manifold corresponding to state sand F(8) is the average such distance. The steepest descent direction this gives is: (4) 3 The Natural Gradient and Policy Iteration We now compare policy improvement under the natural gradient to policy iteration. For an appropriate comparison, consider the case in which Q7r (s, a) is approximated by some compatible function approximator r(s,a;w) parameterized by w [9, 6]. 3.1 Compatible Function Approximation For vectors (), w E ~m, we define: 'IjJ(s,a)7r = \7logn(a;s,()), r(s,a;w) = wT 'ljJ7r(s,a) (5) where [\7logn(a;s, ())]i = 8logn(a;s, ())!8()i. Let w minimize the squared error f(W, n) == L,s,a p7r (s )n(a; s, ())(r (s, a; w) _Q7r (s, a))2. This function approximator is compatible with the policy in the sense that if we use the approximations f7r (s, a; w) in lieu of their true values to compute the gradient (equation 1), then the result would still be exact [9, 6] (and is thus a sensible choice to use in actor-critic schemes). Theorem 1. Let w minimize the squared error f(W, no). Then w = ~1}(()) . Proof. Since w minimizes the squared error, it satisfies the condition 8f!8wi = 0, which implies: LP7r(s)n(a;s,())'ljJ7r (s,a)('ljJ7r (s,a?w - Q7r(s,a)) = O. s,a or equivalently: s,a s,a By definition of 'ljJ7r, \7n(a;s,()) = n(a;s,())'ljJ7r(s,a) and so the right hand side is equal to \71}. Also by definition of 'ljJ7r, F( ()) = L,s,a p7r (s )n( a; s, ())'ljJ7r (s, a )'ljJ7r (s, a) T. Substitution leads to: F(())w = \71}(()) . Solving for w gives w = F(()) - l\71}(()), and the result follows from the definition of the natural gradient. D Thus, sensible actor-critic frameworks (those using f7r(s , a; w)) are forced to use the natural gradient as the weights of a linear function approximator. If the function approximation is accurate, then good actions (ie those with large state-action values) have feature vectors that have a large inner product with the natural gradient. 3.2 Greedy Policy Improvement A greedy policy improvement step using our function approximator would choose action a in state s if a E argmaxa, f7r (s, a'; w). In this section, we show that the natural gradient tends to move toward this best action, rather than just a good action. Let us first consider policies in the exponential family (n(a;s, ()) IX exp(()T¢sa) where ¢sa is some feature vector in ~m). The motivation for the exponential family is because it has affine geometry (ie the flat geometry of a plane), so a translation of a point by a tangent vector will keep the point on the manifold. In general, crudely speaking, the probability manifold of 7r(a; s, 0) could be curved, so a translation of a point by a tangent vector would not necessarily keep the point on the manifold (such as on a sphere). We consider the general (non-exponential) case later. We now show, for the exponential family, that a sufficiently large step in the natural gradient direction will lead to a policy that is equivalent to a policy found after a greedy policy improvement step. Theorem 2. For 7r(a; s, 0) ex: exp(OT 1>sa), assume that ~'TJ(O) is non-zero and that w minimizes the approximation error. Let7roo (a;s) =lima-+oo7r(a;s , O+a~'TJ(O)). Then 7r 00 (a; s) 1- 0 if and only if a E argmaxa, F' (s, a'; w). Proof. By the previous result, F'(s,a;w) = ~'TJ(O)T'lj;7r(s,a). By definition of 7r(a; s, 0) , 'lj;7r (s, a) = 1>sa - E7r(a';s,O) (1)sa'). Since E7r(a';s,O) (1)sa') is not a function of a, it follows that argmaxa, r(s, a'; w) = argmaxa, ~'TJ(Of 1>sa' . After a gradient step, 7r(a; s, 0 + a~'TJ(O)) ex: exp(OT 1>sa + a~'TJ(O)T 1>sa). ~'TJ(O) 1- 0, it is clear that as a -+ 00 the term ~'TJ(O)T 1>sa dominates, 7r 00 (a, s) = 0 if and only if a f{. argmaxa, ~ 'TJ( 0) T 1>sa' . Since and so D It is in this sense that the natural gradient tends to move toward choosing the best action. It is straightforward to show that if the standard non-covariant gradient rule is used instead then 7roo (a; s) will select only a better action (not necessarily the best), ie it will choose an action a such that F'(s,a;w) > E7r(a';s){F'(s,a';w)}. Our use of the exponential family was only to demonstrate this point in the extreme case of an infinite learning rate. Let us return to case of a general parameterized policy. The following theorem shows that the natural gradient is locally moving toward the best action, determined by the local linear approximator for Q7r (s, a). Theorem 3. Assume that w minimizes the approximation error and let the update to the parameter be 0' = 0 + a~'TJ(O). Then 7r(a; s, 0') = 7r(a; s, 0)(1 + r(s, a; w)) + 0(a2 ) Proof. The change in 0, ,6.0, is a~'TJ(O), so by theorem 1, ,6.0 = aw. To first order, 7r(a; s, 0') 7r(a; s, 0) + fJ7r(a~;, O)T ,6.0 + 0(,6.02 ) 7r(a; s, 0)(1 + 'lj;(s, af ,6.0) + 0(,6.02 ) 7r(a; s, 0)(1 + a'lj;(s, af w) + 0(a2 ) 7r(a;s,O)(l + ar(s,a;w)) + 0(a2 ) , where we have used the definition of 'lj; and f. D It is interesting to note that choosing the greedy action will not in general improve the policy, and many detailed studies have gone into understanding this failure [3]. However, with the overhead of a line search, we can guarantee improvement and move toward this greedy one step improvement. Initial improvement is guaranteed since F is positive definite. 4 Metrics and Curvatures Obviously, our choice of F is not unique and the question arises as to whether or not there is a better metric to use than F. In the different setting of parameter estimation, the Fisher information converges to the Hessian, so it is asymptotically efficient [1], ie attains the Cramer-Rao bound. Our situation is more similar to the blind source separation case where a metric is chosen based on the underlying parameter space [1] (of non-singular matrices) and is not necessarily asymptotically efficient (ie does not attain second order convergence). As argued by Mackay [7], one strategy is to pull a metric out of the data-independent terms of the Hessian (if possible), and in fact, Mackay [7] arrives at the same result as Amari for the blind source separation case. Although the previous sections argued that our choice is appropriate, we would like to understand how F relates to the Hessian V 2TJ(B), which, as shown in [5], has the form: sa (6) Unfortunately, all terms in this Hessian are data-dependent (ie are coupled to stateaction values). It is clear that F does not capture any information from these last two terms, due to their VQ7r dependence. The first term might have some relation to F due the factor of V27f. However, the Q values weight this curvature of our policy and our metric is neglecting such weighting. Similar to the blind source separation case, our metric clearly does not necessarily converge to the Hessian and so it is not necessarily asymptotically efficient (ie does not attain a second order convergence rate). However, in general, the Hessian will not be positive definite and so the curvature it provides could be of little use until B is close to a local maxima. Conjugate methods would be expected to be more efficient near a local maximum. 5 Experiments We first look at the performance of the natural gradient in a few simple MDPs before examining its performance in the more challenging MDP of Tetris. It is straightforward to estimate F in an online manner, since the derivatives V log 7f must be computed anyway to estimate VTJ(B). If the update rule f f- f + V log 7f(at; St,B)Vlog7f(at; St,Bf is used in a T-Iength trajectory, then fiT is a consistent estimate of F. In our first two examples, we do not concern ourselves with sampling issues and instead numerically integrate the exact derivative (Bt = Bo + J~ VTJ(BddB). In all of our simulations, the policies tend to become deterministic (V log 7f -+ 0) and to prevent F from becoming singular, we add about 10- 31 at every step in all our simulations. We simulated the natural policy gradient in a simple I-dimensional linear quadratic regulator with dynamics x(t + 1) = .7x(t) + u(t) + E(t) and noise distribution E ~ G(O,l). The goal is to apply a control signal u to keep the system at x = 0, (incurring a cost of X(t)2 at each step). The parameterized policy used was 7f(u; x, B) ex exp(Blx2 + B2X). Figure lA shows the performance improvement when the units of the parameters are scaled by a factor of 10 (see figure text). Notice that the time to obtain a score of about 22 is about three orders of magnitude B ~a C 21 rl D unsealed 8' I '--''''' '~'' $=10 s=1 ...... i 1 r ______ -11 ',05 .. • ~...... 1 2 "E I -'::',$,=1 $2=10 _. -(' ,\2 ~ ":. ~R=O) ~0 '::0 --0~5C------:' -----:-': '5C------::' 2 " \::.:-. h ~21 time x 10 7 /:--------1. Q "\" "::>:" W ~, L-----------.\'; 20 --"-,-, ~ L---::-,::::; ·7J~========-~ L _-=-2 --': -'':::::;0:::=:::' ~2 :=':3=::l4' 0 a 0.5 1 1.5 2 2.5 3 5 8., 10 15 I09 10(time) time Figure 1: A) The cost Vs. 10glo(time) for an LQG (with 20 time step trajectories). The policy used was 7f(u; x, ()) ex: exp(()lslX2 + ()2S2X) where the rescaling constants, Sl and S2, are shown in the legend. Under equivalent starting distributions (()lSl = ()2S2 = -.8) , the right-most three curves are generated using the standard gradient method and the rest use the natural gradient. B) See text. C top) The average reward vs. time (on a 107 scale) of a policy under standard gradient descent using the sigmoidal policy parameterization (7f(I; s, ()i) ex: exp(()i)/(1 + exp(()i)), with the initial conditions 7f(i, 1) = .8 and 7f(j, 1) = .1. C bottom) The average reward vs. time (unscaled) under standard gradient descent (solid line) and natural gradient descent (dashed line) for an early window of the above plot. D) Phase space plot for the standard gradient case (the solid line) and the natural gradient case (dashed line) . faster. Also notice that the curves under different rescaling are not identical. This is because F is not an invariant metric due to the weighting by Ps. The effects of the weighting by p(s) are particularly clear in a simple 2-state MDP (Figure IB), which has self- and cross-transition actions and rewards as shown. Increasing the chance of a self-loop at i decreases the stationary probability of j. Using a sigmoidal policy parameterization (see figure text) and initial conditions corresponding to p(i) = .8 and p(j) = .2, both self-loop action probabilities will initially be increased under a gradient rule (since one step policy improvement chooses the self-loop for each state). Since the standard gradient weights the learning to each parameter by p(s) (see equation 1), the self-loop action at state i is increased faster than the self loop probability at j, which has the effect of decreasing the effective learning-rate to state j even further. This leads to an extremely fiat plateau with average reward 1 (shown in Figure lC top), where the learning for state j is thwarted by its low stationary probability. This problem is so severe that before the optimal policy is reached p(j) drops as low as 10-7 from its initial value of .2, which is disastrous for sampling methods. Figure 1 C bottom shows the performance of the natural gradient (in a very early time window of Figure lC top). Not only is the time to the optimal policy decreased by a factor of 107 , the stationary distribution of state i never drops below .05. Note though the standard gradient does increase the average reward faster at the start, but only to be seduced by sticking at state i. The phase space plot in Figure ID shows the uneven learning to the different parameters, which is at the heart of the problem. In general, if a table lookup Boltzmann policy is used (ie 7f( a; s, ()) ex: exp( () sa)), it is straightforward to show that the natural gradient weights the components of ~'fJ uniformly (instead of using p(s)), thus evening evening out the learning to all parameters. The game of Tetris provides a challenging high dimensional problem. As shown in [3], greedy policy iteration methods using a linear function approximator exhibit drastic performance degradation after providing impressive improvement (see [3] for a description of the game, methods, and results). The upper curve in Figure2A replicates these results. Tetris provides an interesting case to test gradient methods, A 5000,-------------, 4000 ~3000 ·0 a... 2000 1000 1 I09,O(lteralions) 2 B 7000,----------,----, 6000 5000 ~4000 &3000 2000 1000 C 500 1000 1500 2000 Iterations Figure 2: A) Points vs. 10g(Iterations). The top curve duplicates the same results in [3] using the same features (which were simple functions of the heights of each column and the number of holes in the game). We have no explanation for this performance degradation (nor does [3]). The lower curve shows the poor performance of the standard gradient rule. B) The curve on the right shows the natural policy gradient method (and uses the biased gradient method of [2] though this method alone gave poor performance). We found we could obtain faster improvement and higher asymptotes if the robustifying factor of 10- 3 I that we added to F was more carefully controlled (we did not carefully control the parameters). C) Due to the intensive computational power required of these simulations we ran the gradient in a smaller Tetris game (height of 10 rather than 20) to demonstrate that the standard gradient updates (right curve) would eventually reach the same performance of the natural gradient (left curve). which are guaranteed not to degrade the policy. We consider a policy compatible with the linear function approximator used in [3] (ie 7f(a;s, (}) ex: exp((}T¢sa) where ¢sa are the same feature vectors). The features used in [3] are the heights of each column, the differences in height between adjacent columns, the maximum height, and the number of 'holes'. The lower curve in Figure 2A shows the particularly poor performance of the standard gradient method. In an attempt to speed learning, we tried a variety of more sophisticated methods to no avail, such as conjugate methods, weight decay, annealing, the variance reduction method of [2], the Hessian in equation 6, etc. Figure 2B shows a drastic improvement using the natural gradient (note that the timescale is linear). This performance is consistent with our theoretical results in section 3, which showed that the natural gradient is moving toward the solution of a greedy policy improvement step. The performance is somewhat slower than the greedy policy iteration (left curve in Figure 2B) which is to be expected using smaller steps. However, the policy does not degrade with a gradient method. Figure 2 shows that the performance of the standard gradient rule (right curve) eventually reaches the the same performance of the natural gradient, in a scaled down version of the game (see figure text). 6 Discussion Although gradient methods cannot make large policy changes compared to greedy policy iteration, section 3 implies that these two methods might not be that disparate, since a natural gradient method is moving toward the solution of a policy improvement step. With the overhead of a line search, the methods are even more similar. The benefit is that performance improvement is now guaranteed, unlike in a greedy policy iteration step. It is interesting, and unfortunate, to note that the F does not asymptotically converge to the Hessian, so conjugate gradient methods might be more sensible asymptotically. However, far from the converge point, the Hessian is not necessarily informative, and the natural gradient could be more efficient (as demonstrated in Tetris). The intuition as to why the natural gradient could be efficient far from the maximum, is that it is pushing the policy toward choosing greedy optimal actions. Often, the region (in parameter space) far from from the maximum is where large performance changes could occur. Sufficiently close to the maximum, little performance change occurs (due to the small gradient), so although conjugate methods might converge faster near the maximum, the corresponding performance change might be negligible. More experimental work is necessary to further understand the effectiveness of the natural gradient. Acknowledgments We thank Emo Todorov and Peter Dayan for many helpful discussions. Funding is from the NSF and the Gatsby Charitable Foundation. References [I] S. Amari. Natural gradient works efficiently in learning. Neural Computation, 10(2):251- 276, 1998. [2] J. Baxter and P. Bartlett. Direct gradient-based reinforcement learning. Technical report, Australian National University, Research School of Information Sciences and Engineering, July 1999. [3] D. P. Bertsekas and J. N. Tsitsiklis. Neuro-Dynamic Programming. Athena Scientific, 1996. [4] P. Dayan and G. Hinton. Using em for reinforcement learning. Neural Computation, 9:271- 278, 1997. [5] S. Kakade. Optimizing average reward using discounted reward. COLT. in press., 200l. [6] V. Konda and J. Tsitsiklis. Actor-critic algorithms. Advances in Neural Information Processing Systems, 12, 2000. [7] D. MacKay. Maximum likelihood and covariant algorithms for independent component analysis. Technical report, University of Cambridge, 1996. [8] P. Marbach and J . Tsitsiklis. Simulation-based optimization of markov reward processes. Technical report, Massachusetts Institute of Technology, 1998. [9] R. Sutton, D. McAllester, S. Singh, and Y. Mansour. Policy gradient methods for reinforcement learning with function approximation. Neural Information Processing Systems, 13, 2000. [10] L. Xu and M. 1. Jordan. On convergence properties of the EM algorithm for gaussian mixtures. Neural Computation, 8(1):129- 151, 1996.
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Spectral Kernel Methods for Clustering N ello Cristianini BIOwulf Technologies nello@support-vector.net John Shawe-Taylor Jaz Kandola Royal Holloway, University of London {john, jaz} @cs.rhul.ac.uk Abstract In this paper we introduce new algorithms for unsupervised learning based on the use of a kernel matrix. All the information required by such algorithms is contained in the eigenvectors of the matrix or of closely related matrices. We use two different but related cost functions, the Alignment and the 'cut cost'. The first one is discussed in a companion paper [3], the second one is based on graph theoretic concepts. Both functions measure the level of clustering of a labeled dataset, or the correlation between data clusters and labels. We state the problem of unsupervised learning as assigning labels so as to optimize these cost functions. We show how the optimal solution can be approximated by slightly relaxing the corresponding optimization problem, and how this corresponds to using eigenvector information. The resulting simple algorithms are tested on real world data with positive results. 1 Introduction Kernel based learning provides a modular approach to learning system design [2]. A general algorithm can be selected for the appropriate task before being mapped onto a particular application through the choice of a problem specific kernel function. The kernel based method works by mapping data to a high dimensional feature space implicitly defined by the choice of the kernel function. The kernel function computes the inner product of the images of two inputs in the feature space. From a practitioners viewpoint this function can also be regarded as a similarity measure and hence provides a natural way of incorporating domain knowledge about the problem into the bias of the system. One important learning problem is that of dividing the data into classes according to a cost function together with their relative positions in the feature space. We can think of this as clustering in the kernel defined feature space, or non-linear clustering in the input space. In this paper we introduce two novel kernel-based methods for clustering. They both assume that a kernel has been chosen and the kernel matrix constructed. The methods then make use of the matrix's eigenvectors, or of the eigenvectors of the closely related Laplacian matrix, in order to infer a label assignment that approximately optimizes one of two cost functions. See also [4] for use of spectral decompositions of the kernel matrix. The paper includes some analysis of the algorithms together with tests of the methods on real world data with encouraging results. 2 Two partition cost measures All the information needed to specify a clustering of a set of data is contained in the matrix Mij = (cluster(xi) == cluster(xj)), where (A == B) E {-I, +1}. After a clustering is specified, one can measure its cost in many ways. We propose here two cost functions that are easy to compute and lead to efficient algorithms. Learning is possible when some collusion between input distribution and target exists, so that we can predict the target based on the input. Typically one would expect points with similar labels to be clustered and the clusters to be separated. This can be detected in two ways: either by measuring the amount of label-clustering or by measuring the correlation between such variables. In the first case, we need to measure how points of the same class are close to each other and distant from points of different classes. In the second case, kernels can be regarded as oracles predicting whether two points are in the same class. The 'true' oracle is the one that knows the true matrix M. A measure of quality can be obtained by measuring the Pearson correlation coefficient between the kernel matrix K and the true M . Both approaches lead to the same quantity, known as the alignment [3]. We will use the following definition of the inner product between matrices (K1,K2)F = 2:2j=1 K 1(Xi,Xj)K2(Xi,Xj). The index F refers to the Frobenius norm that corresponds to this inner product. Definition 1 Alignment The (empirical) alignment of a kernel kl with a kernel k2 with respect to the sample S is the quantity ...1(S,k1,k2) = (K1,K2)F , yi(K1,K1)F(K2,K2)F where Ki is the kernel matrix for the sample S using kernel ki. This can also be viewed as the cosine of the angle between to bi-dimensional vectors Kl and K 2, representing the Gram matrices. If we consider k2 = yy', where y is the vector of { -1, + I} labels for the sample, then with a slight abuse of notation AA(Sk)= (K,yy')F (K,yY')F. (") 2 , ,y / = mllKllF ' smce yy,yy F = m V (K, K) F (YY' , yy') F Another measure of separation between classes is the average separation between two points in different classes, again normalised by the matrix norm. Definition 2 Cut Cost. The cut cost of a clustering is defined as "' .. -tk(Xi XJ) C(S, k, y) = L..'J:Y;li~IIF' . This quantity is motivated by a graph theoretic concept. If we consider the Kernel matrix as the adjacency matrix of a fully connected weighted graph whose nodes are the data points, the cost of partitioning a graph is given by the total weight of the edges that one needs to cut or remove, and is exactly the numerator of the 'cut cost'. Notice also the relation between alignment and cutcost: '" k(x· x·) - 2C(S k) L..ij " J , = T(S k) - 2C(S k ) myi(K,K)F ' , ,y, ...1(S, k, y) where T(S,k) = ...1(S,k,j), for j the all ones vector. Among other appealing properties of the alignment, is that this quantity is sharply concentrated around its mean, as proven in the companion paper [3]. This shows that the expected alignment can be reliably estimated from its empirical estimate A.(S). As the cut cost can be expressed as the difference of two alignments C(S,k,y) = O.5(T(S,k) - A.(S, k,y)), (1) it will be similarly concentrated around its expected value. 3 Optimising the cost with spectral techniques In this section we will introduce and test two related methods for clustering, as well as their extensions to transduction. The general problem we want to solve is to assign class-labels to datapoints so as to maximize one of the two cost functions given above. By equation (1) the optimal solution to both problems is identical for a fixed data set and kernel. The difference between the approaches is in the two approximation algorithms developed for the different cost functions. The approximation algorithms are obtained by relaxing the discrete problems of optimising over all possible labellings of a dataset to closely related continuous problems solved by eigenvalue decompositions. See [5] for use of eigenvectors in partitioning sparse matrices. 3.1 Optimising the alignment To optimise the alignment, the problem is to find the maximally aligned set of labels A A (K,yy')F A*(S,k)= max A(S,k,y)= max yE{ -1 ,1}= yE{ -l ,l}= mJ(K, K)F Since in this setting the kernel is fixed maximising the alignment reduces to choosing y E {-I, l}m to maximise (K,yy') = y'Ky. If we allow y to be chosen from the larger set IRm subject to the constraint IIyl12 = m, we obtain an approximate maximum-alignment problem that can be solved efficiently. After solving the relaxed problem, we can obtain an approximate discrete solution by choosing a suitable threshold to the entries in the vector y and applying the sign function. Bounds will be given on the quality of the approximations. The solution of the approximate problem follows from the following theorem that provides a variational characterization of the spectrum of symmetric matrices. Theorem 3 (Courant-Fischer Minimax Theorem) If ME IRmxm is symmetric, then for k = 1, ... , m, v'Mv v'Mv Ak(M) = max min -- = min max --, dirn(T)=k OopvET v'v dirn(T)=m - k+lOopvET v'v If we consider the first eigenvector, the first min does not apply and we obtain that the approximate alignment problem is solved by the first eigenvector, so that the maximal alignment is upper bounded by a multiple of the first eigenvalue, Arnax = maxOopvEIR= v:~v. One can now transform the vector v into a vector in {-I, +l}m by choosing the threshold 8 that gives maximum alignment of y = sign(vrnaX - 8). By definition, the value of alignment A.(S, k, y) obtained by this y will be a lower bound of the optimal alignment, hence we have A.(S,k,y):s A.*(S,k):S Amax/IIKIIF. One can hence estimate the quality of a dichotomy by comparing its value with the upper bound. The absolute alignment tells us how specialized a kernel is on a given dataset: the higher this quantity, the more committed to a specific dichotomy. The first eigenvector can be calculated in many ways, for example the Lanczos procedure, which is already effective for large datasets. Search engines like Google are based on estimating the first eigenvector of a matrix with dimensionality more than 109 , so for very large datasets there are approximation techniques. We applied the procedure outlined above to two datasets from the VCI repository. We preprocessed the data by normalising the input vectors in the kernel defined feature space and then centering them by shifting the origin (of the feature space) to their centre of gravity. This can be achieved by the following transformation of the kernel matrix, K +--- K - m - 1jg' - m - 1gj' + m - 2j'KjJ, where j is the all ones vector, J the all ones matrix and 9 the vector of row sums of K. Eigenvalue Number (a) (b) Figure 1: (a) Plot of alignment of the different eigenvectors with the labels ordered by increasing eigenvalue. (b) Plot for Breast Cancer data (linear kernel) of .Amax/llKIIF (straight line), ...1(S, k, y) for y = sign(vmaX (}i ) (bottom curve), and the accuracy of y (middle curve) against threshold number i. The first experiment applied the unsupervised technique to the Breast Cancer data with a linear kernel. Figure l(a) shows the alignmment of the different eigenvectors with the labels. The highest alignment is shown by the last eigenvector corresponding to the largest eigenvalue. For each value (}i of the threshold Figure l(b) shows the upper bound of .Amax/llKIIF (straight line), the alignment ...1(S, k, y) for y = sign( vmax - (}i) (bottom curve), and the accuracy of y (middle curve). Notice that where actual alignment and upper bound on alignment get closest, we have confidence that we have partitioned our data well, and in fact the accuracy is also maximized. Notice also that the choice of the threshold corresponds to maintaining the correct proportion between positives and negatives. This suggests another possible threshold selection strategy, based on the availability of enough labeled points to give a good estimate of the proportion of positive points in the dataset. This is one way label information can be used to choose the threshold. At the end of the experiments we will describe another 'transduction' method. It is a measure of how naturally the data separates that this procedure is able to optimise the split with an accuracy of approximately 97.29% by choosing the threshold that maximises the alignment (threshold number 435) but without making any use of the labels. In Figure 2a we present the same results for the Gaussian kernel (u = 6). In this case the accuracy obtained by optimising the alignment (threshold number 316) of the resulting dichotomy is less impressive being only about 79.65%. Finally, Figure 2b shows the same results for the Ionosphere dataset. Here the accuracy of the split that optimises the alignment (threshold number 158) is approximately (a) (b) Figure 2: Plot for Breast Cancer data (Gaussian kernel) (a) and Ionosphere data (linear kernel) (b) of Amax/ilKIIF (straight line), .4(S, k, y) for y = sign(vmaX - ()i) (bottom curve), and the accuracy of y (middle curve) against threshold number i. 71.37%. We can also use the overall approach to adapt the kernel to the data. For example we can choose the kernel parameters so as to optimize Amax/IIKIIF. Then find the first eigenvector, choose a threshold to maximise the alignment and output the corresponding y. The cost to the alignment of changing a label Yi is 2 Lj Yjk(Xi' xj)/IIKIIF , so that if a point is isolated from the others, or if it is equally close to the two different classes, then changing its label will have only a very small effect. On the other hand, labels in strongly clustered points clearly contribute to the overall cost and changing their label will alter the alignment significantly. The method we have described can be viewed as projecting the data into a 1dimensional space and finding a threshold. The projection also implicitly sorts the data so that points of the same class are nearby in the ordering. We discuss the problem in the 2-class case. We consider embedding the set into the real line, so as to satisfy a clustering criterion. The resulting Kernel matrix should appear as a block diagonal matrix. This problem has been addressed in the case of information retrieval in [1], and also applied to assembling sequences of DNA. In those cases, the eigenvectors of the Laplacian have been used, and the approach is called the Fiedler ordering. Although the Fiedler ordering could be used here as well, we present here a variation based on the simple kernel matrix. Let the coordinate ofthe point Xi on the real line be v(i). Consider the cost function Lij v(i)v(j)K(i,j). It is maximized when points with high similarity have the same sign and high absolute value, and when points with different sign have low similarity. The choice of coordinates v that optimizes this cost is the first eigenvector, and hence by sorting the data according to the value of their entry in this eigenvector one can hope to find a good permutation, that renders the kernel matrix block diagonal. Figure 3 shows the results of this heuristic applied to the Breast cancer dataset. The grey level indicates the size of the kernel entry. The figure on the left is for the unsorted data, while that on the right shows the same plot after sorting. The sorted figure clearly shows the effectivenesss of the method. 3.2 Optimising the cut-cost For a fixed kernel matrix minimising the cut-cost corresponds to mlmmlsmg Ly;#y; k( Xi, X j), that is the sum of the kernel entries between points of two difFigure 3: Gram matrix for cancer data, before and after permutation of data according to sorting order of first eigenvector of K ferent classes. Since we are dealing with normalized kernels, this also controls the expected distance between them. ( ) "'"' 1",", ' 1 , We can express this quantity as ~ Kij ="2 ~ Kij - Y Ky ="2Y Ly, ydy; i,j where L is the Laplacian matrix, defined as L = D-K, where D = diag(dl , ... , dm ) with di = '£';1 k(Xi , Xj). One would like to find y E {-l,+l}m so as to minimize the cut cost subject to the division being even, but this problem is NP-hard. Following the same strategy as with the alignment we can impose a slightly looser constraint on y, y E Rm, '£i yt = m, l:i Yi = O. This gives the problem min y' Ly subject to y E Rm , l: yt = m, l: Yi = O. Since, zero is an eigenvalue of L with eigenvector j, the all ones vector, the problem is equivalent to finding the eigenvector of the smallest non-zero eigenvalue ..\ minO#yl..j y/yY. Hence, this eigenvalue ..\ provides a lower bound on the cut cost . ..\ mm C(S, k, y) ~ IIKII . y E{ - l,l}'" 2 F So the eigenvector corresponding to the eigenvalue ..\ of the Laplacian can be used to obtain a good approximate split and ..\ gives a lower bound on the cut-cost. One can now threshold the entries of the eigenvector in order to obtain a vector with -1 and + 1 entries. We again plot the lower bound, cut-cost, and error rate as a function of the threshold. We applied the procedure to the Breast cancer data with both linear and Gaussian kernels. The results are shown in Figure 4. Now using the cut cost to select the best threshold for the linear kernel sets it at 378 with an accuracy of 67.86%, significantly worse than the results obtained by optimising the alignment. With the Gaussian kernel, on the other hand, the method selects threshold 312 with an accuracy of 80.31 %, a slight improvement over the results obtained with this kernel by optimising the alignment. So far we have presented algorithms that use unsupervised data. We now consider the situation where we are given a partially labelled dataset. This leads to a simple algorithm for transduction or semi-supervised learning. The idea that some labelled data might improve performance comes from observing Figure 4b, where the selection based on the cut-cost is clearly suboptimal. By incorporating some label information, it is hoped that we can obtain an improved threshold selection. 0050'---------c:c------c=--=--~-_=_-_=_-~ (a) (b) Figure 4: Plot for Breast Cancer data using (a) Linear kernel) and (b) Gaussian kernel of C(S,k,y) - ,X/(21IKIIF) (dashed curves), for y = sign(vmaX ()i) and the error of y (solid curve) against threshold number i. Let z be the vector containing the known labels and 0 elsewhere. Set K P = K + Cozz', where Co is a positive constant parameter. We now use the original matrix K to generate the eigenvector, but the matrix K P when measuring the cut-cost of the classifications generated by different thresholds. Taking Co = 1 we performed 5 random selections of 20% of the data and obtained a mean success rate of 85.56% (standard deviation 0.67%) for the Breast cancer data with Gaussian kernel, a marked improvement over the 80.31 % achieved with no label information. 4 Conclusions The paper has considered two partition costs the first derived from the so-called alignment of a kernel to a label vector, and the second from the cut-cost of a label vector for a given kernel matrix. The two quantities are both optimised by the same labelling, but give rise to different approximation algorithms when the discrete constraint is removed from the labelling vector. It was shown how these relaxed problems are solved exactly using spectral techniques, hence leading to two distinct approximation algorithms through a post-processing phase that re-discretises the vector to create a labelling that is chosen to optimise the given criterion. Experiments are presented showing the performance of both of these clustering techniques with some very striking results. For the second algorithm we also gave one preliminary experiment with a transductive version that enables some labelled data to further refine the clustering. References [1] M.W. Berry, B. Hendrickson, and P. Raghavan. Sparse matrix reordering schemes for browsing hypertext. In Th e Matematics of Numerical Analysis, pages 99- 123. AMS, 1996. [2] N. Cristianini and J. Shawe-Taylor. An Introduction to Support Vector Machines. Cambridge University Press, 2000. See also the web site www.support-vector.net. [3] Nello Cristianini, Andre Elisseeff, John Shawe-Taylor, and Jaz Kandola. On kerneltarget alignment. In submitted to Proceedings of Neural Information Processing Systems (NIPS), 200l. [4] Nello Cristianini, Huma Lodhi, and John Shawe-Taylor. Latent semantic kernels for feature selection. Technical Report NC-TR-00-080, NeuroCOLT Working Group, http://www.neurocolt.org, 2000. [5] A. Pothen, H. Simon, and K. Liou. Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Matrix Anal., 11(3):430- 452, 1990.
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Batch Value Function Approximation via Support Vectors Thomas G Dietterich Department of Computet Science Oregon State University Corvallis, OR, 97331 tgd@cs.orst.edu Xin W"ang Department of Computer Science Oregon State University Corvallis, OR, 97331 wangxi@cs. orst. edu Abstract We present three ways of combining linear programming with the kernel trick to find value function approximations for reinforcement learning. One formulation is based on SVM regression; the second is based on the Bellman equation; and the third seeks only to ensure that good moves have an advantage over bad moves. All formulations attempt to minimize the number of support vectors while fitting the data. Experiments in a difficult, synthetic maze problem show that all three formulations give excellent performance, but the advantage formulation is much easier to train. Unlike policy gradient methods, the kernel methods described here can easily 'adjust the complexity of the function approximator to fit the complexity of the value function. 1 Introduction Virtually all existing work on value function approximation and policy-gradient methods starts with a parameterized formula for the value function or policy and then seeks to find the best policy that can be represented in that parameterized form. This can give rise to very difficult search problems for which the Bellman equation is of little or no use. In this paper, we take a different approach: rather than fixing the form of the function approximator and searching for a representable policy, we instead identify a good policy and then search for a function approximator that can represent it. Our approach exploits the ability of mathematical programming to represent a variety of constraints including those that derive from supervised learning, from advantage learning (Baird, 1993), and from the Bellman equation. By combining the kernel trick with mathematical programming, we obtain a function approximator that seeks to find the smallest number of support vectors sufficient to represent the desired policy. This side-steps the difficult problem of searching for a good policy among those policies representable by a fixed function approximator. Our method applies to any episodic MDP, but it works best in domains-such as resource-constrained scheduling and other combinatorial optimization problemsthat are discrete and deterministic. 2 Preliminaries There are two distinct reasons for studying value function approximation methods. The primary reason is to be able to generalize from some set of training experiences to produce a policy that can be applied in new states that were not visited during training. For example, in Tesauro's (1995) work on backgammon, even after training on 200,000 games, the TD-gammon system needed to be able to generalize to new board positions that it had not previously visited. Similarly, in Zhang's (1995) work on space shuttle scheduling, each individual scheduling problem visits only a finite number of states, but the goal is to learn from a series of "training" problems and generalize to new states that arise in "test" problems. Similar MDPs have been studied by Moll, Barto, Perkins & Sutton (1999). The second reason to study function approximation is to support learning in continuous state spaces. Consider a robot with sensors that return continuous values. Even during training, it is unlikely that the same vector of sensor readings will ever be experienced more thaIl once. Hence, generalization is critical during the learning process as well as after learning. The methods described in this paper address only the first of these reason. Specifically, we study the problem of generalizing from a partial policy to construct a complete policy for a Markov Decision Problem (MDP). Formally, consider a discrete time MDP M with probability transition function P(s/ls, a) (probability that state Sl will result from executing action a in state s) and expected reward function R(s/ls, a) (expected reward received from executing action a in state s and entering state Sl). We will assume that, as in backgammon and space shuttle scheduling, P(s/ls, a) and R(s/ls, a) are known and available to the agent, but that the state space is so large that it prevents methods such as value iteration or policy iteration from being applied. Let L be a set of "training" states for which we have an approximation V(s) to the optimal value function V*(s), s E L. In some cases, we will also assume the availability of a policy 'ff consistent with V(s). The goal is to construct a parameterized approximation yes; 8) that can be applied to all states in M to yield a good policy if via one-step lookahead search. In the experiments reported below, the set L contains states that lie along trajectories from a small set of "training" starting states So to terminal states. A successful learning method will be able to generalize to give a good policy for new starting states not in So. This was the situation that arose in space shuttle scheduling, where the set L contained states that were visited while solving "training" problems and the learned value function was applied to solve "test" problems. To represent states for function approximation, let X (s) denote a vector of features describing the state s. Let K(X1,X2 ) be a kernel function (generalized inner product) of the two feature vectors Xl and X 2 • In our experiments, we have employed the gaussian kernel: K(X1,X2;U) == exp(-IIX1 - X 2 11 2/(2 ) with parameter u. 3 Three LP Formulations of Function Approximation We now introduce three linear programming formulations of the function approximationproblem. We first express each of these formulations in terms of a generic fitted function approximator V. Then, we implement V(s) as the dot product of a weight vector W with the feature vector X (s): V(s) == W . X (s). Finally, we apply the "kernel trick" by first rewriting W as a weighted sum of the training points Sj E L, W == ~j ajX(sj), (aj 2: 0), and then replacing all dot products between data points by invocations of the kernel function K. We assume L contains all states along the best paths from So to terminal states and also all states that can be reached from these paths in one step and that have been visited during exploration (so that V is known). In all three formulations we have employed linear objective functions, but quadratic objectives like those employed in standard support vector machines could be used instead. All slack variables in these formulations are constrained- to be non-negative. Formulation 1: Supervised Learning. The first formulation treats the value function approximation problem as a supervised learning problem and applies the standard c-insensitive loss function (Vapnik, 2000) to fit the function approximator. minimize L [u(s) + v(s)] S subject to V(s) + u(s) 2:: V(s) - c; V(s) - v(s) :::; V(s) + c "Is E L In this formulation, u(s) and v(s) are slack variables that are non-zero only if V(s) has an absolute deviation from V(s) of more than c. The objective function seeks to minimize these absolute deviation errors. A key idea of support vector methods is to combine this objective function with a penalty on the norm of the weight vector. We can write this as minimize IIWlll + C L[u(s) + v(s)] S subject to W· X(s) + u(s) 2:: V(s) - c; W· X(s) - v(s) :::; V(s) + c "Is E L The parameter C expresses the tradeoff between fitting the data (by driving the slack variables to zero) and minimizing the norm of the weight vector. We have chosen to minimize the I-norm of the weight vector (11Wlll == Ei IWi!), because this is easy to implement via linear programming. Of course, if the squared Euclidean norm of W is preferred, then quadratic programming methods could be applied to minimize this. Next, we introduce the assumption that W can be written as a weighted sum of the data points themselves. Substituting this into the constraint equations, we obtain minimize L aj + C L[u(s) + v(s)] j 8 subject to E j ajX(sj) . X(s) + u(s) ~ V(s) - c E j ajX(sj) . X(s) - v(s) :::; V(s) + c "Is E L "Is E L Finally, we can apply the kernel trick by replacing each dot product by a call to a kernel function: minimize Laj + CL[u(s) + v(s)] j s subject to E j ajK(X(sj),X(s)) + u(s) 2:: V(s) - c Ej ajK(X(sj), X(s)) - v(s) :::; V(s) + c "Is E L "Is E L Formulation 2: Bellman Learning. The second formulation introduces constraints from the Bellman equation V(s) == maxa ESI P(s'ls, a)[R(s'ls, a) + V(s')]. The standard approach to solving MDPs via linear programming is the following. For each state s and action a, minimize L u(s; a) s,a subject to V(s) == u(s,a) + LP(s'ls,a)[R(s'ls,a) + V(s')] s' The idea is' that for the optimal action a* in state s, the slack variable u(s, a*) can be driven to zero, while for non-optimal actions a_, the slack u(s, a_) will remain non-zero. Hence, the minimization of the slack variables implements the maximization operation of the Bellman equation. We attempted to apply this formulation with function approximation, but the errors introduced by the approximation make the linear program infeasible, because V(s) must sometimes be less than the backed-up value Ls' P(s'ls, a)[R(s'ls, a) + V(s')]. This led us to the following formulation in which we exploit the approximate value function 11 to provide "advice" to the LP optimizer about which constraints should be tight and which ones should be loose. Consider a state s in L. We can group the actions available in s into three groups: (a) the "optimal" action a* == 1f(s) chosen by the approximate policy it, (b) other actions that are tied for optimum (denoted by ao), and (c) actions that are sub-optimal (denoted by a_). We have three different constraint equations, one for each type of action: minimize L[u(s, a*) + v(s, a*)] + LY(s, ao) + L z(s, a_) s s,ao s,a_ subject to 17(s) + u(s, a*) - v(s, a*) == L P(s'ls, a*)[R(s'ls, a*) + V(s')] s' 17(8) + y(s, ao) ~ L P(s'ls, ao)[R(s'ls, ao) + V(s')] s' 17(s) + z(s, a_) ~ L P(s'ls, a_)[R(s'ls, a_) + V(s')] + € s' The first constraint requires V(s) to be approximately equal to the backed-up value of the chosen optimal action a*. The second constraint requires V(s) to be at least as large as the backed-up value of any alternative optimal actions ao. If V(s) is too small, it will be penalized, because the slack variable y(s, ao) will be non-zero. But there is no penalty if V(s) is too large. The main effect of this constraint is to drive the value of V(s') downward as necessary to satisfy the first constraint on a*. Finally, the third constraint requires that V(s) be at least € larger than the backed-up value of all inferior actions a_. If these constraints can be satisfied with all slack variables u, v, y, and z set to zero, then V satisfies the Bellman equation. After applying the kernel trick and introducing the regularization objective, we obtain the following Bellman formulation: minimize ~ aj + C (,~_ u(s, a*) + v(s, a*) + y(s, ao) + z(s, a_)) subject to ~a.j [K(X(Sj),X(S)) - LP(s'ls,a*)K(X(Sj),X(S'))] + J s' u(s, a*) - v(s, a*) == L P(s'ls, a*)R(s'ls, a*) 8' ~aj [K(X(Sj),X(S)) - LP(s'ls,ao)K(X(Sj),X(S'))] +y(s,ao) J ~ ~ LP(s'ls,ao)R(s'ls,ao) s' ~O:j [K(X(Sj),X(S)) - LP(S'IS,a_)K(X(Sj),X(S'))] +z(s,a_) 3 ~ ~ LP(s'ls,a_)R(s'ls,a_) +£ 8/ Formulation 3: Advantage Learning. The third formulation focuses on the minimal constraints that must be satisfied to ensure that the greedy policy computed from V will be identical to the greedy policy computed from V (cf. Utgoff & Saxena, 1987). Specifically, we require that the backed up value of the optimal action a* be greater than the backed up values of all other actions a. minimize L u(s,a*,a) s,a*,a subject to L P(s'ls, a*)[R(s'ls, a*) + V(s')] + u(s, a*, a) 8/ ~ LP(s!ls,a)[R(s!ls,a) + V(s/)] +£ s/ There is one constraint and one slack variable u(s, a*, a) for every action executable in state s except for the chosen optimal action a* = i"(s). The backed-up value of a* must have an advantage of at least € over any other action a, even other actions that, according to V, are just as good as a*. After applying the kernel trick and incorporating the complexity penalty, this becomes minimize Laj+C L u(s,a*,a) j s,a*,a subject to Laj L[P(s'ls,a*) -P(s'ls,a)]K(X(sj),X(s')) +u(s,a*,a) ~ j s/ L P(s'ls, a)R(s'ls, a) - L P(s'ls, a*)R(s'ls, a*) + £ s/ s/ Of course each of these formulations can easily be modified to incorporate a discount factor for discounted cumulative reward. 4 Experimental Results To compare these three formulations, we generated a set of 10 random maze problems as follows. In a 100 by 100 maze, the agent starts in a randomly-chosen square in the left column, (0, y). Three actions are available in every state, east, northeast, and southeast, which deterministically move the agent one square in the indicated direction. The maze is filled with 3000 rewards (each of value -5) generated randomly from a mixture of a uniform distribution (with probability 0.20) and five 2-D gaussians (each with probability 0.16) centered at (80,20), (80,60), (40,20), (40,80), and (20,50) with variance 10 in each dimension. Multiple rewards generated for a single state are accumulated. In addition, in column 99, terminal rewards are generated according to a distribution that varies from -5 to +15 with minima at (99,0), (99,40), and (99,80) and maxima at (99,20) and (99,60). Figure 1 shows one of the generated mazes. These maze problems are surprisingly hard because unlike "traditional" mazes, they contain no walls. In traditional n;tazes, the walls tend to guide the agent to the goal states by reducing what would be a 2-D random walk to a random walk of lower dimension (e.g., 1-D along narrow halls). 10 20 30 40 50 60 70 80 90 100 100 Rewards 90 -5 0 -10 + 80 -15 )( -20 3IE 70 CZl CZl (1) 60 ~ ~ Vi Vi bJJ 50 ~ .s ~ ~ .§ Vi 40 (1) ~ 30 20 10 Figure 1: Example randomly-generated maze. Agent enters at left edge and exits at right edge. We applied the three LP formulations in an incremental-batch method as shown in Table 1. The LPs were solved using the CPLEX package from ILOG. The V giving the best performance on the starting states in So over the 20 iterations was saved and evaluated over all 100 possible starting states to obtain a measure of generalization. The values of C and a were determined by evaluating generalization on a holdout set of 3 start states: (0,30), (0,50), and (0,70). Experimentation showed that C = 100,000 worked well for all three methods. We tuned 0-2 separately for each problem using values of 5, 10, 20, 40, 60, 80, 120, and 160; larger values were preferred in case of ties, since they give better generalization. The results are summarized in Figure 2. The figure shows that the three methods give essentially identical performance, and that after 3 examples, all three methods have a regret per start state of about 2 units, which is less than the cost of a single -5 penalty. However, the three formulations differ in their ease of training and in the information they require. Table 2 compares training performance in terms of (a) the CPU time required for training, (b) the number of support vectors constructed, (c) the number of states in which V prefers a tied-optimal action over the action chosen by n-, (d) the number of states in which V prefers an inferior action, and (e) the number of iterations performed after the best-performing iteration on the training set. A high score on this last measure indicates that the learning algorithm is not converging well, even though it may momentarily attain a good fit to the data. By virtually every measure, the advantage formulation scores better. It requires much less CPU time to train, finds substantially fewer support vectors, finds function approximators that give better fit to .the data, and tends to converge better. In addition, the advantage· Table 1:__ Incremental Batch Reinforcement Learning Repeat 20 times: For each start state So E 80 do Generate 16 f-greedy trajectories using V Record all transitions and rewards to build MDP model if Solve M via value "iteration to obtain V and 7r L=0 For each start state 80 E 80 do Generate trajectory according to -IT Add to L all states visited along this trajectory Apply LP method to L, V, and 7r to find new V Perform Monte Carlo rollouts using greedy policy for V to evaluate each possible start state Report total value of all start states. Table 2: Measures of the quality of the training process (average over 10 MDPs) 1801= 1 1801 = 2 CPU #SV #tie #bad #iter CPU #SV #tie #bad #iter Sup 37.5 29.5 22.4 0.7 5.6 190.7 54.3 49.8 1.9 7.3 Bel 30.4 40.9 18.8 0.9 5.9 92.7 51.1 47.9 0.4 8.2 Adv 11.7 17.2 19.4 0.2 1.6 38.4 39.6 29.1 1.4 2.0 1801= 3 1801 =4 CPU #SV #tie #bad #iter CPU #SV #tie #bad #iter Sup 433.2 105.5 70.5 3.0 10.5 789.1 117.2 90.5 3.3 9.6 Bel 208.0 82.4 62.0 2.2 3.3 379.1 145.7 75.2 1.8 7.3 Adv 74.5 58.6 46.7 0.6 4.0 122.4 74.0 51.9 3.2 2.8 and Bellman formulations do not require the value of V, but only -fr. This makes them suitable for learning to imitate a human-supplied policy. 5 Conclusions This paper has presented three formulations. of batch value function approximation by exploiting the power of linear programming to express a variety of constraints and borrowing the kernel trick from support vector machines. All three formulations were able to learn and generalize well on difficult synthetic maze problems. The advantage formulation is easier and more reliable to train, probably because it places fewer constraints on the value function approximation. Hence, we are now applying the advantage formulation to combinatorial optimization problems in scheduling and protein structure determination~ 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 E1A-9818414. We thank Valentina Zubek and Adam Ashenfelter for their careful reading of the paper. 1200 ~ 1000 u ~00. ~.s 800 a 0 B '"d Q) 600 aa 0 ~ ...... 400 Q.) l-I b1) Q) l-I ~ t (5 200 ~ 0 0 2 3 4 5 Number of Starting States Figure 2: Comparison of the total regret (optimal total reward attained total reward) summed over all 100 starting states for the three formulations as a function of the number of start states in So. The three error bars represent the performance of the supervised, Bellman, and advantage formulations (left-to-right). The bars plot the 25th, 50th, and 75th percentiles computed over 10 randomly generated mazes. Average optimal total reward on these problems is 1306. The random policy receives a total reward of -14,475. References Baird, L. C. (1993). Advantage updating. Tech. rep. 93-1146, Wright-Patterson AFB. Moll, R., Barto, A. G., Perkins, T. J., & Sutton, R. S. (1999). Learning instanceindependent value functions to enhance local search. NIPS-II, 1017-1023. Tesauro, G. (1995). Temporal difference learning and TD-Gammon. CACM, 28(3), 58-68. Utgoff, P. E., & Saxena, S. (1987). Learning a preference predicate. In ICML-87, 115-121. Vapnik, V. (2000). The Nature of Statistical Learning Theory, 2nd Ed. Springer. Zhang, W., & Dietterich, T. G. (1995). A reinforcement learning approach to jobshop scheduling. In IJCAI95, 1114-1120.
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PAC Generalization Bounds for Co-training Sanjoy Dasgupta AT&T Labs–Research dasgupta@research.att.com Michael L. Littman AT&T Labs–Research mlittman@research.att.com David McAllester AT&T Labs–Research dmac@research.att.com Abstract The rule-based bootstrapping introduced by Yarowsky, and its cotraining variant by Blum and Mitchell, have met with considerable empirical success. Earlier work on the theory of co-training has been only loosely related to empirically useful co-training algorithms. Here we give a new PAC-style bound on generalization error which justifies both the use of confidences — partial rules and partial labeling of the unlabeled data — and the use of an agreement-based objective function as suggested by Collins and Singer. Our bounds apply to the multiclass case, i.e., where instances are to be assigned one of labels for . 1 Introduction In this paper, we study bootstrapping algorithms for learning from unlabeled data. The general idea in bootstrapping is to use some initial labeled data to build a (possibly partial) predictive labeling procedure; then use the labeling procedure to label more data; then use the newly labeled data to build a new predictive procedure and so on. This process can be iterated until a fixed point is reached or some other stopping criterion is met. Here we give PAC style bounds on generalization error which can be used to formally justify certain boostrapping algorithms. One well-known form of bootstrapping is the EM algorithm (Dempster, Laird and Rubin, 1977). This algorithm iteratively updates model parameters by using the current model to infer (a probability distribution on) labels for the unlabeled data and then adjusting the model parameters to fit the (distribution on) filled-in labels. When the model defines a joint probability distribution over observable data and unobservable labels, each iteration of the EM algorithm can be shown to increase the probability of the observable data given the model parameters. However, EM is often subject to local minima — situations in which the filled-in data and the model parameters fit each other well but the model parameters are far from their maximum-likelihood values. Furthermore, even if EM does find the globally optimal maximum likelihood parameters, a model with a large number of parameters will over-fit the data. No PAC-style guarantee has yet been given for the generalization accuracy of the maximum likelihood model. An alternative to EM is rule-based bootstrapping of the form used by Yarowsky (1995), in which one assigns labels to some fraction of a corpus of unlabeled data and then infers new labeling rules using these assigned labels as training data. New labels lead to new rules which in turn lead to new labels, and so on. Unlike EM, rule-based bootstrapping typically does not attempt to fill in, or assign a distribution over, labels unless there is compelling evidence for a particular label. One intuitive motivation for this is that by avoiding training on low-confidence filled-in labels one might avoid the self-justifying local optima encountered by EM. Here we prove PAC-style generalization guarantees for rulebased bootstrapping. Our results are based on an independence assumption introduced by Blum and Mitchell (1998) which is rather strong but is used by many successful applications. Consider, for example, a stochastic context-free grammar. If we generate a parse tree using such a grammar then the nonterminal symbol labeling a phrase separates the phrase from its context — the phrase and the context are statistically independent given the nonterminal symbol. More intuitively, in natural language the distribution of contexts into which a given phrase can be inserted is determined to some extent by the “type” of the phrase. The type includes the syntactic category but might also include semantic subclassifications, for instance, whether a noun phrase refers to a person, organization, or location. If we think of each particular occurrence of a phrase as a triple
, where is the phrase itself, is the “type” of the phrase, and is the context, then we expect that is conditionally independent of given . The conditional independence can be made to hold precisely if we generate such triples using a stochastic context free grammar where is the syntactic category of the phrase. Blum and Mitchell introduce co-training as a general term for rule-based bootstrapping in which each rule must be based entirely on or entirely on . In other words, there are two distinct hypothesis classes, which consists of functions predicting from , and which consists of functions predicting from . A co-training algorithm bootstraps by alternately selecting and . The principal assumption made by Blum and Mitchell is that is conditionally independent of given . Under such circumstances, they show that, given a weak predictor in , and given an algorithm which can learn under random misclassification noise, it is possible to learn a good predictor in . This gives some degree of justification for the co-training restriction on rule-based bootstrapping. However, it does not provide a bound on generalization error as a function of empirically measurable quantities. Furthermore, there is no apparent relationship between this PAC-learnability theorem and the iterative co-training algorithm they suggest. Collins and Singer (1999) suggest a refinement of the co-training algorithm in which one explicitly optimizes an objective function that measures the degree of agreement between the predictions based on and those based on . They describe methods for “boosting” this objective function but do not provide any formal justification for the objective function itself. Here we give a PAC-style performance guarantee in terms of this agreement rate. This guarantee formally justifies the Collins and Singer suggestion. In this paper, we use partial classification rules, which either output a class label or output a special symbol indicating no opinion. The error of a partial rule is the probability that the rule is incorrect given that it has an opinion. We work in the co-training setting where we have a pair of partial rules and where (sometimes) predicts from and (sometimes) predicts from . Each of the rules and can be “composite rules”, such as decision lists, where each composite rule contains a large set of smaller rules within it. We give a bound on the generalization error of each of the rules and in terms of the empirical agreement rate between the two rules. This bound formally justifies both the use h1 h2 X X2 1 Y Figure 1: The co-training scenario with rules and . of agreement in the objective function and the use of partial rules. The bound shows the potential power of unlabeled data — low generalization error can be achieved for complex rules with a sufficient quantity of unlabeled data. The use of partial rules is analogous to the use of confidence ratings — a partial rule is just a rule with two levels of confidence. So the bound can also be viewed as justifying the partial labeling aspect of rule-based bootstrapping, at least in the case of co-training where an independence assumption holds. The generalization bound leads naturally to algorithms for optimizing the bound. A simple greedy procedure for doing this is quite similar to the co-training algorithm suggested by Collins and Singer. 2 The Main Result We start with some basic definitions and observations. Let be an i.i.d. sample consisting of individual samples , , . For any statement we let
be the subset . For any two statements and we define the empirical estimate to be
!
. For the co-training bounds proved here we assume data is drawn from some distribution over triples
with #" %$ , and &" , and where and are conditionally independent given , that is, (' and )' . In the co-training framework we are given an unlabeled sample +* of pairs
drawn i.i.d. from the underlying distribution, and possibly some labeled samples +, . We will mainly be interested in making inferences from the unlabeled data. A partial rule on a set " is a mapping from " to %$ - . We will be interested in pairs of partial rules and which largely agree on the unlabeled data. The conditional probability relationships in our scenario are depicted graphically in figure 1. Important intuition is given by the data-processing inequality of information theory (Cover and Thomas, 1991): ./ 10 .2 30 . In other words, any mutual information between and must be mediated through . In particular, if and agree to a large extent, then they must reveal a lot about . And yet finding such a pair requires no labeled data at all. This simple observation is a major motivation for the proof, but things are complicated considerably by partial rules and by approximate agreement. For a given partial rule with (4 ' 6587 define a function 9 on %$ by 9: <;<='?>A@BDC E1F HGJIKGML 'ON P'?;< We want to be a nearly deterministic function of the actual label ; in other words, we want 'Q9: 4 ' to be near one. We would also like to carry the same information as . This is equivalent to saying that 9 should be a permutation of the possible labels A$ . Here we give a condition using only unlabeled data which guarantees, up to high confidence, that 9 is a permutation; this is the best we can hope for using unlabeled data alone. We also bound the error rates D4 ' N :9: ' N 4 ' using only unlabeled data. In the case of ' , if 9 is a permutation then 9 is either the identity function or the function reversing the two possible values. We use the unlabeled data to select and so that 9 is a permutation and has low error rate. We can then use a smaller amount of labeled data to determine which permutation we have found. We now introduce a few definitions related to sampling issues. Some measure of the complexity of rules and is needed; rather than VC dimension, we adopt a clean notion of bit length. We assume that rules are specified in some rule language and write for the number of bits used to specify the rule . We assume that the rule language is prefix-free (no proper prefix of the bit string specifying a rule is itself a legal rule specification). A prefix free code satisfies the Kraft inequality $ . For given partial rules and and N A$ we now define the following functions of the sample . The first, as we will see, is a bound on the sampling error for empirical probabilities conditioned upon 'ON 4 ' . The second is a sampling-adjusted disagreement rate between and .
! #"$
% %$&'% (% )&* ! + , "-% .
0/1 32 5467% 8
9
10/%:;/ ) 2 '46< 9
) 2 0/=%:0/> 2 543?<@"
A( Note that if the sample size is sufficiently large (relative to and % ) then BHI $C3 is near zero. Also note that if and have near perfect agreement when they both are not then D I $C3 is near one. We can now state our main result. Theorem 1 With probability at least $FE5C over the choice of the sample , we have that for all and , if D I GC3
5 7 for $ N then (a) 9 is a permutation and (b) for all $ N , -4 '?N639: ='ON -4 ' -4 '?N6 ' N -4 ' IH5B I GC3 D I GC3 The theorem states, in essence, that if the sample size is large, and and largely agree on the unlabeled data, then 4 ' N6 '?N -4 ' is a good estimate of the error rate -4 '?N639: ='ON -4 ' . The theorem also justifies the use of partial rules. Of course it is possible to convert a partial rule to a total rule by forcing a random choice when the rule would otherwise return . Converting a partial rule to a total rule in this way and then applying the above theorem to the total rule gives a weaker result. An interesting case is when ' , is total and is a perfect copy of , and -4 ' happens to be $1! J . In this case the empirical error rate of the corresponding total rule — the rule that guesses when has no opinion — will be statistically indistinguishable from from 1/2. However, in this case theorem 1 can still establish that the false positive and false negative rate of the partial rule is near zero. 3 The Analysis We start with a general lemma about conditional probability estimation. Lemma 2 For any i.i.d. sample , and any statements and about individual instances in the sample, the following holds with probability at least $KELD over the choice of . M M M A 1E M M M NPO !D
(1) Proof. We have the following where the third step follows by the Chernoff bound. M M M A 1E M M M 5
' L
' M M M 1E M M M 5
> M M M M
' L
' D ' D Therefore, with probability at least $KE 1 C ! , ' N 'ON 4 ' 1E 'ON6 'ON -4 ' B I GC3 (2) for any given . By the union bound and the Kraft inequality, we have that with probability at least $ E C this must hold simultaneously for all and , and all $ N . Lemma 3 Pick any rules and for which equation (2) as well as D I GC35 7 hold for all $ N . Then 9 is a permutation, and moreover, for any N , '?N 39: ='?N -4 ' 5 $1! Proof. Pick any N %$ . We need to show that there exists some such that 9: ='?N . By the definition of D I and condition (2) we know
'?N ' N 4 ' E 4 '?N ' N 4 ' D I $C3 Since D I GC3
587 , it follows that '?N6 '?N 4 ' 5 $! . Rewriting this by conditioning on , we get G GML P' 'ON -4 ' 'ON P'! -4 ' 5 $! The summation is a convex combination; therefore there must exist some such that 'QN ' )4 ' 5 $! . So for each N there must exist a with 9: " %'QN , whereby 9 is a permutation. Lemma 4 Pick any rules and satisfying the conditions of the previous lemma. Then <9: ='ON 'ON -4 ' is at least D I $C3 . Proof. By the previous lemma 9 is a permutation, so 9: has the same information content as . Therefore and are conditionally independent given 9: . For any N , D I GC3 '?N ' N -4 ' 1E -4 '?N6 ' N -4 ' ' <9: ='! 'ON -4 ' 'ON A9: '! -4 ' E -4 ' N6 9: ='! -4 ' <9: ='ON 'ON -4 ' 'ON639: ='?N -4 ' E -4 ' N39: ='?N 4 ' H # $ I 9: ='% 'ON 4 ' 'ON639: =' 4 ' E
'% 39: =' 4 ' where the second step involves conditioning on 9: . Also by the previous lemma, we have ' & 9: ' 4 ' 5 $! so the second term in the above sum must be negative, whereby D I GC3 <9: ='ON6 '?N -4 ' 'ON 9: ='ON 4 ' E 4 '?N6 9: =' N -4 ' <9: ='ON6 '?N -4 ' Under the conditions of these lemmas, we can derive the bounds on error rates: -4 'ON39: =' N -4 ' 4 ' N6 '?N 4 ' 9: ='ON6 ' N 4 ' 4 '?N6 ' N 4 ' IH5BHI GC3 D I GC3 4 Bounding Total Error Assuming that we make a random guess when ' , the total error rate of can be written as follows. =' -4 ' -4 ' 9:
-4 ' =H E8$ ' To give a bound on the total error rate we first define I GC3 to be the bounds on the error rate for label N given in theorem 1. I GC3 ' $ D I GC3 4 'ON 'ON 4 ' IH'BHI GC3 We can now state a bound on the total error rate as a corollary to theorem 1. Corollary 5 With probability at least $ E C over the choice of we have the following for all pairs of rules and such that for all N we have D I GC ! 5 7 and I GC ! E8$ ! . (4 ' E B GC ! C E1F GC ! H E8$ ' IH'B $C ! B $C3 ' NPO H N O ! C Proof. From our main theorem, we know that with probability at least $ E C ! , for all N . #4 ' N-M9: ' N #4 ' is bounded by I $C ! . This implies that with probability at least $KE0C ! , 4 ' C E1F A GC ! IH E8$ $KE 4 ' (3) With probability at least $ E C ! we have that 4 ' E 4 ' is no larger than B GC ! . So by the union bound both of these conditions hold simultaneously with probability at least $KE0C . Since CPE3F% 3 $C ! E8$ ! we have that the upper bound in (3) is maximized by setting -4 ' equal to (4 ' E B GC ! . Corollary 5 can be improved in a variety of ways. One could use a relative Chernoff bound to tighten the uncertainty in 4 ' in the case where this probability is small. One could also use the error rate bounds I $C ! to construct bounds on 9: =' N6 -4 ' . One could then replace the max over I $C ! by a convex combination. Another approach is to use the error rate of a rule that combines and , e.g., the rule outputs if &4 ' , otherwise outputs if )4 ' , and otherwise guesses a random value. This combined rule will have a lower error rate and it is possible to give bounds on the error rate of the combined rule. We will not pursue these refined bounds here. It should be noted, however, that the algorithm described in section 4 can be used with any bound on total error rate. 5 A Decision List Algorithm This section suggests a learning algorithm inspired by both Yarowsky (1995) and Collins and Singer (1999) but modified to take into account theorem 1 and Corollary 5. Corollary 5, or some more refined bound on total error, provides an objective function that can be pursued by a learning algorithm — the objective is to find and so as to minimize the upper bound on the total error rate. Typically, however, the search space is enormous. Following Yarowsky, we consider the greedy construction of decision list rules. Let and be two “feature sets” such that for and " we have 7 $ and for and )" we have 7 $3 . We assume that is to be a decision list over the features in , i.e., a finite sequence of the form
; ; /L L
where /I and 3I A$ . A decision list can be viewed as a right-branching decision tree. More specifically, if is the list
;
; ; /L L
then 1 is if 1 'Q$ and otherwise equals the value of the list
; ; JL L
on . We define an empty decision list to have value . For B in <7 $ we can define as follows where is the number of feature-value pairs in . ' N $ B H N 3 $KE0B It is possible to show that 1 equals the probability that a certain stochastic process generates the rule . This implies the Kraft inequality I $ which is all that is needed in theorem 1 and corollary 5. We also assume that is a decision list over the features and define similarly. Following Yarowsky we suggest growing the decision lists in a greedy manner adding one feature value pair at a time. A natural choice of greedy heuristic might be a bound on the total error rate. However, in many cases the final objective function is not an appropriate choice for the greedy heuristic in greedy algorithms. A* search, for example, might be viewed as a greedy heuristic where the heuristic function estimates the number of steps needed to reach a high-value configuration — a low value configuration might be one step away from a high value configuration. The greedy heuristic used in greedy search should estimate the value of the final configuration. Here we suggest using CPE3F GC ! as a heuristic estimate of the final total error rate — in the final configuration we should have that -4 ' is reasonably large and the important term will be C E1F GC3 . For concreteness, we propose the following algorithm. Many variants of this algorithm also seem sensible. 1. Initialize and to “seed rule” decision lists using domain-specific prior knowledge. 2. Until ' and ' are both zero, or all features have been used in both rules, do the following. (a) Let B denote if ' 65 ' and otherwise. (b) If D I GC ! 7 for some N , then extend B by the pair
which most increases C O I D I $C ! . (c) Otherwise extend B by a single feature-value pair selected to minimize the C E1FA A GC ! . 3. Prune the rules — iteratively remove the pair from the end of either or that greedily minimizes the bound on total error until no such removal reduces the bound. 6 Future Directions We have given some theoretical justification for some aspects of co-training algorithms that have been shown to work well in practice. The co-training assumption we have used in our theorems are is at best only approximately true in practice. One direction for future research is to try to relax this assumption somehow. The co-training assumption states that and are independent given . This is equivalent to the statement that the mutual information between and given is zero. We could relax this assumption by allowing some small amount of mutual information between and given and giving bounds on error rates that involve this quantity of mutual information. Another direction for future work, of course, is the empirical evaluation of co-training and bootstrapping methods suggested by our theory. Acknowledgments The authors wish to acknowledge Avrim Blum for useful discussions and give special thanks to Steve Abney for clarifying insights. Literature cited Blum, A. & Mitchell, T. (1998) Combining labeled and unlabeled data with co-training. COLT. Collins, M. & Singer, Y. (1999) Unsupervised models for named entity classification. EMNLP. Cover, T. & Thomas, J. (1991) Elements of information theory. Wiley. Dempster, A., Laird, N. & Rubin, D. (1977) Maximum-likelihood from incomplete data via the EM algorithm. J. Royal Statist. Soc. Ser. B, 39:1-38. Nigam, K. & Ghani, R. (2000) Analyzing the effectiveness and applicability of co-training. CIKM. Yarowsky, D. (1995) Unsupervised word sense disambiguation rivaling supervised methods. ACL.
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Face Recognition Using Kernel Methods Ming-Hsuan Yang Honda Fundamental Research Labs Mountain View, CA 94041 myang@hra.com Abstract Principal Component Analysis and Fisher Linear Discriminant methods have demonstrated their success in face detection, recognition, and tracking. The representation in these subspace methods is based on second order statistics of the image set, and does not address higher order statistical dependencies such as the relationships among three or more pixels. Recently Higher Order Statistics and Independent Component Analysis (ICA) have been used as informative low dimensional representations for visual recognition. In this paper, we investigate the use of Kernel Principal Component Analysis and Kernel Fisher Linear Discriminant for learning low dimensional representations for face recognition, which we call Kernel Eigenface and Kernel Fisherface methods. While Eigenface and Fisherface methods aim to find projection directions based on the second order correlation of samples, Kernel Eigenface and Kernel Fisherface methods provide generalizations which take higher order correlations into account. We compare the performance of kernel methods with Eigenface, Fisherface and ICA-based methods for face recognition with variation in pose, scale, lighting and expression. Experimental results show that kernel methods provide better representations and achieve lower error rates for face recognition. 1 Motivation and Approach Subspace methods have been applied successfully in numerous visual recognition tasks such as face localization, face recognition, 3D object recognition, and tracking. In particular, Principal Component Analysis (PCA) [20] [13],and Fisher Linear Discriminant (FLD) methods [6] have been applied to face recognition with impressive results. While PCA aims to extract a subspace in which the variance is maximized (or the reconstruction error is minimized), some unwanted variations (due to lighting, facial expressions, viewing points, etc.) may be retained (See [8] for examples). It has been observed that in face recognition the variations between the images of the same face due to illumination and viewing direction are almost always larger than image variations due to the changes in face identity [1]. Therefore, while the PCA projections are optimal in a correlation sense (or for reconstruction" from a low dimensional subspace), these eigenvectors or bases may be suboptimal from the classification viewpoint. Representations of Eigenface [20] (based on PCA) and Fisherface [6] (based on FLD) methods encode the pattern information based on the second order dependencies, i.e., pixelwise covariance among the pixels, and are insensitive to the dependencies among multiple (more than two) pixels in the samples. Higher order dependencies in an image include nonlinear relations among the pixel intensity values, such as the relationships among three or more pixels in an edge or a curve, which can capture important information for recognition. Several researchers have conjectured that higher order statistics may be crucial to better represent complex patterns. Recently, Higher Order Statistics (HOS) have been applied to visual learning problems. Rajagopalan et ale use HOS of the images of a target object to get a better approximation of an unknown distribution. Experiments on face detection [16] and vehicle detection [15] show comparable, if no better, results than other PCA-based methods. The concept of Independent Component Analysis (ICA) maximizes the degree of statistical independence of output variables using contrast functions such as Kullback-Leibler divergence, negentropy, and cumulants [9] [10]. A neural network algorithm to carry out ICA was proposed by Bell and Sejnowski [7], and was applied to face recognition [3]. Although the idea of computing higher order moments in the ICA-based face recognition method is attractive, the assumption that the face images comprise of a set of independent basis images (or factorial codes) is not intuitively clear. In [3] Bartlett et ale showed that ICA representation outperform PCA representation in face recognition using a subset of frontal FERET face images. However, Moghaddam recently showed that ICA representation does not provide significant advantage over PCA [12]. The experimental results suggest that seeking non-Gaussian and independent components may not necessarily yield better representation for face recognition. In [18], Sch6lkopf et ale extended the conventional PCA to Kernel Principal Component Analysis (KPCA). Empirical results on digit recognition using MNIST data set and object recognition using a database of rendered chair images showed that Kernel PCA is able to extract nonlinear features and thus provided better recognition results. Recently Baudat and Anouar, Roth and Steinhage, and Mika et ale applied kernel tricks to FLD and proposed Kernel Fisher Linear Discriminant (KFLD) method [11] [17] [5]. Their experiments showed that KFLD is able to extract the most discriminant features in the feature space, which is equivalent to extracting the most discriminant nonlinear features in the original input space. In this paper we seek a method that not only extracts higher order statistics of samples as features, but also maximizes the class separation when we project these features to a lower dimensional space for efficient recognition. Since much of the important information may be contained in the high order dependences among the pixels of a: face image, we investigate the use of Kernel PCA and Kernel FLD for face recognition, which we call Kernel Eigenface and Kernel Fisherface methods, and compare their performance against the standard Eigenface, Fisherface and ICA methods. In the meanwhile, we explain why kernel methods are suitable for visual recognition tasks such as face recognition. 2 Kernel Principal Component Analysis Given a set of m centered (zero mean, unit variance) samples Xk, Xk == [Xkl, ... ,Xkn]T ERn, PCA aims to find the projection directions that maximize the variance, C, which is equivalent to finding the eigenvalues from the covariance matrix AW=CW (1) for eigenvalues A ~ 0 and eigenvectors W E Rn. In Kernel PCA, each vector x is projected from the input space, Rn, to a high dimensional feature space, Rf, by a nonlinear mapping function: <t> : Rn -+ Rf, f ~ n. Note that the dimensionality of the feature space can be arbitrarily large. In Rf, the corresponding eigenvalue problem is "AW4> = C4>w4> (2) where C4> is a covariance matrix. All solutions weI> with A =I- 0 lie in the span of <t>(x1), ..., <t>(Xm ), and there exist coefficients ai such that m w4> = E ai<t>(xi) i=l Denoting an m x m matrix K by K·· - k(x· x·) - <t>(x·)· <t>(x·) ~1 ~'1 ~ 1 , the Kernel PCA problem becomes mAKa =K2a mAa =Ka (3) (4) (5) (6) where a denotes a column vector with entries aI, ... , am. The above derivations assume that all the projected samples <t>(x) are centered in Rf. See [18] for a ~ethod to center the vectors <t>(x) in Rf. Note that conventional PCA is a special case of Kernel PCA with polynomial kernel of first order. In other words, Kernel PCA is a generalization of conventional PCA since different kernels can be utilized for different nonlinear projections. We can now project the vectors in Rf to a lower dimensional space spanned by the eigenvectors weI>, Let x be a test sample whose projection is <t>(x) in Rf, then the projection of <t>(x) onto the eigenvectors weI> is the nonlinear principal components corresponding to <t>: m m w4> . <t>(x) =E ai (<t>(Xi) . <t>(x)) = E aik(xi, x) i=l i=l (7) In other words, we can extract the first q (1 ~ q ~ m) nonlinear principal components (Le., eigenvectors w4» using the kernel function without the expensive operation that explicitly projects the samples to a high dimensional space Rf" The first q components correspond to the first q non-increasing eigenvalues of (6). For face recognition where each x encodes a face image, we call the extracted nonlinear principal components Kernel Eigenfaces. 3 Kernel Fisher Linear Discriminant Similar to the derivations in Kernel PCA, we assume the projected samples <t>(x) are centered in Rf (See [18] for a method to center the vectors <t>(x) in Rf), we formulate the equations in a way that use dot products for FLD only. Denoting the within-class and between-class scatter matrices by S~ and SiJ, and applying FLD in kernel space, we need to find eigenvalues A and eigenvectors weI> of AS~WeI> = siJweI> (8) (9) , which can be obtained by <P I(W<P)TS~W<P I [<P <P WOPT =argw;x I(Wq,)TS~Wq,1 = W l W 2 ... w;.] where {w[Ii == 1, 2, ... ,m} is the set of generalized eigenvectors corresponding to the m largest generalized eigenvalues {Ai Ii == 1,2, ... ,m}. For given classes t and u and their samples, we define the kernel function by Let K be a m x m matrix defined by the elements (Ktu)~1;:::,cc, where Ktu is a matrix composed of dot products in the feature space Rf, Le., K == (K )=l, ,c where K == (k )r=l, ,lt tu u=l, ,c tu rs s=l, ,I'U (11) Note K tu is a It x Iu matrix, and K is a m x m symmetric matrix. We also define a matrix Z: (12) where (Zt) is a It x It matrix with terms all equal to ~, Le., Z is a m x m block diagonal matrix. The between-class and within-class scatter matrices in a high dimensional feature space Rf are defined as c siJ == L liJ.ti (p/f)T i=l C Ii S~ == L L ep(Xij )~(Xij)T i=l j=l (13) (14) where pi is the mean of class i in Rf, Ii is the number of samples belonging to class i. From the theory of reproducing kernels, any solution w<P E Rf must lie in the span of all training samples in Rf, Le., c Ip w<P == L L cy'pqep(xpq ) p=lq=l It follows that we can get the solution for (15) by solving: AKKa==KZKa Consequently, we can write (9) as <P I(WifJ)T sifJwifJl WOPT == argmaxwifJ I(WifJ)TS!WifJ I laKZKal == argmaxw«p laKKal == [wi ... w~] (15) (16) (17) We can project ~(x) to a lower dimensional space spanned by the eigenvectors w<P in a way similar to Kernel PCA (See Section 2). Adopting the same technique in the Fisherface method (which avoids singularity problems in computing W6PT) for face recognition [6], we call the extracted eigenvectors in (17) Kernel Fisherfaces. 4 Experiments We test both kernel methods against standard rCA, Eigenface, and Fisherface methods using the publicly available AT&T and Yale databases. The face images in these databases have several unique characteristics. While the images in the AT&T database contain the facial contours and vary in pose as well scale, the face images in the Yale database have been cropped and aligned. The face images in the AT&T database were taken under well controlled lighting conditions whereas the images in the Yale database were acquired under varying lighting conditions. We use the first database as a baseline study and then use the second one to evaluate face recognition methods under varying lighting conditions. 4.1 Variation in Pose and Scale The AT&T (formerly Olivetti) database contains 400 images of 40 subjects. To .reduce computational complexity, each face image is downsampled to 23 x 28 pixels. We represent each image by a raster scan vector of the intensity values, .and then normalize them to be zero-mean vectors. The mean and standard deviation of Kurtosis of the face images are 2.08 and 0.41, respectively (the Kurtosis of a Gaussian distribution is 3). Figure 1 shows images of two subjects. In contrast to images of the Yale database, the images include the facial contours, and variation in pose as well as scale. However, the lighting conditions remain constant. Fig~re 1: Face images in the AT&T database (Left) and the Yale database (Right). The experiments are performed using the "leave-one-out" strategy: To classify an image of person, that image is removed from the training set of (m - 1) images and the projection matrix is computed. All the m images in the training set are projected to a reduced space using the computed projection matrix w or weI> and recognition is performed based on a nearest neighbor classifier. The number of principal components or independent components are empirically determined to achieve the lowest error rate by each method. Figure 2 shows the experimental results. Among all the methods, the Kernel Fisherface method with Gaussian kernel and second degree polynomial kernel achieve the lowest error rate. Furthermore, the kernel methods perform better than standard rCA, Eigenface and Fisherface methods. Though our experiments using rCA seem to contradict to the good empirical results reported in [3] [4] [2]' a close look at the data sets reveals a significant difference in pose and scale variation of the face images in the AT&T database, whereas a subset of frontal FERET face images with change of expression was used in [3] [2]. Furthermore, the comparative study on classification with respect to PCA in [4] (pp. 819, Table 1) and the errors made by two rCA algorithms in [2] (pp. 50, Figure 2.18) seem to suggest that lCA methods do not have clear advantage over other approaches in recognizing faces with pose and scale variation. 4.2 Variation in Lighting and Expression The Yale database contains 165 images of 11 subjects that includes variation in both facial expression and lighting. For computational efficiency, each image has been downsampled to 29 x 41 pixels. Likewise, each face image is represented by a Method I rCA Eigenface 30 2.75 (11/400) Fisherface 14 1.50 (6/400) Kernel Eigenface, d==2 50 2.50 (10/400) Kernel Eigenface, d==3 50 2.00 (8/400) Kernel Fisherface (P) 14 1.25 (5/400) Kernel Fisherface (G) 14 1.25 (5/400) Figure 2: Experimental results on AT&T database. centered vector of normalized intensity values. The mean and standard deviation of Kurtosis of the face images are 2.68 and 1.49, respectively. Figure 1 shows 22 closely cropped images of two subjects which include internal facial structures such as the eyebrow, eyes, nose, mouth and chin, but do not contain the facial contours. Using the same leave-one-out strategy, we experiment with the number of principal components and independent components to achieve the lowest error rates for Eigenface and Kernel Eigenface methods. For Fisherface and Kernel Fisherface methods, we project all the samples onto a subspace spanned by the c - 1 largest eigenvectors. The experimental results are shown in Figure 3. Both kernel methods perform better than standard ICA, Eigenface and Fisherface methods. Notice that the improvement by the kernel methods are rather significant (more than 10%). Notice also that kernel methods consistently perform better than conventional methods for both databases. The performance achieved by the ICA method indicates that face representation using independent sources is not effective when the images are taken under varying lighting conditions. Method Eigenface 30 28.48 (47/165) Fisherface 14 8.48 (14/165) Kernel Eigenface, d==2 80 27.27 (45/165) Kernel Eigenface, d==3 60 24.24 (40/165) Kernel Fisherface (P) 14 6.67 (11/165) Kernel Fisherface (G) 14 6.06 (10/165) I lCA 24.24 o. -< < ~~ -<,-.. Q Q,-.. § ~ u ~& ....:l ~& l:l-. ~s ~ Q S ~ 35 30 29.09 28..49 27.27 ~ 25 ~ 20 ~8 15 ~ 10 Figure 3: Experimental results on Yale database. Figure 4 shows the training samples of the Yale database projected onto the first two eigenvectors extracted by the Kernel Eigenface and Kernel Fisherface methods. The projected samples of different classes are smeared by the Kernel Eigenface method whereas the samples projected by the Kernel Fisherface are separated quite welL In fact, the samples belonging to the same class are projected to the same position by the largest two eigenvectors. This example provides an explanation to the good results achieved by the Kernel Fisherface method. The experimental results show that Kernel Eigenface and Fisherface methods are able to extract nonlinear features and achieve lower error rate. Instead of using a nearest neighbor classifier, the performance can potentially be improved by other classifiers (e.g., k-nearest neighbor and perceptron). Another potential improvement is to use all the extracted nonlinear components as features (Le., without projecting to a lower dimensional space) and use a linear Support Vector Machine (SVM) to construct a decision surface. Such a two-stage approach is, in spirit, similar to nonlinear SVMs in which the samples are first projected to a high dimensional feature space where a hyperplane with largest hyperplane is constructed. In fact, one important factor of the recent success in SVM applications for visual recognition is due to the use of kernel methods. ° class 1 ° r1-o+. + : ~::::~ I, ~ : , , ·,··· .. ···1 ~ 1-.......... :· ..~ ...... ·~~:-~"'-e>"0~O·"'....···:-·-¥.-(I·· .. · .. ;······O·.·· .. ; .... ······-:·· ..........:.. ·· ...... ·1 * c1ass4 0 ° ~ ~:::: ~ 1, ;. : : ; ·, .. ·· 1 y ° ~ ° V class? '-' 1:.. .§it'CIlIl''''''~IX'';:) : ; :.'" :.u··.. · ;.,,· .. ·.. · 1 A class8 0 <l 1iiI=-*$*_~" <l class9 I; ;: : ~ : · ·;· ·.. 1 t> 1:* : : ; : : ; ·.. 1 E~i:~~H ~ ~ ° class13 1, , ." 1> , , ·' .. ·· 1 * 5 ~ -21: : ; : : : ; ~~~~::~::~~~ ~ 01, : -: :.° * , , --=.,,,0.:--1 ~ ~ * ! 0 * * '* o 2 4 :0.08 -0.06 -0.04 -0.02 0.02 0.04 0.)6 O.DB (a) Kernel Eigenface method. (b) Kernel Fisherface method. Figure 4: Samples projected by Kernel PCA and Kernel Fisher methods. 5 Discussion and Conclusion The representation in the conventional Eigenface and Fisherface approaches is based on second order statistics of the image set, Le., covariance matrix, and does not use high order statistical dependencies such as the relationships among three or more pixels. For face recognition, much of the important information may be contained in the high order statistical relationships among the pixels. Using the kernel tricks that are often used in SVMs, we extend the conventional methods to kernel space where we can extract nonlinear features among three or more pixels. We have investigated Kernel Eigenface and Kernel Fisherface methods, and demonstrate that they provide a more effective representation for face recognition. Compared to other techniques for nonlinear feature extraction, kernel methods have the advantages that they do not require nonlinear optimization, but only the solution of an eigenvalue problem. Experimental results on two benchmark databases show that Kernel Eigenface and Kernel Fisherface methods achieve lower error rates than the ICA, Eigenface and Fisherface approaches in face recognition. The performance achieved by the ICA method also indicates that face representation using independent basis images is not effective when the images contain pose, scale or lighting variation. Our future work will focus on analyzing face recognition methods using other kernel methods in high dimensional space. We plan to investigate and compare the performance of other face recognition methods [14] [12] [19]. References [1] Y. Adini, Y. Moses, and S. Ullman. Face recognition: The problem of compensating for changes in illumination direction. IEEE PAMI, 19(7):721-732, 1997. [2] M. S. Bartlett. Face Image Analysis by Unsupervised Learning and Redundancy Reduction. PhD thesis, University of California at San Diego, 1998. [3] M. S. Bartlett, H. M. Lades, and T. J. Sejnowski. Independent component representations for face recognition. In Proc. of SPIE, volume 3299, pages 528-539, 1998. [4] M. S. Bartlett and T. J. Sejnowski. Viewpoint invariant face recognition using independent component analysis and attractor networks. In NIPS 9, page 817, 1997. [5] G. Baudat and F. Anouar. Generalized discriminant analysis using a kernel approach. Neural Computation, 12:2385-2404,2000. [6] P. Belhumeur, J. Hespanha, and D. Kriegman. Eigenfaces vs. Fisherfaces: Recognition using class specific linear projection. IEEE PAMI, 19(7):711-720, 1997. [7] A. J. Bell and T. J. Sejnowski. An information - maximization approach to blind separation and blind deconvolution. Neural Computation, 7(6):11291159, 1995. [8] C. 1\1. Bishop. fleural fretworks for .J.Dattern Recognition. Oxford University Press, 1995. [9] P. Comon. Independent component analysis: A new concept? Signal Processing, 36(3):287-314-, 1994. [10] A. Hyviirinen, J. Karhunen, and E. Oja. Independent Component Analysis. Wiley-Interscience, 2001. [11] S. Mika, G. Riitsch, J. Weston, B. Sch6lkopf, A. Smola, and K.-R. Muller. Invariant feature extraction and classification in kernel spaces. In NIPS 12, pages 526-532, 2000. [12] B. Moghaddam. Principal manifolds and bayesian subspaces for visual recognition. In Proc. IEEE Int'l Conf. on Computer Vision, pages 1131-1136,1999. [13] B. Moghaddam and A. Pentland. Probabilistic visual learning for object recognition. IEEE PAMI, 19(7):696-710, 1997. [14] P. J. Phillips. Support vector machines applied to face recognition. In NIPS 11, pages 803-809, 1998. [15] A. N. Rajagopalan, P. Burlina, and R. Chellappa. Higher order statistical learning for vehicle detection in images. In Proc. IEEE Int'l Con!. on Computer Vision, volume 2, pages 1204-1209,1999. [16] A. N. Rajagopalan, K. S. Kumar, J. Karlekar, R. Manivasakan, and M. M. Patil. Finding faces in photographs. In Proc. IEEE Int'l Conf. on Computer Vision, pages 640-645, 1998. [17] V. Roth and V. Steinhage. Nonlinear discriminant analysis using kernel functions. In NIPS 12, pages 568-574,2000. [18] B. Sch6lkopf, A. Smola, and K.-R. Muller. Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10(5):1299-1319,1998. [19] Y. W. Teh and G. E. Hinton. Rate-coded restricted Boltzmann machines for face recognition. In NIPS 13, pages 908-914, 2001. [20] M. Turk and A. Pentland. Eigenfaces for recognition. J. of Cognitive Neuroscience, 3(1):71-86, 1991.
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Keywords: portfolio management, financial forecasting, recurrent neural networks. Active Portfolio-Management based on Error Correction Neural Networks Hans Georg Zimmermann , Ralph Neuneier and Ralph Grothmann Siemens AG Corporate Technology D-81730 M¨unchen, Germany Abstract This paper deals with a neural network architecture which establishes a portfolio management system similar to the Black / Litterman approach. This allocation scheme distributes funds across various securities or financial markets while simultaneously complying with specific allocation constraints which meet the requirements of an investor. The portfolio optimization algorithm is modeled by a feedforward neural network. The underlying expected return forecasts are based on error correction neural networks (ECNN), which utilize the last model error as an auxiliary input to evaluate their own misspecification. The portfolio optimization is implemented such that (i.) the allocations comply with investor’s constraints and that (ii.) the risk of the portfolio can be controlled. We demonstrate the profitability of our approach by constructing internationally diversified portfolios across different financial markets of the G7 contries. It turns out, that our approach is superior to a preset benchmark portfolio. 1 Introduction: Portfolio-Management We integrate the portfolio optimization algorithm suggested by Black / Litterman [1] into a neural network architecture. Combining the mean-variance theory [5] with the capital asset pricing model (CAPM) [7], this approach utilizes excess returns of the CAPM equilibrium to define a neutral, well balanced benchmark portfolio. Deviations from the benchmark allocation are only allowed within preset boundaries. Hence, as an advantage, there are no unrealistic solutions (e. g. large short positions, huge portfolio changes). Moreover, there is no need of formulating return expectations for all assets. In contrast to Black / Litterman, excess return forecasts are estimated by time-delay recurrent error correction neural networks [8]. Investment decisions which comply with given allocation constraints are derived from these predictions. The risk exposure of the portfolio is implicitly controlled by a parameter-optimizing task over time (sec. 3 and 5). Our approach consists of the following three steps: (i.) Construction of forecast models
on the basis of error correction neural networks (ECNN) for all assets (sec. 2). To whom correspondence should be addressed: Georg.Zimmermann@mchp.siemens.de. (ii.) Computation of excess returns by a higher-level feedforward network (sec. 3 and 4). By this, the profitability of an asset with respect to all others is measured. (iii.) Optimization of the investment proportions on the basis of the excess returns. Allocation constraints ensure, that the investment proportions may deviate from a given benchmark only within predefined intervals (sec. 3 and 4). Finally, we apply our neural network based portfolio management system to an asset allocation problem concerning the G7 countries (sec. 6). 2 Forecasting by Error Correction Neural Networks Most dynamical systems are driven by a superposition of autonomous development and external influences [8]. For discrete time grids, such a dynamics can be described by a recurrent state transition
and an output equation (Eq. 1).
state transition eq. output eq. (1) The state transition
is a mapping from the previous state , external influences and a comparison between the model output and observed data . If the last model error is zero, we have a perfect description of the dynamics. However, due unknown external influences or noise, our knowledge about the dynamics is often incomplete. Under such conditions, the model error quantifies the model’s misfit and serves as an indicator of short-term effects or external shocks [8]. Using weight matrices "! # of appropriate dimensions corresponding to $ , $ and $%& $ , a neural network approach of Eq. 1 can be formulated as
'"(*)+ , . / .# '"(*)+ 0! * ! 1 (2) In Eq. 2, the output is recomputed by ! and compared to the observation 2 . Different dimensions in $ are adjusted by # . The system identification (Eq. 3) is a parameter optimization task of appropriate sized weight matrices "! # [8]: 3 54 6 7 98 * : ;1<,= >@? ) ACB DEB F:B G (3) For an overview of algorithmic solution techniques see [6]. We solve the system identification task of Eq. 3 by finite unfolding in time using shared weights. For details see [3, 8]. Fig. 1 depicts the resulting neural network solution of Eq. 3. zt−2 D C D C D t−1 s zt−1 ts zt st+2 C st+3 C t+3 y t+2 y st+1 C t+1 y A yt−2 d t−2 u ydt−1 ydt B ut ut−1 A A A B −Id −Id −Id B Figure 1. Error correction neural network (ECNN) using unfolding in time and overshooting. Note, that HJILK is the fixed negative of an appropriate sized identity matrix, while MON0PRQ are output clusters with target values of zero in order to optimize the error correction mechanism. The ECNN (Fig. 1) is best to comprehend by analyzing the dependencies of , , ! and
. The ECNN has two different inputs: the externals directly influencing the state transition and the targets . Only the difference between and has an impact on
[8]. At all future time steps - , we have no compensation of the internal expectations
$ and thus, the system offers forecasts
$ !
$ . A forecast of the ECNN is based on a modeling of the recursive structure of a dynamical system (coded in ), external influences (coded in ) and the error correction mechanism which is also acting as an external input (coded in ! , # ). Using finite unfolding in time, we have by definition an incomplete formulation of accumulated memory in the leftmost part of the network and thus, the autoregressive modeling is handicapped [9]. Due to the error correction, the ECNN has an explicit mechanism to handle the initialization shock of the unfolding [8]. The autonomous part of the ECNN is extended into the future by the iteration of matrices and ! . This is called overshooting [8]. Overshooting provides additional information about the system dynamics and regularizes the learning. Hence, the learning of false causalities might be reduced and the generalization ability of the model should be improved [8]. Of course, we have to support the additional output clusters
$ by target values. However, due to shared weights, we have the same number of parameters [8]. 3 The Asset Allocation Strategy Now, we explain how the forecasts are transformed into an asset allocation vector with investment proportions ( ). For simplicity, short sales (i. e. R
) are not allowed. We have to consider, that the allocation (i.) pays attention to the uncertainty of the forecasts and (ii.) complies with given investment constraints. In order to handle the uncertainty of the asset forecasts , we utilize the concept of excess return. An excess return L is defined as the difference between the expected returns and of two assets and , i. e. L . . The investment proportions of assets which have a superior excess return should be enlarged, because they seem to be more valuable. Further on, let us define the cumulated excess return as a weighted sum of the excess returns for one asset over all other assets , : 6 7 with (4) The forbiddance of short sales ( ) and the constraint, that investment proportions sum up to one ( C ), can be easily satisfied by the transformation R: 7
(5) The market share constraints are given by the asset manager in form of intervals, which have a mean value of 5 . 5 is the benchmark allocation. The admissible spread defines how much the allocation may deviate from : R"! 5 # (6) Since we have to level the excess returns around the mean of the intervals, Eq. 4 is adjusted by a bias vector $ $ $ % corresponding to the benchmark allocation: :&$* 6 7 (7) The bias $* forces the system to put funds into asset , even if the cumulated excess return does not propose an investment. The vector $ can be computed before-hand by solving the system of nonlinear equations which results by setting the excess returns (Eq. 7) to zero: $ $ ... ... ... $ $ (8) Since the allocation represents the benchmark portfolio, the pre-condition 7 leads to a non-unique solution (Eq. 9) of the latter system (Eq. 8) $* O) - (9) for any real number . In the following, we choose . The interval 5: 5 defines constraints for the parameters B B because the latter quantifies the deviation of from the benchmark . Thus, the return maximization task can be stated as a constraint optimization problem with B as the actual return of asset at time : 3 4 6 7 6 7 1 B B B E= > ( R"! # (10) This problem can be solved as a penalized maximization task 3 4 6 7 6 7 B R B B R 5
= > ( (11) with is defined as a type of -insensitive error function: if L# otherwise (12) Summarizing, the construction of the allocation scheme consists of the following two steps: (i.) Train the error correction sub-networks and compute the excess returns : . (ii.) Optimize the allocation parameters using the forecast models with respect to the market share constraints : 3 4 6 7 6 7 1 B R O) 5 6 7 B B E= > ( (13) As we will explain in sec. 5, Eq. 13 also controls the portfolio risk. 4 Modeling the Asset Allocation Strategy by Neural Networks A neural network approach of the allocation scheme (sec. 3) is shown in Fig. 2. The first layer of the portfolio optimization neural network (Fig. 2) collects the predictions from the underlying ECNNs. The matrix entitled ’unfolding’ computes the excess returns for assets as a contour plot. White spaces indicate weights with a value of , while grey equals and black stands for . The layer entitled ’excess returns’ is designed as an output cluster, i. e. it is associated with an error function which computes error signals for each training pattern. By this, we can identify inter-market dependencies, since the neural network is forced to learn cross-market relationships. asset 1 ECNN forecasts of k assets asset k−1 asset k t+6 y (...) asset 2 w = t+6 y y t+6 y t+6 0 w 0 k(k−1) 2 k(k−1) excess returns 1 w 2 ln(m ), ..., ln(m ) k k market shares id (fixed) folding (fixed) unfolding (fixed) k forecasts [ 0 ... 0 1 ] [ 1 0 ... 0 ] assets excess return 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 20 40 60 80 100 120 140 160 180 200 weigthed excess returns k asset allocations id (fixed) 1 unfolding matrix Figure 2. Arranged on the top of the ECNN sub-networks, a higher-level neural network models the portfolio optimization algorithm on basis of the excess returns H . The diagonal matrix which computes the weighted excess returns includes the only tunable parameters . All others are fixed. Next, each excess return is weighted by a particular via a diagonal connection to the layer entitled ’weighted excess returns’. Afterwards, the weighted excess returns are folded using the transpose of the sparse matrix called ’unfolding’ (see Fig. 2). By this, we calculate the sum of the weighted excess returns for each asset . According to the predictions of the excess returns , the layer ’asset allocation’ computes profitable investment decisions. In case that all excess returns are zero the benchmark portfolio is reproducedby the offset $ $ ) O) . Otherwise, funds are allocated within the preset investment boundaries , while simultaneously complying with the constraints 2 and R . In order to prevent short selling, we assume for each investment proportion. Further on, we have to guarantee that the sum of the proportions invested in the securities equals , i. e. . Both constraints are satisfied by the activation function of the ’asset allocation’ layer’, which implements the non-linear transformation of Eq. 5 using softmax. The return maximization task of Eq. 10 is also solved by this cluster by generating error signals utilizing the prof-max error function (Eq. 14). 3 4 6 7 6 7 B B
B E= > ( (14) The layer ’market shares’ takes care of the allocation constraints. The error function of Eq. 15 is implemented to ensure, that the investments do not violate preset constraints. 3 4 6 7 6 7 5 = > ( (15) The ’market shares’ cluster generates error signals for the penalized optimization problem stated in Eq. 11. By this, we implement a penalty for exceeding the allocation intervals . The error signals of Eq. 10 and Eq. 11 are subsequently used for computing the gradients in order to adapt the parameters . 5 Risk Analysis of the Neural Portfolio-Management In Tab. 1 we compare the mean-variance framework of Markowitz with our neural network based portfolio optimization algorithm. Markowitz: Neural Network: input: for each decision: forecasts , accepted risk exposure prediction models , benchmark allocation , deviation interval optimization: 7 0 R = > ( with B 1 B R
E= > ( with implicit risk control output: for each decision: vector k decision schemes
Table 1. Comparison of the portfolio optimization algorithm of Markowitz with our approach. The most crucial difference between the mean-variance framework and our approach is the handling of the risk exposure (see Tab. 1). The Markowitz algorithm optimizes the expected risk explicitly by quadratic programming. Assuming that it is not possible to forecast the expected returns of the assets (often referred to as random walk hypothesis), the forecasts are determined by an average of most recent observed returns, while the risk-covariance matrix B 7 BB is estimated by the historical volatility of the assets. Hence, the risk of the portfolio is determined by the volatility of the time series of the assets. However, insisting on the existence of useful forecast models, we propose to derive the covariance matrix from the forecast model residuals, i. e. the risk-matrix is determined by the covariances of the model errors. Now, the risk of the portfolio is due to the nonforecastability of the assets only. Since our allocation scheme is based on the model uncertainty, we refer to this approach as causal risk. Using the covariances of the model errors as a measurement of risk still allows to apply the Markowitz optimization scheme. Here, we propose to substitute the quadratic optimization problem of the Markowitz approach by the objective function of Eq. 16. 3 4 6 7 6 7 B O) 6 7 B B = > ( (16) The error function of Eq. 16 is optimized over time 3 with respect to the parameters , which are used to evaluate the certainty of the excess return forecasts. By this, it is possible to construct an asset allocations strategy which implicitly controls the risk exposure of the portfolio according to the certainty of the forecasts . Note, that Eq. 16 can be extended by a time delay parameter 4 $ in order to focus on more recent events. If the predicted excess returns B B are reliable, then the weights are greater than zero, because the optimization algorithm emphasizes the particular asset in comparison to other assets with less reliable forecasts. In contrast, unreliable predictions are ruled out by pushing the associated weights towards zero. Therefore, Eq. 16 implicitly controls the risk exposure of the portfolio although it is formulated as a return maximization task. Eq. 16 has to be optimized with respect to the allocation constraints . This allows the definition of an active risk parameter ! quantifying the readiness to deviate from the benchmark portfolio within the allocation constraints: R ! 5 (17) The weights B and the allocations are now dependent on the risk level . If , then the benchmark is recovered, while allows deviations from the benchmark within the bounds . Thus, the active risk parameter analysis the risk sensitivity of the portfolios with respect to the quality of the forecast models. 6 Empirical Study Now, we apply our approach to the financial markets of the G7 countries. We work on the basis of monthly data in order to forecast the semi-annual development of the stock, cash and bond markets of the G7 countries Spain, France, Germany, Italy, Japan, UK and USA. A separate ECNN is constructed for each market on the basis of country specific economic data. Due to the recurrent modeling, we only calculated the relative change of each input. The transformed inputs are scaled such that they have a mean of zero and a variance of one [8]. The complete data set (Sept. 1979 to May 1995) is divided into three subsets: (i.) Training set (Sept. 1979 to Jan. 1992). (ii.) Validation set (Feb. 1992 to June 1993), which is used to learn the allocation parameters . (iii.) Generalization set (July 1993 to May 1995). Each ECNN was trained until convergence by using stochastical vario-eta learning, which includes re-normalization of the gradients in each step of the backpropagation algorithm [9]. We evaluate the performance of our approach by a comparison with the benchmark portfolio < which is calculated with respect to the market shares . The comparison of our strategy and the benchmark portfolio is drawn on the basis of the accumulated return of investment (Fig. 3). Our strategy is able to outperform the benchmark portfolio on the generalization set by nearly . A further enhancement of the portfolio performance can only be achieved if one relaxes the market share constraints. This indicates, that the tight allocation boundaries, which prevent huge capital transactions from non-profitable to booming markets, narrow additional gains. In Fig. 4 we compare the risk of our portfolio to the risk of the benchmark portfolio. Here, the portfolio risk is defined analogous to the mean-variance framework. However, in contrast to this approach, the expected (co-)variances are replaced by the residuals 1 of the underlying forecast models. The risk level which is induced by our strategy is comparable to the benchmark (Fig. 4), while simultaneously increasing the portfolio return (Fig. 3). Fig. 5 compares the allocations of German bonds and stocks across the generalization set: A typical reciprocal investment behavior is depicted, e. g. enlarged positions in stocks often occur in parallel with smaller investments in bonds. This effect is slightly disturbed by international diversification. Not all countries show such a coherent investment behavior. 7 Conclusions and Future Work We described a neural network approach which adapts the Black / Litterman portfolio optimization algorithm. Here, funds are allocated across various securities while simultaneously complying with allocation constraints. In contrast to the mean-variance theory, the risk exposure of our approach focuses on the uncertainty of the underlying forecast models. July 1993 May 1995 −2 0 2 4 6 8 10 12 14 Date accumulated return ECNN Benchmark Figure 3. July 1993 May 1995 date risk portfolio risk benchmark risk Figure 4. July 1993 May 1995 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 date allcocation german stocks german bonds Figure 5. Fig.3. Comparison of accumulated return of investment (generalization set). Fig.4. Comparison of portfolio risk (generalization set). Fig.5. Investments in German bond and stocks (generalization set). The underlying forecasts are generated by ECNNs, since our empirical results indicate, that this is a very promising framework for financial modeling. Extending the ECNN by using techniques like overshooting, variants-invariants separation or unfolding in space and time, one is able to include additional prior knowledge of the dynamics into the model [8, 9]. Future work will include the handling of a larger universe of assets. In this case, one may extend the neural network by a bottleneck which selects the most promising assets. References [1] Black, F., Litterman, R.:Global Portfolio Optimization, Financial Analysts Journal, Sep. 1992. [2] Elton, E. J., Gruber, M. J.: Modern Portfolio Theory and Investment Analysis, J. Wiley & Sons. [3] Haykin S.: Neural Networks. A Comprehensive Foundation., ed., Macmillan, N. Y. 1998. [4] Lintner, J.:The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets, in: Review of Economics and Statistics, Feb. 1965. [5] Markowitz, H. M.: Portfolio Selection, in: Journal of Finance, Vol. 7, 1952, p. 77-91. [6] Pearlmatter, B.:Gradient Calculations for Dynamic Recurrent Neural Networks: A survey, In IEEE Transactions on Neural Networks, Vol. 6, 1995. [7] Sharpe, F.:A Simplified Model for Portfolio Analysis, Management Science, Vol. 9, 1963. [8] Zimmermann, H. G., Neuneier, R., Grothmann, R.: Modeling of Dynamical Systems by Error Correction Neural Networks, in: Modeling and Forecasting Financial Data, Techniques of Nonlinear Dynamics, Eds. Soofi, A. and Cao, L., Kluwer 2001. [9] Zimmermann, H.G., Neuneier, R.:Neural Network Architectures for the Modeling of Dynamical Systems, in: A Field Guide to Dynamical Recurrent Networks, Eds. Kremer, St. et al., IEEE.
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Generalization Performance of Some Learning Problems in Hilbert Functional Spaces Tong Zhang IBM T.J. Watson Research Center Yorktown Heights, NY 10598 tzhang@watson.ibm.com Abstract We investigate the generalization performance of some learning problems in Hilbert functional Spaces. We introduce a notion of convergence of the estimated functional predictor to the best underlying predictor, and obtain an estimate on the rate of the convergence. This estimate allows us to derive generalization bounds on some learning formulations. 1 Introduction 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 machine learning, we assume that the data are drawn from an underlying distribution which is not known. Our goal is to find so that the expected true loss of given below is as small as possible: !"$#&%' ( ) In order to estimate a good predictor from a set of training data randomly drawn from , it is necessary to start with a model of the functional relationship. In this paper, we consider models that are subsets in some Hilbert functional spaces * . Denote by + +-, the norm in * , we consider models in the set ./10 324*657+ 8+ ,:9;=< , where ; is a parameter that can be used to control the size of the underlying model family. We would like to find the best model in . which is given by: > ?@ ACB7DFEHG IKJML NH !8"O
() (1) By introducing a non-negative Lagrangian multiplier PQSR , we may rewrite the above problem as: > T@ ACB7DFE G IKJ , U NH !8"V
( W P X + Y+Z ,\[ ) (2) We shall only consider this equivalent formulation in this paper. In addition, for technical reasons, we also assume that ^] C_ is a convex function of ] . Given training examples K 8)8)Y)& , we consider the following estimation method to approximate the optimal predictor > '2 . : > ?@ ACB7DFEHG IKJ , U
W P X + 8+ Z , [ ) (3) The goal of this paper is to show that as , >
> in probability under appropriate regularity conditions. Furthermore, we obtain an estimate on the rate of convergence. Consequences of this result in some specific learning formulations are examined. 2 Convergence of the estimated predictor Assume that input belongs to a set . We make the reasonable assumption that is pointwise continuous under the + + , topology: 2 , EHD I I ? where is in the sense that + +8, R . This assumption is equivalent to ! I #"%$ '& W' ( 2) . The condition implies that each data point can be regarded as a bounded linear functional * on * such that 2 * : *
= . Since a Hilbert space * is self-dual, we can represent * by an element in * . For notational simplicity, we shall let * 2 * defined as * for all 2 * , where denotes the inner product of * . It is clear that * can be regarded as a representing feature vector of in * . This representation can be computed as follows. Let + @MA B7D @-, I " , then it is not difficult to see that + .* 0/ +* +, and + ?+* +, . It follows that * ?1+ 2+ . Note that this method of computing * is not important for the purpose of this paper. Since * can now be considered as a linear functional using the feature space representation * of , we can use the idea from [6] to analyze the convergence behavior of > in * . Following [6], using the linear representation of , we differentiate (2) at the optimal solution > , which leads to the following first order condition: M ! 43 > * 5* W P > TR (4) where 3 ^] C_8 is the derivative of ^] C_ with respect to ] if is smooth; it denotes a subgradient (see [4]) otherwise. Since we have assumed that ^] _ is a convex function of ] , we know that ^] _W ^] Z ] 8 3 ^] K C_ 9 ^] Z _ . This implies the following inequality: > W 3 > > > 9 > ( which is equivalent to: > ( &W P X > Z W U 3 > ( > > W4P > M > > [ W P X > 6 > Z 9 > ( W P X > Z ) Note that we have used Z to denote + + Z , . Also note that by the definition of > , we have > ( W P X > Z Q > ( W P X > Z ) Therefore by comparing the above two inequalities, we obtain: P X > > Z 9 U 43 > ( * > > W P > M > > [ 9 + 3 > ( 5* W4P > + , + > 6 > + , ) This implies that + > 6 > &+8, 9 X P + 3 > * W P > +, X P + 3 > * ! 3 > ( 5* + , ) (5) Note that the last equality follows from the first order condition (4). This is the only place the condition is used. In (5), we have already bounded the convergence of > to > in terms of the convergence of the empirical expectation of a random vector 3 > ( * to its mean. The latter is often easier to estimate. For example, if its variance can be bounded, then we may use the Chebyshev inequality to obtain a probability bound. In this paper, we are interested in obtaining an exponential probability bound. In order to do so, similar to the analysis in [6], we use the following form of concentration inequality which can be found in [5], page 95: Theorem 2.1 ([5]) Let be zero-mean independent random vectors in a Hilbert space * . If there exists R such that for all natural numbers Q X : + + , 9 Z
. Then for all R : + +,Q 9 X , Z Z / _ Z W . We may now use the following form of Jensen’s inequality to bound the moments of the zero-mean random vector 3 > ( * M ! 3 > ( 5* : 9 X W 9 X W4 7 X ) From inequality (5) and Theorem 2.1, we immediately obtain the following bound: Theorem 2.2 If there exists R such that for all natural numbers Q X : M ! 3 > ( Y+* +, 9 Z . Then for all R : + > > +8, Q 9 X , 5 P Z Z / ! _ Z W4P" ) Although Theorem 2.2 is quite general, the quantity _ and on the right hand side of the bound depend on the optimal predictor > which requires to be estimated. In order to obtain a bound that does not require any knowledge of the true distribution , we may impose the following assumptions: both 3 ^] _ and +* +8, are bounded. Observe that +* + , IKJ ,$# I " , we obtain the following result: Corollary 2.1 Assume that 2 , 5 IKJ ,%# I " 9'& . Also assume that the loss function ^] C_ satisfies the condition that 3 ^] C_( 9 , then )* R : + > > + , Q+ 9 X, , P Z Z / ! & Z Z W & P" () 3 Generalization performance We study some consequences of Corollary 2.1, which bounds the convergence rate of the estimated predictor to the best predictor. 3.1 Regression We consider the following type of Huber’s robust loss function: 7 Z
Z if 6 9 6 Z if 6 Q ) (6) It is clear that is continuous differentiable and 3 ( 9 for all and . It is also not hard to check that Z and : Z
K
Z 3 9 X Z 8ZM) Using this inequality and (4), we obtain: U M ! > ( W P X + > +Z , [ U M ! > ( W P X + > +8Z , [ \ ! > ( > ( 3 > ( > > W P X + > > +Z , 9 ! X > > Z W P X + > > +Z , ) If we assume that + > > +8, 9 and IKJ ,$# I " 9 & , then M ! > 9 M ! > W P X + > + Z , + > + Z , W Z X & Z W4P() (7) This gives the following inequality: M ! > ( ! > ( 9 P" + > &+, W Z & Z X W P() It is clear that the right-hand side of the above inequality does not depend on the unobserved function > . Using Corollary 2.1, we obtain the following bound: Theorem 3.1 Using loss function (6) in (3). Assume that IKJ ,%# I " 9 & , then R &+ 9 & / P , with probability of at least X , 5 P Z Z / ! R & Z Z , we have M ! > ( 9 M ! > ( W4P" + > &+, W Z & Z X W P() Theorem 3.1 compares the performance of the computed function with that of the optimal predictor > 2 . * in (1). This style of analysis has been extensively used in the literature. For example, see [3] and references therein. In order to compare with their results, we can rewrite Theorem 3.1 in another form as: with probability of at least , M ! > 9 M ! > W HG ) In [3], the authors employed a covering number analysis which led to a bound of the form (for squared loss) M ! > ( 9 M ! > ( W
HG HG for finite dimensional problems. Note that the constant in their depends on the pseudodimension, which can be infinity for problems considered in this paper. It is possible to employ their analysis using some covering number bounds for general Hilbert spaces. However, such an analysis would have led to a result of the following form for our problems: M ! > ( 9 M ! > ( W
G G () It is also interesting to compare Theorem 3.1 with the leave-one-out analysis in [7]. The generalization error averaged over all training examples for squared loss can be bounded as ! > 9 W P M ! > ( W4P + > +Z , ) This result is not directly comparable with Theorem 3.1 since the right hand side includes an extra term of P+ > + Z , . Using the analysis in this paper, we may obtain a similar result from (7) which leads to an average bound of the form: ! > ( 9 M ! > W P + > + Z , &W P Z () It is clear that the term resulted in our paper is not as good as from [7]. However analysis in this paper leads to probability bounds while the leave-one-out analysis in [7] only gives average bounds. It is also worth mentioning that it is possible to refine the analysis presented in this section to obtain a probability bound which when averaged, gives a bound with the correct term of , rather than in the current analysis. However due to the space limitation, we shall skip this more elaborated derivation. In addition to the above style bounds, it is also interesting to compare the generalization performance of the computed function to the empirical error of the computed function. Such results have occurred, for example, in [1]. In order to obtain a comparable result, we may use a derivation similar to that of (7), together with the first order condition of (3) as follows: 3 > * 5* W P > R ) This leads to a bound of the form: > ( 9 > ( W P X + > &+ Z , + > + Z , W Z X & Z W P ) Combining the above inequality and (7), we obtain the following theorem: Theorem 3.2 Using loss function (6) in (3). Assume that IKJ ,%# I " 9 & , then R &+ 9 & / P , with probability of at least X , 5 P Z Z / ! R & Z Z , we have ! > > ( 9 Z & Z W P W U M ! > ( > ( [ ) Unlike Theorem 3.1, the bound given in Theorem 3.2 contains a term U M ! > ( > [ which relies on the unknown optimal predictor > . From Theorem 3.1, we know that this term does not affect the performance of the estimated function > when compared with the performance of > . In order for us to compare with the bound in [1] obtained from an algorithmic stability point of view, we make the additional assumption that > 9 for all . Note that this assumption is also required in [1]. Using Hoeffding’s inequality, we obtain that with probability of at most , 5 P Z Z / ! R & Z Z , ! > > ( % P / & ) Together with Theorem 3.2, we have with probability of at least , M ! > > ( 9 & ZW P ! R & Z Z HG P Z W Z HG ) This compares very favorably to the following bound in [1]:1 M ! > > ( 9 X & Z Z P W X Z X & P Z W ! & Z P W X HG Z ) 3.2 Binary classification In binary classification, the output value 20 < is a discrete variable. Given a continuous model , we consider the following prediction rule: predict if QTR , and predict otherwise. The classification error (we shall ignore the point 1R , which is assumed to occur rarely) is ( if 9 R R if 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. We shall thus consider the following soft-margin SVM style loss as an illustration:
7DF@-, R ) (8) Note that the separable case of this loss was investigated in [6]. In this case, 3
denotes a subgradient rather than gradient since
is non-smooth: at , 3
2 U CR [ ; 3
when & and 3 7R when . Since , Z 9 and , Z 9
, we know that if + > > +, 9 , then M ! > ( 9 ! > & ( > & 9 > X & () Using the standard Hoeffding’s inequality, we have with probability of at most , 5 P Z Z / ! R & Z , M ! > & > & W P / ! & ) 1In [1], there was a small error after equation (11). As a result, the original bound in their paper was in a form equivalent to the one we cite here with
replaced by
. When > & R , it is usually better to use a different (multiplicative) form of Hoeffding’s inequality, which implies that with probability of at most , 5 P Z Z / ! R & Z , M ! > & D @ , X > & ( P Z Z / & Z () Together with Corollary 2.1, we obtain the following margin-percentile result: Theorem 3.3 Using loss function (8) in (3). Assume that IKJ ,%# I " 9 & , then R &+ 9 & / P , with probability of at least , 5 P Z Z / ! R & Z , we have M ! > ( 9 > X & W P / ! & ) We also have with probability of at least , P Z Z / ! R & Z , M ! > 9 DF@-, X > X & ( CP Z KZ / & Z8 () We may obtain from Theorem 3.3 the following result: with probability of at least , ! > 9 > ! R G P Z & Z &W G X ) It is interesting to compare this result with margin percentile style bounds from VC analysis. For example, Theorem 4.19 in [2] implies that there exists a constant such that with probability of at least : for all we have ! > ( 9 > W & Z HG Z P Z W HG ) We can see that if we assume that P is small and the margin ! R & Z is also small, then the above bound with this choice of is inferior to the bound in Theorem 3.3. Clearly, this implies that our analysis has some advantages over VC analysis due to the fact that we directly analyze the numerical formulation of support vector classification. 4 Conclusion In this paper, we have introduced a notion of the convergence of the estimated predictor to the best underlying predictor for some learning problems in Hilbert spaces. This generalizes an earlier study in [6]. We derived generalization bounds for some regression and classification problems. We have shown that results from our analysis compare favorably with a number of earlier studies. This indicates that the concept introduced in this paper can lead to valuable insights into certain numerical formulations of learning problems. References [1] Olivier Bousquet and Andr´e Elisseeff. Algorithmic stability and generalization performance. In Todd K. Leen, Thomas G. Dietterich, and Volker Tresp, editors, Advances in Neural Information Processing Systems 13, pages 196–202. MIT Press, 2001. [2] Nello Cristianini and John Shawe-Taylor. An Introduction to Support Vector Machines and other Kernel-based Learning Methods. Cambridge University Press, 2000. [3] Wee Sun Lee, Peter L. Bartlett, and Robert C. Williamson. The importance of convexity in learning with squared loss. IEEE Trans. Inform. Theory, 44(5):1974–1980, 1998. [4] R. Tyrrell Rockafellar. Convex analysis. Princeton University Press, Princeton, NJ, 1970. [5] Vadim Yurinsky. Sums and Gaussian vectors. Springer-Verlag, Berlin, 1995. [6] Tong Zhang. Convergence of large margin separable linear classification. In Advances in Neural Information Processing Systems 13, pages 357–363, 2001. [7] Tong Zhang. A leave-one-out cross validation bound for kernel methods with applications in learning. In 14th Annual Conference on Computational Learning Theory, pages 427–443, 2001.
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Analog Soft-Pattern-Matching Classifier using Floating-Gate MOS Technology Toshihiko YAMASAKI and Tadashi SHIBATA* Department of Electronic Engineering, School of Engineering *Department of Frontier Informatics, School of Frontier Science The University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan yamasaki@if.t.u-tokyo.ac.jp, shibata@ee.t.u-tokyo.ac.jp Abstract A flexible pattern-matching analog classifier is presented in conjunction with a robust image representation algorithm called Principal Axes Projection (PAP). In the circuit, the functional form of matching is configurable in terms of the peak position, the peak height and the sharpness of the similarity evaluation. The test chip was fabricated in a 0.6-µm CMOS technology and successfully applied to hand-written pattern recognition and medical radiograph analysis using PAP as a feature extraction pre-processing step for robust image coding. The separation and classification of overlapping patterns is also experimentally demonstrated. 1 Introduction Pattern classification using template matching techniques is a powerful tool in implementing human-like intelligent systems. However, the processing is computationally very expensive, consuming a lot of CPU time when implemented as software running on general-purpose computers. Therefore, software approaches are not practical for real-time applications. For systems working in mobile environment, in particular, they are not realistic because the memory and computational resources are severely limited. The development of analog VLSI chips having a fully parallel template matching architecture [1,2] would be a promising solution in such applications because they offer an opportunity of low-power operation as well as very compact implementation. In order to build a real human-like intelligent system, however, not only the pattern representation algorithm but also the matching hardware itself needs to be made flexible and robust in carrying out the pattern matching task. First of all, two-dimensional patterns need to be represented by feature vectors having substantially reduced dimensions, while at the same time preserving the human perception of similarity among patterns in the vector space mapping. For this purpose, an image representation algorithm called Principal Axes Projection (PAP) has been de veloped [3] and its robust nature in pattern recognition has been demonstrated in the applications to medical radiograph analysis [3] and hand-written digits recognition [4]. However, the demonstration so far was only carried out by computer simulation. Regarding the matching hardware, high-flexibility analog template matching circuits have been developed for PAP vector representation. The circuits are flexible in a sense that the matching criteria (the weight to elements, the strictness in matching) are configurable. In Ref. [5], the fundamental characteristics of the building block circuits were presented, and their application to simple hand-written digits was presented in Ref. [6]. The purpose of this paper is to demonstrate the robust nature of the hardware matching system by experiments. The classification of simple hand-written patterns and the cephalometric landmark identification in gray-scale medical radiographs have been carried out and successful results are presented. In addition, multiple overlapping patterns can be separated without utilizing a priori knowledge, which is one of the most difficult problems at present in artificial intelligence. 2 Image representation by PAP PAP is a feature extraction technique using the edge information. The input image (64x64 pixels) is first subjected to pixel-by-pixel spatial filtering operations to detect edges in four directions: horizontal (HR); vertical (VR); +45 degrees (+45); and –45 degrees (-45). Each detected edge is represented by a binary flag and four edge maps are generated. The two-dimensional bit array in an edge map is reduced to a one-dimensional array of numerals by projection. The horizontal edge flags are accumulated in the horizontal direction and projected onto vertical axis. The vertical, +45-degree and –45-degree edge flags are similarly projected onto horizontal, -45-degree and +45-degree axes, respectively. Therefore the method is called “Principal Axes Projection (PAP)” [3,4]. Then each projection data set is series connected in the order of HR, +45, VR, -45 to form a feature vector. Neighboring four elements are averaged and merged to one element and a 64-dimensional vector is finally obtained. This vector representation very well preserves the human perception of similarity in the vector space. In the experiments below, we have further reduced the feature vector to 16 dimensions by merging each set of four neighboring elements into one, without any significant degradation in performance. 3 Circuit configurations VGG IOUT VIN RST A B 1 C 2 4 1 VGG IOUT VIN A B 1 C 2 4 13 RST Figure 1: Schematic of vector element matching circuit: (a) pyramid (gain reduction) type; (b) plateau (feedback) type. The capacitor area ratio is indicated in the figure. The basic functional form of the similarity evaluation is generated by the shortcut current flowing in a CMOS inverter as in Refs. [7,8,9]. However, their circuits were utilized to form radial basis functions and only the peak position was programmable. In our circuits, not only the peak position but also the peak height and the sharpness of the peak response shape are made configurable to realize flexible matching operations [5]. Two types of the element matching circuit are shown in Fig. 1. They evaluate the similarity between two vector elements. The result of the evaluation is given as an output current (IOUT) from the pMOS current mirror. The peak position is temporarily memorized by auto-zeroing of the CMOS inverter. The common-gate transistor with VGG stabilizes the voltage supply to the inverter. By controlling the gate bias VGG, the peak height can be changed. This corresponds to multiplying a weight factor to the element. The sharpness of the functional form is taken as the strictness of the similarity evaluation. In the pyramid type circuit (Fig. 1(a)), the sharpness is controlled by the gain reduction in the input. In the plateau type (Fig. 1(b)), the output voltage of the inverter is fed back to input nodes and the sharpness changes in accordance with the amount of the feedback.
! "! # # # #$ %#&"# ' ' ' '( ')"' Figure 2: Schematic of n-dimensional vector matching circuit utilizing the pyramid type vector element circuits. 16-dimension
15-vector
matching circuit Decoder Time-domain Winner-Take-All 4.5mm The total matching score between input and template vectors is obtained by taking the wired sum of all IOUT’s from the element matching circuits as shown in Fig. 2. A multiplier circuit as utilized in Ref. [8] was eliminated because the radial basis function is not suitable for the template matching using PAP vectors. ISUM, the sum of IOUT’s, is then sunk through the nMOS with the VRAMP input. This forms a current comparator circuit, which compares ISUM and the sink current in the nMOS with VRAMP. The VOUT nodes are connected to a time-domain Winner-Take-All circuit [9]. A common ramp down voltage is applied to the VRAMP nodes of all vector matching circuits. When VRAMP is ramped down from VDD to 0V, the vector matching circuit yielding the maximum ISUM firstly upsets and its output voltage (VOUT) shows a 0-to-1 transition. The time-domain WTA circuit senses the first upsetting signal and memorizes the location in the open-loop OR-tree architecture [10]. In this manner, the maximum-likelihood template vector is easily identified. Figure 3: Photomicrograph of soft-pattern-matching classifier circuit. The circuits were designed and fabricated in a 0.6-µm double-poly triple-metal CMOS technology. Fig. 3 shows the photomicrograph of a pattern classifier circuit for 16-dimensional vectors. It contains 15 vector matching circuits. One element matching circuit occupies the area of 150µm x 110µm. In the latest design, however, the area is reduced to 54 µm x 68 µm in the same technology by layout optimization. Further area reduction is anticipated by employing high-K dielectric films for capacitors since the capacitors occupy a large area. The full functioning of the chip was experimentally confirmed [6]. In the following experiments, the simple vector matching circuit in Fig. 2 was utilized to investigate the response from each template vector instead of just detecting the winner using the full chip. 4 Experimental results and discussion 4.1 Vector-element matching circuit
µ ! ! " ! ! " µ ! "
µ µ ! " $#% & & &'( ( ( ) ) )*,+ + + -/.0 -1. 0 Figure 4: Measured characteristics: (a) pyramid type; (b) plateau type. VGG was varied from 3.0V to 4.5V, and control signals A~C from 000 to 111 for sharpness control. Fig. 4 shows the measured characteristics of vector-element matching circuits in both linear and log plots. The peak position was set at 1.05V by auto-zeroing. The peak height was altered by VGG. Also, the operation mode was altered from the above-threshold region to the sub-threshold region by VGG. In the plateau type circuit (Fig. 4(b)), IOUT becomes constant around the peak position and the flat region widens in proportion to the amount of feedback. This is because the inverter operates so as to keep the floating gate potential constant in the high-gain region of the inverter as in the case of virtual ground of an operational amplifier. 4.2 Matching of simple hand-written patterns Fig. 5 demonstrates the matching results for the simple input patterns. 16 templates were stored in the matching circuit and several hand-written pattern vectors were presented to the circuit as inputs. A slight difference in the matching score is observed between the pyramid type and the plateau type, but the answers are correct for both types. Fig. 6 shows the effect of sharpness variation. As the sharpness gets steeper, all the scores decrease. However, the score ratios between the winner and loosers are increased, thus enhancing the winner discrimination margins. The matching results with varying operational regimes of the circuit are given in Fig. 7. The circuit functions properly even in the sub-threshold regime, demonstrating the opportunity of extremely low power operation. Presented
Patterns Best
Matched Best
Matched Template Patterns Template Patterns Template # 1 2 3 4 5 6 7 8 9 10 111213141516 Template # 1 2 3 4 5 6 7 8 9 10 111213141516 Figure 5: Result of simple pattern matching: (a) pyramid type (left) where gain reduction level was set with ABC=010; (b) plateau type (right) where feedback ratio was set with ABC=101.
µ ! " # $ % $ & $$ & % $ $ $ ' ( $ ' ) $ ' * $ ' + & ,.-0/10243450647, 8 , , , -9, /9, 19, 29, 3 :<; =?>@ A B ; C ,D-0/14203454647E, 8, , , -9, /, 19, 29, 3 :; = >@ A B ; C Figure 6: Effect of sharpness variation in the pyramid type with ABC=010. VGG=4.0V VGG=3.5V VGG=3.0V VGG=2.5V Input Pattern Best Matched Template#
1 2 3 4 5 6 7 8 9 10111213141516 Template #4 Figure 7: Matching results as a function of VGG. Correct results are obtained in the sub-threshold regime as well as in the above-threshold regime (the pyramid type was utilized). 4.3 Application to gray-scale medical radiograph analysis In Fig. 8, are presented the result of cephalometric landmark identification experiments, where the Sella (pituitary gland) pattern search was carried out using the same matching circuit. Since the 64-dimension PAP representation is essential for grayscale image recognition, the 64-dimension vector was divided into four 16-dimension vectors and the matching scores were measured separately and then summed up by off-chip calculation. The correct position was successfully identified both in the above-threshold (Fig. 8(b)) and the sub-threshold (Fig. 8(c)) regimes using the 14 learned vectors as templates. In the previous work [3], successful search was demonstrated by the computer simulation.
! µ"$# ! µ"$# %'&)(+*-,/.* 01-2 3-4658792 3*+: ; <8=>@?BAC9DFE D µ G %'&)(+*-,/.* 01-2 3-4658792 3*+: ; <8=>@?HDDFE I µ G
Figure 8: Matching results of Sella search using pyramid type with ABC=000: (a) input image; (b) above-threshold regime; (c) sub-threshold regime. 4.4 Separation of overlapping patterns Suppose an unknown pattern is presented to the matching circuit. The pattern might consist of a single or multiple overlapping patterns. Let X represent the input vector and W1st the winner (best matched) vector obtained by the matching circuit. Let the first matching trial be expressed as follows: 1st trial: matching → 1st X W Then, the residue vector (X-W1st) is generated. The subtraction is perfomed in the vector space. When an element in the residue vector becomes negative, the value is set to 0. Such operation is easily implemented using the floating gate technique. Here, the residue was obtained by off-line calculation. If the input pattern is single, the residue vector is meaningless: only the leftover edge information remains in the residue vector. If the input consists of overlapping patterns, the edge information of other patterns remains. If the residue vector is very small, we can expect that the input is single. But in many cases, the residue vector is not so small due to the distortion in hand-written patterns. Thus, it is almost impossible to judge which is the case only from the magnitude of the residue vector. Therefore, we proceed to the second trial to find the second winner: 2nd trial: matching − → 1st 2nd X W W With the same sequence, the second residue vector (X-W1st-W2nd), the third (X-W1st-W2nd-W3rd) and so forth are generated by repeating the winner subtraction after each trial. Then, new template vectors are generated such as W1st+W2nd, W1st+W2nd+W3rd, and so forth. If the input vector is that of a single pattern, the matching score is the highest at W1st and the scores are lower at W1st+W2nd and W1st+W2nd+W3rd. On the other hand, if the input vector is that of two overlapping patterns, the score is the highest at W1st+W2nd. This procedure can be terminated automatically when the new template composed of n overlapping patterns yields lower score than that of n-1 overlapping patterns. In this manner, we are able to know how many patterns are overlapping and what patterns are overlapping without a priori knowledge. An example of separating multiple overlapping patterns is illustrated in Fig. 9. + + Presented Patterns Best Matched Template Patterns ? ? ? ? 1st try: 2nd try: 3rd try: Final try: + + Figure 9: Experimental result illustrating the algorithm for separating overlapping patterns. The solid black bars indicate the winner locations. B A C B A C D D E E F F + + + + + + + + + + + + + + Presented
Patterns Best
Matched Template Patterns #1 #2 #3 (a) (b) Figure 10: Measured results demonstrating separation of multiple overlapping patterns: (a) result of separation and classification (A~F are depicted in (b)); (b) newly created templates such as W1st+W2nd, W1st+W2nd+W3rd, and so on. Several other examples are shown in Fig. 10. Pattern #1 is correctly classified as a single rectangle by yielding the higher score for single template than that for W1st+W2nd. Pattern #3 consists of three overlapping patterns, but is erroneously recognized as four overlapping patterns. However, the result is not against human perception. When we look at pattern #3, a triangle is visible in the pattern. This mistake is quite similar to that made by humans. 5 Conclusions A soft-pattern matching circuit has been demonstrated in conjunction with a robust image representation algorithm called PAP. The circuit has been successfully applied to hand-written pattern recognition and medical radiograph analysis. The recognition of overlapping patterns similar to human perception has been also experimentally demonstrated. Acknowledgments Test circuits were fabricated in the VDEC program (The Univ. of Tokyo), in collaboration with Rohm Corp. and Toppan Printing Corp. The work is partially supported by the Ministry of Education, Science, Sports and Culture under the Grant-in-Aid for Scientific Research (No. 11305024) and by JST in the program of CREST. References [1] G.T. Tuttle, S. Fallahi, and A.A. Abidi. (1993) An 8b CMOS Vector A/D Converter. in ISSCC Tech. Digest, vol. 36, pp. 38-39. IEEE Press. [2] G. Cauwenberghs and V. Pedroni. (1995) A Charge-Based CMOS Parallel Analog Vector Quantizer. In G. Tesauro, D. S. Touretzky and T.K. Leen (eds.), Advances in Neural Information Processing Systems 7, pp. 779-786. Cambridge, MA: MIT Press. [3] M. Yagi, M. Adachi, and T. Shibata. (2000) A Hardware-Friendly Soft-Computing Algorithm for Image Recognition. X European Signal Processing Conf., Sept. 4-8, 2000 (EUSIPCO 2000), Vol. 2, pp. 729-732, Tampere, Finland. [4] M. Adachi and T. Shibata. (2001) Image Representation Algorithm Featuring Human Perception of Similarity for Hardware Recognition Systems. In Proc. of the Int. Conf. on Artificial Intelligence (IC-AI'2001), Ed. by H. R. Arabnia, Vol. I, 229-234 (CSREA Press, ISDBN: 1-892512-78-5), Las Vegas, Nevada, USA, June 25-28, 2001. [5] T. Yamasaki and T. Shibata. (2001) An Analog Similarity Evaluation Circuit Featuring Variable Functional Forms. In Proc. IEEE Int. Symp. Circuits Syst. (ISCAS 2001), Vol. 3, pp. III-561-564, Sydney, Australia, May. 7-9, 2001. [6] T. Yamasaki, K. Yamamoto and T. Shibata. (2001) Analog Pattern Classifier with Flexible Matching Circuitry Based on Principal-Axis-Projection Vector Representation. In Proc. 27th European Solid-State Circ. Conf. (ESSCIRC 2001), Ed. by F. Dielacher and H. Grunbacher, pp. 212-215 (Frontier Group), Villach, Austria, September 18-20, 2001. [7] J. Anderson, J. C. Platt, and D. B. Kirk. (1993) An Analog VLSI Chip for Radial Basis Functions. In S. J. Hanson, J. D. Cowan, and C. L. Giles Eds., Advances in Neural Information Processing Systems 5, pp. 765-772., San Maetro, CA; Morgan Kaufmann. [8] L. Theogarajan and L. A. Akers. (1996) A Multi-Dimentional Analog Gaussian Radial Basis Circuit. In Proc. IEEE Int. Symp. Circuits Syst. (ISCAS ’96), Vol. 3, pp. III-543 -546 Atlanta, GA, USA, May, 1996. [9] L. Theogarajan and L. A. Akers. (1997) A scalable low voltage analog Gaussian radial basis circuit. IEEE Trans. on Circuits and Systems II, Volume 44, No. 11, pp. 977 –979, 1997. [10] K. Ito, M. Ogawa and T. Shibata. (2001) A High-Performance Time-Domain Winner-Take-All Circuit Employing OR-Tree Architecture. In Proc. 2001 Int. Conf. on Solid State Devices and Materials (SSDM2001), pp. 94-95, Tokyo, Japan, Sep. 26-28, 2001.
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Generalizable Relational Binding from Coarse-coded Distributed Representations Randall C. O’Reilly Department of Psychology University of Colorado Boulder 345 UCB Boulder, CO 80309 oreilly@psych.colorado.edu Richard S. Busby Department of Psychology University of Colorado Boulder 345 UCB Boulder, CO 80309 Richard.Busby@Colorado.EDU Abstract We present a model of binding of relationship information in a spatial domain (e.g., square above triangle) that uses low-order coarse-coded conjunctive representations instead of more popular temporal synchrony mechanisms. Supporters of temporal synchrony argue that conjunctive representations lack both efficiency (i.e., combinatorial numbers of units are required) and systematicity (i.e., the resulting representations are overly specific and thus do not support generalization to novel exemplars). To counter these claims, we show that our model: a) uses far fewer hidden units than the number of conjunctions represented, by using coarse-coded, distributed representations where each unit has a broad tuning curve through high-dimensional conjunction space, and b) is capable of considerable generalization to novel inputs. 1 Introduction The binding problem as it is classically conceived arises when different pieces of information are processed by entirely separate units. For example, we can imagine there are neurons that separately code for the shape and color of objects, and we are viewing a scene having a red triangle and a blue square (Figure 1). Because color and shape are encoded separately in this system, the internal representations do not discriminate this situation from one where we are viewing a red square and a blue triangle. This is the problem. Broadly speaking, there are two solutions to it. Perhaps the most popular solution is to imagine that binding is encoded by some kind of transient signal, such as temporal synchrony (e.g., von der Malsburg, 1981; Gray, Engel, Konig, & Singer, 1992; Hummel & Holyoak, 1997). Under this solution, the red and triangle units should fire together, as should the blue and square units, with each group firing out of phase with the other. The other solution can be construed as solving the problem by questioning its fundamental assumption — that information is encoded completely separately in the first place (which is so seductive that it typically goes unnoticed). Instead, one can imagine that color and shape information are encoded together (i.e., conjunctively). In the red-triangle blue-square example, some neurons encode the conjunction of red and triangle, while others encode the conjunction of blue and square. Because these units are explicitly sensitive to these Red Blue Square Triangle ? ? Red Blue Square Triangle a) Input activates features b) But rest of brain doesn’t know which features go with each other Figure 1: Illustration of the binding problem. a) Visual inputs (red triangle, blue square) activate separate representations of color and shape properties. b) However, just the mere activation of these features does not distinguish for the rest of the brain the alternative scenario of a blue triangle and a red square. Red is indicated by dashed outline and blue by a dotted outline. RC RC GS GS obj1 obj2 R G B S C T BT obj1 obj2 R G B S C T BT RS GC 1 1 0 1 1 0 0 RT GC 1 1 0 0 1 1 0 RC GS 1 1 0 1 1 0 1 RC BT 1 0 1 0 1 1 1 RS GT 1 1 0 1 0 1 0 RT BC 1 0 1 0 1 1 0 RT GS 1 1 0 1 0 1 1 GS BC 0 1 1 1 1 0 1 RS BC 1 0 1 1 1 0 0 GC BS 0 1 1 1 1 0 0 RC BS 1 0 1 1 1 0 1 GS BT 0 1 1 1 0 1 1 RS BT 1 0 1 1 0 1 1 GT BS 0 1 1 1 0 1 0 RT BS 1 0 1 1 0 1 0 GC BT 0 1 1 0 1 1 1 RC GT 1 1 0 0 1 1 1 GT BC 0 1 1 0 1 1 0 Table 1: Solution to the binding problem by using representations that encode combinations of input features (i.e., color and shape), but achieve greater efficiency by representing multiple such combinations. Obj1 and obj2 show the features of the two objects (R = Red, G = Green, B = Blue, S = Square, C = Circle, T = Triangle), and remaining columns show 6 localist units and one coarsecoded conjunctive unit. Adding this one conjunctive unit is enough to disambiguate the inputs. conjunctions, they will not fire to a red square or a blue triangle, and thereby avoid the binding problem. The obvious problem with this solution, and one reason it has been largely rejected in the literature, is that it would appear to require far too many units to cover all of the possible conjunctions that need to be represented — a combinatorial explosion. However, the combinatorial explosion problem is predicated on another seductive notion — that separate units are used for each possible conjunction. In short, both the binding problem itself and the problem with the conjunctive solution derive from localist assumptions about neural coding. In contrast, these problems can be greatly reduced by simply thinking in terms of distributed representations, where each unit encodes some possibly-difficult to describe amalgam of input features, such that individual units are active at different levels for different inputs, and many such units are active for each input (Hinton, McClelland, & Rumelhart, 1986). Therefore, the input is represented by a complex distributed pattern of activation over units, and each unit can exhibit varying levels of sensitivity to the featural conjunctions present in the input. The binding problem is largely avoided because a different pattern of activation will be present for a red-triangle, blue-square input as compared to a red-square, blue-triangle input. These kinds of distributed representations can be difficult to understand. This is probably a significant reason why the ability of distributed representations to resolve the binding problem goes under-appreciated. However, we can analyze special cases of these representations to gain some insight. One such special case is shown in Table 1 from O’Reilly and Munakata (2000). Here, we add one additional distributed unit to an otherwise localist featural encoding like that shown in Figure 1. This unit has a coarse-coded conjunctive representation, meaning that instead of coding for a single conjunction, it codes for several possible conjunctions. The table shows that if this set of conjunctions is chosen wisely, this single unit can enable the distributed pattern of activation across all units to distinguish between any two possible combinations of stimulus inputs. A more realistic system will have a larger number of partially redundant coarse-coded conjunctive units that will not require such precise representations from each unit. A similar demonstration was recently provided by Mel and Fiser (2000) in an analysis of distributed, low-order conjunctive representations (resembling “Wickelfeatures”; Wickelgren, 1969; Seidenberg & McClelland, 1989) in the domain of textual inputs. However, they did not demonstrate that a neural network learning mechanism would develop these representations, or that they could support systematic generalization to novel inputs. 2 Learning Generalizable Relational Bindings We present here a series of models that test the ability of existing neural network learning mechanisms to develop low-order coarse-coded conjunctive representations in a challenging binding domain. Specifically, we focus on the problem of relational binding, which provides a link to higher-level cognitive function, and speaks to the continued use of structured representations in these domains. Furthermore, we conduct a critical test of these models in assessing their ability to generalize to novel inputs after moderate amounts of training. This is important because conjunctive representations might appear to limit generalization as these representations are more specific than purely localist representations. Indeed the inability to generalize is considered by some the primary limitation of conjunctive binding mechanisms (Holyoak & Hummel, 2000). 2.1 Relational Binding, Structured Representations, and Higher-level Cognition A number of existing models rely on structured representations because they are regarded as essential for encoding complex relational information and other kinds of data structures that are used in symbolic models (e.g., lists, trees, sequences) (e.g., Touretzky, 1986; Shastri & Ajjanagadde, 1993; Hummel & Holyoak, 1997). A canonical example of a structured representation is a propositional encoding (e.g., LIKES cats milk) that has a main relational term (LIKES) that operates on a set of slot-like arguments that specify the items entering into the relationship. The primary advantages of such a representation are that it is transparently systematic or productive (anything can be put in the slots), and it is typically easy to compose more elaborate structures from these individual propositions (e.g., this proposition can have other propositions in its slots instead of just basic symbols). The fundamental problem with structured representations, regardless of what implements them, is that they cannot be easily learned. To date, there have been no structured representation models that exhibit powerful learning of the form typically associated with neural networks. There are good reasons to believe that this reflects basic tradeoffs between complex structured representations and powerful learning mechanisms (Elman, 1991; St John & McClelland, 1990; O’Reilly & Munakata, 2000). Essentially, structured representations are discrete and fragile, and therefore do not admit to gradual changes over learning. In contrast, neural networks employ massively-parallel, graded processing that can search out many possible solutions at the same time, and optimize those that seem to make graded improvements in performance. In contrast, the discrete character of structured representations requires exhaustive combinatorial search in high-dimensional spaces. To provide an alternative to these structured representations, our models test a simple example of relational encoding, focusing on easily-visualized spatial relationships, which can be thought of in propositional terms as for example (LEFT-OF square triangle). Input Location Question Hidden where? what? relation−obj? relation−loc? Relation Object right left above below Figure 2: Spatial relationship binding model. Objects are represented by distributed patterns of activation over 8 features per location within a 4x4 array of locations. Inputs have two objects, arranged vertically or horizontally. The network answers questions posed by the Question input (“what”, “where”, and “what relationship?”) —the answers require binding of object, location, and relationship information. 3 Spatial Relationship Binding Model The spatial relationship binding model is shown in Figure 2. The overall framework for training the network is to present it with input patterns containing objects in different locations, and ask it various questions about these input displays. These questions ask about the identity and location of objects (i.e., “what?” and “where?”), and the relationships between the two objects (e.g., “where is object1 relative to object2?”). To answer these questions correctly, the hidden layer must bind object, location, and relationship information accurately in the hidden layer. Otherwise, it will confuse the two objects and their locations and relationships. Furthermore, we encoded the objects using distributed representations over features, so these features must be correctly bound into the same object. Specifically, objects are represented by distributed patterns of activation over 8 features per location, in a 4x4 location array. Inputs have two different objects, arranged either vertically or horizontally. The network answers different questions about the objects posed by the Question input. For the “what?” question, the location of one of the objects is activated as an input in the Location layer, and the network must produce the correct object features for the object in that location. We also refer to this target object as the agent object. For the “where?” question, the object features for the agent object are activated in the Object layer, and the network must produce the correct location activation for that object. For the “relation-obj?” question, the object features for the agent object are activated, and the network must activate the relationship between this object and the other object (referred to as the patient object), in addition to activating the location for the agent object. For the “relation-loc?” question, the location of the agent object is activated, and the network must activate the relationship between this object and the patient object, in addition to activating the object features for the agent object. This network architecture has a number of nice properties. For example, it has only one object and location encoding layer, both of which can act as either an input or an output. This is better than an alternative architecture having separate slots representing the agent and patient objects, because such slot-based encodings solve the binding problem by having separate role-specific units, which becomes implausible as the number of different roles and objects multiply. Note that supporting the dual input/output roles requires an interactive (recurrent, bidirectionally-connected) network (O’Reilly, 2001, 1998). a) b) Input Location Question Hidden Object Relation R L A B What? Where? Rel-Obj? Rel-Loc? Input Location Question Hidden Object Relation R L A B What? Where? Rel-Obj? Rel-Loc? c) d) Input Location Question Hidden Object Relation R L A B What? Where? Rel-Obj? Rel-Loc? Input Location Question Hidden Object Relation R L A B What? Where? Rel-Obj? Rel-Loc? Figure 3: Hidden unit representations (values are weights into a hidden unit from all other layers) showing units (a & b) that bind object, location, & relationship information via low-order conjunctions, and other units that have systematic representations of location (c) and object features (d). There are four levels of questions we can ask about this network. First, we can ask if standard neural network learning mechanisms are capable of solving this challenging binding problem. They are. Second, we can ask whether the network actually develops coarsecoded distributed representations. It does. Third, we can ask if these networks can generalize to novel inputs (both novel objects and novel locations for existing objects). They can. Finally, we can ask whether there are differences in how well different kinds of learning algorithms generalize, specifically comparing the Leabra algorithm with purely error-driven networks, as was recently done in other generalization tests with interactive networks (O’Reilly, 2001). This paper showed that interactive networks generalize significantly worse than comparable feedforward networks, but that good generalization can be achieved by adding additional biases or constraints on the learning mechanisms in the form of inhibitory competition and Hebbian learning in the Leabra algorithm. These results are replicated here, with Leabra generalization being roughly twice as good as other interactive algorithms. 0 10 20 30 40 No. of Patients Per Agent, Location 0.0 0.2 0.4 0.6 0.8 1.0 Generalization Error Spat Rel Generalization (Fam Objs) 200 Agent, Locs 300 Agent, Locs 400 Agent, Locs a) 0 10 20 30 40 No. of Patients Per Agent, Location 0.0 0.2 0.4 0.6 0.8 1.0 Generalization Error Spat Rel Generalization (Nov Objs) 200 Agent, Locs 300 Agent, Locs 400 Agent, Locs b) Figure 4: Generalization results (proportion errors on testing set) for the spatial relationship binding model using the Leabra algorithm as a function of the number of training items, specified as number of agent, location combinations and number of patient, locations per each agent, location. a) shows results for testing on familiar objects in novel locations. b) shows results for testing on novel objects that were never trained before. 3.1 Detailed Results First, we examined the representations that developed in the network’s hidden layer (Figure 3). Many units encoded low-order combinations (conjunctions) of object, location, and relationship features (Figure 3a & b). This is consistent with our hypothesis. Other units also encoded more systematic representations of location without respect to objects (Figure 3c) and object features without respect to location (Figure 3d). To test the generalization capacity of the networks, we trained on only 26 of the 28 possible objects that can be composed out of 8 features with two units active, and only a subset of all 416 possible agent object x location combinations. We trained on 200, 300, and 400 such combinations. For each agent object-location input, there are 150 different patient object-location combinations per agent object-location, and we trained on 4, 10, 20, and 40, selected at random, for each different level of agent object-location combination training. At the most (400x40)there were a total of 16000 unique inputs trained out of a total possible of 62400, which amounts to about 1/4 of the training space. At the least (200x4) only roughly 1.3% of the training space was covered. The ability of the network to generalize to the 26 familiar objects in novel locations was tested by measuring performance on a random sample of 640 of the untrained agent objectlocation combinations. The results for the Leabra algorithm are shown in Figure 4a. As one would expect, the number of training patterns improves generalization in a roughly proportional manner. Importantly, the network is able to generalize to a high level of performance, getting roughly 95% correct after training on only 25% of the training space (400x40), and achieving roughly 80% correct after training on only roughly 10% of the space (300x20). The ability of the network to generalize to novel objects was tested by simply presenting the two novel objects as agents in all possible locations, with a random sampling of 20 different patients (which were the familiar objects), for a total of 640 different testing items (Figure 4b). Generalization on these novel objects was roughly comparable to the familiar objects, except there was an apparent ceiling point at roughly 15% generalization error where the generalization did not improve even with more training. Overall, the network performed remarkably well on these novel objects, and future work will explore generalization with fewer training objects. To evaluate the extent to which the additional biologically-motivated biases in the Leabra algorithm are contributing to these generalization results, we ran networks using the contrastive Hebbian learning algorithm (CHL) and the Almeida-Pineda (AP) recurrent back10 20 No. of Patients Per Agent, Location 0.00 0.10 0.20 0.30 0.40 Generalization Error Spat Rel Generalization (Fam Objs) Leabra CHL Figure 5: Generalization results for different algorithms on the spatial relationship binding task (see previous figure for details on measures) in the 400 x 10 or 20 conditions. propagation algorithm, as in O’Reilly (2001). Both of these algorithms work in interactive, bidirectionally-connected networks, which are required for this task. Standard AP was unable to learn the task, we suspected because it does not preserve the symmetry of the weights as is required for stable settling. Attempts to to rectify this problem by enforcing symmetric weight changes did not succeed either. The results for CHL (Figure 5) replicated earlier results (O’Reilly, 2001) in showing that the additional biases in Leabra produced roughly twice as good of generalization performance compared to CHL. 4 Discussion These networks demonstrate that existing, powerful neural network learning algorithms can learn representations that perform complex relational binding of information. Specifically, these networks had to bind together object identity, location, and relationship information to answer a number of questions about input displays containing two objects. This supports our contention that rich distributed representations containing coarse-coded conjunctive encodings can effectively perform binding. It is critical to appreciate that these distributed representations are highly efficient, encoding over 62400 unique input configurations with only 200 hidden units. Furthermore, these representations are systematic, in that they support generalization to novel inputs after training on a fraction of the input space. Despite these initial successes, more work needs to be done to extend this approach to other kinds of domains that require binding. One early example of such an application is the St John and McClelland (1990) sentence gestalt model, which was able to sequentially process words in a sentence and construct a distributed internal representation of the meaning of the sentence (the sentence gestalt). This model was limited in that it required extremely large numbers of training trials and an elaborate training control mechanism. However, these limitations were eliminated in a recent replication of this model based on the Leabra algorithm (O’Reilly & Munakata, 2000). We plan to extend this model to handle a more complex corpus of sentences to more fully push the relational binding capacities of the model. Finally, it is important to emphasize that we do not think that these low-order conjunctive representations are entirely sufficient to resolve the binding problems that arise in the cortex. One important additional mechanism is the use of selective attention to focus neural processing on coherent subsets of information present in the input (e.g., on individual objects, people, or conversations). The interaction between such a selective attentional system and a complex object recognition system was modeled in O’Reilly and Munakata (2000). In this model, selective attention was an emergent process deriving from excitatory interactions between a spatial processing pathway and the object processing pathway, combined with surround inhibition as implemented by inhibitory interneurons. The resulting model was capable of sequentially processing individual objects when multiple such objects were simultaneously present in the input. Acknowledgments This work was supported by ONR grant N00014-00-1-0246 and NSF grant IBN-9873492. 5 References Elman, J. L. (1991). Distributed representations, simple recurrent networks, and grammatical structure. Machine Learning, 7, 195–225. Gray, C. M., Engel, A. K., Konig, P., & Singer, W. (1992). Synchronization of oscillatory neuronal responses in cat striate cortex —temporal properties. Visual Neuroscience, 8, 337–347. Hinton, G. E., McClelland, J. L., & Rumelhart, D. E. (1986). Distributed representations. In D. E. Rumelhart, J. L. McClelland, & PDP Research Group (Eds.), Parallel distributed processing. Volume 1: Foundations (Chap. 3, pp. 77–109). Cambridge, MA: MIT Press. Holyoak, K. J., & Hummel, J. E. (2000). The proper treatment of symbols in a connectionist architecture. In E. Dietrich, & A. Markman (Eds.), Cognitive dynamics: Conceptual and representational change in humans and machines. Mahwah, NJ: Lawrence Erlbaum Associates. Hummel, J. E., & Holyoak, K. J. (1997). Distributed representations of structure: A theory of analogical access and mapping. Psychological Review, 104(3), 427–466. Mel, B. A., & Fiser, J. (2000). Minimizing binding errors using learned conjunctive features. Neural Computation, 12, 731–762. O’Reilly, R. C. (1998). Six principles for biologically-based computational models of cortical cognition. Trends in Cognitive Sciences, 2(11), 455–462. O’Reilly, R. C. (2001). Generalization in interactive networks: The benefits of inhibitory competition and Hebbian learning. Neural Computation, 13, 1199–1242. O’Reilly, R. C., & Munakata, Y. (2000). Computational explorations in cognitive neuroscience: Understanding the mind by simulating the brain. Cambridge, MA: MIT Press. Seidenberg, M. S., & McClelland, J. L. (1989). A distributed, developmental model of word recognition and naming. Psychological Review, 96, 523–568. Shastri, L., & Ajjanagadde, V. (1993). From simple associations to systematic reasoning: A connectionist representation of rules, variables, and dynamic bindings using temporal synchrony. Behavioral and Brain Sciences, 16, 417–494. St John, M. F., & McClelland, J. L. (1990). Learning and applying contextual constraints in sentence comprehension. Artificial Intelligence, 46, 217–257. Touretzky, D. S. (1986). BoltzCONS: Reconciling connectionism with the recursive nature of stacks and trees. Proceedings of the 8th Annual Conference of the Cognitive Science Society (pp. 522– 530). Hillsdale, NJ: Lawrence Erlbaum Associates. von der Malsburg, C. (1981). The correlation theory of brain function. MPI Biophysical Chemistry, Internal Report 81-2. In E. Domany, J. L. van Hemmen, & K. Schulten (Eds.), Models of neural networks, II (1994). Berlin: Springer. Wickelgren, W. A. (1969). Context-sensitive coding, associative memory, and serial order in (speech) behavior. Psychological Review, 76, 1–15.
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A hierarchical model of complex cells in visual cortex for the binocular perception of motion-in-depth Silvio P. Sabatini, Fabio Solari, Giulia Andreani, Chiara Bartolozzi, and Giacomo M. Bisio Department of Biophysical and Electronic Engineering University of Genoa, 1-16145 Genova, ITALY silvio@dibe.unige.it Abstract A cortical model for motion-in-depth selectivity of complex cells in the visual cortex is proposed. The model is based on a time extension of the phase-based techniques for disparity estimation. We consider the computation of the total temporal derivative of the time-varying disparity through the combination of the responses of disparity energy units. To take into account the physiological plausibility, the model is based on the combinations of binocular cells characterized by different ocular dominance indices. The resulting cortical units of the model show a sharp selectivity for motion-indepth that has been compared with that reported in the literature for real cortical cells. 1 Introduction The analysis of a dynamic scene implies estimates of motion parameters to infer spatio-temporal information about the visual world. In particular, the perception of motion-in-depth (MID), i.e. the capability of discriminating between forward and backward movements of objects from an observer, has important implications for navigation in dynamic environments. In general, a reliable estimate of motionin-depth can be gained by considering the dynamic stereo correspondence problem in the stereo image signals acquired by a binocular vision system. Fig. 1 shows the relationships between an object moving in the 3-D space and its geometrical projections in the right and left retinas. In a first approximation, the positions of corresponding points are related by a 1-D horizontal shift, the disparity, along the direction of the epipolar lines. Formally, the left and right observed intensities from the two eyes, respectively JL(X) and JR(x), result related as JL(X) = JR[x + 8(x)], where 8(x) is the horizontal binocular disparity. If an object moves from P to Q its disparity changes and projects different velocities (VL' VR) on the retinas. .............. .9J ............ t+~t ( ) a 8(t+lit) = (XQL-XQR) "" a(D-ZQ)/D2 V "" li8 D2/a z M li8 = 8(t+lit)-&(t) = lit lit _ (XQL -XPL)-(XQR -XPR) "" lit VZ "" (VL-vR)D2/a Figure 1: The dynamic stereo correspondence problem. A moving object in the 3-D space projects different trajectories onto the left and right retinas. The differences between the two trajectories carry information about motion-in-depth. Thus, the Z component of the object's motion (i.e., its motion-in-depth) Vz can be approximated in two ways [1]: (1) by the rate of change of disparity, and (2) by the difference between retinal velocities, as it is evidenced in the box in Fig. l. The predominance of one measure on the other one corresponds to different hypotheses on the architectural solutions adopted by visual cortical cells to encode dynamic 3-D visual information. Recently, numerous experimental and computational studies (see e.g., [2] [3] [4] [5]) addressed this issue, by analyzing the binocular spatio-temporal properties of simple and complex cells. The fact that the resulting disparity tuning does not vary with time, and that most of the cells in the primary visual cortex have the same motion preference for the two eyes, led to the conclusion that these cells are not tuned to motion-in-depth. In this paper, we demonstrate that, within a phase-based disparity encoding scheme, such cells relay phase temporal derivative components that can be combined, at a higher level, to yield a specific motion-in-depth selectivity. The rationale of this statement relies upon analytical considerations on phase-based dynamic stereopsis, as a time extension of the well-known phase-based techniques for disparity estimation [6] [7]. The resulting model is based on the computation of the total temporal derivative of the disparity through the combination of the outputs of binocular disparity energy units [4] [5] characterized by different ocular dominance indices. Since each energy unit is just a binocular Adelson and Bergen's motion detector, this establishes a link between the information contained in the total rate of change of the binocular disparity and that held by the interocular velocity differences. 2 Phase-based dynamic stereopsis In the last decades, a computational approach for stereopsis, that rely on the phase information contained in the spectral components of the stereo image pair, has been proposed [6] [7]. Spatially-localized phase measures on the left and right images can be obtained by filtering operations with a complex-valued quadrature pair of Gabor 2; 2 'k filters h(x, ko) = e- X "et ox, where ko is the peak frequency of the filter and a relates to its spatial extension. The resulting convolutions with the left and right binocular signals can be expressed as Q(x) = p(x)ei¢(x) = C(x) + is(x) where p(x) = ylC2(X) + S2(X) and ¢(x) = arctan (S(x)/C(x)) denote their amplitude and phase components, respectively, and C(x) and S(x) are the responses of the quadrature pair of filters. Hence, binocular disparity can be predicted by 8(x) = [¢L(X) - ¢R(x)]/k(x) where k(x) = [¢~(x) + ¢;Z(x)]/2, with ¢x spatial derivative of phase ¢, is the average instantaneous frequency of the bandpass signal, that, under a linear phase model, can be approximated by the peak frequency of the Gabor filter ko. Extending to time domain, the disparity of a point moving with the motion field can be estimated by: 5:[ () ] _ ¢L[X(t), t] - ¢R[x(t), t] uxt ,tko (1) where phase components are computed from the spatiotemporal convolutions of the stereo image pair Q(x, t) = C(x, t) + is(x, t) with directionally tuned Gabor filters with a central frequency p = (ko, wo). For spatiotemporal locations where linear phase approximation still holds (¢ ~ kox + wot), the phase differences in Eq. (1) provide only spatial information, useful for reliable disparity estimates. 2.1 Motion-in-depth If disparity is defined with respect to the spatial coordinate XL, by differentiating with respect to time, its total rate of variation can be written as d8 = 88 VL (A.L _ A.R) dt 8t + ko 'l'x 'l'x (2) where VL is the horizontal component of the velocity signal on the left retina. Considering the conservation property of local phase measurements [8], image velocities can be computed from the temporal evolution of constant phase contours, and thus: and (3) with ¢t = ~. Combining Eq. (3) with Eq. (2) we obtain d8/dt = (VR - VL)¢;Z /ko, where (v R - V L) is the phase-based interocular velocity difference along the epipolar lines. When the spatial tuning frequency of the Gabor filter ko approaches the instantaneous spatial frequency of the left and right convolution signals one can derive the following approximated expressions: d8 88 ¢t - ¢f ~ = ~VR-VL dt 8t ko (4) The partial derivative of the disparity can be directly computed by convolutions (S, C) of stereo image pairs and by their temporal derivatives (St, Ct): a8 at [StCL - SLCt s[lcR - SRC[l] 1 (SL)2 + (CL)2 (SR)2 + (CR)2 ko (5) thus avoiding explicit calculation and differentiation of phase, and the attendant problem of phase unwrapping. Considering that, at first approximation (SL)2 + (CL)2 ::: (SR)2 + (CR)2 and that these terms are scantly discriminant for motionin-depth, we can formulate the cortical model taking into account the numerator terms only. 2.2 The cortical model If one prefilters the image signal to extract some temporal frequency sub-band, S(x, t) ::: 9 * S(x, t) and C(x, t) ::: 9 * C(x, t) , and evaluates the temporal changes in that sub-band, differentiation can be attained by convolutions on the data with appropriate bandpass temporal filters: S'(x, t) ::: g' * S(x, t) ; C'(x, t) ::: g' * C(x, t) . S' and C' approximate St and Ct, respectively, if 9 and g' are a quadrature pair of temporal filters, e.g.: g(t) = e- t/ T sinwot and g'(t) = e- t / T coswot. From a modeling perspective, that approximation allows us to express derivative operations in terms of convolutions with a set of spatio-temporal filters, whose shapes resemble those of simple cell receptive fields (RFs) of the primary visual cortex. Though, it is worthy to note that a direct interpretation of the computational model is not biologically plausible. Indeed, in the computational scheme (see Eq. (5)), the temporal variations of phases are obtained by processing monocular images separately and then the resulting signals are binocularly combined to give at an estimate of motionin-depth in each spatial location. To employ binocular RFs from the beginning, as they exist for most of the cells in the visual cortex, we manipulated the numerator by rewriting it as the combination of terms characterized by a dominant contribution for the ipsilateral eye and a non-dominant contribution for the controlateral eye. These contributions are referable to binocular disparity energy units [5] built from two pairs of binocular direction selective simple cells with left and right RFs weighted by an ocular dominance index a E [0,1]. The "tilted" spatio-temporal RFs of simple cells of the model are obtained by combining separable RFs according to an Adelson and Bergen's scheme [9]. It can be demonstrated that the information about motion-in-depth can be obtained with a minimum number of eight binocular simple cells, four with a left and four with a right ocular dominance, respectively (see Fig. 2): Sl = (1 - a)(Cf + SL) - a(CR - sf") S3 = (1 - a)(Cf - SL) - a(CR + sf") S5 = a(Cf + SL) - (1 - a)(CR - sf") S7 = a(Cf - SL) - (1 - a)(CR + sf") C11 = si + S~ ; C12 = S5 + S~ S2 = (1 - a)(CL + Sf) + a(Cf" + SR) S4 = (1 - a)(CL + Sf) + a(Cf" - SR) S6 = a(CL - Sf) + (1 - a)(Cf" + SR) S8 = a(CL + Sf) + (1 - a)(Cf" - SR) C13 = S~ + S~ ; C14 = S¥ + S~ C21 = C12 - C11 ; C22 = C13 - C14 C3 = (1 - 20:) (stcL - sLCt - s[lcR + sRc[l) . The output of the higher complex cell in the hierarchy (C3 ) truly encodes motionin-depth information. It is worthy to note that for a balanced ocular dominance (0: = 0.5) the cell looses its selectivity. 3 Results To assess model performances we derived cells' responses to drifting sinusoidal gratings with different speeds in the left and right eye. The spatial frequency of the gratings has been chosen as central to the RF's bandwidth. For each layer, the tuning characteristics of the cells are analyzed as sensitivity maps in the (XL - XR) and (VL - VR) domains for the static and dynamic properties, respectively. The (XL - XR) represents the binocular RF [5] of a cell, evidencing its disparity tuning. The (v L - v R) response represents the binocular tuning curve of the velocities along the epipolar lines. To better evidence motion-in-depth sensitivity, we represent as polar plots, the responses of the model cells with respect to the interocular velocities ratio for 12 different motion trajectories in depth (labeled 1 to 12) [10]. The cells of the cortical model exhibit properties and typical profiles similar to those observed in the visual cortex [5] [10]. The middle two layers (see insets A and B in Fig. 2) exhibit a strong selectivity to static disparity, but no specific tuning to motion-in-depth. On the contrary, the output cell C3 shows a narrow tuning to the Z direction of the object's motion, while lacking disparity tuning (see inset C in Fig. 2). To consider more biologically plausible RFs for the simple cells, we included a coefficient f3 in the scheme used to obtain tilted RFs in the space-time domain (e.g. C + f3St). This coefficient takes into account the simple cell response to the nonpreferred direction. We analytically demonstrated (results not shown here) that the resulting effect is a constant term that multiplies the cortical model output. In this way, the model is based on more realistic simple cells without lacking its functionality, provided that the basic direction selective units maintain a significant direction selective index. To analyze the effect of the architectural parameters on the model performance, we systematically varied the ocular dominance index 0: and introduced a weight I representing the inhibition strength of the afferent signals to the complex cells in layer 2. The resulting direction-in-depth polar plots are shown in Fig. 3. The 0: parameter yields a strong effect on the response profile: if 0: = 0.5 there is no direction-in-depth selectivity; according that 0: > 0.5 or 0: < 0.5 cells exhibit a tuning to opposite directions in depth. As 0: approaches the boundary values 0 or 1 the binocular model turns to a monocular one. A decrease of the inhibition strength I yields cells characterized by a less selective response to direction-in-depth, whereas an increase of I diminishes their response amplitude. 4 Discussion and conclusions There are at least two binocular cues that can be used to determine the MID [1]: binocular combination of monocular velocity signals or the rate of change of retinal disparity. Assuming a phase-based disparity encoding scheme [6], we demonstrated that information held in the interocular velocity difference is the same of A , "" S, EB- ( " / ,,-.,. ...... ...c 01) ·c ,' "" s '--' § EB_2 ( ;:::l , / "0 12 VR )' ~ \ ' VL XR ~" )2---{] : u <l) c u , :::: "" S3 ro :::: ·s EB- ( 0 " / '"0 · 12 8 3 ', ,,,, S u 0 EB_4 ( , / • ~ C2' : 6 • : X R )2 . . : : ~3 , "" S EB5 ( " / 6 Figure 2: Functional representation of the proposed cortical architecture. Each branch groups cells belonging to an ocular dominance column. The afferent signals from left and right ocular dominance columns are combined in layer 3. The basic units are binocular simple cells tuned to motion directions (S1, . . . ,S8). The responses of the complex cells in layers 1, 2 and 3 are obtained by linear and nonlinear combinations of the outputs of those basic units. See text. White squares denote excitatory synapses whereas black squares denote inhibitory ones. a = 0.3 a = 0.7 a = 0.9 , = 0.5 , = 1.0 , = 2.0 12 12 12 9~ 3 9 3 9 3 6 6 6 12 9 --~~-- 3 9 --~I!'--- 3 9 ------':111:"--- 3 6 12 I 9 --~~-3 6 6 12 ~ I v 9 A~ 3 6 6 12 9 ------7,i!k--- 3 6 Figure 3: Effects on the direction-in-depth selectivity of the systematic variation of the model's parameters a and f. The responses are normalized to the largest amplitude value. that derived by the evaluation of the total derivative of the binocular disparity. The resulting computation relies upon spatio-temporal differentials of the left and right retinal phases that can be approximated by linear filtering operations with spatiotemporal RFs. Accordingly, we proposed a cortical model for the generation of binocular motion-in-depth selective cells as a hierarchical combination of binocular energy complex cells. It is worth noting that the phase response and the associated characteristic disparity of simple and complex cells in layers 1 and 2 do not change with time, but the amplitudes of their responses carry information on temporal phase derivatives, that can be related to both retinal velocities and temporal changes in disparity. Moreover, the model evidences the different roles of simple and complex cells. Simple cells provide a Gabor-like spatio-temporal transformation of the visual space, on which to base a variety of visual functions (perception ofform, depth, motion). Complex cells, by proper combinations ofthe same signals provided by simple cells, actively eliminate sensitivity to a selected set of parameters, thus becoming specifically tuned to different features, such as disparity but not motion-in-depth (layer 1 and 2), motion-in-depth but not disparity (layer 3). Acknowledgments This work was partially supported by the UNIGE-2000 Project "Spatio-temporal Operators for the Analysis of Motion in Depth from Binocular Images". References [1] J. Harris and S. N.J. Watamaniuk. Speed discrimination of Motion-in depth using binocular cues. Vision Research, 35(7):885- 896, 1995. [2] N. Qian and S. Mikaelian. Relationship between phase and energy methods for disparity computation. Neural Comp., 12(2):279- 292, 2000. [3] Y. Chen, Y. Wang, and N. Qian. Modelling VI disparity tuning to time-varying stimuli. J. Neurophysiol., pages 504- 600, 2001. [4] D. J. Fleet, H. Wagner, and D. J. Heeger. Neural encoding of binocular diparity: energy models, position shift and phase shift. Vision Research, 17:345- 398, 1996. [5] 1. Ohzawa, G.C. DeAngelis, and R.D. Freeman. Encoding of binocular disparity by complex cells in the cat's visual cortex. J. Neurophysiol., 77:2879- 2909, 1997. [6] T.D. Sanger. Stereo disparity computation using Gabor filters. BioI. Cybern., 59:405- 418, 1988. [7] D.J. Fleet, A.D. Jepson, and M. Jenkin. Phase-based disparity measurements. CVGIP: Image Understanding, 53:198- 210, 1991. [8] D. J. Fleet and A. D. Jepson. Computation of component image velocity from local phase information. International Journal of Computer Vision, 1 :77- 104, 1990. [9] E.H. Adelson and J.R. Bergen. Spatiotemporal energy models for the perception of motion. J. Opt. Soc. Amer., 2:284-321, 1985. [10] W. Spileers, G.A. Orban, B. Gulyas, and H. Maes. Selectivity of cat area 18 neurons for direction and speed in depth. J. Neurophysiol. , 63(4):936- 954, 1990.
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On the Concentration of Spectral Properties John Shawe-Taylor Royal Holloway, University of London john@cs.rhul.ac.uk Jaz Kandola N ella Cristianini BIOwulf Technologies nello@support-vector. net Royal Holloway, University of London jaz@cs.rhul.ac.uk Abstract We consider the problem of measuring the eigenvalues of a randomly drawn sample of points. We show that these values can be reliably estimated as can the sum of the tail of eigenvalues. Furthermore, the residuals when data is projected into a subspace is shown to be reliably estimated on a random sample. Experiments are presented that confirm the theoretical results. 1 Introduction A number of learning algorithms rely on estimating spectral data on a sample of training points and using this data as input to further analyses. For example in Principal Component Analysis (PCA) the subspace spanned by the first k eigenvectors is used to give a k dimensional model of the data with minimal residual, hence forming a low dimensional representation of the data for analysis or clustering. Recently the approach has been applied in kernel defined feature spaces in what has become known as kernel-PCA [5]. This representation has also been related to an Information Retrieval algorithm known as latent semantic indexing, again with kernel defined feature spaces [2]. Furthermore eigenvectors have been used in the HITS [3] and Google's PageRank [1] algorithms. In both cases the entries in the eigenvector corresponding to the maximal eigenvalue are interpreted as authority weightings for individual articles or web pages. The use of these techniques raises the question of how reliably these quantities can be estimated from a random sample of data, or phrased differently, how much data is required to obtain an accurate empirical estimate with high confidence. Ng et al. [6] have undertaken a study of the sensitivity of the estimate of the first eigenvector to perturbations of the connection matrix. They have also highlighted the potential instability that can arise when two eigenvalues are very close in value, so that their eigenspaces become very difficult to distinguish empirically. The aim of this paper is to study the error in estimation that can arise from the random sampling rather than from perturbations of the connectivity. We address this question using concentration inequalities. We will show that eigenvalues estimated from a sample of size m are indeed concentrated, and furthermore the sum of the last m - k eigenvalues is subject to a similar concentration effect, both results of independent mathematical interest. The sum of the last m - k eigenvalues is related to the error in forming a k dimensional PCA approximation, and hence will be shown to justify using empirical projection subspaces in such algorithms as kernel-PCA and latent semantic kernels. The paper is organised as follows. In section 2 we give the background results and develop the basic techniques that are required to derive the main results in section 3. We provide experimental verification of the theoretical findings in section 4, before drawing our conclusions. 2 Background and Techniques We will make use of the following results due to McDiarmid. Note that lEs is the expectation operator under the selection of the sample. TheoreIll 1 (McDiarmid!4}) Let Xl, ... ,Xn be independent random variables taking values in a set A, and assume that f : An -+~, and fi : An- l -+ ~ satisfy for l:::;i:::;n Xl,··· , Xn TheoreIll 2 (McDiarmid!4}) Let Xl, ... ,Xn be independent random variables taking values in a set A, and assume that f : An -+ ~, for 1 :::; i :::; n sup If(xI, ... , xn) - f(XI, ... , Xi- I, Xi, Xi+!,···, xn)1 :::; Ci, We will also make use of the following theorem characterising the eigenvectors of a symmetric matrix. TheoreIll 3 (Courant-Fischer MiniIllax TheoreIll) If M E ~mxm is symmetric, then for k = 1, ... , m, v'Mv v'Mv Ak(M) = max min -- = min max dim(T) = k O#v ET vlv dim(T) = m - k+IO#v ET vlv ' with the extrama achieved by the corresponding eigenvector. The approach adopted in the proofs of the next section is to view the eigenvalues as the sums of squares of residuals. This is applicable when the matrix is positive semidefinite and hence can be written as an inner product matrix M = XI X, where XI is the transpose of the matrix X containing the m vectors Xl, . . . , Xm as columns. This is the finite dimensional version of Mercer's theorem, and follows immediately if we take X = V VA, where M = VA VI is the eigenvalue decomposition of M. There may be more succinct ways of representing X, but we will assume for simplicity (but without loss of generality) that X is a square matrix with the same dimensions as M. To set the scene, we now present a short description of the residuals viewpoint. The starting point is the singular value decomposition of X = U~V', where U and V are orthonormal matrices and ~ is a diagonal matrix containing the singular values (in descending order). We can now reconstruct the eigenvalue decomposition of M = X' X = V~U'U~V' = V AV', where A = ~2. But equally we can construct a matrix N = XX' = U~V'V~U' = UAU' , with the same eigenvalues as M. As a simple example consider now the first eigenvalue, which by Theorem 3 and the above observations is given by A1(M) v'Nv v'XX'v max -- = max O,t:vEIR= v'v O,t:vEIR= v'v m m max O,t:vEIR= v'v m max L IIPv(xj)11 2 = L IIxjl12 min L IIP;-(xj)112 O,t:vEIR= O,t:vEIR= j=l j=l j=l where Pv(x) (Pv..l (x)) is the projection of x onto the space spanned by v (space perpendicular to v), since IIxI12 = IIPv(x)112 + IIPv..l(x)112. It follows that the first eigenvector is characterised as the direction for which sum of the squares of the residuals is minimal. Applying the same line of reasoning to the first equality of Theorem 3, delivers the following equality m Ak = max min L IlPv(xj)112. dim(V)= k O,t:vEV . J=l (1) Notice that this characterisation implies that if v k is the k-th eigenvector of N, then m Ak = L IlPv k (xj)112, (2) j=l which in turn implies that if Vk is the space spanned by the first k eigenvectors, then k m m m L Ai = L IIPVk (Xj) 112 = L IIXj W - L IIP'* (Xj) 112, (3) i=l j=l j=l j=l where Pv(x) (PV(x)) is the projection of x into the space V (space perpendicular to V). It readily follows by induction over the dimension of V that we can equally characterise the sum of the first k and last m - k eigenvalues by m m m i= l max L IIPv(xj)11 2 = L IIxjl12 min L IIPv(xj)112, dim(V) = k . . dim(V) = k . J=l ) = 1 ) = 1 m m k m L IIXjl12 - L Ai = min L IlPv(xj)112. . . dim(V)=k . J=l .=1 J=l (4) Hence, as for the case when k = 1, the subspace spanned by the first k eigenvalues is characterised as that for which the sum of the squares of the residuals is minimal. Frequently, we consider all of the above as occurring in a kernel defined feature space, so that wherever we have written Xj we should have put ¢>(Xj), where ¢> is the corresponding projection. 3 Concentration of eigenvalues The previous section outlined the relatively well-known perspective that we now apply to obtain the concentration results for the eigenvalues of positive semi-definite matrices. The key to the results is the characterisation in terms of the sums of residuals given in equations (1) and (4). Theorem 4 Let K(x,z) be a positive semi-definite kernel function on a space X, and let J-t be a distribution on X. Fix natural numbers m and 1 :::; k < m and let S = (Xl"'" xm) E xm be a sample of m points drawn according to J-t. Then for all f > 0, P{I~ )..k(S) -lEs [~ )..k(S)ll 2: f} :::; 2exp ( -~:m) , where )..k (S) is the k-th eigenvalue of the matrix K(S) with entries K(S)ij K(Xi,Xj) and R2 = maxxEx K(x,x). Proof: The result follows from an application of Theorem 1 provided 1 1 2 sup 1- )..k(S) - - )..k(S \ {xd)1 :::; Rim. s m m Let S = S \ {Xi} and let V (11) be the k dimensional subspace spanned by the first k eigenvectors of K(S) (K(S)). Using equation (1) we have m m D Surprisingly a very similar result holds when we consider the sum of the last m - k eigenvalues. Theorem 5 Let K(x, z) be a positive semi-definite kernel function on a space X, and let J-t be a distribution on X. Fix natural numbers m and 1 :::; k < m and let S = (Xl, ... , Xm) E xm be a sample of m points drawn according to J-t. Then for all f > 0, P{I~ )..>k(S) -lEs [~ )..>k(S)ll 2: f} :::; 2 exp ( -~:m) , where )..>k(S) is the sum of all but the largest k eigenvalues of the matrix K(S) with entries K(S)ij = K(Xi,Xj) and R2 = maxxEX K(x,x). Proof: The result follows from an application of Theorem 1 provided sup 1~)..>k(S) ~)..>k(S \ {xd)1 :::; R2/m. s m m Let S = S \ {xd and let V (11) be the k dimensional subspace spanned by the first k eigenvectors of K(S) (K(S)). Using equation (4) we have m j=l #i m #i j=l D Our next result concerns the concentration of the residuals with respect to a fixed subspace. For a subspace V and training set S , we introduce the notation 1 m Fv(S) = - L IIPV(Xi )112. m i=l TheoreIll 6 Let J-t be a distribution on X. Fix natural numbers m and a subspace V and let S = (Xl, .. . ,Xm) E xm be a sample of m points drawn according to J-t. Then for all t > 0, P{IFv(S) -lEs [Fv(S)ll ~ t} ::::: 2exp (~~r;) . Proof: The result follows from an application of Theorem 2 provided sup IFv(S) - F(S \ {xd U {xi)1 ::::: R2/m. S,Xi Clearly the largest change will occur if one of the points Xi and Xi is lies in the subspace V and the other does not. In this case the change will be at most R2/m. D 4 Experiments In order to test the concentration results we performed experiments with the Breast cancer data using a cubic polynomial kernel. The kernel was chosen to ensure that the spectrum did not decay too fast. We randomly selected 50% of the data as a 'training' set and kept the remaining 50% as a 'test' set. We centered the whole data set so that the origin of the feature space is placed at the centre of gravity of the training set. We then performed an eigenvalue decomposition of the training set. The sum of the eigenvalues greater than the k-th gives the sum of the residual squared norms of the training points when we project onto the space spanned by the first k eigenvectors. Dividing this by the average of all the eigenvalues (which measures the average square norm of the training points in the transformed space) gives a fraction residual not captured in the k dimensional projection. This quantity was averaged over 5 random splits and plotted against dimension in Figure 1 as the continuous line. The error bars give one standard deviation. The Figure la shows the full spectrum, while Figure 1 b shows a zoomed in subwindow. The very tight error bars show clearly the very tight concentration of the sums of tail of eigenvalues as predicted by Theorem 5. In order to test the concentration results for subsets we measured the residuals of the test points when they are projected into the subspace spanned by the first k eigenvectors generated above for the training set. The dashed lines in Figure 1 show the ratio of the average squares of these residuals to the average squared norm of the test points. We see the two curves tracking each other very closely, indicating that the subspace identified as optimal for the training set is indeed capturing almost the same amount of information in the test points. 5 Conclusions The paper has shown that the eigenvalues of a positive semi-definite matrix generated from a random sample is concentrated. Furthermore the sum of the last m - k eigenvalues is similarly concentrated as is the residual when the data is projected into a fixed subspace. 0.7,------,-------,-------,------,-------,-------,------,,------, 0.6 0.5 0.2 0.1 Projection Dimensionality (a) 0.14,-----,-----,-----,-----,-----,-----,-----,-----,-----,-----, 0.12 0.1 e 0.08 W Cii OJ :g en &! 0.06 0.04 0.02 1 \ '1- - I - -:E-- -I- --:1:- _ '.[ __ O~--L---~--~--~---L-=~~~~======~~ o 10 20 30 40 50 60 70 80 90 100 Projection Dimensionality (b) Figure 1: Plots ofthe fraction of the average squared norm captured in the subspace spanned by the first k eigenvectors for different values of k. Continuous line is fraction for training set, while the dashed line is for the test set. (a) shows the full spectrum, while (b) zooms in on an interesting portion. Experiments are presented that confirm the theoretical predictions on a real world dataset. The results provide a basis for performing PCA or kernel-PCA from a randomly generated sample, as they confirm that the subset identified by the sample will indeed 'generalise' in the sense that it will capture most of the information in a test sample. Further research should look at the question of how the space identified by a subsample relates to the eigenspace of the underlying kernel operator. References [1] S. Brin and L. Page. The anatomy of a large-scale hypertextual (web) search engine. In Proceedings of the Seventh International World Wide Web Conference, 1998. [2] Nello Cristianini, Huma Lodhi, and John Shawe-Taylor. Latent semantic kernels for feature selection. Technical Report NC-TR-00-080, NeuroCOLT Working Group, http://www.neurocolt.org, 2000. [3] J. Kleinberg. Authoritative sources in a hyperlinked environment. In Proceedings of 9th ACM-SIAM Symposium on Discrete Algorithms, 1998. [4] C. McDiarmid. On the method of bounded differences. In Surveys in Combinatorics 1989, pages 148- 188. Cambridge University Press, 1989. [5] S. Mika, B. SchCilkopf, A. Smola, K.-R. MUller, M. Scholz, and G. Ratsch. Kernel PCA and de-noising in feature spaces. In Advances in Neural Information Processing Systems 11, 1998. [6] Andrew Y. Ng, Alice X. Zheng, and Michael 1. Jordan. Link analysis, eigenvectors and stability. In To appear in the Seventeenth International Joint Conference on Artificial Intelligence (UCAI-Ol), 2001.
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Small-World Phenomena and the Dynamics of Information Jon Kleinberg Department of Computer Science Cornell University Ithaca NY 14853 1 Introduction The problem of searching for information in networks like the World Wide Web can be approached in a variety of ways, ranging from centralized indexing schemes to decentralized mechanisms that navigate the underlying network without knowledge of its global structure. The decentralized approach appears in a variety of settings: in the behavior of users browsing the Web by following hyperlinks; in the design of focused crawlers [4, 5, 8] and other agents that explore the Web’s links to gather information; and in the search protocols underlying decentralized peer-to-peer systems such as Gnutella [10], Freenet [7], and recent research prototypes [21, 22, 23], through which users can share resources without a central server. In recent work, we have been investigating the problem of decentralized search in large information networks [14, 15]. Our initial motivation was an experiment that dealt directly with the search problem in a decidedly pre-Internet context: Stanley Milgram’s famous study of the small-world phenomenon [16, 17]. Milgram was seeking to determine whether most pairs of people in society were linked by short chains of acquaintances, and for this purpose he recruited individuals to try forwarding a letter to a designated “target” through people they knew on a firstname basis. The starting individuals were given basic information about the target — his name, address, occupation, and a few other personal details — and had to choose a single acquaintance to send the letter to, with goal of reaching the target as quickly as possible; subsequent recipients followed the same procedure, and the chain closed in on its destination. Of the chains that completed, the median number of steps required was six — a result that has since entered popular culture as the “six degrees of separation” principle [11]. Milgram’s experiment contains two striking discoveries — that short chains are pervasive, and that people are able to find them. This latter point is concerned precisely with a type of decentralized navigation in a social network, consisting of people as nodes and links joining pairs of people who know each other. From an algorithmic perspective, it is an interesting question to understand the structure of networks in which this phenomenon emerges — in which message-passing with purely local information can be efficient. Networks that Support Efficient Search. A model of a “navigable” network requires a few basic features. It should contain short paths among all (or most) pairs of nodes. To be non-trivial, its structure should be partially known and partially unknown to its constituent nodes; in this way, information about the known parts can be used to construct paths that make use of the unknown parts as well. This is clearly what was taking place in Milgram’s experiments: participants, using the information available to them, were estimating which of their acquaintances would lead to the shortest path through the full social network. Guided by these observations, we turned to the work of Watts and Strogatz [25], which proposes a model of “small-world networks” that very concisely incorporates these features. A simple variant of their basic model can be described as follows. One starts with a p-dimensional lattice, in which nodes are joined only to their nearest neighbors. One then adds k directed long-range links out of each node v, for a constant k; the endpoint of each link is chosen uniformly at random. Results from the theory of random graphs can be used to show that with high probability, there will be short paths connecting all pairs of nodes (see e.g. [3]); at the same time, the network will locally retain a lattice-like structure. Asymptotically, our criterion for “shortness” of paths is what one obtains from this and similar random constructions: there should be paths whose lengths are bounded by a polynomial in log n, where n is the number of nodes. We will refer to such a function as polylogarithmic. This network model, a superposition of a lattice and a set of long-range links, is a natural one in which to study the behavior of a decentralized search algorithm. The algorithm knows the structure of the lattice; it starts from a node s, and is told the coordinates of a target node t. It successively traverses links of the network so as to reach the target as quickly as possible; but, crucially, it does not know the long-range links out of any node that it has not yet visited. In addition to moving forward across directed links, the algorithm may travel in reverse across any link that it has already followed in the forward direction; this allows it to back up when it does not want to continue exploring from its current node. One can view this as hitting the “back button” on a Web browser — or returning the letter to its previous holder in Milgram’s experiments, with instructions that he or she should try someone else. We say that the algorithm has search time Y (n) if, on a randomly generated n-node network with s and t chosen uniformly at random, it reaches the target t in at most Y (n) steps with probability at least 1 −ε(n), for a function ε(·) going to 0 with n. The first result in [15] is that no decentralized algorithm can achieve a polylogarithmic search time in this network model — even though, with high probability, there are paths of polylogarithmic length joining all pairs of nodes. However, if we generalize the model slightly, then it can support efficient search. Specifically, when we construct a long-range link (v, w) out of v, we do not choose w uniformly at random; rather, we choose it with probability proportional to d−α, where d is the lattice distance from v to w. In this way, the long-range links become correlated to the geometry of the lattice. We show in [15] that when α is equal to p, the dimension of the underlying lattice, then a decentralized greedy algorithm achieves search time proportional to log2 n; and for any other value of α, there is no decentralized algorithm with a polylogarithmic search time. Recent work by Zhang, Goel, and Govindan [26] has shown how the distribution of links associated with the optimal value of α may lead to improved performance for decentralized search in the Freenet peer-to-peer system. Adamic, Lukose, Puniyani, and Huberman [2] have recently considered a variation of the decentralized search problem in a network that has essentially no known underlying structure; however, when the number of links incident to nodes follows a power-law distribution, then a search strategy that seeks high-degree nodes can be effective. They have applied their results to the Gnutella system, which exhibits such a structure. In joint work with Kempe and Demers [12], we have studied how distributions that are inverse-polynomial in the distance between nodes can be used in the design of gossip protocols for spreading information in a network of communicating agents. The goal of the present paper is to consider more generally the problem of decentralized search in networks with partial information about the underlying structure. While a lattice makes for a natural network backbone, we would like to understand the extent to which the principles underlying efficient decentralized search can be abstracted away from a lattice-like structure. We begin by considering networks that are generated from a hierarchical structure, and show that qualitatively similar results can be obtained; we then discuss a general model of group structures, which can be viewed as a simultaneous generalization of lattices and hierarchies. We refer to k, the number of out-links per node, as the out-degree of the model. The technical details of our results — both in the statements of the results and the proofs — are simpler when we allow the out-degree to be polylogarithmic, rather than constant. Thus we describe this case first, and then move on to the case in which each node has only a constant number of out-links. 2 Hierarchical Network Models In a number of settings, nodes represent objects that can be classified according to a hierarchy or taxonomy; and nodes are more likely to form links if they belong to the same small sub-tree in the hierarchy, indicating they are more closely related. To construct a network model from this idea, we represent the hierarchy using a complete b-ary tree T, where b is a constant. Let V denote the set of leaves of T; we use n to denote the size of V , and for two leaves v and w, we use h(v, w) to denote the height of the least common ancestor of v and w in T. We are also given a monotone non-increasing function f(·) that will determine link probabilities. For each node v ∈V , we create a random link to w with probability proportional to f(h(v, w)); in other words, the probability of choosing w is equal to f(h(v, w))/ P x̸=v f(h(v, x)). We create k links out of each node v this way, choosing the endpoint w each time independently and with repetition allowed. This results in a graph G on the set V . For the analysis in this section, we will take the out-degree to be k = c log2 n, for a constant c. It is important to note that the tree T is used only in the generation process of G; neither the edges nor the non-leaf nodes of T appear in G. (By way of contrast, the lattice model in [15] included both the long-range links and the nearest-neighbor links of the lattice itself.) When we use the term “node” without any qualification, we are referring to nodes of G, and hence to leaves of T; we will use “internal node” to refer to non-leaf nodes of T. We refer to the process producing G as a hierarchical model with exponent α if the function f(h) grows asymptotically like b−αh: lim h→∞ f(h) b−α′h = 0 for all α′ < α and lim h→∞ b−α′′h f(h) = 0 for all α′′ > α. There are several natural interpretations for a hierarchical network model. One is in terms of the World Wide Web, where T is a topic hierarchy such as www.yahoo.com. Each leaf of T corresponds to a Web page, and its path from the root specifies an increasingly fine-grained description of the page’s topic. Thus, a particular leaf may be associated with Science/Computer Science/Algorithms or with Arts/Music/Opera. The linkage probabilities then have a simple meaning — they are based on the distance between the topics of the pages, as measured by the height of their least common ancestor in the topic hierarchy. A page on Science/Computer Science/Algorithms may thus be more likely to link to one on Science/Computer Science/Machine Learning than to one on Arts/Music/Opera. Of course, the model is a strong simplification, since topic structures are not fully hierarchical, and certainly do not have uniform branching and depth. It is worth noting that a number of recent models for the link structure of the Web, as well as other relational structures, have looked at different ways in which similarities in content can affect the probability of linkage; see e.g. [1, 6, 9]. Another interpretation of the hierarchical model is in terms of Milgram’s original experiment. Studies performed by Killworth and Bernard [13] showed that in choosing a recipient for the letter, participants were overwhelmingly guided by one of two criteria: similarity to the target in geography, or similarity to the target in occupation. If one views the lattice as forming a simple model for geographic factors, the hierarchical model can similarly be interpreted as forming a “topic hierarchy” of occupations, with individuals at the leaves. Thus, for example, the occupations of “banker” and “stock broker” may belong to the same small sub-tree; since the target in one of Milgram’s experiments was a stock broker, it might therefore be a good strategy to send the letter to a banker. Independently of our work here, Watts, Dodds, and Newman have recently studied hierarchical structures for modeling Milgram’s experiment in social networks [24]. We now consider the search problem in a graph G generated from a hierarchical model: A decentralized algorithm has knowledge of the tree T, and knows the location of a target leaf that it must reach; however, it only learns the structure of G as it visits nodes. The exponent α determines how the structures of G and T are related; how does this affect the navigability of G? In the analysis of the lattice model [15], the key property of the optimal exponent was that, from any point, there was a reasonable probability of a long-range link that halved the distance to the target. We make use of a similar idea here: when α = 1, there is always a reasonable probability of finding a long-range link into a strictly smaller sub-tree containing the target. As mentioned above, we focus here on the case of polylogarithmic outdegree, with the case of constant out-degree deferred until later. Theorem 2.1 (a) There is a hierarchical model with exponent α = 1 and polylogarithmic out-degree in which a decentralized algorithm can achieve search time O(logn). (b) For every α ̸= 1, there is no hierarchical model with exponent α and polylogarithmic out-degree in which a decentralized algorithm can achieve polylogarithmic search time. Due to space limitations, we omit proofs from this version of the paper. Complete proofs may be found in the extended version, which is available on the author’s Web page (http://www.cs.cornell.edu/home/kleinber/). To prove (a), we show that when the search is at a node v whose least common ancestor with the target has height h, there is a high probability that v has a link into the sub-tree of height h−1 containing the target. In this way, the search reaches the target in logarithmically many steps. To prove (b), we exhibit a sub-tree T ′ containing the target such that, with high probability, it takes any decentralized algorithm more than a polylogarithmic number of steps to find a link into T ′. 3 Group Structures The analysis of the search problem in a hierarchical model is similar to the analysis of the lattice-based approach in [15], although the two types of models seem superficially quite different. It is natural to look for a model that would serve as a simultaneous generalization of each. Consider a collection of individuals in a social network, and suppose that we know of certain groups to which individuals belong — people who live in the same town, or work in the same profession, or have some other affiliation in common. We could imagine that people are more likely to be connected if they both belong to the same small group. In a lattice-based model, there may be a group for each subset of lattice points contained in a common ball (grouping based on proximity); in a hierarchical model, there may be a group for each subset of leaves contained in a common sub-tree. We now discuss the notion of a group structure, to make this precise; we follow a model proposed in joint work with Kempe and Demers [12], where we were concerned with designing gossip protocols for lattices and hierarchies. A technically different model of affiliation networks, also motivated by these types of issues, has been studied recently by Newman, Watts, and Strogatz [18]. A group structure consists of an underlying set V of nodes, and a collection of subsets of V (the groups). The collection of groups must include V itself; and it must satisfy the following two properties, for constants λ < 1 and β > 1. (i) If R is a group of size q ≥2 containing a node v, then there is a group R′ ⊆R containing v that is strictly smaller than R, but has size at least λq. (ii) If R1, R2, R3, . . . are groups that all have size at most q and all contain a common node v, then their union has size at most βq. The reader can verify that these two properties hold for the collection of balls in a lattice, as well as for the collection of sub-trees in a hierarchy. However, it is easy to construct examples of group structures that do not arise in this way from lattices or hierarchies. Given a group structure (V, {Ri}), and a monotone non-increasing function f(·), we now consider the following process for generating a graph on V . For two nodes v and w, we use q(v, w) to denote the minimum size of a group containing both v and w. (Note that such a group must exist, since V itself is a group.) For each node v ∈V , we create a random link to w with probability proportional to f(q(v, w)); repeating this k times independently yields k links out of v. We refer to this as a group-induced model with exponent α if f(q) grows asymptotically like q−α: lim h→∞ f(q) q−α′ = 0 for all α′ < α and lim h→∞ q−α′′ f(q) = 0 for all α′′ > α. A decentralized search algorithm in such a network is given knowledge of the full group structure, and must follow links of G to a designated target t. We now state an analogue of Theorem 2.1 for group structures. Theorem 3.1 (a) For every group structure, there is a group-induced model with exponent α = 1 and polylogarithmic out-degree in which a decentralized algorithm can achieve search time O(logn). (b) For every α < 1, there is no group-induced model with exponent α and polylogarithmic out-degree in which a decentralized algorithm can achieve polylogarithmic search time. Notice that in a hierarchical model, the smallest group (sub-tree) containing two nodes v and w has size bh(v,w), and so Theorem 3.1(a) implies Theorem 2.1(a). Similarly, on a lattice, the smallest group (ball) containing two nodes v and w at lattice distance d has size Θ(dp), and so Theorem 3.1(a) implies a version of the result from [15], that efficient search is possible in a lattice model when nodes form links with probability d−p. (In the version of the lattice result implied here, there are no nearest-neighbor links at all; but each node has a polylogarithmic number of out-links.) The proof of Theorem 3.1(a) closely follows the proof of Theorem 2.1(a). We consider a node v — the current point in the search — for which the smallest group containing v and the target t has size q. Using group structure properties (i) and (ii), we show there is a high probability that v has a link into a group containing t of size between λ2q and λq. In this way, the search passes through groups containing t of sizes that diminish geometrically, and hence it terminates in logarithmic time. Note that Theorem 3.1(b) only considers exponents α < 1. This is because there exist group-induced models with exponents α > 1 in which decentralized algorithms can achieve polylogarithmic search time. For example, consider an undirected graph G∗in which each node has 3 neighbors, and each pair of nodes can be connected by a path of length O(log n). It is possible to define a group structure satisfying properties (i) and (ii) in which each edge of G∗appears as a 2-node group; but then, a graph G generated from a group-induced model with a very large exponent α will contain all edges of G∗with high probability, and a decentralized search algorithm will be able to follow these edges directly to construct a short path to the target. However, a lower bound for the case α > 1 can be obtained if we place one additional restriction on the group structure. Give a group structure (V, {Ri}), and a cut-off value q, we define a graph H(q) on V by joining any two nodes that belong to a common group of size at most q. Note that H(q) is not a random graph; it is defined simply in terms of the group structure and q. We now argue that if many pairs of nodes are far apart in H(q), for a suitably large value of q, then a decentralized algorithm cannot be efficient when α > 1. Theorem 3.2 Let (V, {Ri}) be a group structure. Suppose there exist constants γ, δ > 0 so that a constant fraction of all pairs of nodes have shortest-path distance Ω(nδ) in H(nγ). Then for every α > 1, there is no group-induced model on (V, {Ri}) with exponent α and a polylogarithmic number of out-links per node in which a decentralized algorithm can achieve polylogarithmic search time. Notice this property holds for group structures arising from both lattices and hierarchies; in a lattice, a constant fraction of all pairs in H(n1/2p) have distance Ω(n1/2p), while in a hierarchy, the graph H(nγ) is disconnected for every γ < 1. 4 Nodes with a Constant Number of Out-Links Thus far, by giving each node more than a constant number of out-links, we have been able to design very simple search algorithms in networks generated according to the optimal exponent α. From each node, there is a way to make progress toward the target node t, and so the structure of the graph G funnels the search towards its destination. When the out-degree is constant, however, things get much more complicated. First of all, with high probability, many nodes will have all their links leading “away” from the target in the hierarchy. Second, there is a constant probability that the target t will have no in-coming links, and so the whole task of finding t becomes ill-defined. This indicates that the statement of the results themselves in this case will have to be somewhat different. In this section, we work with a hierarchical model, and construct graphs with constant out-degree k; the value of k will need to be sufficiently large in terms of other parameters of the model. It is straightforward to formulate an analogue of our results for group structures, but we do not go into the details of this here. To deal with the problem that t itself may have no incoming links, we relax the search problem to that of finding a cluster of nodes containing t. In a topic-based model of Web pages, for example, we can consider t as a representative of a desired type of page, with goal being to find any page of this type. Thus, we are given a complete b-ary tree T, where b is a constant; we let L denote the set of leaves of T, and m denote the size of L. We place r nodes at each leaf of T, forming a set V of n = mr nodes total. We then define a graph G on V as in Section 2: for a non-increasing function f(·), we create k links out of each node v ∈V , choosing w as an endpoint with probability proportional to f(h(v, w)). As before, we refer to this process as a hierarchical model with exponent α, for the appropriate value of α. We refer to each set of r nodes at a common leaf of T as a cluster, and define the resolution of the hierarchical model to be the value r. A decentralized algorithm is given knowledge of T, and a target node t; it must reach any node in the cluster containing t. Unlike the previous algorithms we have developed, which only moved forward across links, the algorithm we design here will need to make use of the ability to travel in reverse across any link that it has already followed in the forward direction. Note also that we cannot easily reduce the current search problem to that of Section 2 by collapsing clusters into “super-nodes,” since there are not necessarily links joining nodes within the same cluster. The search task clearly becomes easier as the resolution of the model (i.e. the size of clusters) becomes larger. Thus, our goal is to achieve polylogarithmic search time in a hierarchical model with polylogarithmic resolution. Theorem 4.1 (a) There is a hierarchical model with exponent α = 1, constant out-degree, and polylogarithmic resolution in which a decentralized algorithm can achieve polylogarithmic search time. (b) For every α ̸= 1, there is no hierarchical model with exponent α, constant outdegree, and polylogarithmic resolution in which a decentralized algorithm can achieve polylogarithmic search time. The search algorithm used to establish part (a) operates in phases. It begins each phase j with a collection of Θ(log n) nodes all belonging to the sub-tree Tj that contains the target t and whose root is at depth j. During phase j, it explores outward from each of these nodes until it has discovered a larger but still polylogarithmicsized set of nodes belonging to Tj. From among these, there is a high probability that at least Θ(log n) have links into the smaller sub-tree Tj+1 that contains t and whose root is at depth j + 1. At this point, phase j + 1 begins, and the process continues until the cluster containing t is found. Acknowledgments My thinking about models for Web graphs and social networks has benefited greatly from discussions and collaboration with Dimitris Achlioptas, Avrim Blum, Duncan Callaway, Michelle Girvan, John Hopcroft, David Kempe, Ravi Kumar, Tom Leighton, Mark Newman, Prabhakar Raghavan, Sridhar Rajagopalan, Steve Strogatz, Andrew Tomkins, Eli Upfal, and Duncan Watts. The research described here was supported in part by a David and Lucile Packard Foundation Fellowship, an ONR Young Investigator Award, NSF ITR/IM Grant IIS-0081334, and NSF Faculty Early Career Development Award CCR-9701399. References [1] D. Achlioptas, A. Fiat, A. Karlin, F. McSherry, “Web search via hub synthesis,” Proc. 42nd IEEE Symp. on Foundations of Computer Science, 2001. [2] L. Adamic, R. Lukose, A. Puniyani, B. Huberman, “Search in Power-Law Networks,” Phys. Rev. E, 64 46135 (2001) [3] B. Bollob´as, F. Chung, “The diameter of a cycle plus a random matching,” SIAM J. Disc. Math. 1(1988). [4] S. Chakrabarti, M. van den Berg, B. Dom, “Focused crawling: A new approach to topic-specific Web resource discovery,” Proc. 8th Intl. World Wide Web Conf., 1999. [5] J. Cho, H. Garcia-Molina, L. Page, “Efficient Crawling Through URL Ordering,” Proc. 7th Intl. World Wide Web Conf., 1998. [6] D. Cohn and T. Hofmann, “The Missing Link – A Probabilistic Model of Document Content and Hypertext Connectivity,” Adv. Neural Inf. Proc. Sys. (NIPS) 13, 2000. [7] I. Clarke, O. Sandberg, B. Wiley, T. Hong, “Freenet: A Distributed Anonymous Information Storage and Retrieval System,” International Workshop on Design Issues in Anonymity and Unobservability, 2000. [8] M. Diligenti, F.M. Coetzee, S. Lawrence, C.L. Giles, M. Gori, “Focused Crawling Using Context Graphs,” Proc. 26th Intl. Conf. on Very Large Databases (VLDB), 2000. [9] L. Getoor, N. Friedman, D. Koller, and B. Taskar. “Learning Probabilistic Models of Relational Structure,” Proc. 18th International Conference on Machine Learning, 2001. [10] Gnutella. http://gnutella.wego.com. [11] J. Guare, Six Degrees of Separation: A Play (Vintage Books, New York, 1990). [12] D. Kempe, J. Kleinberg, A. Demers. “Spatial gossip and resource location protocols,” Proc. 33rd ACM Symp. on Theory of Computing, 2001. [13] P. Killworth, H. Bernard, “Reverse small world experiment,” Social Networks 1(1978). [14] J. Kleinberg. “Navigation in a Small World.” Nature 406(2000). [15] J. Kleinberg. “The small-world phenomenon: An algorithmic perspective.” Proc. 32nd ACM Symposium on Theory of Computing, 2000. Also appears as Cornell Computer Science Technical Report 99-1776 (October 1999). [16] M. Kochen, Ed., The Small World (Ablex, Norwood, 1989). [17] S. Milgram, “The small world problem,” Psychology Today 1(1967). [18] M. Newman, D. Watts, S. Strogatz, “Random graph models of social networks,” Proc. Natl. Acad. Sci., to appear. [19] A. Oram, editor, Peer-to-Peer: Harnessing the Power of Disruptive Technologies O’Reilly and Associates, 2001. [20] A. Puniyani, R. Lukose, B. Huberman, “Intentional Walks on Scale Free Small Worlds,” HP Labs Information Dynamics Group, at http://www.hpl.hp.com/shl/. [21] S. Ratnasamy, P. Francis, M. Handley, R. Karp, S. Shenker, “A Scalable ContentAddressable Network,” Proc. ACM SIGCOMM, 2001 [22] A. Rowstron, P. Druschel, “Pastry: Scalable, distributed object location and routing for large-scale peer-to-peer systems,” Proc. 18th IFIP/ACM International Conference on Distributed Systems Platforms (Middleware 2001), 2001. [23] I. Stoica, R. Morris, D. Karger, F. Kaashoek, H. Balakrishnan, “Chord: A Scalable Peer-to-peer Lookup Service for Internet Applications,” Proc. ACM SIGCOMM, 2001 [24] D. Watts, P. Dodds, M. Newman, personal communication, December 2001. [25] D. Watts, S. Strogatz, “Collective dynamics of small-world networks,” Nature 393(1998). [26] H. Zhang, A. Goel, R. Govindan, “Using the Small-World Model to Improve Freenet Performance,” Proc. IEEE Infocom, 2002.
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Stochastic Mixed-Signal VLSI Architecture for High-Dimensional Kernel Machines Roman Genov and Gert Cauwenberghs Department of Electrical and Computer Engineering Johns Hopkins University, Baltimore, MD 21218 roman,gert @jhu.edu Abstract A mixed-signal paradigm is presented for high-resolution parallel innerproduct computation in very high dimensions, suitable for efficient implementation of kernels in image processing. At the core of the externally digital architecture is a high-density, low-power analog array performing binary-binary partial matrix-vector multiplication. Full digital resolution is maintained even with low-resolution analog-to-digital conversion, owing to random statistics in the analog summation of binary products. A random modulation scheme produces near-Bernoulli statistics even for highly correlated inputs. The approach is validated with real image data, and with experimental results from a CID/DRAM analog array prototype in 0.5 m CMOS. 1 Introduction Analog computational arrays [1, 2, 3, 4] for neural information processing offer very large integration density and throughput as needed for real-time tasks in computer vision and pattern recognition [5]. Despite the success of adaptive algorithms and architectures in reducing the effect of analog component mismatch and noise on system performance [6, 7], the precision and repeatability of analog VLSI computation under process and environmental variations is inadequate for some applications. Digital implementation [10] offers absolute precision limited only by wordlength, but at the cost of significantly larger silicon area and power dissipation compared with dedicated, fine-grain parallel analog implementation, e.g., [2, 4]. The purpose of this paper is twofold: to present an internally analog, externally digital architecture for dedicated VLSI kernel-based array processing that outperforms purely digital approaches with a factor 100-10,000 in throughput, density and energy efficiency; and to provide a scheme for digital resolution enhancement that exploits Bernoulli random statistics of binary vectors. Largest gains in system precision are obtained for high input dimensions. The framework allows to operate at full digital resolution with relatively imprecise analog hardware, and with minimal cost in implementation complexity to randomize the input data. The computational core of inner-product based kernel operations in image processing and pattern recognition is that of vector-matrix multiplication (VMM) in high dimensions:
(1) with -dimensional input vector , -dimensional output vector , and matrix elements . In artificial neural networks, the matrix elements correspond to weights, or synapses, between neurons. The elements also represent templates in a vector quantizer [8], or support vectors in a support vector machine [9]. In what follows we concentrate on VMM computation which dominates inner-product based1 kernel computations for high vector dimensions. 2 The Kerneltron: A Massively Parallel VLSI Computational Array 2.1 Internally Analog, Externally Digital Computation The approach combines the computational efficiency of analog array processing with the precision of digital processing and the convenience of a programmable and reconfigurable digital interface. The digital representation is embedded in the analog array architecture, with inputs presented in bit-serial fashion, and matrix elements stored locally in bit-parallel form: "! $# (2) % & & (' ! & # (3) decomposing (1) into: ) *
+ , - % & & . ! / & # (4) with binary-binary VMM partials: ! / & #
! 0# ' ! & # 21 (5) The key is to compute and accumulate the binary-binary partial products (5) using an analog VMM array, and to combine the quantized results in the digital domain according to (4). Digital-to-analog conversion at the input interface is inherent in the bit-serial implementation, and row-parallel analog-to-digital converters (ADCs) are used at the output interface to quantize ! / & # . A 512 128 array prototype using CID/DRAM cells is shown in Figure 1 (a). 2.2 CID/DRAM Cell and Array The unit cell in the analog array combines a CID computational element [12, 13] with a DRAM storage element. The cell stores one bit of a matrix element ! 0# , performs a one-quadrant binary-binary multiplication of ! $# and ' ! & # in (5), and accumulates 1Radial basis kernels with 354 -norm can also be formulated in inner product format. (a) RS(i) Vout(i) RS(i) Vout(i) M1 M2 M3 0 Vdd/2 Vdd DRAM CID Write Compute 0 Vdd/2 Vdd 0 Vdd/2 Vdd x(j) w(i) x(j) mn n n m m m m (b) Figure 1: (a) Micrograph of the Kerneltron prototype, containing an array of CID/DRAM cells, and a row-parallel bank of flash ADCs. Die size is in 0.5 m CMOS technology. (b) CID computational cell with integrated DRAM storage. Circuit diagram, and charge transfer diagram for active write and compute operations. the result across cells with common and indices. The circuit diagram and operation of the cell are given in Figure 1 (b). An array of cells thus performs (unsigned) binary multiplication (5) of matrix ! $# and vector ' ! & # yielding ! / & # , for values of in parallel across the array, and values of in sequence over time. The cell contains three MOS transistors connected in series as depicted in Figure 1 (b). Transistors M1 and M2 comprise a dynamic random-access memory (DRAM) cell, with switch M1 controlled by Row Select signal
! $# . When activated, the binary quantity ! $# is written in the form of charge (either or 0) stored under the gate of M2. Transistors M2 and M3 in turn comprise a charge injection device (CID), which by virtue of charge conservation moves electric charge between two potential wells in a non-destructive manner [12, 13, 14]. The charge left under the gate of M2 can only be redistributed between the two CID transistors, M2 and M3. An active charge transfer from M2 to M3 can only occur if there is non-zero charge stored, and if the potential on the gate of M2 drops below that of M3 [12]. This condition implies a logical AND, i.e., unsigned binary multiplication, of ! $# and ' ! & # . The multiply-and-accumulate operation is then completed by capacitively sensing the amount of charge transferred onto the electrode of M3, the output summing node. To this end, the voltage on the output line, left floating after being pre-charged to , is observed. When the charge transfer is active, the cell contributes a change in voltage "! where # "! is the total capacitance on the output line across cells. The total response is thus proportional to the number of actively transferring cells. After deactivating the input ' ! & # , the transferred charge returns to the storage node M2. The CID computation is non-destructive and intrinsically reversible [12], and DRAM refresh is only required to counteract junction and subthreshold leakage. The bottom diagram in Figure 1 (b) depicts the charge transfer timing diagram for write and compute operations in the case when both ! 0# and ' ! & # are of logic level 1. 2.3 System-Level Performance Measurements on the 512 128-element analog array and other fabricated prototypes show a dynamic range of 43 dB, and a computational cycle of 10 s with power consumption of 50 nW per cell. The size of the CID/DRAM cell is 8 45 with 1 . The overall system resolution is limited by the precision in the quantization of the outputs from the analog array. Through digital postprocessing, two bits are gained over the resolution of the ADCs used [15], for a total system resolution of 8 bits. Larger resolutions can be obtained by accounting for the statistics of binary terms in the addition, the subject of the next section. 3 Resolution Enhancement Through Stochastic Encoding Since the analog inner product (5) is discrete, zero error can be achieved (as if computed digitally) by matching the quantization levels of the ADC with each of the discrete levels in the inner product. Perfect reconstruction of ! / & # from the quantized output, for an overall resolution of
. bits, assumes the combined effect of noise and nonlinearity in the analog array and the ADC is within one LSB (least significant bit). For large arrays, this places stringent requirements on analog precision and ADC resolution,
. . The implicit assumption is that all quantization levels are (equally) needed. A straightforward study of the statistics of the inner product, below, reveals that this is poor use of available resources. 3.1 Bernoulli Statistics In what follows we assume signed, rather than unsigned, binary values for inputs and weights, ' ! & # and ! 0# . This translates to exclusive-OR (XOR), rather than AND, multiplication on the analog array, an operation that can be easily accomplished with the CID/DRAM architecture by differentially coding input and stored bits using twice the number of columns and unit cells. For input bits ' ! & # that are Bernoulli distributed (i.e., fair coin flips), the (XOR) product terms ! $# ' ! & # in (5) are Bernoulli distributed, regardless of ! $# . Their sum ! / & # thus follows a binomial distribution ! / & # ! #" %$'&)( & ( (6) with & # 1 , #+* 1$101 * , which in the Central Limit -,/. approaches a normal distribution with zero mean and variance . In other words, for random inputs in high dimensions the active range (or standard deviation) of the inner-product is 10 . , a factor 10 . smaller than the full range . In principle, this allows to relax the effective resolution of the ADC. However, any reduction in conversion range will result in a small but non-zero probability of overflow. In practice, the risk of overflow can be reduced to negligible levels with a few additional bits in the ADC conversion range. An alternative strategy is to use a variable resolution ADC which expands the conversion range on rare occurences of overflow.2 2Or, with stochastic input encoding, overflow detection could initiate a different random draw. 0.2 0.4 0.6 0.8 −20 −10 0 10 20 Output Voltage (V) Inner Product 0.2 0.4 0.6 0.8 0 10 20 30 40 50 Output Voltage (V) Count (a) (b) Figure 2: Experimental results from CID/DRAM analog array. (a) Output voltage on the sense line computing exclusive-or inner product of 64-dimensional stored and presented binary vectors. A variable number of active bits is summed at different locations in the array by shifting the presented bits. (b) Top: Measured output and actual inner product for 1,024 samples of Bernoulli distributed pairs of stored and presented vectors. Bottom: Histogram of measured array outputs. 3.2 Experimental Results While the reduced range of the analog inner product supports lower ADC resolution in terms of number of quantization levels, it requires low levels of mismatch and noise so that the discrete levels can be individually resolved, near the center of the distribution. To verify this, we conducted the following experiment. Figure 2 shows the measured outputs on one row of 128 CID/DRAM cells, configured differentially to compute signed binary (exclusive-OR) inner products of stored and presented binary vectors in 64 dimensions. The scope trace in Figure 2 (a) is obtained by storing all bits, and shifting a sequence of input bits that differ with the stored bits by bits. The left and right segment of the scope trace correspond to different selections of active bit locations along the array that are maximally disjoint, to indicate a worst-case mismatch scenario. The measured and actual inner products in Figure 2 (b) are obtained by storing and presenting 1,024 pairs of random binary vectors. The histogram shows a clearly resolved, discrete binomial distribution for the observed analog voltage. For very large arrays, mismatch and noise may pose a problem in the present implementation with floating sense line. A sense amplifier with virtual ground on the sense line and feedback capacitor optimized to the 10 . range would provide a simple solution. 3.3 Real Image Data Although most randomly selected patterns do not correlate with any chosen template, patterns from the real world tend to correlate, and certainly those that are of interest to kernel computation 3. The key is stochastic encoding of the inputs, as to randomize the bits presented to the analog array. 3This observation, and the binomial distribution for sums of random bits (6), forms the basis for the associative recall in a Kanerva memory. −1000 −500 0 500 1000 0 100 200 300 400 500 600 Inner Product Count −1000 −500 0 500 1000 0 100 200 300 400 500 600 Inner Product Count −1000 −500 0 500 1000 0 1 2 3 4 5 6 7 8 9 10 11 Inner Product Count −1000 −500 0 500 1000 0 1 2 3 4 5 6 7 8 9 10 11 Inner Product Count Figure 3: Histograms of partial binary inner products ! / & # for 256 pairs of randomly selected 32 32 pixel segments of Lena. Left: with unmodulated 8-bit image data for both vectors. Right: with 12-bit modulated stochastic encoding of one of the two vectors. Top: all bit planes and . Bottom: most significant bit (MSB) plane, . Randomizing an informative input while retaining the information is a futile goal, and we are content with a solution that approachesthe ideal performance within observable bounds, and with reasonable cost in implementation. Given that “ideal” randomized inputs relax the ADC resolution by . bits, they necessarily reduce the wordlenght of the output by the same. To account for the lost bits in the range of the output, it is necessary to increase the range of the “ideal” randomized input by the same number of bits. One possible stochastic encoding scheme that restores the range is 10 . -fold oversampling of the input through (digital) delta-sigma modulation. This is a workable solution; however we propose one that is simpler and less costly to implement. For each -bit input component , pick a random integer in the range 10 . , and subtract it to produce a modulated input with . additional bits. It can be shown that for worst-case deterministic inputs the mean of the inner product for is off at most by 10 . from the origin. The desired inner products for are retrieved by digitally adding the inner products obtained for and . The random offset can be chosen once, so its inner product with the templates can be pre-computed upon initializing or programming the array. The implementation cost is thus limited to component-wise subtraction of and , achieved using one full adder cell, one bit register, and ROM storage of the ! $# bits for every column of the array. Figure 3 provides a proof of principle, using image data selected at random from Lena. 12-bit stochastic encoding of the 8-bit image, by subtracting a random variable in a range 15 times larger than the image, produces the desired binomial distribution for the partial bit inner products, even for the most significant bit (MSB) which is most highly correlated. 4 Conclusions We presented an externally digital, internally analog VLSI array architecture suitable for real-time kernel-based neural computation and machine learning in very large dimensions, such as image recognition. Fine-grain massive parallelism and distributed memory, in an array of 3-transistor CID/DRAM cells, provides a throughput of . binary MACS (multiply accumulates per second) per Watt of power in a 0.5 m process. A simple stochastic encoding scheme relaxes precision requirements in the analog implementation by one bit for each four-fold increase in vector dimension, while retaining full digital overall system resolution. Acknowledgments This research was supported by ONR N00014-99-1-0612, ONR/DARPA N00014-00-C0315, and NSF MIP-9702346. Chips were fabricated through the MOSIS service. References [1] A. Kramer, “Array-based analog computation,”IEEE Micro, vol. 16 (5), pp. 40-49, 1996. [2] G. Han, E. Sanchez-Sinencio, “A general purpose neuro-image processor architecture,” Proc. of IEEE Int. Symp. on Circuits and Systems (ISCAS’96), vol. 3, pp 495-498, 1996 [3] F. Kub, K. Moon, I. Mack, F. Long, “Programmable analog vector-matrix multipliers,”IEEE Journal of Solid-State Circuits, vol. 25 (1), pp. 207-214, 1990. [4] G. Cauwenberghs and V. Pedroni, “A Charge-Based CMOS Parallel Analog Vector Quantizer,” Adv. 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Wawrzynek, et al., “SPERT-II: A Vector Microprocessor System and its Application to Large Problems in Backpropagation Training,” in Advances in Neural Information Processing Systems, Cambridge, MA: MIT Press, vol. 8, pp 619-625, 1996. [11] A. Chiang, “A programmable CCD signal processor,” IEEE Journal of Solid-State Circuits, vol. 25 (6), pp. 1510-1517, 1990. [12] C. Neugebauer and A. Yariv, “A Parallel Analog CCD/CMOS Neural Network IC,”Proc. IEEE Int. Joint Conference on Neural Networks (IJCNN’91), Seattle, WA, vol. 1, pp 447-451, 1991. [13] V. Pedroni, A. Agranat, C. Neugebauer, A. Yariv, “Pattern matching and parallel processing with CCD technology,” Proc. IEEE Int. Joint Conference on Neural Networks (IJCNN’92), vol. 3, pp 620-623, 1992. [14] M. Howes, D. Morgan, Eds., Charge-Coupled Devices and Systems, John Wiley & Sons, 1979. [15] R. Genov, G. Cauwenberghs “Charge-Mode Parallel Architecture for Matrix-Vector Multiplication,”IEEE T. Circuits and Systems II, vol. 48 (10), 2001.
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The Emergence of Multiple Movement Units in the Presence of Noise and Feedback Delay Michael Kositsky Andrew G. Barto Department of Computer Science University of Massachusetts Amherst, MA 01003-4610 kositsky,barto @cs.umass.edu Abstract Tangential hand velocity profiles of rapid human arm movements often appear as sequences of several bell-shaped acceleration-deceleration phases called submovements or movement units. This suggests how the nervous system might efficiently control a motor plant in the presence of noise and feedback delay. Another critical observation is that stochasticity in a motor control problem makes the optimal control policy essentially different from the optimal control policy for the deterministic case. We use a simplified dynamic model of an arm and address rapid aimed arm movements. We use reinforcement learning as a tool to approximate the optimal policy in the presence of noise and feedback delay. Using a simplified model we show that multiple submovements emerge as an optimal policy in the presence of noise and feedback delay. The optimal policy in this situation is to drive the arm’s end point close to the target by one fast submovement and then apply a few slow submovements to accurately drive the arm’s end point into the target region. In our simulations, the controller sometimes generates corrective submovements before the initial fast submovement is completed, much like the predictive corrections observed in a number of psychophysical experiments. 1 Introduction It has been consistently observed that rapid human arm movements in both infants and adults often consist of several submovements, sometimes called “movement units” [21]. The tangential hand velocity profiles of such movements show sequences of several bellshaped acceleration-deceleration phases, sometimes overlapping in the time domain and sometimes completely separate. Multiple movement units are observed mostly in infant reaching [5, 21] and in reaching movements by adult subjects in the face of difficult timeaccuracy requirements [15]. These data provide clues about how the nervous system efficiently produces fast and accurate movements in the presence of noise and significant feedback delay. Most modeling efforts concerned with movement units have addressed only the kinematic aspects of movement, e.g., [5, 12]. We show that multiple movement units might emerge as the result of a control policy that is optimal in the face of uncertainty and feedback delay. We use a simplified dynamic model of an arm and address rapid aimed arm movements. We use reinforcement learning as a tool to approximate the optimal policy in the presence of noise and feedback delay. An important motivation for this research is that stochasticity inherent in the motor control problem has a significant influence on the optimal control policy [9]. We are following the preliminary work of Zelevinsky [23] who showed that multiple movement units emerge from the stochasticity of the environment combined with a feedback delay. Whereas he restricted attention to a finite-state system to which he applied dynamic programming, our model has a continuous state space and we use reinforcement learning in a simulated realtime learning framework. 2 The model description The model we simulated is sketched in Figure 1. Two main parts of this model are the “RL controller” (Reinforcement Learning controller) and the “plant.” The controller here represents some functionality of the central nervous system dealing with the control of reaching movements. The plant represents a simplified arm together with spinal circuitry. The controller generates the control signal, , which influences how the state, , of the plant changes over time. To simulate delayed feedback the state of the plant is made available to the controller after a delay period , so at time the controller can only observe . To introduce stochasticity, we disturbed by adding noise to it, to produce a corrupted control . The controller learns to move the plant state as quickly as possible into a small region about a target state
. The reward structure block in Figure 1 provides a negative unit reward when the plant’s state is out of the target area of the state space, and it provides zero reward when the plant state is within the target area. The reinforcement learning controller tries to maximize the total cumulative reward for each movement. With the above mentioned reward structure, the faster the plant is driven into the target region, the less negative reward is accumulated during the movement. Thus this reward structure specifies the minimum time-to-goal criterion. s target state state s s delay noise reward structure T plant target u~ RL controller u efferent copy u r r reward Figure 1: Sketch of the model used in our simulations. “RL controller” stands for a Reinforcement Learning controller. 2.1 The plant To model arm dynamics together with the spinal reflex mechanisms we used a fractionalpower damping dynamic model [22]. The simplest model that captures the most critical dynamical features is a spring-mass system with a nonlinear damping: "!$# Here, is the position of the mass attached to the spring, and are respectively the velocity and the acceleration of the object, is the mass of the object (the mass of the spring is assumed equal to zero), is the damping coefficient, is the stiffness coefficient, and is the control signal which determines the resting, or equilibrium, position. Later in this paper, we call activation, referring to the activation level of a muscle pair. The Table 1: Parameter values used in the simulations. description value description value the basic simulation time step 1 ms threshold velocity radius 0.1 cm/s the feedback delay, 200 ms standard deviation of the noise 1 cm initial position 0 cm value function learning rate 0.5 initial velocity 0 cm/s preferences learning rate 1 target position 5 cm discount factor, 0.9 target velocity 5 cm bootstrapping factor, 0.9 target position radius 0.5 cm values for the mass, the damping coefficient, and the stiffness coefficient were taken from Barto et al. [3]: kg,
, ! . These values provide movement trajectories qualitatively similar to those observed in human wrist movements [22]. The fractional-power damping in this model is motivated by both biological evidence [8, 14] and computational considerations. Controlling a system with such a concave damping function is an easier control problem than for a system with apparently simpler linear damping. Fractional-power damping creates a qualitatively novel dynamical feature called a stiction region, a region in the position space around the equilibrium position consisting of pseudo-stable states, where the velocity of the plant remains very close to zero. Such states are stable states for all practical purposes. For the parameter magnitudes used in our simulations, the stiction region is a region of radius 2.5 cm about the true equilibrium in the position space. Another essential feature of the neural signal transmission can be accounted for by using a cascade of low-pass temporal filters on the activation level [16]. We used a second-order low-pass filter with the time constant of 25 ms. 2.2 The reinforcement learning controller We used the version of the actor-critic algorithm described by Sutton and Barto [20]. A possible model of how an actor-critic architecture might be implemented in the nervous system was suggested by Barto [2] and Houk et al. [10]. We implemented the actor-critic algorithm for a continuous state space and a finite set of actions, i.e., activation level magnitudes evenly spaced every 1 cm between 0 cm and 10 cm. To represent functions defined over the continuous state space we used a CMAC representation [1] with 10 tilings, each tiling spans all three dimensions of the state space and has 10 tiles per dimension. The tilings have random offsets drawn from the uniform distribution. Learning is done in episodes. At the beginning of each episode the plant is at a fixed initial state, and the episode is complete when the plant reaches the target region of the state space. Table 1 shows the parameter values used in the simulations. Refer to ref. [20] for algorithm details. 2.3 Clocking the control signal For the controller to have sufficient information about the current state of the plant, the controller internal representation of the state should be augmented by a vector of all the actions selected during the last delay period. To keep the dimension of the state space at a feasible level, we restrict the set of available policies and make the controller select a new activation level, , in a clocked manner at time intervals equal to the delay period. One step of the reinforcement learning controller is performed once a delay period, which corresponds to many steps of the underlying plant simulation. To simulate a stochastic plant we added Gaussian noise to . A new Gaussian disturbance was drawn every time a new activation level was selected. Apart from the computational motivation, there is evidence of intermittent motor control by human subjects [13]. In our simulations we use an oversimplified special kind of intermittent control with a piecewise constant control signal whose magnitude changes at equal time intervals, but this is done for the sake of acceleration of the simulations and overall clarity. Intermittent control does not necessarily assume this particular kind of the control signal; the most important feature is that control segments are selected at particular points in time, and each control segment determines the control signal for an extended time interval. The time interval until selection of the next control segment can itself be one of the parameters [11]. 3 Results The model learned to move the mass quickly and accurately to the target in approximately 1,000 episodes. Figure 2 shows the corresponding learning curve. Figure 3 shows a typical movement accomplished by the controller after learning. The movement shown in Figure 3 has two acceleration-deceleration phases called movement units or submovements. 0 100 200 300 400 500 600 700 800 900 1000 0 500 1000 1500 2000 2500 3000 3500 4000 episode # time per episode, ms Figure 2: The learning curve averaged over 100 trials. The performance is measured in time-per-episode. Corrective submovements may occur before the plant reaches zero velocity. The controller generates this corrective submovement “on the fly,” i.e., before the initial fast submovement is completed. Figure 4 shows a sample movement accomplished by the controller after learning where such overlapping submovements occur. This can be seen clearly in panel (b) of Figure 4 where the velocity profile of the movement is shown. Each of the submovements appears as a bell-shaped unit in the tangential velocity plot. Sometimes the controller accomplishes a movement with a single smooth submovement. A sample of such a movement is shown in Figure 5. 4 Discussion The model learns to produce movements that are fast and accurate in the presence of noise and delayed sensory feedback. Most of the movements consist of several submovements. The first submovement is always fast and covers most of the distance from the initial po0 200 400 600 800 1000 1200 0 2 4 6 t, ms position, cm (a) 0 200 400 600 800 1000 1200 −10 0 10 20 30 t, ms velocity, cm/s (b) 0 200 400 600 800 1000 1200 0 5 10 15 t, ms activation, cm (c) 0 1 2 3 4 5 6 −5 0 5 10 15 20 25 position, cm velocity, cm/s (d) Figure 3: A sample movement accomplished by the controller after learning. Panels (a) and (b) show the position and velocity time course respectively. Panel (c) shows the activation time courses. The thin solid line shows the activation selected by the controller. The thick solid line shows the disturbed activation which is sent as the control signal to the plant. The dashed line shows the activation after the temporal filtering is applied. Panel (d) shows the phase trajectory of the movement. The thick bar at the lower-right corner is the target region. sition to the target. All of the subsequent submovements are much slower and cover much shorter segments in the position space. This feature stands in good agreement with the dual control model [12, 17], where the initial part of a movement is conducted in a ballistic manner, and the final part is conducted under closed-loop control. Some evidence for this kind of dual control strategy comes from experiments in which subjects were given visual feedback only during the initial stage of movement. Subjects did not show significant improvement under these conditions compared to trials in which they were deprived of visual feedback during the entire movement [4, 6]. In another set of experiments, proprioceptive feedback was altered by stimulations of muscle tendons. Movement accuracy decreased only when the stimulation was applied at the final stages of movement [18]. Note, however, that the dual control strategy though is not explicitly designed into our model, but naturally emerges from the existing constraints and conditions. The reinforcement learning controller is encouraged by the reward structure to accomplish each movement as quickly as possible. On the other hand, it faces high uncertainty in the plant behavior. In states with low velocities the information available to the controller determines the actual state of the plant quite accurately as opposed to states with high 0 200 400 600 800 1000 0 2 4 6 t, ms position, cm (a) 0 200 400 600 800 1000 0 5 10 15 20 t, ms velocity, cm/s (b) 0 200 400 600 800 1000 0 5 10 15 t, ms activation, cm (c) 0 1 2 3 4 5 6 −5 0 5 10 15 20 position, cm velocity, cm/s (d) Figure 4: A sample movement accomplished by the controller after learning with a well expressed predictive correction. velocities. If the controller were to adopt a policy in which it attempts to directly hit the target in one fast submovement, then very often it would miss the target and spend long additional time to accomplish the task. The optimal policy in this situation is to move the arm close to the target by one fast submovement and then apply a few slow submovements to accurately move arm into the target region. The model learns to produce control sequences consisting of pairs of high activation steps followed by low activation steps. This feature stands in good agreement with pulse-step models of motor control [7, 19]. Each of the pulse-step combinations produces a submovement characterized by a bell-shaped unit in the velocity profile. In biological motor control corrective submovements are observed very consistently, including both the overlapping and separate submovements. In the case of overlapping submovements, the corrective movement is called a predictive correction. Multiple submovements are observed mostly in infant reaching [5]. Adults perform routine everyday reaching movements ordinarily with a single smooth submovement, but in case of tight time constraints or accuracy requirements they revert to multiple submovements [15]. The suggested model sometimes accomplishes movements with a single smooth submovement (see Figure 5), but in most cases it produces multiple submovements much like an infant or an adult subject trying to move quickly and accurately. The suggested model is also consistent with theories of basal ganglia information processing for motor control [10]. Some of these theories suggest that dopamine neurons in the basal ganglia carry information similar to the secondary reinforcement (or temporal difference) in the actor-critic controller, i.e., information about how the expected perfor0 200 400 600 800 0 2 4 6 t, ms position, cm (a) 0 200 400 600 800 0 5 10 15 20 t, ms velocity, cm/s (b) 0 200 400 600 800 0 5 10 15 t, ms activation, cm (c) 0 1 2 3 4 5 6 −5 0 5 10 15 20 position, cm velocity, cm/s (d) Figure 5: A sample movement accomplished by the controller after learning with a single smooth submovement. mance (time-to-target) changes over time during a movement. A possible use of this kind of information is for initiating corrective submovements before the current movement is completed. This kind of behavior is exhibited by our model (Figure 4). Acknowledgments This work was supported by NIH Grant MH 48185–09. We thank Andrew H. Fagg and Michael T. Rosenstein for helpful comments. References [1] J. S. Albus. A new approach to manipulator control: the cerebellar model articulation controller (CMAC). Journal of Dynamics, Systems, Measurement and Control, 97:220–227, 1975. [2] A. G. Barto. Adaptive critics and the basal ganglia. In J. C. Houk, J. L. Davis, and D. G. Beiser, editors, Models of Information Processing in the Basal Ganglia, pages 215–232. MIT Press, Cambridge, MA, 1995. [3] A. G. Barto, A. H. Fagg, N. Sitkoff, and J. C. Houk. A cerebellar model of timing and prediction in the control of reaching. Neural Computation, 11:565–594, 1999. [4] D. Beaubaton and L. Hay. Contribution of visual information to feedforward and feedback processes in rapid pointing movements. Human Movement Science, 5:19–34, 1986. [5] N. E. Berthier. Learning to reach: a mathematical model. Developmental Psychology, 32:811– 832, 1996. [6] L. G. Carlton. Processing of visual feedback information for movement control. Journal of Experimental Psychology: Human Perception and Performance, 7:1019–1030, 1981. [7] C. Ghez. Contributions of central programs to rapid limb movement in the cat. In H. Asanuma and V. J. Wilson, editors, Integration in the Nervous System, pages 305–320. Igaku-Shoin, Tokyo, 1979. [8] C. C. A. M. Gielen and J. C. Houk. A model of the motor servo: incorporating nonlinear spindle receptor and muscle mechanical properties. Biological Cybernetics, 57:217–231, 1987. [9] C. M. Harris and D. M. Wolpert. Signal-dependent noise determines motor planning. Nature, 394:780–784, 1998. [10] J. C. Houk, J. L. Adams, and A. G. Barto. A model of how the basal ganglia generates and uses neural signals that predict reinforcement. In J. C. Houk, J. L. Davis, and D. G. Beiser, editors, Models of Information Processing in the Basal Ganglia, pages 249–270. MIT Press, Cambridge, MA, 1995. [11] M. Kositsky. Motor Learning and Skill Acquisition by Sequences of Elementary Actions. PhD thesis, The Weizmann Institute of Science, Israel, October 1998. [12] D. E. Meyer, S. Kornblum, R. A. Abrams, C. E. Wright, and J. E. K. Smith. Optimality in human motor performance: ideal control of rapid aimed movements. Psychological Review, 95(3):340–370, 1988. [13] R. C. Miall, D. J. Weir, and J. F. Stein. Intermittency in human manual tracking tasks. Journal of Motor Behavior, 25:53–63, 1993. [14] L. E. Miller. Reflex stiffness of the human wrist. Master’s thesis, Department of Physiology, Northwestern University, Evanston, IL, 1984. [15] K. E. Novak, L. E. Miller, and J. C. Houk. Kinematic properties of rapid hand movements in a knob turning task. Experimental Brain Research, 132:419–433, 2000. [16] L. D. Partridge. Integration in the central nervous system. In J. H. U. Brown and S. S. Gann, editors, Engineering Principles in physiology, pages 47–98. Academic Press, New York, 1973. [17] R. Plamondon and A. M. Alimi. Speed/accuracy trade-offs in target-directed movements. Behavioral and Brain Science, 20:279–349, 1997. [18] C. Redon, L. Hay, and J.-L. Velay. Proprioceptive control of goal directed movements in man studied by means of vibratory muscle tendon stimulation. Journal of Motor Behavior, 23:101– 108, 1991. [19] D. A. Robinson. Oculomotor control signals. In G. Lennerstrand and P. B. y Rita, editors, Basic Mechanisms of Ocular Mobility and Their Clinical Implications, pages 337–374. Pergamon Press, Oxford, 1975. [20] R. S. Sutton and A. G. Barto. Reinforcement Learning: An Introduction. MIT Press, Cambridge, MA, 1998. [21] C. von Hofsten. Structuring of early reaching movements: A longitudinal study. Journal of Motor Behavior, 23:280–292, 1991. [22] C. H. Wu, J. C. Houk, K. Y. Young, and L. E. Miller. Nonlinear damping of limb motion. In J. M. Winters and S. L.-Y. Woo, editors, Multiple Muscle Systems: Biomechanics and Movement Organization, pages 214–235. Springer-Verlag, New York, 1990. [23] L. Zelevinsky. Does time-optimal control of a stochastic system with sensory delay produce movement units? Master’s thesis, University of Massachusetts, Amherst, 1998.
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Pranking with Ranking Koby Crammer and Yoram Singer School of Computer Science & Engineering The Hebrew University, Jerusalem 91904, Israel {kobics,singer}@cs.huji.ac.il Abstract We discuss the problem of ranking instances. In our framework each instance is associated with a rank or a rating, which is an integer from 1 to k. Our goal is to find a rank-prediction rule that assigns each instance a rank which is as close as possible to the instance's true rank. We describe a simple and efficient online algorithm, analyze its performance in the mistake bound model, and prove its correctness. We describe two sets of experiments, with synthetic data and with the EachMovie dataset for collaborative filtering. In the experiments we performed, our algorithm outperforms online algorithms for regression and classification applied to ranking. 1 Introduction The ranking problem we discuss in this paper shares common properties with both classification and regression problems. As in classification problems the goal is to assign one of k possible labels to a new instance. Similar to regression problems, the set of k labels is structured as there is a total order relation between the labels. We refer to the labels as ranks and without loss of generality assume that the ranks constitute the set {I, 2, .. . , k} . Settings in which it is natural to rank or rate instances rather than classify are common in tasks such as information retrieval and collaborative filtering. We use the latter as our running example. In collaborative filtering the goal is to predict a user's rating on new items such as books or movies given the user's past ratings of the similar items. The goal is to determine whether a movie fan will like a new movie and to what degree, which is expressed as a rank. An example for possible ratings might be, run-to-see, very-good, good, only-if-you-must, and do-not-bother. While the different ratings carry meaningful semantics, from a learning-theoretic point of view we model the ratings as a totally ordered set (whose size is 5 in the example above). The interest in ordering or ranking of objects is by no means new and is still the source of ongoing research in many fields such mathematical economics, social science, and computer science. Due to lack of space we clearly cannot cover thoroughly previous work related to ranking. For a short overview from a learning-theoretic point of view see [1] and the references therein. One of the main results of [1] underscores a complexity gap between classification learning and ranking learning. To sidestep the inherent intractability problems of ranking learning several approaches have been suggested. One possible approach is to cast a ranking problem as a regression problem. Another approach is to reduce a total order into a set of prefCorrect interval #l \ I Figure 1: An Illustration of the update rule. erences over pairs [3, 5]. The first case imposes a metric on the set of ranking rules which might not be realistic, while the second approach is time consuming since it requires increasing the sample size from n to O(n2 ). In this paper we consider an alternative approach that directly maintains a totally ordered set via projections. Our starting point is similar to that of Herbrich et. al [5] in the sense that we project each instance into the reals. However, our work then deviates and operates directly on rankings by associating each ranking with distinct sub-interval of the reals and adapting the support of each sub-interval while learning. In the next section we describe a simple and efficient online algorithm that manipulates concurrently the direction onto which we project the instances and the division into sub-intervals. In Sec. 3 we prove the correctness of the algorithm and analyze its performance in the mistake bound model. We describe in Sec. 4 experiments that compare the algorithm to online algorithms for classification and regression applied to ranking which demonstrate the merits of our approach. 2 The PRank Algorithm This paper focuses on online algorithms for ranking instances. We are given a sequence (Xl, yl), ... , (xt, yt) , ... of instance-rank pairs. Each instance xt is in IRn and its corresponding rank yt is an element from finite set y with a total order relation. We assume without loss of generality that y = {I, 2, ... ,k} with ">" as the order relation. The total order over the set Y induces a partial order over the instances in the following natural sense. We say that xt is preferred over X S if yt > yS. We also say that xt and x S are not comparable if neither yt > yS nor yt < yS. We denote this case simply as yt = yS. Note that the induced partial order is of a unique form in which the instances form k equivalence classes which are totally orderedl . A ranking rule H is a mapping from instances to ranks, H : IRn -+ y. The family of ranking rules we discuss in this paper employs a vector w E IRn and a set of k thresholds bl :::; ... :::; bk- l :::; bk = 00. For convenience we denote by b = (bl , . .. ,bk-d the vector of thresholds excluding bk which is fixed to 00. Given a new instance x the ranking rule first computes the inner-product between w and x . The predicted rank is then defined to be the index of the first (smallest) threshold br for which w . x < br . This type of ranking rules divide the space into parallel equally-ranked regions: all the instances that satisfy br - l < W· x < br are assigned the same rank r. Formally, given a ranking rule defined by wand b the predicted rank of an instance x is, H(x) = minrE{l, ... ,k}{r : w . x - br < O}. Note that the above minimum is always well defined since we set bk = 00. The analysis that we use in this paper is based on the mistake bound model for online learning. The algorithm we describe works in rounds. On round t the learning algorithm gets an instance xt. Given xt, the algorithm outputs a rank, il = minr {r : W· x - br < O}. It then receives the correct rank yt and updates its ranking rule by modifying wand b. We say that our algorithm made a ranking mistake if il f:. yt. IFor a discussion of this type of partial orders see [6]. Initialize: Set wI = 0 , b~ , ... , bLl = 0, bl = 00 . Loop: Fort=1 ,2, ... ,T • Get a new rank-value xt E IRn. • Predict fl = minr E{I, ... ,k} {r: w t . xt b~ < o}. • Get a new label yt. • If fl t- yt update w t (otherwise set w t+! = w t , \;fr : b~+! = bn : 1. For r = 1, ... , k - 1 If yt :::; r Then y~ = -1 Else y~ = 1. 2. For r = 1, ... , k - 1 If (wt . xt b~)y~ :::; 0 Then T; = y~ Else T; = o. 3. Update w t+! f- w t + CLr T;)xt. For r = 1, . .. , k - 1 update: b~+1 f- b~ - T; Output: H(x) = minr E{1, ... ,k} {r : w T +! . x - b;+! < O}. Figure 2: The PRank algorithm. We wish to make the predicted rank as close as possible to the true rank. Formally, the goal of the learning algorithm is to minimize the ranking-loss which is defined to be the number of thresholds between the true rank and the predicted rank. Using the representation of ranks as integers in {I ... k}, the ranking-loss after T rounds is equal to the accumulated difference between the predicted and true rank-values, '£'[=1 W - yt I. The algorithm we describe updates its ranking rule only on rounds on which it made ranking mistakes. Such algorithms are called conservative. We now describe the update rule of the algorithm which is motivated by the perceptron algorithm for classification and hence we call it the PRank algorithm (for Perceptron Ranking). For simplicity, we omit the index of the round when referring to an input instance-rank pair (x, y) and the ranking rule wand h. Since b1 :::; b2 :::; ... :::; bk- 1 :::; bk then the predicted rank is correct if w . x > br for r = 1, ... ,y - 1 and w . x < br for y, . .. , k - 1. We represent the above inequalities by expanding the rank y into into k - 1 virtual variables Yl , ... ,Yk-l. We set Yr = +1 for the case W· x > br and Yr = -1 for w . x < br. Put another way, a rank value y induces the vector (Yl, ... , Yk-d = (+1, ... , +1, -1, ... , -1) where the maximal index r for which Yr = +1 is y-1. Thus, the prediction of a ranking rule is correct if Yr(w· x - br) > 0 for all r. If the algorithm makes a mistake by ranking x as fj instead of Y then there is at least one threshold, indexed r, for which the value of W· x is on the wrong side of br, i.e. Yr(w· x - br) :::; O. To correct the mistake, we need to "move" the values of W· x and br toward each other. We do so by modifying only the values of the br's for which Yr (w . x - br) :::; 0 and replace them with br - Yr. We also replace the value of w with w + ('£ Yr)x where the sum is taken over the indices r for which there was a prediction error, i.e., Yr (w . x - br) :::; o. An illustration of the update rule is given in Fig 1. In the example, we used the set Y = {I ... 5}. (Note that b5 = 00 is omitted from all the plots in Fig 1.) The correct rank of the instance is Y = 4, and thus the value of w . x should fall in the fourth interval, between b3 and b4 . However, in the illustration the value of w . x fell below b1 and the predicted rank is fj = 1. The threshold values b1 , b2 and b3 are a source of the error since the value of b1 , b2 , b3 is higher then W· x. To mend the mistake the algorithm decreases b1 , b2 and b3 by a unit value and replace them with b1 -1, b2 -1 and b3 -1. It also modifies w to be w+3x since '£r:Yr(w.x- br):SO Yr = 3. Thus, the inner-product W· x increases by 311x11 2 . This update is illustrated at the middle plot of Fig. 1. The updated prediction rule is sketched on the right hand side of Fig. 1. Note that after the update, the predicted rank of x is Y = 3 which is closer to the true rank y = 4. The pseudocode of algorithm is given in Fig 2. To conclude this section we like to note that PRank can be straightforwardly combined with Mercer kernels [8] and voting techniques [4] often used for improving the performance of margin classifiers in batch and online settings. 3 Analysis Before we prove the mistake bound of the algorithm we first show that it maintains a consistent hypothesis in the sense that it preserves the correct order of the thresholds. Specifically, we show by induction that for any ranking rule that can be derived by the algorithm along its run, (w1 , b 1 ) , ... , (wT+1 , b T +1) we have that b~ :S ... :S bL1 for all t. Since the initialization of the thresholds is such that b~ :S b~ :S ... :S bL1' then it suffices to show that the claim holds inductively. For simplicity, we write the updating rule of PRank in an alternative form. Let [7f] be 1 if the predicate 7f holds and 0 otherwise. We now rewrite the value of T; (from Fig. 2) as T; = y~[(wt . xt bny~ :S 0]. Note that the values of b~ are integers for all r and t since for all r we initialize b; = 0, and b~+l b~ E {-1, 0, +1}. Lemma 1 (Order Preservation) Let w t and b t be the current ranking rule, where bi :S .. . :S bL1' and let (xt,yt) be an instance-rank pair fed to PRank on round t. Denote by wt+1 and bt+1 the resulting ranking rule after the update of PRank, then bi+1 :S ... :S bt~ll· Proof: In order to show that PRank maintains the order of the thresholds we use the definition of the algorithm for y~, namely we define y~ = +1 for r < yt and y~ = -1 for r 2:: yt. We now prove that b~t~ 2:: b~+l for all r by showing that b~+l b~ 2:: y~+1[(wt . xt b~+1)Y;+l :S 0] - y;[(wt . xt - b;)y; :S 0], (1) which we obtain by substituting the values of bt+1. Since b~+1 :S b~ and b~ ,b~+1 E Z we get that the value of b~+1 b~ on the left hand side of Eq. (1) is a non-negative integer. Recall that y~ = 1 if yt > r and y~ = -1 otherwise, and therefore, y~+l :S y~. We now analyze two cases. We first consider the case y~+1 :j:. y~ which implies that y~+l = -1, y~ = +1. In this case, the right hand-side of Eq. (1) is at most zero, and the claim trivially holds. The other case is when y~+1 = y~. Here we get that the value of the right hand-side Eq. (1) cannot exceed 1. We therefore have to consider only the case where b~ = b~+1 and y~+1 = y~. But given these two conditions we have that y~+1[(wt. xt b~+1)Y~+1 < 0] and y~[(wt. xt b~)y~ < 0] are equal. The right hand side of Eq. (1) is now zero and the inequality holds with equality. • In order to simplify the analysis of the algorithm we introduce the following notation. Given a hyperplane wand a set of k -1 thresholds b we denote by v E ~n+k-1 the vector which is a concatenation of wand b that is v = (w, b). For brevity we refer to the vector vas a ranking rule. Given two vectors v' = (w', b') and v = (w, b) we have v' . v = w' . w + b' . b and IIvl12 = IIwl12 + IlbW. Theorem 2 (Mistake bound) Let (xl, y1), ... , (xT , yT) be an input sequence for PRank where xt E ~n and yt E {l. .. k}. Denote by R2 = maxt Ilxtl12. Assume that there is a ranking rule v* = (w* , b*) with br :S ... :S bk- 1 of a unit norm that classifies the entire sequence correctly with margin "( = minr,t{ (w* . xt - b;)yn > o. Then, the rank loss of the algorithm '£;=1 Iyt - yt I, is at most (k - 1) (R2 + 1) / "(2. Proof: Let us fix an example (xt, yt) which the algorithm received on round t. By definition the algorithm ranked the example using the ranking rule vt which is composed of w t and the thresholds b t. Similarly, we denote by vt+l the updated rule (wt+l bt+l) after round t That is wt+l = wt + (" Tt)xt and bt+l = bt - Tt , . , ur r r r r for r = 1, 2, ... , k - 1. Let us denote by n t = W - yt 1 the difference between the true rank and the predicted rank. It is straightforward to verify that nt = 2:=r ITn Note that if there wasn't a ranking mistake on round t then T; = ° for r = 1, ... , k-1, and thus also nt = 0. To prove the theorem we bound 2:=t nt from above by bounding IIvtl12 from above and below. First, we derive a lower bound on IIvtl12 by bounding v* . vH1 . Substituting the values of wH1 and bH1 we get, k-1 v* . vt+l = v* . vt + 2:= T; (w* . xt - b;) r=1 (2) We further bound the right term by considering two cases. Using the definition of T; from the pseudocode in Fig. 2 we need to analyze two cases. If (wt ·xt b~)y; :::; ° then T; = y;. Using the assumption that v* ranks the data correctly with a margin of at least "( we get that T;(W* . xt - b;) ~ "(. For the other case for which (wt . xt - b;)y; > ° we have T; = ° and thus T;(W* . xt - b;) = 0. Summing now over r we get, k-1 2:= T; (w* . xt - b;) ~ nt"( . (3) r = 1 Combining Eq. (2) and Eq. (3) we get v* . vt+l ~ v* . vt + nt"(. Unfolding the sum, we get that after T rounds the algorithm satisfies, v* . vT+1 ~ 2:=t nt"( = "( 2:=t nt. Plugging this result into Cauchy-Schwartz inequality, (1IvT+11121Iv* 112 ~ (vT+l . v*) 2) and using the assumption that v* is of a unit norm we get the lower bound, IIvT+ll12 ~ (2:=t nt)2 "(2. Next, we bound the norm of v from above. As before, assume that an example (xt, yt) was ranked using the ranking rule vt and denote by vt+l the ranking rule after the round. We now expand the values ofwt+1 and bt+l in the norm ofvH1 and get, IIvH1 112 = IIwtl12 + IIbt l12 + 2 2:=r T; (wt . xt - b;) + (2:=r T;)21IxtI12 + 2:=r (T;)2. Since T; E {-1,0,+1} we have that (2:=rT;)2 :::; (nt)2 and 2:=r(T;)2 = nt and we therefore get, IIvH1 112 :::; IIvtl12 + 22:= T; (wt . xt b~) + (nt)21IxtW + nt . (4) r We further develop the second term using the update rule of the algorithm and get, 2:= T; (wt . xt b~) = 2:=[(wt . xt b~)y; :::; 0] ((wt . xt b~)y~) :::; ° . (5) r r Plugging Eq. (5) into Eq. (4) and using the bound IIxtl12 :::; R2 we get that IlvH1112:::; IIvtl12 + (nt)2R2 + nt. Thus, the ranking rule we obtain after T rounds of the algorithm satisfies the upper bound, IlvT+l W :::; R2 2:=t(nt)2 + 2:=t nt. Combining the lower bound IlvT+l W ~ (2:=t nt)2 "(2 with the upper bound we have that, (2:=tnt)2"(2:::; IlvT+1112:::; R2 2:=t(nt)2 + 2:=t nt. Dividing both sides by "(2 2:=tnt we finally get, 2:= nt :::; R2 [2:=t(nt)2] f [2:=t ntl + 1 . (6) t "( By definition, nt is at most k - 1, which implies that 2:=t(nt)2 :::; 2:=t nt(k - 1) = (k -1) 2:=t nt. Using this inequality in Eq. (6) we get the desired bound, 2:=;=1 Igt ytl = 2:=;=1 nt :::; [(k - 1)R2 + 1lh2 :::; [(k - 1)(R2 + 1)lh2 . • i" I ... ~ Figure 3: Comparison of the time-averaged ranking-loss of PRank, WH, and MCP on synthetic data (left). Comparison of the time-averaged ranking-loss of PRank, WH, and MCP on the EachMovie dataset using viewers who rated and at least 200 movies (middle) and at least 100 movies (right). 4 Experiments In this section we describe experiments we performed that compared PRank with two other online learning algorithms applied to ranking: a multiclass generalization of the perceptron algorithm [2], denoted MCP, and the Widrow-Hoff [9] algorithm for online regression learning which we denote by WHo For WH we fixed its learning rate to a constant value. The hypotheses the three algorithms maintain share similarities but are different in their complexity: PRank maintains a vector w of dimension n and a vector of k - 1 modifiable thresholds b, totaling n + k - 1 parameters; MCP maintains k prototypes which are vectors of size n, yielding kn parameters; WH maintains a single vector w of size n. Therefore, MCP builds the most complex hypothesis of the three while WH builds the simplest. Due to the lack of space, we only describe two sets of experiments with two different datasets. The dataset used in the first experiment is synthetic and was generated in a similar way to the dataset used by Herbrich et. al. [5]. We first generated random points x = (Xl, X2) uniformly at random from the unit square [0,1 f. Each point was assigned a rank y from the set {I, ... , 5} according to the following ranking rule, y = maxr{r : lO((XI - 0.5)(X2 - 0.5)) + ~ > br } where b = (-00, -1, -0.1,0.25,1) and ~ is a normally distributed noise of a zero mean and a standard deviation of 0.125. We generated 100 sequences of instance-rank pairs each of length 7000. We fed the sequences to the three algorithms and obtained a prediction for each instance. We converted the real-valued predictions of WH into ranks by rounding each prediction to its closest rank value. As in ~5] we used a non-homogeneous polynomial of degree 2, K(XI' X2) = ((Xl· X2) + 1) as the inner-product operation between each input instance and the hyperplanes the three algorithms maintain. At each time step, we computed for each algorithm the accumulated ranking-loss normalized by the instantaneous sequence length. Formally, the time-averaged loss after T rounds is, (liT) 'L,i Iyt _ytl. We computed these losses for T = 1, ... ,7000. To increase the statistical significance of the results we repeated the process 100 times, picking a new random instance-rank sequence of length 7,000 each time, and averaging the instantaneous losses across the 100 runs. The results are depicted on the left hand side of Fig. 3. The 95% confidence intervals are smaller then the symbols used in the plot. In this experiment the performance of MPC is constantly worse than the performance of WH and PRank. WH initially suffers the smallest instantaneous loss but after about 500 rounds PRank achieves the best performance and eventually the number of ranking mistakes that PRank suffers is significantly lower than both WH and MPC. In the second set of experiments we used the EachMovie dataset [7]. This dataset is used for collaborative filtering tasks and contains ratings of movies provided by 61, 265 people. Each person in the dataset viewed a subset of movies from a collection of 1623 titles. Each viewer rated each movie that she saw using one of 6 possible ratings: 0, 0.2, 0.4, 0.6, 0.8,1. We chose subsets of people who viewed a significant amount of movies extracting for evaluation people who have rated at least 100 movies. There were 7,542 such viewers. We chose at random one person among these viewers and set the person's ratings to be the target rank. We used the ratings of all the rest of the people who viewed enough movies as features. Thus, the goal is to learn to predict the "taste" of a random user using the user's past ratings as a feedback and the ratings of fellow viewers as features. The prediction rule associates a weight with each fellow viewer an therefore can be seen as learning correlations between the tastes of different viewers. Next, we subtracted 0.5 from each rating and therefore the possible ratings are -0.5, -0.3, -0.1, 0.1, 0.3, 0.5. This linear transformation enabled us to assign a value of zero to movies which have not been rated. We fed these feature-rank pairs one at a time, in an online fashion. Since we picked viewer who rated at least 100 movies, we were able to perform at least 100 rounds of online predictions and updates. We repeated this experiment 500 times, choosing each time a random viewer for the target rank. The results are shown on the right hand-side of Fig. 3. The error bars in the plot indicate 95% condfidence levels. We repeated the experiment using viewers who have seen at least 200 movies. (There were 1802 such viewers.) The results of this experiment are shown in the middle plot of Fig. 3. Along the entire run of the algorithms, PRank is significantly better than WH, and consistently better than the multiclass perceptron algorithm, although the latter employs a bigger hypothesis. Finally, we have also evaluated the performance of PRank in a batch setting, using the experimental setup of [5]. In this experiment, we ran PRank over the training data as an online algorithm and used its last hypothesis to rank unseen test data. Here as well PRank came out first, outperforming all the algorithms described in [5]. Acknowledgments Thanks to Sanjoy Dagupta and Rob Schapire for numerous discussions on ranking problems and algorithms. Thanks also to Eleazar Eskin and Uri Maoz for carefully reading the manuscript. References [1] William W. Cohen, Robert E. Schapire, and Yoram Singer. Learning to order things. Journal of Artificial Intelligence Research, 10:243- 270, 1999. [2] K. Crammer and Y. Singer. Ultraconservative online algorithms for multiclass problems. Proc. of the Fourteenth Annual ConI on Computational Learning Theory, 200l. [3] Y. Freund, R. Iyer, R. E. Schapire, and Y. Singer. An efficient boosting algorithm for combining preferences. Machine Learning: Proc. of the Fifteenth Inti. ConI, 1998. [4] Y. Freund and R. E. Schapire. Large margin classification using the perceptron algorithm. Machine Learning, 37(3): 277-296, 1999. [5] R. Herbrich, T. Graepel, and K. Obermayer. Large margin rank boundaries for ordinal regression. Advances in Large Margin Classifiers. MIT Press, 2000. [6] J. Kemeny and J . Snell. Mathematical Models in the Social Sciences. MIT Press, 1962. [7] Paul McJones. EachMovie collaborative filtering data set. DEC Systems Research Center, 1997. http://www.research.digital.com/SRC/eachmoviej. [8] Vladimir N. Vapnik. Statistical Learning Theory. Wiley, 1998. [9] Bernard Widrow and Marcian E. Hoff. Adaptive switching circuits. 1960 IRE WESCON Convention Record, 1960. Reprinted in Neurocomputing (MIT Press, 1988).
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Variance Reduction Techniques for Gradient Estimates in Reinforcement Learning Evan Greensmith Australian National University evan@csl.anu.edu.au Peter L. Bartlett∗ BIOwulf Technologies Peter.Bartlett@anu.edu.au Jonathan Baxter∗ WhizBang! Labs, East jbaxter@whizbang.com Abstract We consider the use of two additive control variate methods to reduce the variance of performance gradient estimates in reinforcement learning problems. The first approach we consider is the baseline method, in which a function of the current state is added to the discounted value estimate. We relate the performance of these methods, which use sample paths, to the variance of estimates based on iid data. We derive the baseline function that minimizes this variance, and we show that the variance for any baseline is the sum of the optimal variance and a weighted squared distance to the optimal baseline. We show that the widely used average discounted value baseline (where the reward is replaced by the difference between the reward and its expectation) is suboptimal. The second approach we consider is the actor-critic method, which uses an approximate value function. We give bounds on the expected squared error of its estimates. We show that minimizing distance to the true value function is suboptimal in general; we provide an example for which the true value function gives an estimate with positive variance, but the optimal value function gives an unbiased estimate with zero variance. Our bounds suggest algorithms to estimate the gradient of the performance of parameterized baseline or value functions. We present preliminary experiments that illustrate the performance improvements on a simple control problem. 1 Introduction, Background, and Preliminary Results In reinforcement learning problems, the aim is to select a controller that will maximize the average reward in some environment. We model the environment as a partially observable Markov decision process (POMDP). Gradient ascent methods (e.g., [7, 12, 15]) estimate the gradient of the average reward, usually using Monte Carlo techniques to cal∗Most of this work was performed while the authors were with the Research School of Information Sciences and Engineering at the Australian National University. culate an average over a sample path of the controlled POMDP. However such estimates tend to have a high variance, which means many steps are needed to obtain a good estimate. GPOMDP [4] is an algorithm for generating an estimate of the gradient in this way. Compared with other approaches, it is suitable for large systems, when the time between visits to a state is large but the mixing time of the controlled POMDP is short. However, it can suffer from the problem of producing high variance estimates. In this paper, we investigate techniques for variance reduction in GPOMDP. One generic approach to reducing the variance of Monte Carlo estimates of integrals is to use an additive control variate (see, for example, [6]). Suppose we wish to estimate the integral of f : X →R, and we know the integral of another function ϕ : X →R. Since R X f = R X(f −ϕ) + R X ϕ, the integral of f −ϕ can be estimated instead. Obviously if ϕ = f then the variance is zero. More generally, Var(f −ϕ) = Var(f) −2Cov(f, ϕ) + Var(ϕ), so that if φ and f are strongly correlated, the variance of the estimate is reduced. In this paper, we consider two approaches of this form. The first (Section 2) is the technique of adding a baseline. We find the optimal baseline and we show that the additional variance of a suboptimal baseline can be expressed as a weighted squared distance from the optimal baseline. Constant baselines, which do not depend on the state or observations, have been widely used [13, 15, 9, 11]. In particular, the expectation over all states of the discounted value of the state is a popular constant baseline (where, for example, the reward at each step is replaced by the difference between the reward and the expected reward). We give bounds on the estimation variance that show that, perhaps surprisingly, this may not be the best choice. The second approach (Section 3) is the use of an approximate value function. Such actorcritic methods have been investigated extensively [3, 1, 14, 10]. Generally the idea is to minimize some notion of distance between the fixed value function and the true value function. In this paper we show that this may not be the best approach: selecting the fixed value function to be equal to the true value function is not always the best choice. Even more surprisingly, we give an example for which the use of a fixed value function that is different from the true value function reduces the variance to zero, for no increase in bias. We give a bound on the expected squared error (that is, including the estimation variance) of the gradient estimate produced with a fixed value function. Our results suggest new algorithms to learn the optimum baseline, and to learn a fixed value function that minimizes the bound on the error of the estimate. In Section 5, we describe the results of preliminary experiments, which show that these algorithms give performance improvements. POMDP with Reactive, Parameterized Policy A partially observable Markov decision process (POMDP) consists of a state space, S, a control space, U, an observation space, Y, a set of transition probability matrices {P(u) : u ∈U}, each with components pij(u) for i, j ∈S, u ∈U, an observation process ν : S →PY, where PY is the space of probability distributions over Y, and a reward function r : S →R. We assume that S, U, Y are finite, although all our results extend easily to infinite U and Y, and with more restrictive assumptions can be extended to infinite S. A reactive, parameterized policy for a POMDP is a set of mappings {µ(·, θ) : Y →PU|θ ∈RK}. Together with the POMDP, this defines the controlled POMDP (S, U, Y, P, ν, r, µ). The joint state, observation and control process, {Xt, Yt, Ut}, is Markov. The state process, {Xt}, is also Markov, with transition probabilities pij(θ) = P y∈Y,u∈U νy(i)µu(y, θ)pij(u), where νy(i) denotes the probability of observation y given the state i, and µu(y, θ) denotes the probability of action u given parameters θ and observation y. The Markov chain M(θ) = (S, P(θ)) then describes the behaviour of the process {Xt}. Assumption 1 The controlled POMDP (S, U, Y, P, ν, r, µ) satisfies: For all θ ∈RK there exists a unique stationary distribution satisfying π′(θ) P(θ) = π′(θ). There is an R < ∞such that, for all i ∈S, |r(i)| ≤R. There is a B < ∞such that, for all u ∈U, y ∈Y and θ ∈RK the derivatives ∂µu(y, θ)/∂θk (1 ≤k ≤K) exist, and the vector of these derivatives satisfies ∥∇µu(y, θ)/µu(y, θ)∥≤B, where ∥· ∥denotes the Euclidean norm on RK. We consider the average reward, η(θ) def = limT →∞E h 1 T PT −1 t=0 r(Xt) i . Assumption 1 implies that this limit exists, and does not depend on the start state X0. The aim is to select a policy to maximize this quantity. Define the discounted value function, Jβ(i, θ) def = limT →∞E hPT −1 t=0 βtr(Xt) X0 = i i . Throughout the rest of the paper, dependences upon θ are assumed, and dropped in the notation. For a random vector A, we denote Var(A) = E h (A −E [A])2i , where a2 denotes a′a, and a′ denotes the transpose of the column vector a. GPOMDP Algorithm The GPOMDP algorithm [4] uses a sample path to estimate the gradient approximation ∇βη def = E h ∇µu(y) µu(y) Jβ(j) i . As β →1, ∇βη approaches the true gradient ∇η, but the variance increases. We consider a slight modification [2]: with Jt def = P2T s=t βs−tr(Xs), ∆T def = 1 T T −1 X t=0 ∇µUt(Yt) µUt(Yt) Jt+1. (1) Throughout this paper the process {Xt, Yt, Ut, Xt+1} is generally understood to be generated by a controlled POMDP satisfying Assumption 1, with X0∼π (ie the initial state distributed according to the stationary distribution). That is, before computing the gradient estimates, we wait until the process has settled down to the stationary distribution. Dependent Samples Correlation terms arise in the variance quantities to be analysed. We show here that considering iid samples gives an upper bound on the variance of the general case. The mixing time of a finite ergodic Markov chain M = (S, P) is defined as τ def = min t > 1 : max i,j dT V P t i , P t j ≤e−1 , where [P t]i denotes the ith row of P t and dT V is the total variation distance, dT V (P, Q) = P i |P(i) −Q(i)|. Theorem 1 Let M = (S, P) be a finite ergodic Markov chain, with mixing time τ, and let π be its stationary distribution. There are constants L < p 2|S|e and 0 ≤α < exp(−1/(2τ)), which depend only on M, such that, for all f : S →R and all t, Covπ f (t) ≤LαtVarπ(f), where Varπ(f) is the variance of f under π, and Covπ f (t) is the auto-covariance of the process {f(Xt)}, where the process {Xt} is generated by M with initial distribution π. Hence, for some constant Ω∗≤4Lτ, Var 1 T T −1 X t=0 f(Xt) ! ≤Ω∗ T Varπ(f). We call (L, τ) the mixing constants of the Markov chain M (or of the controlled POMDP D; ie the Markov chain (S, P)). We omit the proof (all proofs are in the full version [8]). Briefly, we show that for a finite ergodic Markov chain M, Covπ f (t) ≤Rt(M)Varπ(f), for some Rt(M). We then show that Rt(M)2 < 2|S| exp(− t τ ). In fact, for a reversible chain, we can choose L = 1 and α = |λ2|, the second largest magnitude eigenvalue of P. 2 Baseline We consider an alteration of (1), ∆T def = 1 T T −1 X t=0 ∇µUt(Yt) µUt(Yt) (Jt+1 −Ar(Yt)) . (2) For any baseline Ar : Y →R, it is easy to show that E [∆T ] = E [∆T ]. Thus, we select Ar to minimize variance. The following theorem shows that this variance is bounded by a variance involving iid samples, with Jt replaced by the exact value function. Theorem 2 Suppose that D = (S, U, Y, P, ν, r, µ) is a controlled POMDP satisfying Assumption 1, D has mixing constants (L, τ), {Xt, Yt, Ut, Xt+1} is a process generated by D with X0∼π ,Ar : Y →R is a baseline that is uniformly bounded by M, and J (j) has the distribution of P∞ s=0 βsr(Xt), where the states Xt are generated by D starting in X0 = j. Then there are constants C ≤5B2R(R + M) and Ω≤4Lτ ln(eT ) such that Var 1 T T −1 X t=0 ∇µUt(Yt) µUt(Yt) (Jt+1−Ar(Yt)) ! ≤Ω T Varπ ∇µu(y) µu(y) (Jβ(j)−Ar(y)) + Ω T E ∇µu(y) µu(y) (J (j) −Jβ(j)) 2 + Ω T + 1 C (1 −β)2 βT , where, as always, (i, y, u, j) are generated iid with i∼π, y∼ν(i), u∼µ(y) and j∼Pi(u). The proof uses Theorem 1 and [2, Lemma 4.3]. Here we have bounded the variance of (2) with the variance of a quantity we may readily analyse. The second term on the right hand side shows the error associated with replacing an unbiased, uncorrelated estimate of the value function with the true value function. This quantity is not dependent on the baseline. The final term on the right hand side arises from the truncation of the discounted reward— and is exponentially decreasing. We now concentrate on minimizing the variance σ2 r = Varπ ∇µu(y) µu(y) (Jβ(j) −Ar(y)) , (3) which the following lemma relates to the variance σ2 without a baseline, σ2 = Varπ ∇µu(y) µu(y) Jβ(j) . Lemma 3 Let D = (S, U, Y, P, ν, r, µ) be a controlled POMDP satisfying Assumption 1. For any baseline Ar : Y →R, and for i∼π, y∼ν(i), u∼µ(y) and j∼Pi(u), σ2 r = σ2 + E " A2 r(y) E "∇µu(y) µu(y) 2 y # −2Ar(y)E "∇µu(y) µu(y) 2 Jβ(j) y ## . From Lemma 3 it can be seen that the task of finding the optimal baseline is in effect that of minimizing a quadratic for each observation y ∈Y. This gives the following theorem. Theorem 4 For the controlled POMDP as in Lemma 3, min Ar σ2 r = σ2 −E E "∇µu(y) µu(y) 2 Jβ(j) y #!2 /E "∇µu(y) µu(y) 2 y # , and this minimum is attained with the baseline A∗ r(y) = E "∇µu(y) µu(y) 2 Jβ(j) y # /E "∇µu(y) µu(y) 2 y # . Furthermore, the optimal constant baseline is A∗ r = E "∇µu(y) µu(y) 2 Jβ(j) # /E ∇µu(y) µu(y) 2 . The following theorem shows that the variance of an estimate with an arbitrary baseline can be expressed as the sum of the variance with the optimal baseline and a certain squared weighted distance between the baseline function and the optimal baseline function. Theorem 5 If Ar : Y →R is a baseline function, A∗ r is the optimal baseline defined in Theorem 4, and σ2 r∗is the variance of the corresponding estimate, then σ2 r = σ2 r∗+ E "∇µu(y) µu(y) 2 (Ar(y) −A∗ r(y))2 # , where i∼π, y∼ν(i), and u∼µ(y). Furthermore, the same result is true for the case of constant baselines, with Ar(y) replaced by an arbitrary constant baseline Ar, and A∗ r(y) replaced by A∗ r, the optimum constant baseline defined in Theorem 4. For the constant baseline Ar = E i∼π[Jβ(i)], Theorem 5 implies that σ2 r is equal to min Ar∈R σ2 r + E ∇µu(y) µu(y) 2 E [Jβ(j)] −E "∇µu(y) µu(y) 2 Jβ(j) #!2 /E ∇µu(y) µu(y) 2 . Thus, its performance depends on the random variables (∇µu(y)/µu(y))2 and Jβ(j); if they are nearly independent, E [Jβ] is a good choice. 3 Fixed Value Functions: Actor-Critic Methods We consider an alteration of (1), ˜∆T def = 1 T T −1 X t=0 ∇µUt(Yt) µUt(Yt) ˜Jβ(Xt+1), (4) for some fixed value function ˜Jβ : S →R. Define Aβ(j) def = E " ∞ X k=0 βkd(Xk, Xk+1) X0 = j # , where d(i, j) def = r(i) + β˜Jβ(j) −˜Jβ(i) is the temporal difference. Then it is easy to show that the estimate (4) has a bias of ∇βη −E h ˜∆T i = E ∇µu(y) µu(y) Aβ(j) . The following theorem gives a bound on the expected squared error of (4). The main tool in the proof is Theorem 1. Theorem 6 Let D = (S, U, Y, P, ν, r, µ) be a controlled POMDP satisfying Assumption 1. For a sample path from D, that is, {X0∼π, Yt∼ν(Xt), Ut∼µ(Yt), Xt+1∼PXt(Ut)}, E ˜∆T −∇βη 2 ≤Ω∗ T Varπ ∇µu(y) µu(y) ˜Jβ(j) + E ∇µu(y) µu(y) Aβ(j) 2 , where the second expectation is over i∼π, y∼ν(i), u∼µ(y), and j∼Pi(u). If we write ˜Jβ(j) = Jβ(j) + v(j), then by selecting v = (v(1), . . . , v(|S|))′ from the right null space of the K × |S| matrix G, where G def = P i,y,u πiνy(i)∇µu(y)P ′ i(u), (4) will produce an unbiased estimate of ∇βη. An obvious example of such a v is a constant vector, (c, c, . . . , c)′ : c ∈R. We can use this to construct a trivial example where (4) produces an unbiased estimate with zero variance. Indeed, let D = (S, U, Y, P, ν, r, µ) be a controlled POMDP satisfying Assumption 1, with r(i) = c, for some 0 < c ≤R. Then Jβ(j) = c/(1 −β) and ∇βη = 0. If we choose v = (−c/(1 −β), . . . , −c/(1 −β))′ and ˜Jβ(j) = Jβ(j) + v(j), then ∇µu(y) µu(y) ˜Jβ(j) = 0 for all y, u, j, and so (4) gives an unbiased estimate of ∇βη, with zero variance. Furthermore, for any D for which there exists a pair y, u such that µu(y) > 0 and ∇µu(y) ̸= 0, choosing ˜Jβ(j) = Jβ(j) gives a variance greater than zero—there is a non-zero probability event, (Xt = i, Yt = y, Ut = u, Xt+1 = j), such that ∇µu(y) µu(y) Jβ(j) ̸= 0. 4 Algorithms Given a parameterized class of baseline functions Ar(·, θ) : Y →R θ ∈RL , we can use Theorem 5 to bound the variance of our estimates. Computing the gradient of this bound with respect to the parameters θ of the baseline function allows a gradient optimization of the baseline. The GDORB Algorithm produces an estimate ∆S of these gradients from a sample path of length S. Under the assumption that the baseline function and its gradients are uniformly bounded, we can show that these estimates converge to the gradient of σ2 r. We omit the details (see [8]). GDORB Algorithm: Given: Controlled POMDP (S, U, Y, P, ν, r, µ), parameterized baseline Ar. set z0 = 0 (z0 ∈RL), ∆0 = 0 (∆0 ∈RL) for all {is, ys, us, is+1, ys+1} generated by the POMDP do zs+1 = βzs + ∇Ar(ys) ∇µus(ys) µus(ys) 2 ∆s+1 = ∆s + 1 s+1 ((Ar(ys) −βAr(ys+1) −r(xs+1)) zs+1 −∆s) end for For a parameterized class of fixed value functions {˜Jβ(·, θ) : S →R θ ∈RL }, we can use Theorem 6 to bound the expected squared error of our estimates, and compute the gradient of this bound with respect to the parameters θ of the baseline function. The GBTE Algorithm produces an estimate ∆S of these gradients from a sample path of length S. Under the assumption that the value function and its gradients are uniformly bounded, we can show that these estimates converge to the true gradient. GBTE Algorithm: Given: Controlled POMDP (S, U, Y, P, ν, r, µ), parameterized fixed value function ˜Jβ. set z0 = 0 (z0 ∈RK), ∆A0 = 0 (∆A0 ∈R1×L), ∆B0 = 0 (∆B0 ∈RK), ∆C0 = 0 (∆C0 ∈RK) and ∆D0 = 0 (∆D0 ∈RK×L) for all {is, ys, us, is+1, is+2} generated by the POMDP do zs+1 = βzs + ∇µus(ys) µus(ys) ∆As+1 = ∆As + 1 s+1 ∇µus(ys) µus(ys) ˜Jβ(is+1) ′ ∇µus(ys) µus(ys) ∇˜Jβ(is+1) ′ −∆As ∆Bs+1 = ∆Bs + 1 s+1 ∇µus(ys) µus(ys) ˜Jβ(is+1) −∆Bs ∆Cs+1 = ∆Cs + 1 s+1 r(is+1) + β˜Jβ(is+2) −˜Jβ(is+1) zs+1 −∆Cs ∆Ds+1 = ∆Ds + 1 s+1 ∇µus(ys) µus(ys) ∇˜Jβ(is+1) ′ −∆Ds ∆s+1 = Ω∗ T ∆As+1 −Ω∗ T ∆B′ s+1∆Ds+1 −∆C′ s+1∆Ds+1 ′ end for 5 Experiments Experimental results comparing these GPOMDP variants for a simple three state MDP (described in [5]) are shown in Figure 1. The exact value function plots show how different choices of baseline and fixed value function compare when all algorithms have access to the exact value function Jβ. Using the expected value function as a baseline was an improvement over GPOMDP. Using the optimum, or optimum constant, baseline was a further improvement, each performing comparably to the other. Using the pre-trained fixed value function was also an improvement over GPOMDP, showing that selecting the true value function was indeed not the best choice in this case. The trained fixed value function was not optimal though, as Jβ(j) −A∗ r is a valid choice of fixed value function. The optimum baseline, and fixed value function, will not normally be known. The online plots show experiments where the baseline and fixed value function were trained using online gradient descent whilst the performance gradient was being estimated, using the same data. Clear improvement over GPOMDP is seen for the online trained baseline variant. For the online trained fixed value function, improvement is seen until T becomes—given the simplicity of the system—very large. References [1] L. Baird and A. Moore. Gradient descent for general reinforcement learning. In Advances in Neural Information Processing Systems 11, pages 968–974. MIT Press, 1999. [2] P. L. Bartlett and J. Baxter. Estimation and approximation bounds for gradient-based reinforcement learning. Journal of Computer and Systems Sciences, 2002. To appear. [3] A. G. Barto, R. S. Sutton, and C. W. Anderson. Neuronlike adaptive elements that can solve difficult learning control problems. IEEE Transactions on Systems, Man, and Cybernetics, SMC-13:834–846, 1983. [4] J. Baxter and P. L. Bartlett. Infinite-horizon gradient-based policy search. Journal of Artificial Intelligence Research, 15:319–350, 2001. [5] J. Baxter, P. L. Bartlett, and L. Weaver. Infinite-horizon gradient-based policy search: II. Gradient ascent algorithms and experiments. Journal of Artificial Intelligence Research, 15:351–381, 2001. [6] M. Evans and T. Swartz. Approximating integrals via Monte Carlo and deterministic methods. Oxford University Press, 2000. FVF-pretrain BL-A∗ r BL-A∗ r(y) BL [Jβ] GPOMDP-Jβ Exact Value Function—Mean Error T Relative Norm Difference 1e+07 1e+06 100000 10000 1000 100 10 1 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 FVF-pretrain BL-A∗ r BL-A∗ r(y) BL- [Jβ] GPOMDP-Jβ Exact Value Function—One Standard Deviation T Relative Norm Difference 1e+07 1e+06 100000 10000 1000 100 10 1 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 FVF-online BL-online GPOMDP Online—Mean Error T Relative Norm Difference 1e+07 1e+06 100000 10000 1000 100 10 1 1 0.8 0.6 0.4 0.2 0 FVF-online BL-online GPOMDP Online—One Standard Deviation T Relative Norm Difference 1e+07 1e+06 100000 10000 1000 100 10 1 1 0.8 0.6 0.4 0.2 0 Figure 1: Three state experiments—relative norm error ∥∆est −∇η∥/ ∥∇η∥. Exact value function plots compare mean error and standard deviations for gradient estimates (with knowledge of Jβ) computed by: GPOMDP [GPOMDP-Jβ]; with baseline Ar = [Jβ] [BL [Jβ]]; with optimum baseline [BL-A∗ r(y)]; with optimum constant baseline [BL-A∗ r]; with pre-trained fixed value function [FVF-pretrain]. Online plots do a similar comparison of estimates computed by: GPOMDP [GPOMDP]; with online trained baseline [BL-online]; with online trained fixed value function [FVFonline]. The plots were computed over 500 runs (1000 for FVF-online), with β = 0.95. Ω∗/T was set to 0.001 for FVF-pretrain, and 0.01 for FVF-online. [7] P. W. Glynn. Likelihood ratio gradient estimation for stochastic systems. Communications of the ACM, 33:75–84, 1990. [8] E. Greensmith, P. L. Bartlett, and J. Baxter. Variance reduction techniques for gradient estimates in reinforcement learning. Technical report, ANU, 2002. [9] H. Kimura, K. Miyazaki, and S. Kobayashi. Reinforcement learning in POMDPs with function approximation. In D. H. Fisher, editor, Proceedings of the Fourteenth International Conference on Machine Learning (ICML’97), pages 152–160, 1997. [10] V. R. Konda and J. N. Tsitsiklis. Actor-Critic Algorithms. In Advances in Neural Information Processing Systems 12, pages 1008–1014. MIT Press, 2000. [11] P. Marbach and J. N. Tsitsiklis. Simulation-Based Optimization of Markov Reward Processes. Technical report, MIT, 1998. [12] R. Y. Rubinstein. How to optimize complex stochastic systems from a single sample path by the score function method. Ann. Oper. Res., 27:175–211, 1991. [13] R. S. Sutton and A. G. Barto. Reinforcement Learning: An Introduction. MIT Press, Cambridge MA, 1998. ISBN 0-262-19398-1. [14] R. S. Sutton, D. McAllester, S. Singh, and Y. Mansour. Policy Gradient Methods for Reinforcement Learning with Function Approximation. In Advances in Neural Information Processing Systems 12, pages 1057–1063. MIT Press, 2000. [15] R. J. Williams. Simple Statistical Gradient-Following Algorithms for Connectionist Reinforcement Learning. Machine Learning, 8:229–256, 1992.
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Asymptotic Universality for Learning Curves of Support Vector Machines M.Opperl R. Urbanczik2 1 Neural Computing Research Group School of Engineering and Applied Science Aston University, Birmingham B4 7ET, UK. opperm@aston.ac.uk 2Institut Fur Theoretische Physik, Universitiit Wurzburg Am Rubland, D-97074 Wurzburg, Germany robert@physik.uni-wuerzburg.de. Abstract Using methods of Statistical Physics, we investigate the rOle of model complexity in learning with support vector machines (SVMs). We show the advantages of using SVMs with kernels of infinite complexity on noisy target rules, which, in contrast to common theoretical beliefs, are found to achieve optimal generalization error although the training error does not converge to the generalization error. Moreover, we find a universal asymptotics of the learning curves which only depend on the target rule but not on the SVM kernel. 1 Introduction Powerful systems for data inference, like neural networks implement complex inputoutput relations by learning from example data. The price one has to pay for the flexibility of these models is the need to choose the proper model complexity for a given task, i.e. the system architecture which gives good generalization ability for novel data. This has become an important problem also for support vector machines [1]. The main advantage of SVMs is that the learning task is a convex optimization problem which can be reliably solved even when the example data require the fitting of a very complicated function. A common argument in computational learning theory suggests that it is dangerous to utilize the full flexibility of the SVM to learn the training data perfectly when these contain an amount of noise. By fitting more and more noisy data, the machine may implement a rapidly oscillating function rather than the smooth mapping which characterizes most practical learning tasks. Its prediction ability could be no better than random guessing in that case. Rence, modifications of SVM training [2] that allow for training errors were suggested to be necessary for realistic noisy scenarios. This has the drawback of introducing extra model parameters and spoils much of the original elegance of SVMs. Surprisingly, the results of this paper show that the picture is rather different in the important case of high dimensional data spaces. Using methods of Statistical Physics, we show that asymptotically, the SVM achieves optimal generalization ability for noisy data already for zero training error. Moreover, the asymptotic rate of decay of the generalization error is universal, i.e. independent of the kernel that defines the SVM. These results have been published previously only in a physics journal [3]. As is well known, SVMs classify inputs y using a nonlinear mapping into a feature vector w(y) which is an element of a Hilbert space. Based on a training set of m inputs xl' and their desired classifications 71' , SVMs construct the maximum margin hyperplane P in the feature space. P can be expressed as a linear combination of the feature vectors w(xl'), and to classify an input y, that is to decide on which side of P the image W (y) lies, one basically has to evaluate inner products W (xl') . W (y). For carefully chosen mappings wand Hilbert spaces, inner products w(x) . w(y) can be evaluated efficiently using a kernel function k(x, y) = w(x) . w(y), without having to individually calculate the feature vectors w(x) and w(y). In this manner it becomes computationally feasible to use very high and even infinite dimensional feature vectors. This raises the intriguing question whether the use of a very high dimensional feature space may typically be helpful. So far, recent results [4, 5] obtained by using methods of Statistical Mechanics (which are naturally well suited for analysing stochastic models in high dimensional spaces), have been largely negative in this respect. They suggest (as one might perhaps expect) that it is rather important to match the complexity of the kernel to the target rule. The analysis in [4] considers the case of N-dimensional inputs with binary components and assumes that the target rule giving the correct classification 7 of an input x is obtained as the sign of a function t(x) which is polynomial in the input components and of degree L. The SVM uses a kernel which is a polynomial of the inner product x . y in input space of degree K ;::: L, and the feature space dimension is thus O(NK ). In this scenario it is shown, under mild regularity condition on the kernel and for large N, that the SVM generalizes well when the number of training examples m is on the order of N L . So the scale of the learning curve is determined by the complexity of the target rule and not by the kernel. However, considering the rate with which the generalization error approaches zero one finds the optimal N L 1m decay only when K is equal to L and the convergence is substantially slower when K > L. So it is important to match the complexity of the kernel to the target rule and using a large value of K is only justified if L is assumed large and if one can use on the order of N L examples for training. In this paper we show that the situation is very different when one considers the arguably more realistic scenario of a target rule corrupted by noise. Now one can no longer use K = L since no separating hyperplane P will exist when m is sufficiently large compared to N L. However when K > L, this plane will exist and we will show that it achieves optimal generalization performance in the limit that N L 1m is small. Remarkably, the asymptotic rate of decay of the generalization error is independent of the kernel in this case and a general characterization of the asymptote in terms of properties of the target rule is possible. In a second step we show that under mild regularity conditions these findings also hold when k(x, y) is an arbitrary function of x . y or when the kernel is a function of the Euclidean distance Ix - YI. The latter type of kernels is widely used in practical applications of SVMs. 2 Learning with Noise: Polynomial Kernels We begin by assuming a polynomial kernel k(x, y) = J(x· y) where J(z) = Lf=o Ck zk is of degree K. Denoting by P a multi-index P = (PI , ... ,PN) with Pi E No, we set xp = JTPTfnf:l %.r and the degree of xp is Ipi = Lf:l Pi· The kernel can then be described by features wp(x) = JCiPTxp since k(x,y) = Lp wp(x)wp(y), where the summation runs over all multi-indices of degree up to K. To assure that the features are real, we assume that the coefficients Ck in the kernel are nonnegative. A hyperplane in feature space is parameterized by a weight vector w with components wp, and if 0 < TI'W . W (xl'), a point (xl', TI') of the training set lies on the correct side of the plane. To express that the plane P has maximal distance to the points of the training set, we choose an arbitrary positive stability parameter /'i, and require that the weight vector w* of P minimize w . w subject to the constraints /'i, < TI'W' w(xl'), for f.l = 1, ... ,m. 2.1 The Statistical Mechanics Formulation Statistical Mechanics allows to analyze a variety of learning scenarios exactly in the "thermodynamic limit" of high input dimensionality, when the data distributions are simple enough. In this approach, one computes a partition function which serves as a generating function for important averages such as the generalization error. To define the partition problem for SVMs one first analyzes a soft version of the optimization problem characterized by an inverse temperature f3. One considers the partition function z = f dwe- ~f3w.w IT 8(TI'W' w(xl') /'i,), 1'=1 (1) where the SVM constraints are enforced strictly using the Heaviside step function 8. Properties of w* can be computed from In Z and taking the limit f3 -t 00. To model the training data, we assume that the random and independent input components have zero mean and variance liN. This scaling assures that the variance of w . w(xl') stays finite in the large N limit. For the target rule we assume that its deterministic part is given by the polynomial t(x) = Lp JCiPTBpxp with real parameters Bp. The magnitude of the contribution of each degree k to the value of t(x) is measured by the quantities 1 '"' 2 Tk = Ck Nk ~ Bp p,lpl=k (2) where Nk = (N+;- I) is the number of terms in the sum. The degree of t(x) is L and lower than K, so TL > 0 and TL+l = ... = TK = O. Note, that this definition of t(x) ensures that any feature necessary for computing t(x) is available to the SVM. For brevity we assume that the constant term in t(x) vanishes (To = 0) and the normalization is Lk Tk = 1. 2.2 The Noise Model In the deterministic case the label of a point x would simply be the sign of t(x). Here we consider a nondeterministic rule and the output label is obtained using a random variable Tu E {-1, 1} parameterized by a scalar u. The observable instances of the rule, and in particular the elements of the training set, are then obtained by independently sampling the random variable (x, Tt(x))' Simple examples are additive noise, Tt(x) = sgn(t(x) + 77), or multiplicative noise, Tt(x) = sgn(t(x)77), where 77 is a noise term independent of x. In general, we will assume that the noise does not systematically corrupt the deterministic component, formally 1 1> Prob(Tu = sgn(u)) >"2 for all u. (3) So sgn( t( x)) is the best possible prediction of the output label of x, and the minimal achievable generalization error is fmin = (8( -t(X)Tt(x)))x. In the limit of many input dimensions N, a central limit argument yields that for a typical target rule fmin = 2(8( -u)0(u))u , where u is zero mean and unit variance Gaussian. The function 0 will play a considerable role in the sequel. It is a symmetrized form of the probability p(u) that Tu is equal to 1, 0(u) = ~(p(u) + 1 - p( -u)). 2.3 Order Parameter Equations One now evaluates the average of In Z (Eq. 1) over random drawings of training data for large N in terms of the order parameters Q (((W.1]i(X))2)Jw' q=((w)w·1]i(X))2)x and r Q-! «w ·1]i(x))w B ·1]i(x))x . (4) Here the thermal average over w refers to the Gibbs distribution (1). For the large N limit, a scaling of the training set size m must be specified, for which we make the generic Ansatz m = aNt, where I = 1, ... ,L. Focusing on the limit of large j3, where the density on the weight vectors converges to a delta peak and q -+ Q, we introduce the rescaled order parameter X = j3( Q - q) / St, with t St = i (1) - L Ci . (5) i=O Note that this scaling with St is only possible since the degree K of the kernel i(x, y) is greater than I, and thus St ¥- O. Finally, we obtain an expression for it = lim,B--+oo limN --+00 «In Z)) St / (j3Nt) , where the double brackets denote averaging over all training sets of size m. The value of it results from extremizing, with respect to r, q and X, the function it(r,q,X) = -aq /0(-u)G (ru + ~v -~)) X \ v0 u,v ~ (~: - X ~ 1) (1- -(X -1)TzS~;Ct + L~=l TJ (6) where G(z) = 8(z)z2, and u, v are independent Gaussian random variables with zero mean and unit variance. Since the stationary value of it is finite, « w* . w*)) is of the order Nt. So the higher order components of w* are small, (W;)2 « 1 for Ipi > I, although these components playa crucial role in ensuring that a hyperplane separating the training points exists even for large a. But the key quantity obtained from Eq. (6) is the stationary value of r which determines the generalization error of the SVM via fg = (0(-u)8(ru + ~v))u,v, and in particular fg = fmin for r = 1. 2.4 Universal Asymptotics We now specialize to the case that l equals L, the degree of the polynomial t(x) in the target rule. So m = aNL and for large a, after some algebra, Eq. (6) yields r = 1 A(q*) ~ (7) 4B(q*)2 a where B(q) (e(Y)8(-Y+Ii/yrj))}y and A(q) (e(Y)8(-Y+Ii/y7i) (_Y+Ii/y7i)2}y. Further q* = argminqqA(q), and considering the derivatives of qA(q) for q --+ 0 and q --+ 00, one may show that condition (3) assures that qA(q) does have a minimum. Equation (7) shows that optimal generalization performance is achieved on this scale in the limit of large a. Remarkably, as long as K > L, the asymptote is invariant to the choice of the kernel since A(q) and B(q) are defined solely in terms of the target rule. 3 Extension to other Kernels Our next goal is to understand cases where the kernel is a general function of the inner product or of the distance between the vectors. We still assume that the target rule is of finite complexity, i.e. defined by a polynomial and corrupted by noise and that the number of examples is polynomial in N. Remarkably, the more general kernels then reduce to the polynomial case in the thermodynamic limit. Since it is difficult to find a description of the Hilbert space for k( x, y) which is useful for a Statistical Physics calculation, our starting point is the dual representation: The weight vector w* defining the maximal margin hyperplane P can be written as a linear combination of the feature vectors w(xM) and hence w* . w(y) = IJ(Y), where m (8) M=l By standard results of convex optimization theory the AM are uniquely defined by the Kuhn-Tucker conditions AM ::::: 0, TMIJ(XM) ::::: Ii (for all patterns), further requiring that for positive AM the second of the two inequalities holds as an equality. One also finds that w* . w* = 2:;=1 AM and for a polynomial kernel we thus obtain a bound on 2:;=1 AM since w* . w* is O(m). We first consider kernels ¢(x· y), with a general continuous function ¢ of the inner product, and assume that ¢ can be approximated by a polynomial J in the sense that ¢(1) = J(l) and ¢(z) - J(z) = O(ZK) for z --+ O. Now, with a probability approaching 1 with increasing N, the magnitude of xM·xl/ is smaller than, say, N-1/3 for all different indices {t and v as long as m is polynomial in N. So, considering Eq. (8), for large N the functions ¢(z) and J(z) will only be evaluated in a small region around zero and at z = 1 when used as kernels of a SVM trained on m = aN L examples. Using the fact that 2:;=1 AM = O(m) we conclude that for large Nand K > 3L the solution of the K uhn-Tucker conditions for J converges to the one for ¢. So Eqs. (6,7) can be used to calculate the generalization error for ¢ by setting ttl = ¢(l) (O)/l! for l = 1, ... , L, when ¢ is an analytic function. Note that results in [4] assure that ttl ::::: 0 if the kernel ¢( X· y) is positive definite for all input dimensions N. Further, the same reduction to the polynomial case will hold in many instances where ¢ is not analytical but just sufficiently smooth close to O. 3.1 RBF Kernels We next turn to radial basis function (RBF) kernels where k( x, y) depends only on the Euclidean distance between two inputs, k(x,y) = <I>(lx - YI2). For binary input components (Xi = ±N- 1/ 2 ) this is just the inner product case since <I>(lx Y12) = <I>(2 - 2x· y). However, for more general input distributions, e.g. Gaussian input components, the fluctuations of Ixl around its mean value 1 have the same magnitude as x . y even for large N, and an equivalence with inner product kernels is not evident. Our starting point is the observation that any kernel <I>(lx - Y12) which is positive definite for all input dimensions N is a positive mixture of Gaussians [6]. More precisely <I>(z) = fooo e-ez da(k) where the transform a(k) is nondecreasing. For the special case of a single Gaussian one easily obtains features 'IT p by rewriting <I>(lx Y12) = e-lx-vI2/2 = e-1x12 /2ex've-lvI2 /2. Expanding the kernel eX 'v into polynomial features, yields the features 'IT p(x) = e- 1x12 /2xpl M for <I>(lx _ YI2). But, for a generic weight vector w in feature space, w· 'IT(x) = ~Wp'ITp(x) = e-~lxI2 ~wp M (9) is of order 1, and thus for large N the fluctuations of Ixl can be neglected. This line of argument can be extended to the case that the kernel is a finite mixture of Gaussians, <I>(z) = L~=l aie-'Y7z/2 with positive coefficients ai. Applying the reasoning for a single Gaussian to each term in the sum, one obtains a doubly indexed feature vector with components 'lTp,i(X) = e-'Y7IxI2/2(ai/';lpl/lpll)1/2xp. It is then straightforward to adapt the calculation of the partition function (Eq. 16) to the doubly indexed features, showing that the kernel <I>(lx - Y12) yields the same generalization behavior as the inner product kernel <I> (2 - 2x . y). Based on the calculation, we expect the same equivalence to hold for general radial basis function kernels, i.e. an infinite mixture of Gaussians, even if it would be involved to prove that the limit of many Gaussians commutes with the large N limit. 4 Simulations To illustrate the general results we first consider a scenario where a linear target rule, corrupted by additive Gaussian noise, is learned using different transcendental RBF kernels (Fig. 1). While Eq. (7) predicts that the asymptote of the generalization error does not depend on the kernel, remarkably, the dependence on the kernel is very weak for all values of a. In contrast, the generalization error depends substantially on the nature of the noise process. Figure 2 shows the finding for a quadratic rule with additive noise for the cases that the noise is Gaussian and binary. In the Gaussian case a 1/a decay of Eg to Emin is found, whereas for binary noise the decay is exponential in a. Note that in both cases the order parameter r approaches 1 as 1/a. 5 Summary The general characterization of learning curves obtained in this paper demonstrates that support vector machines with high order or even transcendental kernels have definitive advantages when the training data is noisy. Further the calculations leading to Eq. (6) show that maximizing the margin is an essential ingredient of the approach: If one just chooses any hyperplane which classifies the training data correctly, the scale of the learning curve is not determined by the target rule and far more examples are needed to achieve good generalization. Nevertheless our findings are at odds with many of the current theoretical motivations for maximizing the margin which argue that this minimizes the effective Vapnik Chervonenkis dimension of the classifier and thus ensures fast convergence of the training error to the generalization error [1, 2]. Since we have considered hard margins, in contrast to the generalization error, the training error is zero, and we find no convergence between the two quantities. But close to optimal generalization is achieved since maximizing the margin biases the SVM to explain as much as possible of the data in terms of a low order polynomial. While the Statistical Physics approach used in this paper is only exactly valid in the thermodynamic limit, the numerical simulations indicate that the theory is already a good approximation for a realistic number of input dimensions. We thank Rainer Dietrich for useful discussion and for giving us his code for the simulations. The work of M.O. was supported by the EPSRC (grant no. GR/M81601) and the British Council (ARC project 1037); R.U. was supported by the DFG and the DAAD. References [1] C. Cortes and V. Vapnik. , Machine Learning, 20:273-297, 1995. [2] N. Cristianini and J. Shawe-Taylor. Support Vector Machines. Cambridge U niversity Press, 2000. [3] M. Opper and R. Urbanczik. Phys. Rev. Lett., 86:4410- 4413, 200l. [4] R. Dietrich, M. Opper, and H. Sompolinsky. Phys. Rev. Lett., 82:2975 - 2978, 1999. [5] S. Risau-Gusman and M. Gordon. Phys. Rev. E, 62:7092- 7099,2000. [6] I. Schoenberg. Anal. Math, 39:811-841, 1938. tOg 0.3 ,-----r------------------, 0.2 0.1 o (A) (8) (C) (D) (E) - trllin 5 10 a=P/N D 6. <> 0 15 20 Figure 1: Linear target rule corrupted by additive Gaussian noise rJ ((rJ) = 0, \rJ2 ) = 1/16) and learned using different kernels. The curves are the theoretical prediction; symbols show simulation results for N = 600 with Gaussian inputs and error bars are approximately the size of the symbols. (A) Gaussian kernel, <I>(z) = e-kz with k = 2/3. (B) Wiener kernel given by the non analytic function <I>(z) = e - e..jZ. We chose c ~ 0.065 so that the theoretical prediction for this case coincides with (A). (C) Gaussian kernel with k = 1/20, the influence of the parameter change on the learning curve is minimal. (D) Perceptron, ¢(z) = z. Above a e ~ 7.5 vanishing training error cannot be achieved and the SVM is undefined. (E) Kernel invariant asymptote for (A,B,C). 0.1 -E~ino w-______ ~ ______ ~ _____ ~ ____ __w o 2 4 6 8 a = P/N2 Figure 2: A noisy quadratic rule (Tl = 0, T2 = 1) learned using the Gaussian kernel with k = 1/20. The upper curve (simulations.) is for additive Gaussian noise as in Fig. 1. The lower curve (simulations .) is for binary noise, rJ ± s with equal probability. We chose s ~ 0.20 so that the value of fmin is the same for the two noise processes. The inset shows that f9 decays as l/a for Gaussian noise, whereas an exponential decay is found in the binary case. The dashed curves are the kernel invariant asymptotes. The simulations are for N = 90 with Gaussian inputs, and standard errors are approximately the size of the symbols.
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Active Information Retrieval Tommi Jaakkola MIT AI Lab Cambridge, MA tommi@ai.mit.edu Abstract Hava Siegelmann MIT LIDS Cambridge, MA hava@mit.edu In classical large information retrieval systems, the system responds to a user initiated query with a list of results ranked by relevance. The users may further refine their query as needed. This process may result in a lengthy correspondence without conclusion. We propose an alternative active learning approach, where the system responds to the initial user's query by successively probing the user for distinctions at multiple levels of abstraction. The system's initiated queries are optimized for speedy recovery and the user is permitted to respond with multiple selections or may reject the query. The information is in each case unambiguously incorporated by the system and the subsequent queries are adjusted to minimize the need for further exchange. The system's initiated queries are subject to resource constraints pertaining to the amount of information that can be presented to the user per iteration. 1 Introduction An IR system consists of a collection of documents and an engine that retrieves documents described by users queries. In large systems, such as the Web, queries are typically too vague, and hence, an iterative process in which the users refine their queries gradually has to take place. Since much dissatisfaction of IR users stems from long, tedious repetitive search sessions, our research is targeted at shortening the search session. We propose a new search paradigm of active information retrieval in which the user initiates only one query, and the subsequent iterative process is led by the engine/system. The active process exploits optimum experiment design to permit minimal effort on the part of the user. Our approach is related but not identical to the interactive search processes called relevance feedback. The primary differences pertain to the way in which the feedback is incorporated and queried from the user. In relevance feedback, the system has to deduce a set of "features" (words, phrases, etc.) that characterize the set of selected relevant documents, and use these features in formulating a new query (e.g., [5,6]) . In contrast, we cast the problem as a problem of estimation and the goal is to recover the unknown document weights or relevance assessments. Our system also relates to the Scatter/Gather algorithm of browsing information systems [2], where the system initially scatters the document collection into a fixed number k of clusters whose summaries are presented to the user. The user select clusters from a new sub-collection, to be scattered again into k clusters, and so forth, until enumerating single documents. In our approach, documents are not discarded but rather their weighting is updated appropriately. Like many other clustering methods, the scatter/gather is based on hierarchical orderings. Overlapping clusters were recently proposed to better match real-life grouping and allow natural summarizing and viewing [4]. This short paper focuses on the underlying methodology of the active learning approach. 2 Active search Let X be the set of documents (elements) in the database and C = {GI , ... , Gm } a set of available clusters of documents for which appropriate summaries can be generated. The set of clusters typically includes individual documents and may come from a fiat, hierarchical, or overlapping clustering method. The clustering need not be static, however, and could be easily defined dynamically in the search process. Given the set of available clusters, we may choose a query set, a limited set of clusters that are presented to the user for selection at each iteration of the search process. The user is expected to choose the best matching cluster in this set or, alternatively, annotate the clusters with relevant/irrelevant labels (select the relevant ones). We will address both modes of operation. The active retrieval algorithm proceeds as follows: (1) it finds a small subset S of clusters to present, along with their summaries, to the user; (2) waits until the user selects none, one or more of the presented clusters; (3) uses the evidence from the user's selections to update the distribution over documents or relevance assessments; (4) outputs the top documents so far, ranked by their weights, and the iteration continues until terminated by the user or the system (based on any remaining uncertainty about the relevant documents or the implied ranking). The following sections address three primary issues: the user model, how to incorporate the information from user selections, and how to optimize the query set presented to the user. All the algorithms should scale linearly with the database size (and the size of the query set). 3 Contrastive selection model We start with a contrastive selection model where the user is expected to choose only the best matching cluster in the query set. In case of multiple selections, we will interpret the marked clusters as a redefined cluster of the query set. While this interpretation will result in sub-optimal choices for the query set assuming the user consistently selects multiple clusters, the interpretation nevertheless obviates the need for modeling user's selection biases in this regard. An empty selection, on the other hand, suggests that the clusters outside the query set are deemed more likely. 9 8 7 6 5 4 3 , b) 10 / 10 databasesize(log-sca~) 09 06 c) 10' database size (log-scale) Figure 1: a) A three level hierarchical transform of a flat Dirichlet; b) dependence of mean retrieval time on the database size (log-scale); c) median ratio of retrieval times corresponding to doubling the query set size. To capture the ranking implied by the user selections, we define weights (distribution) {Bx}, L:XEX Bx = lover the underlying documents. We assume that the user behavior is (probabilistic ally) consistent with one such weighting B;. The goal of a retrieval algorithm is therefore to recover this underlying weighting through interactions with the user. The resulting (approximation to) B; can be used to correctly rank the documents or, for example, to display all the documents with sufficiently large weight (cf. coverage). Naturally, B; changes from one retrieval task to another and has to be inferred separately in each task. We might estimate a user specific prior (model) over the document weights to reflect consistent biases that different users have across retrieval tasks. We express our prior belief about the document weights in terms of a Dirichlet distribution: P(B) = liZ· rrxExB~ ' -l, where Z = [f1 x Exr(ax)l/r(L:~= l ax). 3.1 Inference Suppose a fiat Dirichlet distribution P(B) over the document weights and a fixed query set S = {CS1, .. . ,CSk }. We evaluate here the posterior distribution P(Bly) given the user response y. The key is to transform P(B) into a hierarchical form so as to explicate the portion of the distribution potentially affected by the user response. The hierarchy, illustrated in Figure 1a), contains three levels: selection of S or X \ S; choices within the query set S (of most interest to us) and those under X \ S; selections within the clusters CS1 in S. For simplicity, the clusters are assumed to be either nested or disjoint, i.e., can be organized hierarchically. We use B?) , i = 1,2 to denote the top level parameters, B;f{, j = 1, ... , k for the cluster choices within the query set whereas B~~~, x ~ S gives the document choices outside S. Finally, B~~j for x E CSj indicate the parameters associated with the cluster CSj E S. The original flat Dirichlet P(B) can be written as a product p(B(l) )P(BW )P(BW) [rr~=l P( B( I~)) ] with the appropriate normalization constraints. If clusters in S overlap, the expansion is carried out in terms of the disjoint subsets. The parameters governing the Dirichlet component distributions are readily obtained by gathering the appropriate parameters ax of the original Dirichlet (cf. [3]). For example, a~l) = L:xES ax; am = L:xECs j ax, for j = 1, ... , k; a~~~ = ax, for x ~ S; a~~j = ax, whenever x E CSj , j = 1, ... , k. If user selects cluster CSy , we will update P( 8W) which reduces to adjusting the counts a~~i f- a~~i + 1. The resulting new parameters give rise to the posterior distribution P(8W IY) and, by including the other components, to the overall posterior P(8IY). If the user selects "none of these items," only the first level parameters 8~1) will be updated. 3.2 Query set optimization Our optimization criterion for choosing the query set S is the information that we stand to gain from querying the user with it. Let y indicate the user choice, the mutual information between y and the parameters 8 is given by (derivation omitted) J(Yi 8) (1) (2) where P(y) = a~~iI (I::=l a~~~) defines our current expectation about user selection from Si H(y) = I:~=l P(y) log P(y) is the entropy of the selections y, and w(·) is the Di-gamma function, defined as w(z) = djdzlogr(z). Extending the criterion to "no selection" is trivial. To simplify, we expand the counts aW in terms of the original (flat) counts ax, and define for all clusters (whether or not they appear in the query set) the weights ai = I:xECi ax, bi = aiw(ai + 1) - ailogai. The mutual information criterion now depends only on as = I:~=l aSi = I:xES ax, the overall weight of the query set and bs = I:~= l bSi which provides an overall measure of how informative the individual clusters in S are. With these changes, we obtain: (2) bs J(Yi 8. ll) = + log(as) - w(as + 1) as (3) We can optimize the choice of S with a simple greedy method that successively finds the next best cluster index i to include in the information set. This algorithm scales as O(km), where m is the number of clusters in our database and k is the the maximal query set size in terms of the number of clusters. Note that this simple criterion excludes nested or overlapping clusters from S. In a more general context, the bookkeeping problem associated with the overlapping clusters is analogous to that of the Kikuchi expansion in statistical physics (cf. [7]) . 3.3 Projection back to a flat Dirichlet The hierarchical posterior is not a flat Dirichlet anymore. To maintain simplicity, we project it back into a flat Dirichlet in the KL-divergence sense: P~ I Y = argminQo KL(Pe1yIIQe), where P(8Iy) is the hierarchical posterior expressed in terms of the original flat variables 8x ,x E X (but no longer a flat Dirichlet). The transformation from hierarchical to flat variables is given by: 8x = 8~1 ) 8JN 8~~j for x E CSj , j = 1, ... ,k, and Ox = O~l) o~~L for x E X \ S. As a result, when x E CSj for some j = 1, ... , k we get (derivation omitted) (4) where y denotes the user selection. For x E X\S, EO ly log Ox = w(ax) - W(LzEX a z) If we define rx = Eely log Ox for all x E X, then the counts f3x corresponding to the flat approximation Qo can be found by minimizing (5) xEX xEX where we have omitted any terms not depending on f3x . This is a strictly convex optimization problem over the convex set f3x ~ 0, x E X and therefore admits a unique solution. Furthermore, we can efficiently apply second order methods such as Newton-Raphson in this context due to the specific structure of the Hessian: 1i = D - el1 T, where D is a diagonal matrix containing the derivatives of the di-gamma function l w'(f3x) = d/df3x w(f3x) and e = W'(LXEX f3x ). Each NewtonRaphson iteration requires only O(m) space/time. 3.4 Decreasing entropy Since the query set was chosen to maximize the mutual information between the user selection and the parameters 0, we get the maximal reduction in the expected entropy of the parameters: J(y; 0) = H(Po) - Ey H(Pely) As discussed in the previous section, we cannot maintain the true posterior but have to settle for a projection. It is therefore no longer obvious that the expected entropy of the projected posterior possesses any analogous guarantees; indeed, projections of this type typically increase the entropy. We can easily show, however, that the expected entropy is non-increasing: since P~ y is the minimizing argument. It is possible to make a stronger statement indicating that the expected entropy of the projected distribution decreases monotonically after each iteration. Theorem 1 For any 10 > 0, Ey {H(Qo IY) } :::; H(Pe) - f(k -l)/as + 0(102 ), where k is the size of the query set and as = L zEs a z. While this result is not tight, it does demonstrate that the projection back into a flat Dirichlet still permits a semi-linear decrease in the entropy2. The denominator of the first order term, i.e., as, can increase only by 1 at each iteration. IThese derivatives can be evaluated efficiently on the basis of the highly accurate approximation to the di-gamma function. 2Note that the entropy of a Dirichlet distribution is not bounded from below (it is bounded from above). The manner in which the Dirichlet updates are carried out (how a x change) still keeps the entropy a meaningful quantity. 4 Annotation model The contrastive selection approach discussed above operates a priori in a single topic mode3 . The expectation that the user should select the best matching cluster in the query set also makes an inefficient use of the query set. We provide here an analogous development of the active learning approach under the assumption that the user classifies rather than contrasts the clusters. The user responses are now assumed to be consistent with a noisy-OR model P(Ye = 1Ir*) = 1 - (1 - qo) II (1 - qr: (7) xEe where Ye is the binary relevance annotation (outcome) for a cluster c in the query, r; E {O, 1 }, x E X are the underlying task specific relevance assignments to the elements in the database, q is the probability that a relevant element in the cluster is caught by the user, and qo is the probability that a cluster is deemed "relevant" in the absence of any relevant elements. While the parameters qo and q could easily be inferred from past searches, we assume here for simplicity that they are known to the search algorithm. The user annotations of different clusters in the query set are independent of each other, even for overlapping clusters. To ensure that we can infer the unknown relevance assignments from the observabIes (cluster annotations), we require identifiability: the annotation probabilities P(Ye = 1Ir*), for all c E C, should uniquely determine {r;}. Equivalently, knowing the number of relevant documents in each cluster should enable us to recover the underlying relevance assignments. This is a property of the cluster structure and holds trivially for any clustering with access to individual elements. The search algorithm maintains a simple independent Bernoulli model over the unknown relevance assignments: P(rIB) = TIxEx B;' (1 - Bx) l - r • . This gives rise to a marginal noisy-OR model over cluster annotations: P(Ye = liB) = L P(Ye = 1Ir)P(rIB) = 1 - (1 - qo) II (1- Bxq) (8) r x Ee The uncertainty about the relevance assignments {rx} makes the system beliefs about the cluster annotations dependent on each other. The parameters (relevance probabilities) {Bx} are, of course, specific to each search task. 4.1 Inference and projection Given fie E {O, 1} for a single cluster c, we can evaluate the posterior P(rlfie, B) over the relevance assignments. Similarly to noisy-OR graphical models, this posterior can be (exponentially) costly to maintain and we instead sequentially project the posterior back into the set of independent Bernoulli distributions. The projection here is in the moments sense (m- projection): Pr;(I' = argminQr KL(Pr IVc,(lIIQr), where Qr is an independent Bernoulli model. The m-projection preserves the posterior expectations B~ ; vc = ErlYc {rx} used for ranking the documents. 3Dynamic redefinition of clusters partially avoids this problem. The projection yields simple element-wise updates for the parameters4 : for x E c, (9) where Po = P(yc = OIB) = (l-qo) ITxEc(l-Bxq) is the only parameter that depends on the cluster as a whole. 4.2 Query set optimization The best single cluster c E C to query has the highest mutual information between the expected user response Yc = {O, I} and the underlying relevance assignments l' = {rx}xEx, maximizing I(yc; r iB) = EYe {KL( PrIO,Ye II PrIO)}' This mutual information cannot be evaluated in closed form, however. We use a lower bound: I(yc; r iB) ::::: EYe { l: D(Bx;Ye II Bx) } d~ Ip(yc; riB) (10) xEc where BX;Ye' x E X are the parameters of the m-projected posterior and KL(Bx;yJBx) is the KL-divergence between two Bernoulli distributions with mean parameters BX;Ye and Bx, respectively. To alleviate the concern that the lower bound would prematurely terminate the search, we note that if Ip(r; B) = 0 for all c E C, then Bx E {O, I} for all x E X. In other words, the search terminates only if we are already fully certain about the underlying relevance assignments. The best k clusters to query are those maximizing Finding the optimal query set under this criterion (even with the m-projections) involves O(nk2k) operations. We select the clusters sequentially while maintaining an explicit dependence on the hypothetical outcome (classification) of only the previous cluster choice. More precisely, we combine the cluster selection with conditional projections: for k > 1, Ck = argmaxclp(Yc,Yck;rIBk- l), B~.y = , ek E{ B~;!k_l ,Yek I YCk }. The mutual information terms do not, however, decompose additively with the elements in the clusters. The desired O(kn) scaling of the selection algorithm requires a cached spline reconstruction5 . 4.3 Sanity check results Figure 1 b) gives the mean number of iterations of the query process as function of the database size. Each point represents an average over 20 runs with parameters 4The parameters 8x;fiq ,fie2 , ... ,fiek resulting from k successive projections define a martingale process Eyq ,Ye2 , . .. ,Yek {8x;yq ,Ye2 , . . . ,Yek } = 8x, x EX, where the expectation is taken w.r.t . to the posterior approximation. 5The mutual information terms for select fixed values of po can be cached additively relative to the cluster structure. The actual Po dependence is reconstructed (quadratically) from the cached values (Ip is convex in po) . k = 5, qo = 0.05, and q = 0.95. The user responses were selected on the basis of the same parameters and a randomly chosen (single) underlying element of interest. The search is terminated when the sought after element in the database has the highest rank according to {Ox} , x E X. The randomized cluster structures were relatively balanced and hierarchical. Similarly to the theoretically optimal system, the performance scales linearly with the log-database size. Results for random choice of the clusters in the query are far outside the figure. Figure lc), on the other hand, demonstrates that increasing the query set size appropriately reduces the interaction time. Note that since all the clusters in the query set have to be chosen prior to getting feedback from any of the clusters, doubling the query set size cannot theoretically reduce the retrieval time to a half. 5 Discussion The active learning approach proposed here provides the basic methodology for optimally querying the user at multiple levels of abstraction. There are a number of extensions to the approach presented in this short paper. For example, we can encourage the user to provide confidence rated selections/annotations among the presented clusters. Both user models can be adapted to handle such selections. Analyzing the fundamental trade-offs between the size of the query set (resource constraints) and the expected completion time of the retrieval process will also be addressed in later work. References [1] A. C. Atkinson and A. N. Donev, Optimum experimental designs, Clarendon Press, 1992. [2] D. R. Cutting, D. R. Karger, J. O. Pederson, J. W. Tukey, Scatter/Gather: A cluster Based Approach to Browse Document Collections, In Proceedings of the Fifteenth Annual International ACM SIGIR Conference, Denmark, June 1996. [3] D. Heckerman, D. Geiger, and D. M. Chickering, Learning Bayesian Networks: The Combination of Knowledge and Statistical Data, Machine Learning, Vol 20, 1995. [4] H. Lipson and H.T. Siegelmann, Geometric Neurons for Clustering, Neural Computation 12(10), August 2000 [5] J. J. Jr. Rocchio, Relevance Feedback in Information Retrieval, In The Smart System - experiments in automatic document processing, 313-323, Englewood Cliffs, NJ: Prentice Hall Inc. [6] G. Salton and C. Buckley, Improving Retrieval Performance by Relevance Feedback, Journal ofthe American Society for Information Science, 41(4): 288-297, 1990. [7] J.S. Yedidia, W.T. Freeman, Y. Weiss, Generalized Belief Propagation, Neural Information Processing Systems 13, 2001.
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Tempo Tracking Rhythm by Sequential Monte Ali Taylan Ce:mgil and Bert Kappen SNN, University of Nijmegen NL 6525 EZ Nijmegen The Netherlands {cemgil,bert}@mbfys.kun.nl Abstract We present a probabilistic generative model for timing deviations in expressive music. performance. The structure of the proposed model is equivalent to a switching state space model. We formulate two well known music recognition problems, namely tempo tracking and automatic transcription (rhythm quantization) as filtering and maximum a posteriori (MAP) state estimation tasks. The inferences are carried out using sequential Monte Carlo integration (particle filtering) techniques. For this purpose, we have derived a novel Viterbi algorithm for Rao-Blackwellized particle filters, where a subset of the hidden variables is integrated out. The resulting model is suitable for realtime tempo tracking and transcription and hence useful in a number of music applications such as adaptive automatic accompaniment and score typesetting. 1 Introduction Automatic music transcription refers to extraction of a high level description from musical performance, for example in form of a music notation. Music notation can be viewed as a list of the pitch levels and corresponding timestamps. Ideally, one would like to recover a score directly frOID: sound. Such a representation of the surface structure of music would be very useful in music information retrieval (Music-IR) and content description of musical material in large audio databases. However, when operating on sampled audio data from polyphonic acoustical signals, extraction of a score-like description is a very challenging auditory scene analysis task [13]. In this paper, we focus on a subproblem in music-ir, where we assume that exact timing information of notes is available, for example as a stream of MIDI! events 1Musical Instruments Digital Interface. A standard communication protocol especially designed for digital instruments such as keyboards. Each time a key is pressed, a MIDI keyboard generates a short message containing pitch and key velocity. A computer can tag each received message by a timestamp for real-time processing and/or "recording" into a file. from a digital keyboard. A model for tempo tracking and transcription is useful in a broad spectrum of applications. One example is automatic score typesetting, the musical analog of word processing. Almost all score typesetting applications provide a means of automatic generation of a conventional music notation from MIDI data. In conventional music notation, onset time of each note is implicitly represented by the cumulative sum of durations of previous notes. Durations are encoded by simple rational numbers (e.q. quarter note, eight note), consequently all events in music are placed on a discrete grid. So the basic task in MIDI transcription is to associate discrete grid locations with onsets, Le. quantization. However, unless the music is performed with mechanical precision, identification of the correct association becomes difficult. Consequently resulting scores have often very poor quality. This is due to the fact that musicians introduce intentional (and unintentional) deviations from a mechanical prescription. For example timing of events can be deliberately delayed or pushed. Moreover, the tempo can fluctuate by slowing down or accelerating. In fact, such deviations are natural aspects of expressive performance; in the absence of these, music tends to sound rather dull. Robust and fast quantization and tempo tracking is also an important requirement in interactive performance systems. These are emerging applications that "listen" to the performance for generating an accompaniment or improvisation in real time [10, 12]. At last, such models are also useful in musicology for systematic study and characterization of express~ve timing by principled analysis of existing performance data. 2 Model Consider the following generative model for timing deviations in music Ck Ck-1 + "/k-1 (1) Wk Wk-1 + (k (2) Tk Tk-1 + 2 Wk (Ck Ck-1) (3) 'Yk Tk +€k (4) In Eq. 1, Ck denotes the grid location of k'th onset in a score. The interval between two consecutive onsets in the score is denoted by "/k-1 . .For example consider the notation j n which encodes ,,/1:3 == [1 0.5 0.5], hence C1:4 == [0 1 1.5 2]. We assign a prior of form p(Ck) ex exp(-d(Ck)) where d(Ck) is the number of significant digits in the binary expansion of the fraction of Ck [1]. One can check that such a prior prefers simpler notations, e.g. p( J]~ITJ ) < p( j n ). We note that Ck are drawn from an infinite (but discrete) set and are increasing in k, i.e Ck 2:: Ck-1. To allow for different time signatures and alternative rhythmic subdivisions, one can introduce additional hidden variables [1], but this is not addressed in this paper. Eq. 2 defines a prior over possible tempo deviations. We denote the logarithm of the period (inverse tempo) by w. For example if the tempo is 60 beats per minute (bpm), w == log 1sec == O. Since tempo appears as a scale variable in mapping grid locations on a score to the actual performance time, we have chosen to represent it in the logarithmic scale (eventually a gamma distribution can also be used). This representation is both perceptually plausible and mathematically convenient since a symmetric noise model on w assigns equal probabilities to equal relative c~anges in tempo. We take (k to be a Gaussian random variable with N(O, A2'kQ). Depending upon the interval between consecutive onsets, the model scales the noise covariance; longer jumps in the score allow for more freedom in fluctuating the tempo. Given the W sequence, Eq. 3 defines a model of noiseless onsets with variable tempo. We will denote the pair of hidden continuous variables by Zk == (Tk' Wk). Eq. 4 defines the observation model. Here Yk is the observed onset time of the k'th onset in the performance. The noise term tk models small scale expressive deviations in timing of individual notes and has a Gaussian distribution parameterized by N(tt("(k-l), "E("(k-l)).Such a parameterization is useful for appropriate quantization of phrases (short sequences of notes) that are shifted or delayed as a whole [1]. ill reality, a random walk model for tempo such as in Eq. 2 is not very realistic. Tempo deviations are usually more smooth. ill the dynamical model framework such smooth deviations can be allowed by increasing the dimensionality of W by include higher order "inertia" variables [2]. ill this case we simply rewrite Eq. 2 as Wk == AWk-l + (k and take a diagonal Q. Accordingly, the observation model (Eq. 4) changed such that Wk is replaced by CWk where C == [1 0 ... 0]. The graphical model is shown in Figure 1. The model is similar to a switching state space model, that has been recently applied in the context of music transcription [11]. The differences are in parameterization and more importantly in the inference method. Figure 1: Graphical Model. The pair of continuous hidden variables (Tk' Wk) is denoted by Zk. Both C and Z are hidden; only the onsets Y are observed. We define tempo tracking as a filtering problem == argmax LP(Ck,ZkIYl:k) Zk and rhythm transcription as a MAP state estimation problem argmaxp(Cl:KIY1:K) Cl:K (5) (6) p(Cl:K IY1:K) (7) The exact computation of the quantities in Eq. 6 and Eq. 5 is intractable due to the explosion in the number of mixture components required to represent the exact posterior at each step k. Consequently we will use Monte Carlo approximation techniques. (9) 3 Sequential Monte Carlo Sampling Sequential Monte Carlo sampling (a.k.a. particle filtering) is an integration method especially powerful for inference in dynamical systems. See [4] for a detailed review of state of the art. At each step k, the exact marginal posterior over hidden states Xk is approximated by an empirical distribution of form N p(Xk IY1:k) ~ L wi i )<5(Xk x~i)) (8) i==l where x~i) are a set of points obtained by sampling from a proposal distribution and wii ) are associated importance weights such that 2::::1 wii ) == 1. Particles at step k are evolved to k + 1 by sequential importance sampling and resampling methods [6]. Once a set of discrete sample points is obtained during the forward phase by sampling, particle approximations to quantities such as the smoothed marginal posterior p(Xk IYl:K) or the maximum a posteriori state sequence (Viterbi path) xr:K can be obtained efficiently. Due to the discrete nature of the approximate representation, resulting algorithms are closely related to standard smoothing and Viterbi algorithms in Hidden Markov models [9, 7, 6]. Unfortunately, if the hidden state space is of high dimensionality, sampling can be inefficient. Hence increasingly many particles are needed to accurately represent the posterior. Consequently, the estimation of "off-line" quantities such as p(Xk IY1:K) and x~:K becomes very costly since one has to store all past trajectories. For some models, including the one proposed here, one can identify substructures where integrations, conditioned on certain nodes can be computed analytically [5]. Conditioned on C1:k, the model reduces to the (extended) 2 Kalman filter. In this case the joint marginal posterior is represented as a mixture N """" (i) (i) (i) p(Ck' Zk IY1:k) ~ L...J W k p(Zk ICk ,Y1:k)<5(Ck - Ck ) i==l The particular case of Gaussian p(ZkIci i ) ,Y1:k) is extensively used in diverse applications [8] and reported to give superior results when compared to standard particle filtering [3, 6]. 3.1 Particle Filtering We assume that we have obtained a set- of particles from filtered posterior p(Ck IY1:k). Due to lack of space we do not give the details of the particle filtering algorithm but refer the reader to [6]. One important point to note is that we have to use the optimal proposal distribution given as p(cklc~i~l' Y1:k) oc JdZk- 1:k p(Yklzk' Ck, c~i~l) C) C) p(Zk' Ck IZk-1, ck~_1)P(Zk-1Ick~_1'Y1:k-1) (10) Since the state-space of Ck is effectively infinite, this step is crucial for efficiency. Evaluation of the proposal distribution amounts to looking forward and selecting a set of high probability candidate grid locations for quantization. Once cii ) are obtained we can use standard Kalman filtering algorithms to update the Gaussian potentials p(zklcii) ,Y1:k). Thus tempo tracking problem as stated in Eq. 5 is ~eadily solved. 2We linearize the nonlinear observation model 2Wk (en - Ck-l) around the expectation (Wk). 3.2 Modified Viterbi algorithlll The quantization problem in Eq. 6 can only be solved approximately. Since Z is integrated over, in general all Ck become coupled and the Markov property is lost, i.e. p(Cl:K IYl:K) is in general not a chain. One possible approximation, that we adapt also here, is to assume smoothed estimates are not much different from filtered estimates [8] i.e. p(Ck' zklck-l, Zk-l, Yl:K) ~ p(Ck' zklck-l, Zk-1, Yl:k) (11) and to write K p(Cl:KIYl:K) R:j f dZ1:KP(CIZIIYl) P(Ck,Zkh-l,Zk-l,Yl:k) k==2 K <X rdZ1:KP(YIIZl, Cl)p(Zl, Cl) p(Yk IZk, ck, Ck-l)p(Zk, Ck, IZk-l, Ck-l) v k==2 IT we plug in the mixture approximation in Eq. 9 and take the argmaxlog on both sides we obtain a sum that can be stepwise optimized using the Viterbi algorithm [9J. The standard Viterbi algorithm for particle filters [7] defines a transition matrix Tk - 1 == f(c~) IC~i~l) between each pair of particles at consecutive time slices. Here, f is a state transition distribution that can be evaluated pointwise and Tk- 1 can be computed on the fly by evaluating f at (c~), c~i~l)' In contrast, the modified Viterbi algorithm replaces the pointwise evaluation by an expectation under p(Zk' Zk-llc~), c~i~l' Yl:k) where the transition matrix is defined as Tk- 1(j, i) == p(c~) Ic~i~l'Yl:k). In this case, each entry of T is computed by one step Kalman likelihood evaluation. 1. Initialization. For i == 1 : N ~1 (i) == logp(cii)) + IOgp(Yllcii)) 2. Recursion. For j == 1 : N, k == 2 : K Tk- 1(j, i) logp(c~) Icii~l' Y1:k) (See Eq. 10) ~k(j) m?X{dk-l (i) + Tk-1 (j, i)} ~ 'l/Jk(j) arg m?X{~k-1 (i) + Tk-l (j, i)} ~ 3. Termination. rK argm9X~K(i) ~ 4. Backtracking. For k == K - 1 : -1 : 1 rk 'l/Jk+1 (rk+l) ck == cirk ) Since the tempo trajectories can be integrated out online, we need to only store the links '¢k and quantization locations cii ). Consequently, the random walk tempo prior can be replaced by a richer model as in Eq. 5, virtually without additional computational or storage cost. Ail outline of the algorithm is shown in Figure 2. Of course, the efficiency and accuracy of our approach depends heavily onto the assumption in Eq. 11, that the T matrix based on filtered estimates is accurate. -0.1J.5 1.5 T Figure 2: Outline of the algorithm. Left: forward filtering phase. The ellipses correspond to the conditionals p(zklcin) ,Yl:k). Vertical dotted lines denote the observations Yk. At each step k, particles with low likelihood are discarded. Surviving particles are linked to their parents. Right: The transition matrix Tk- 1 between each generation (forall pairs of c~) ,cii~l) is computed by standard Kalman filter likelihood equations. Note that Tk - 1 can be discarded once the forward messages ~k are computed and only the backward links 'l/Jl:K and corresponding Ck' need to be stored. When all onsets Yl:K are observed, the MAP sequence ci:K is computed by backtracking. 4 Simulation Results We have tested tempo tracking and quantization performance of the model on two different examples. The first example is a repeating "son-clave" pattern I/:! j j ! /J .J. j :11 (c == [1 2 4 5.5 7... ]) with fluctuating tempo 3. Such syncopated rhythms are usually hard to transcribe and make it difficult to track the tempo even for experienced human listeners. Moreover, since onsets are absent at prominent beat locations, standard beat tracking algorithms usually loose track. We observe' that for various realistic tempo fluctuations and observation noise level, the particle.filter is able to identify the correct tempo trajectory and the corresponding quantization (Figure 3, above). The second example is a piano arrangement of the Beatles song (Yesterday) performed by a professional classical pianist on a MIDI grand piano. This is a polyphonic piece, Le. the arrangement contains chords and events occurring at the same time. We model polyphony by allowing Ck -Ck-l == O. In this case, since the original arrangement is known, we estimate the true tempo trajectory by Kalman filtering after clamping Cl:K. As shown in Figure 3, the particle filter estimate and the true tempo trajectory are almost identical. 5 Discussion and Conclusion There are several advantages offered by particle filtering approach. The algorithm is suitable for real time implementation. Since the implementation is easy, this provides an important flexibility in the models one can employ. Although we have not 3We modulate the tempo deterministically according to Wk = 0.3 sin(27rck/32). The observation noise variance is R = 0.0005. 16 14 12 10 0.8 0 0.6 0.4 0.2 * 3 0 M . . -0.2 -0.4 -0.6 -0.8 -1 0 2 4 6 8 3 Figure 3: Above: Tempo tracking results for the clave pattern with 4 particles. Each circle denotes the mean (T~n), w~n)). The diameter of each particle is proportional to the normalized importance weight at each generation. '*' denote the true (T, w) pairs. Below: Tracking results for "Yesterday". '*' denote the mean of the filtered Zl:K after clamping to true Cl:K. Small circles denote the mean Zl:K corresponding to the estimated MAP trajectory Cr:K using 10 particles. addressed issues such as learning and online adaptation in this- paper, parameters of the model can also treated as hidden variables and can be eventually integrated out similar to the tempo trajectories. Especially in real time music applications fine tuning and careful allocation of computational resources is of primary importance. Particle filtering is suitable since one can simply reduce the number of particles when computational resources become overloaded. Motivated by the advantages of the particle filtering approach, we are currently working on a real time implementation of the particle filter based tempo tracker for eventual automatic' accompaniment generation such as an adaptive drum machine. Consequently, the music is quantized such that it can be typeset in a notation program. Acknowledgements This research is supported by the Technology Foundation STW, applied science division of NWO and the technology programme of the Dutch Ministry of Economic Affairs. References [1] A. T. Cemgil, P. Desain, and H. Kappen. Rhythm quantization for transcription. Computer Music Journal, 24:2:60-76, 2000. [2] A. T. Cemgil, H. Kappen, P. Desain, and H. Honing. On tempo tracking: Tempogram representation and kalman filtering. Journal of New Music Research, Accepted for Publication. [3] R. Chen and J. S. Liu. Mixture kalman filters. J. R. Statist. Soc., 10, 2000. [4] A. Douchet, N. de Freitas, and N. J. Gordon, editors. Sequential Monte Carlo Methods in Practice. Springer-Verlag, New York, 2000. [5] A. Douchet, N. de Freitas, K. Murphy, and S. Russell. Rao-blackwellised particle filtering for dynamic bayesian networks. In Uncertainty in Artificial Intelligence, 2000. [6] A. Douchet, S. Godsill, and C. Andrieu. On sequential monte carlo sampling methods for bayesian filtering. Statistics and Computing, 10(3):197-208, 2000. [7] S. Godsill, A. Douchet, and M. West. Maximum a posteriori sequence estimation using monte carlo particle filters. Annals of the Institute of Statistical Mathematics., 2000. [8] K. P. Murphy. Switching kalman filters. Technical report, Dept. of Computer Science, University of California, Berkeley, 1998. [9] L. R. Rabiner. A tutorial in hidden markov models and selected applications in speech recognation. Proc. of the IEEE, 77(2):257-286, 1989. [10] C. Raphael. A probabilistic expert system for automatic musical accompaniment. Journal of Computational and Graphical Statistics, Accepted for Publication, 1999. [11] C. Raphael. A mixed graphical model for rhythmic parsing. In to appear in Proc. of Uncertainty in AI, 2001. [12] B. Thorn. Unsuper~ised learning and interactive jazz/blues improvisation. In Proceedings of the AAAI2000. AAAI Press, 2000. [13] Barry L. Vercoe, William G. Gardner, and Eric D. Scheirer. Structured audio: Creation, transmission, and rendering of parametric sound representations. Proc. IEEE, 86:5:922-940, May 1998.
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Learning a Gaussian Process Prior for Automatically Generating Music Playlists John C. Platt Christopher J. C. Burges Steven Swenson Christopher Weare Alice Zheng Microsoft Corporation 1 Microsoft Way Redmond, WA 98052 jplatt,cburges,sswenson,chriswea @microsoft.com, alicez@cs.berkeley.edu Abstract This paper presents AutoDJ: a system for automatically generating music playlists based on one or more seed songs selected by a user. AutoDJ uses Gaussian Process Regression to learn a user preference function over songs. This function takes music metadata as inputs. This paper further introduces Kernel Meta-Training, which is a method of learning a Gaussian Process kernel from a distribution of functions that generates the learned function. For playlist generation, AutoDJ learns a kernel from a large set of albums. This learned kernel is shown to be more effective at predicting users’ playlists than a reasonable hand-designed kernel. 1 Introduction Digital music is becoming very widespread, as personal collections of music grow to thousands of songs. One typical way for a user to interact with a personal music collection is to specify a playlist, an ordered list of music to be played. Using existing digital music software, a user can manually construct a playlist by individually choosing each song. Alternatively, playlists can be generated by the user specifying a set of rules about songs (e.g., genre = rock), and the system randomly choosing songs that match those rules. Constructing a playlist is a tedious process: it takes time to generate a playlist that matches a particular mood. It is also difficult to construct a playlist in advance, as a user may not anticipate all possible music moods and preferences he or she will have in the future. AutoDJ is a system for automatically generating playlists at the time that a user wants to listen to music. The playlist plays with minimal user intervention: the user hears music that is suitable for his or her current mood, preferences and situation. AutoDJ has a simple and intuitive user interface. The user selects one or more seed songs for AutoDJ to play. AutoDJ then generates a playlist with songs that are similar to the seed songs. The user may also review the playlist and add or remove certain songs, if they don’t fit. Based on this modification, AutoDJ then generates a new playlist. AutoDJ uses a machine learning system that finds a current user preference function over a feature space of music. Every time a user selects a seed song or removes a song from the Current address: Department of Electrical Engineering and Computer Science, University of California at Berkeley playlist, a training example is generated. In general, a user can give an arbitrary preference value to any song. By default, we assume that selected songs have target values of 1, while removed songs have target values of 0. Given a training set, a full user preference function is inferred by regression. The for each song owned by the user is evaluated, and the songs with the highest are placed into the playlist. The machine learning problem defined above is difficult to solve well. The training set often contains only one training example: a single seed song that the user wishes to listen to. Most often, AutoDJ must infer an entire function from 1–3 training points. An appropriate machine learning method for such small training sets is Gaussian Process Regression (GPR) [14], which has been shown empirically to work well on small data sets. Technical details of how to apply GPR to playlist generation are given in section 2. In broad detail, GPR starts with a similarity or kernel function between any two songs. We define the input space to be descriptive metadata about the song. Given a training set of user preferences, a user preference function is generated by forming a linear blend of these kernel functions, whose weights are solved via a linear system. This user preference function is then used to evaluate all of the songs in the user’s collection. This paper introduces a new method of generating a kernel for use in GPR. We call this method Kernel Meta-Training (KMT). Technical details of KMT are described in section 3. KMT improves GPR by adding an additional phase of learning: meta-training. During meta-training, a kernel is learned before any training examples are available. The kernel is learned from a set of samples from meta-training functions. These meta-training functions are drawn from the same function distribution that will eventually generate the training function. In order to generalize the kernel beyond the meta-training data set, we fit a parameterized kernel to the meta-training data, with many fewer parameters than data points. The kernel is parameterized as a non-negative combination of base Mercer kernels. These kernel parameters are tuned to fit the samples across the meta-training functions. This constrained fit leads to a simple quadratic program. After meta-training, the kernel is ready to use in standard GPR. To use KMT to generate playlists, we meta-train a kernel on a large number of albums. The learned kernel thus reflects the similarity of songs on professionally designed albums. The learned kernel is hardwired into AutoDJ. GPR is then performed using the learned kernel every time a user selects or removes songs from a playlist. The learned kernel forms a good prior, which enables AutoDJ to learn a user preference function with a very small number of user training examples. 1.1 Previous Work There are several commercial Web sites for playing or recommending music based on one seed song. The algorithms behind these sites are still unpublished. This work is related to Collaborative Filtering (CF) [9] and to building user profiles in textual information retrieval [11]. However, CF does not use metadata associated with a media object, hence CF will not generalize to new music that has few or no user votes. Also, no work has been published on building user profiles for music. The ideas in this work may also be applicable to text retrieval. Previous work in GPR [14] learned kernel parameters through Bayesian methods from just the training set, not from meta-training data. When AutoDJ generates playlists, the user may select only one training example. No useful similarity metric can be derived from one training example, so AutoDJ uses meta-training to learn the kernel. The idea of meta-training comes from the “learning to learn” or multi-task learning literature [2, 5, 10, 13]. This paper is most similar to Minka & Picard [10], who also suggested fitting a mean and covariance for a Gaussian Process based on related functions. However, in [10], in order to generalize the covariance beyond the meta-training points, a Multi-Layer Perceptron (MLP) is used to learn multiple tasks, which requires non-convex optimization. The Gaussian Process is then extracted from the MLP. In this work, using a quadratic program, we fit a parameterized Mercer kernel directly to a meta-training kernel matrix in order to generalize the covariance. Meta-training is also related to algorithms that learn from both labeled and unlabeled data [3, 6]. However, meta-training has access to more data than simply unlabeled data: it has access to the values of the meta-training functions. Therefore, meta-training may perform better than these other algorithms. 2 Gaussian Process Regression for Playlist Generation AutoDJ uses GPR to generate a playlist every time a user selects one or more songs. GPR uses a Gaussian Process (GP) as a prior over functions. A GP is a stochastic process over a multi-dimensional input space . For any , if vectors are chosen in the input space, and the corresponding samples are drawn from the GP, then the are jointly Gaussian. There are two statistics that fully describe a GP: the mean and the covariance . In this paper, we assume that the GP over user preference functions is zero mean. That is, at any particular time, the user does not want to listen to most of the songs in the world, which leads to a mean preference close enough to zero to approximate as zero. Therefore, the covariance kernel simply turns into a correlation over a distribution of functions :
. In section 3, we learn a kernel which takes music metadata as and . In this paper, whenever we refer to a music metadata vector, we mean a vector consisting of 7 categorical variables: genre, subgenre, style, mood, rhythm type, rhythm description, and vocal code. This music metadata vector is assigned by editors to every track of a large corpus of music CDs. Sample values of these variables are shown in Table 1. Our kernel function thus computes the similarity between two metadata vectors corresponding to two songs. The kernel only depends on whether the same slot in the two vectors are the same or different. Specific details about the kernel function are described in section 3.2. Metadata Field Example Values Number of Values Genre Jazz, Reggae, Hip-Hop 30 Subgenre Heavy Metal, I’m So Sad and Spaced Out 572 Style East Coast Rap, Gangsta Rap, West Coast Rap 890 Mood Dreamy, Fun, Angry 21 Rhythm Type Straight, Swing, Disco 10 Rhythm Description Frenetic, Funky, Lazy 13 Vocal Code Instrumental, Male, Female, Duet 6 Table 1: Music metadata fields, with some example values Once we have defined a kernel, it is simple to perform GPR. Let be the metadata vectors for the songs for which the user has expressed a preference by selecting or removing them from the playlist. Let be the expressed user preference. In general, can be any real value. If the user does not express a real-valued preference, is assumed 1 if the user wants to listen to the song and 0 if the user does not. Even if the values are binary, we do not use Gaussian Process Classification (GPC), in order to maintain generality and because GPC requires an iterative procedure to estimate the posterior [1]. Let be the underlying true user preference for the th song, of which is a noisy measurement, with Gaussian noise of variance . Also, let be a metadata vector of any song that will be considered to be on a playlist: is the (unknown) user preference for that song. Before seeing the preferences , the vector forms a joint prior Gaussian derived from the GP. After incorporating the information, the posterior mean of is (1) where
and
(2) Thus, the user preference function for a song s, , is a linear blend of kernels that compare the metadata vector for song with the metadata vectors for the songs that the user expressed a preference. The weights are computed by inverting an by matrix. Since the number of user preferences tends to be small, inverting this matrix is very fast. Since the kernel is learned before GPR, and the vector is supplied by the user, the only free hyperparameter is the noise value . This hyperparameter is selected via maximum likelihood on the training set. The formula for the log likelihood of the training data given is ! #" $ %'&)(*&,+ . / ! #" + 0+)123/)465 4 123/ 7"98: (3) Every time a playlist is generated, different values of are evaluated and the that generates the highest log likelihood is used. In order to generate the playlist, the matrix is computed, and the user preference function is computed for every song that the user owns. The songs are then ranked in descending order of . The playlist consists of the top songs in the ranked list. The playlist can cut off after a fixed number of songs, e.g., 30. It can also cut off if the value of gets too low, so that the playlist only contains songs that the user will enjoy. The order of the playlist is the order of the songs in the ranked list. This is empirically effective: the playlist typically starts with the selected seed songs, proceeds to songs very similar to the seed songs, and then gradually drifts away from the seed songs towards the end of the list, when the user is paying less attention. We explored neural networks and SVMs for determining the order of the playlist, but have not found a clearly more effective ordering algorithm than simply the order of . Here, “effective” is defined as generating playlists that are pleasing to the authors. 3 Kernel Meta-Training (KMT) This section describes Kernel Meta-Training (KMT) that creates the GP kernel used in the previous section. As described in the introduction, KMT operates on samples drawn from a set of ; functions 7< . This set of functions should be related to a final trained function, since we derive a similarity kernel from the meta-training set of functions. In other words, we learn a Gaussian prior over the space of functions by computing Gaussian statistics on a set of functions related to a function we wish to learn. We express the kernel as a covariance components model [12]: = > @? >@AB> (4) where A > are pre-defined Mercer kernels and ? >DC - . We then fit ? > to the samples drawn from the meta-training functions. We use the simpler model instead of an empirical covariance matrix, in order to generalize the GPR beyond points that are in the meta-training set. The functional form of the kernel A and 0E can be chosen via cross-validation. In our application, both the form of A and E are determined by the available input data (see section 3.2, below). One possible method to fit the ? > is to maximize the likelihood in (3) over all samples drawn from all meta-training functions [7]. However, solving for the optimal ? > requires an iterative algorithm whose inner loop requires Cholesky decomposition of a matrix of dimension the number of meta-training samples. For our application, this matrix would have dimension 174,577, which makes maximizing the likelihood impractical. Instead of maximizing the likelihood, we fit a covariance components model to an empirical covariance computed on the meta-training data set, using a least-square distance function: & " 8
1 = > @? >'AB>
(5) where and index all of the samples in the meta-training data set, and where
is the empirical covariance
; < < <
(6) In order to ensure that the final kernel in (4) is Mercer, we apply ? > C - as a constraint in optimization. Solving (5) subject to non-negativity constraints results in a fast quadratic program of size E . Such a quadratic program can be solved quickly and robustly by standard optimization packages. The cost function in equation (5) is the square of the Frobenius norm of the difference between the empirical matrix
and the fit kernel ? . The use of the Frobenius norm is similar to the Ordinary Least Squares technique of fitting variogram parameters in geostatistics [7]. However, instead of summing variogram estimates within spatial bins, we form covariance estimates over all meta-training data pairs . Analogous to [8], we can prove that the Frobenius norm is consistent: as the amount of training data goes to infinity, the empirical Frobenius norm, above, approaches the Frobenius norm of the difference between the true kernel and our fit kernel. (The proof is omitted to save space). Finally, unlike the cost function presented in [8], the cost function in equation (5) produces an easy-to-solve quadratic program. 3.1 KMT for Music Playlist Generation In this section, we consider the application of the general KMT technique to music playlist generation. We decided to use albums to generate a prior for playlist generation, since albums can be considered to be professionally designed playlists. For the meta-training function , we use album indicator functions that are 1 for songs on an album , and 0 otherwise. Thus, KMT learns a similarity metric that professionals use when they assemble albums. This same similarity metric empirically makes consonant playlists. Using a small E in equation (4) forces a smoother, more general similarity metric. If we had simply used the meta-training kernel matrix
without fitting ? , the playlist generator would exactly reproduce one or more albums in the meta-training database. This is the meta-training equivalent of overfitting. Because the album indicator functions are uniquely defined for songs, not for metadata vectors, we cannot simply generate a kernel matrix according to (6). Instead, we generate a meta-training kernel matrix using meta-training functions that depend on songs:
; (7) where is 1 if song belongs to album , 0 otherwise. We then fit the ? > according to (5), where the A > Mercer kernels depend on music metadata vectors that are defined in Table 1. The resulting kernel is still defined by (4), with a specific A9> that will be defined in section 3.2, below. We used 174,577 songs and 14,198 albums to make up the meta-training matrix
, which is dimension 174,577x174,577. However, note that the
meta-training matrix is very sparse, since most songs only belong to 1 or 2 albums. Therefore, it can be stored as a sparse matrix. We use a quadratic programming package in Matlab that requires the constant and linear parts of the gradient of the cost function in (5): ? <
1 > ? >@A >
A <
(8)
A <
> ? >
A >
A <
(9) where the first (constant) term is only evaluated on those indicies in the set of nonzero
. The second (linear) term requires a sum over all and , which is impractical. Instead, we estimate the second term by sampling a random subset of pairs (100 random for each ). 3.2 Kernels for Categorical Data The kernel learned in section 3 must operate on categorical music metadata. Up until now, kernels have been defined to operate on continuous data. We could convert the categorical data to a vector space by allocating one dimension for every possible value of each categorical variable, using a 1-of-N sparse code. This would lead to a vector space of dimension 1542 (see Table 1) and would produce a large number of kernel parameters. Hence, we have designed a new kernel that operates directly on categorical data. We define a family of Mercer kernels: A >
if > - or ; otherwise, (10) where > is defined to be the binary representation of the number . The > vector serves as a mask: when > is 1, then the th component of the two vectors must match in order for the output of the kernel to be 1. Due to space limitations, proof of the Mercer property of this kernel is omitted. For playlist generation, the A > operate on music metadata vectors that are defined in Table 1. These vectors have 7 fields, thus runs from 1 to 7 and runs from 1 to 128. Therefore, there are 128 free parameters in the kernel which are fit according to (5). The sum of 128 terms in (4) can be expressed as a single look-up table, whose keys are 7-bit long binary vectors, the th bit corresponding to whether . Thus, the evaluation of from equation (1) on thousands of pieces of music can be done in less than a second on a modern PC. 4 Experimental Results We have tested the combination of GPR and KMT for the generation of playlists. We tested AutoDJ on 60 playlists manually designed by users in Microsoft Research. We compared the full GPR + KMT AutoDJ with simply using GPR with a pre-defined kernel, and without using GPR and with a pre-defined kernel (using (1) with all equal). We also compare to a playlist which are all of the user’s songs permuted in a random order. As a baseline, we decided to use Hamming distance as the pre-defined kernel. That is, the similarity between two songs is the number of metadata fields that they have in common. We performed tests using only positive training examples, which emulates users choosing seed songs. There were 9 experiments, each with a different number of seed songs, from 1 to 9. Let the number of seed songs for an experiment be . Each experiment consisted of 1000 trials. Each trial chose a playlist at random (out of the playlists that consisted of at least songs), then chose songs at random out of the playlist as a training set. The test set of each trial consisted of all of the remaining songs in the playlist, plus all other songs owned by the designer of the playlist. This test set thus emulates the possible songs available to the playlist generator. To score the produced playlists, we use a standard collaborative filtering metric, described in [4]. The score of a playlist for trial is defined to be
!
8 (11) where
is the user preference of the th element of the th playlist (1 if th element is on playlist , 0 otherwise), ? is a “half-life” of user interest in the playlist (set here to be 10), and
are the number of test songs for playlist . This score is summed over all 1000 trials, and normalized: -#
(12) where
is the score from (11) if that playlist were perfect (i.e., all of the true playlist songs were at the head of the list). Thus, an score of 100 indicates perfect prediction. Number of Seed Songs Playlist Method 1 2 3 4 5 6 7 8 9 KMT + GPR 42.9 46.0 44.8 43.8 46.8 45.0 44.2 44.4 44.8 Hamming + GPR 32.7 39.2 39.8 39.6 41.3 40.0 39.5 38.4 39.8 Hamming + No GPR 32.7 39.0 39.6 40.2 42.6 41.4 41.5 41.7 43.2 Random Order 6.3 6.6 6.5 6.2 6.5 6.6 6.2 6.1 6.8 Table 2: Scores for Different Playlist Methods. Boldface indicates best method with statistical significance level $ / . The results for the 9 different experiments are shown in Table 2. A boldface result shows the best method based on pairwise Wilcoxon signed rank test with a significance level of 0.05 (and a Bonferroni correction for 6 tests). There are several notable results in Table 2. First, all of the experimental systems perform much better than random, so they all capture some notion of playlist generation. This is probably due to the work that went into designing the metadata schema. Second, and most importantly, the kernel that came out of KMT is substantially better than the handdesigned kernel, especially when the number of positive examples is 1–3. This matches the hypothesis that KMT creates a good prior based on previous experience. This good prior helps when the training set is extremely small in size. Third, the performance of KMT + GPR saturates very quickly with number of seed songs. This saturation is caused by the fact that exact playlists are hard to predict: there are many appropriate songs that would be valid in a test playlist, even if the user did not choose those songs. Thus, the quantitative results shown in Table 2 are actually quite conservative. Playlist 1 Playlist 2 Seed Eagles, The Sad Cafe Eagles, Life in the Fast Lane 1 Genesis, More Fool Me Eagles, Victim of Love 2 Bee Gees, Rest Your Love On Me Rolling Stones, Ruby Tuesday 3 Chicago, If You Leave Me Now Led Zeppelin, Communication Breakdown 4 Eagles, After The Thrill Is Gone Creedence Clearwater, Sweet Hitch-hiker 5 Cat Stevens, Wild World Beatles, Revolution Table 3: Sample Playlists To qualitatively test the playlist generator, we distributed a prototype version of it to a few individuals in Microsoft Research. The feedback from use of the prototype has been very positive. Qualitative results of the playlist generator are shown in Table 3. In that table, two different Eagles songs are selected as single seed songs, and the top 5 playlist songs are shown. The seed song is always first in the playlist and is not repeated. The seed song on the left is softer and leads to a softer playlist, while the seed song on the right is harder rock and leads to a more hard rock play list. 5 Conclusions We have presented an algorithm, Kernel Meta-Training, which derives a kernel from a set of meta-training functions that are related to the function that is being learned. KMT permits the learning of functions from very few training points. We have applied KMT to create AutoDJ, which is a system for automatically generating music playlists. However, the KMT idea may be applicable to other tasks. Experiments with music playlist generation show that KMT leads to better results than a hand-built kernel when the number of training examples is small. The generated playlists are qualitatively very consonant and useful to play as background music. References [1] D. Barber and C. K. I. Williams. Gaussian processes for Bayesian classification via hybrid Monte Carlo. In M. C. Mozer, M. I. Jordan, and T. Petsche, editors, NIPS, volume 9, pages 340–346, 1997. [2] J. Baxter. A Bayesian/information theoretic model of bias learning. Machine Learning, 28:7–40, 1997. [3] K. P. Bennett and A. Demiriz. Semi-supervised support vector machines. In M. S. Kearns, S. A. Solla, and D. A. Cohn, editors, NIPS, volume 11, pages 368–374, 1998. [4] J. S. Breese, D. Heckerman, and C. Kadie. Empirical analysis of predictive algorithms for collaborative filtering. In Uncertainty in Artificial Intelligence, pages 43–52, 1998. [5] R. Caruana. Learning many related tasks at the same time with backpropagation. In NIPS, volume 7, pages 657–664, 1995. [6] V. Castelli and T. M. Cover. The relative value of labeled and unlabled samples in pattern recognition with an unknown mixing parameter. IEEE Trans. Info. Theory, 42(6):75–85, 1996. [7] N. A. C. Cressie. Statistics for Spatial Data. Wiley, New York, 1993. [8] N. Cristianini, A. Elisseeff, and J. Shawe-Taylor. On optimizing kernel alignment. Technical Report NC-TR-01-087, NeuroCOLT, 2001. [9] D. Goldberg, D. Nichols, B. M. Oki, and D. Terry. Using collaborative filtering to weave an information tapestry. CACM, 35(12):61–70, 1992. [10] T. Minka and R. Picard. Learning how to learn is learning with points sets. http:// wwwwhite.media.mit.edu/ tpminka/papers/learning.html, 1997. [11] M. Pazzani and D. Billsus. Learning and revising user profiles: The identification of interesting web sites. Machine Learning, 27:313–331, 1997. [12] P. S. R. S. Rao. Variance Components Estimation: Mixed models, methodologies and applications. Chapman & Hill, 1997. [13] S. Thrun. Is learning the n-th thing any easier than learning the first? In NIPS, volume 8, pages 640–646, 1996. [14] C. K. I. Williams and C. E. Rasmussen. Gaussian processes for regression. In NIPS, volume 8, pages 514–520, 1996.
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Motivated Reinforcement Learning Peter Dayan Gatsby Computational Neuroscience Unit 17 Queen Square, London, England, WClN 3AR. dayan@gatsby.ucl.ac.uk Abstract The standard reinforcement learning view of the involvement of neuromodulatory systems in instrumental conditioning includes a rather straightforward conception of motivation as prediction of sum future reward. Competition between actions is based on the motivating characteristics of their consequent states in this sense. Substantial, careful, experiments reviewed in Dickinson & Balleine, 12,13 into the neurobiology and psychology of motivation shows that this view is incomplete. In many cases, animals are faced with the choice not between many different actions at a given state, but rather whether a single response is worth executing at all. Evidence suggests that the motivational process underlying this choice has different psychological and neural properties from that underlying action choice. We describe and model these motivational systems, and consider the way they interact. 1 Introduction Reinforcement learning (RL 28) bears a tortuous relationship with historical and contemporary ideas in classical and instrumental conditioning. Although RL sheds important light in some murky areas, it has paid less attention to research concerning the motivation of stimulus-response (SR) links. RL methods are mainly concerned with preparatory Pavlovian (eg secondary) conditioning, and, in instrumental conditioning, the competition between multiple possible actions given a particular stimulus or state, based on the future rewarding or punishing consequences of those actions. These have been used to build successful and predictive models of the activity of monkey dopamine cells in conditioning. 22,24 By contrast, SR research starts from the premise that, in many circumstances, given an unconditioned stimulus (US; such as a food pellet), there is only one natural set of actions (the habit of approaching and eating the food), and the main issue is whether this set is worth executing (yes, if hungry, no if sated). This is traditionally conceived as a question of consummatory motivation. SR research goes on to study how these habits, and also the motivation associated with them, are 'attached' in an appropriately preparatory sense to conditioned stimuli (CSs) that are predictive of the USs. The difference between RL's competition between multiple actions and SR's motivation of a single action might seem trivial, particularly if an extra, nUll, action is included in the action competition in RL, so the subject can actively choose to do nothing. However, there is substantial evidence from experiments in which drive states (eg hunger, thirst) are manipulated, that motivation in the SR sense works in a sophisticated, intrinsically goal-sensitive, way and can exert unexpected effects on instrumental conditioning. By comparison with RL, psychological study of multiple goals within single environments is quite advanced, particularly in experiments in which one goal or set of goals is effective during learning, and another during performance. Based on these and other studies, (and earlier theoretical ideas from, amongst others, Konorski, 18,19 Dickinson, Balleine and their colleagues13 have suggested that there are really two separate motivational systems, one associated with Pavlovian motivation, as in SR, and one associated with instrumental action choice. They further suggest, partly based on related suggestions by Berridge and his colleagues,? that only the Pavlovian system involves dopamine. Neither the Pavlovian nor the instrumental system maps cleanly onto the standard view of RL, and the suggestion about dopamine would clearly Significantly damage the RL interpretation of the involvement of this neuromodulatory system in conditioning. In this paper, we describe some of the key evidence supporting the difference between instrumental and Pavlovian motivation (see also Balkenius3 and Spier25), and expand the model of RL in the brain to incorporate SR motivation and concomitant evidence on intrinsic goal sensitivity (as well as intrinsic habits). Some of the computational properties of this new model turn out to be rather strange - but this is a direct consequence of equivalently strange observable behavior. 2 Theoretical and Experimental Background Figure 1 shows a standard view of the involvement of the dopamine system in RL.22,24 Dopamine neurons in the ventral tegmental area (VTA) and substantia nigra pars compacta (SN c) report the temporal difference (TD) error 8 (t). In the simplest version of the theory, this is calculated as 8 (t) = r (t) + V 1T(x( t + 1» V 1T(x( t) ), where r (t) is the value of the reward at time t, x( t) is an internal representation of the state at time t, V 1T(x( t» is the expectation of the sum total future reward expected by the animal based on starting from that state, following policy IT, and the transition from x( t) to x( t + 1) is occasioned by the action a selected by the subject. In the actor-critic6 version of the dopamine theory, this TD error signal is put to two uses. One is adapting parameters that underlie the actual predictions V 1T(x(t». For this, 8(t) > 0 if the prediction from the state at time t, V 1T(x(t», is overly pessimistic with respect to the sum of the actual reward, r (t), and the estimated future reward, V 1T(x( t + 1», from the subsequent state. The other use for 8 (t) is criticizing the action a adopted at time t. For this, 8(t) > 0 implies that the action chosen is worth more than the average worth of x(t), and that the overall policy IT of the subject can therefore be improved by choosing it more often. In a Q-Iearning31 version of the theory, Q 1T(X, a) values are learned using an analogous quantity to 8 (t), for each pair of states x and actions a, and can directly be used to choose between the actions to improve the policy. Even absent an account for intrinsic habits, three key paradigms show the incompleteness of this view of conditioning: appetitive Pavlovian-instrumental transfer,15 intrinsic drive preference under speCific deprivation states,8 and incentive learning, as in the control of chains of instrumental behavior. 5 The SR view of conditioning places its emphasis on motivational control of a prepotent action. That is, the natural response associated with a stimulus (presumably as output by an action specification mechanism) is only elicited if A x state ~muli amygdala TO prediction accumbens~--+--'--=--l error OFC r reward '" B state stimuli x '" TO predictio error action 8 competition stRkfu~ a Figure 1: Actor-critic version of the standard RL model. A) Evaluator: A TD error signal 8 to learn V 1T(x) to match the sum of future rewards r, putatively via the basolateral nuclei of the amygdala, the orbitofrontal cortex and the nucleus accumbens. B) Instrumental controller: The TD error 8 is used to choose, and teach the choice of, appropriate actions a to maximize future reward based on the current state and stimuli, putatively via the dorsolateral prefrontal cortex and the dorsal striatum. it is motivationally appropriate, according to the current goals of the animal. The suggestion is that this is mediated by a separate motivational system. USs have direct access to this system, and CSs have learned access. A conclusion used to test this structure for the control of actions is that this motivational system could be able to energize any action being executed by the animal. Appetitive Pavlovian-instrumental transfer15 shows exactly this. Animals executing an action for an outcome under instrumental control, will perform more quickly when a CS predictive of reward is presented, even if the CS predicts a completely different reward from the instrumental outcome. This effect is abolished by lesions of the shell of the nucleus accumbens,10 one of the main targets of DA from the VTA. The standard RL model offers no account of the speed or force of action (though one could certainly imagine various possible extensions), and has no natural way to accommodate this finding. * The second challenge to RL comes from experiments on the effects of changing speCific and general needs for animals. For instance, Berridge & Schulkin8 first gave rats sucrose and saline solutions with one of a bitter (quinine) and a sour (citric) taste. They then artificially induced a strong physiological requirement for salt, for the first time in the life of the animal. Presented with a choice between the two flavors (in plain water, ie in extinction), the rats preferred to drink the flavor associated with the salt. Furthermore, the flavor paired with the salt was awarded positive hedonic reactions, whereas before pairing (and if it had been paired with sucrose instead) it was treated as being aversive. The key feature of this experiment is that this preference is evident without the opportunity for learning. Whereas the RL system could certainly take the physiological lack of salt as helping determine part of the state x(t), this could only exert an effect on behavior through learning, contrary to the evidence. The final complexity for standard RL comes from incentive learning. One paradigm involves a sequential chain of two actions (a1 and a2) that rats had to execute in order to get a reward.5 The subjects were made hungry, and were first trained to perform action a2 to get a particular reward (a Noyes pellet), and then to perform the chain of actions a1 - a2 to get the reward. In a final test phase, the animals were offered the chance of executing a1 and a2 in extinction, for half of them when they were still hungry; for the other half when they were sated on their normal diet. Figure 2A shows what happens. Sated animals perform a1 at the same rate as hungry animals, but perform a2 sig*Note that aversive Pavlovian instrumental transfer, in the form of the suppression of appetitive instrumental responding, is the conventional method for testing aversive Pavlovian conditioning. There is an obvious motivational explanation for this as well as the conventional view of competition between appetitive and protective actions. 100 .~ 80 ieo s a 40 ~ E20 hungry al al al al Figure 2: Incentive learning. A) Mean total actions al and a 2 for an animal trained on the chain schedule al - a2 -Noyes pellets. Hungry and sated rats perform al at the same rate, but sated animals fail to perform a 2. B) Mean total actions when sated following prior re-exposure to the Noyes pellets when hungry ('hungry-sated') or when sated (,sated-sated'). Animals re-exposed when sated are significantly less willing to perform a 2. Note the change in scale between A and B. Adapted from Balieine et a/.5 nificantly less frequently. Figure 2B shows the basic incentive learning effect. Here, before the test, animals were given a limited number of the Noyes pellets (without the availability of the manipulanda) either when hungry or when sated. Those who experienced them hungry ('hungry-sated') show the same results as the 'sated' group of figure 2A; whereas those who experienced them sated (,sated-sated') now declined to perform action al either. This experiment makes two points about the standard RL model. First, the action nearest to the reward (a2) is affected by the deprivation state without additional learning. This is like the effect of specific deprivation states discussed above. Second is that a change in the willingness to execute al happens after re-exposure to the Noyes pellets whilst sated; this learning is believed to involve insular cortex (part of gustatory neocortex4). That re-exposure directly affects the choice of al suggests that the instrumental act is partly determined by an evaluation of its ultimate consequence, a conclusion that relates to a long-standing psychological debate about the 'cognitive' evaluation of actions. Dickinson & Balleine13 suggest that the execution of a2 is mainly controlled by Pavlovian contingencies, and that Pavlovian motivation is instantly sensitive to goal devaluation via satiation. At this stage in the experiment, however, al is controlled by instrumental contingencies. By comparison with Pavlovian motivation, instrumental motivation is powerful (since it can depend on response-outcome expectancies), but dumb (since, without re-exposure, the animal works hard doing al when it wouldn't be interested in the food in any case). Ultimately, after extended training,14 in the birth of a new habit, al becomes controlled by Pavlovian contingencies too, and so becomes directly sensitive to devaluation. t 3 New Model These experiments suggest some major modifications to the standard RL view. Figure 3 shows a sketch of the new model, whose key principles include • Pavlovian motivation (figure 3A) is associated with prediction error 8(t) = r(t) + VTT(X(t + 1)) - VTT(x(t) for long term expected future rewards VTT(X, a), given a policy IT. Adopting this makes the model account for the classical conditioning paradigms explained by the standard RL model. tIt is not empirically clear whether actions that have become habits are completely automatic1 or are subject to Pavlovian motivational influences. A stimuli B stimuli C stimuli X X prior X habit CS prior CS bias CS ! , bias sIs sIs S S ~ amygdala ," ~ US shell US ~~.~ : 8 a sta~e~ __ >~ vrT(X) I s~a~: Figure 3: Tripartite model. A) Evaluator: USs are evaluated by a hard-wired evaluation system (HE) which is intrinsically sensitive to devaluation. USs can also be evaluated via a plastic route, as in figure 1, but which nevertheless has prior biases. CSs undergo Pavlovian stimulus substitution with the USs they predict, and can also be directly evaluated through the learned route. The two sources of information for VTT(x) compete, forcing the plastic route to adjust to the hard-wired route. B) Habit system: The SR mapping suggests an appropriate action based on the state X; the vigor of its execution is controlled by dopaminergic 8, putatively acting via the shell of the accumbens. C) Instrumental controller: Action choice is based on advantages, which are learned, putatively via the core of the accumbens. Prefrontal working memory is used to unfold the consequences of chosen actions. • ret) is determined by a devaluation-sensitive, 'hard-wired', US evaluator that provides direct value information about motivationally inappropriate USs. • 8(t), possibly acting through the shell of the accumbens, provides Pavlovian motivation for pre-wired and new habits (figure 3B), as in Pavlovian instrumental transfer. • V 1T(x(t) is determined by two competing sources: one as in the standard model (involving the basolateral nuclei of the amygdala and the orbitofrontal cortex (OFC),16,23 and including prior biases (sweet tasting foods are appetitive) expressed in the connections from primary taste cortex to oFC and the amygdala; the other, which is primary, dependent largely on a stimulus substitution20 relationship between CSs and USs, that is also devaluationdependent. The latter is important for ultimate Pavlovian control over actions; the former for phenomena such as secondary conditioning, which are known to be devaluation independentY Figure 4A (dashed) shows the contribution of the hard-wired evaluation route, via stimulus-substitution, on the prediction of value in classical conditioning. Here, stimulus-substitution was based on a form of Hebbian learning with a synaptic trace, so the shorter the CS-US interval, the greater the HE component. This translates into greater immediate sensitivity to devaluation, the main characteristic of the hard-wired route. The plastic route via the amygdala takes responsibility for the remainder of the prediction; and the sum prediction is always correct (solid line). • Short-term storage of predictive stimuli in prefrontal working memory is gated9 by 8(t), so can also be devaluation dependent. • Instrumental motivation depends on policy-based advantages (3C; Baird2) A 1T(x,a) = Q 1T(x,a) - V1T(X) trained by the error signal 8A (t) = 8(t) -A1T(x,a) Over the course of policy improvement, the advantage of a sub-optimal action becomes negative, and of an optimal action tends to O. The latter offers a natural model of the transition from instrumental action selection to an SR habit. Note that, in this actor-critic scheme, some aspects of advantages are not necessary, such as the normalizing updates. A B 0.5 1:: sum bO \ 0[j \ HE ;s: 0.5 \ component 79 ' advantages , , C value V7T(xo) 1 .5 ,----"'-"-''-----''-''''-'---, sum , , ~4 ,------:.A-,-7T--,(:..:: XO,-,-, .:..:. a ,-) -----, successor 0.2 , -reprn --1 :o 8 , 'HE ~ ~ ~ ~ -0.5 A 7T(X , b) , ~ ~ ~ , component -0.2 °0~--1-0 ~~20 0 50 100 50 101 0~--50 ---' 1 00 ··. plastic cs-us interval iteration iteration iteration Figure 4: A) Role of the hard-wired route (dashed line), via stimulus-substitution, in predicting future reward (r = 1) as a function of the CS-US interval. The solid line shows that the net prediction is always correct. B) Advantages of useful (a) and worthless (b) actions at the start state Xo. C) Evolution of the value of Xo over learning. The solid line shows the mean value; the dashed line the hard-wired component, providing immediate devaluation sensitivity. 0) Construction of A7T(xo, a) via a successor representation componentll (dashed) and a conventionally learned component (dotted). The former is sensitive to re-exposure devaluation, as in figure 2B. B-D) Action a produces reward r = 1 with probability 0.9 after 3 timesteps; curves are averages over 2000 runs. Figure 4B;C show two aspects of instrumental conditioning. Two actions compete at state xo, one, a, with a small cost and a large future payoff; the other, b, with no cost and no payoff. Figure 4B shows the development of the advantages of these actions over learning. Action a starts looking worse, because it has a greater immediate cost; its advantage increases as the worth of a grows greater than the mean value of xo, and then goes to 0 (the birth of the habit) as the subject learns to choose it every time. Figure 4C shows the value component of state Xo. This comes to be responsible for the entire prediction (as A1T(XO, a) ~ 0). As in figure 4A, there is a hard-wired component to this value which would result in the immediate decrement of response evident in figure 2A. • On-line action choice is dependent on 8A (t) as in learned klinokinesis.21 Incentive learning in chains suggests that the representation underlying the advantage of an action includes information about its future consequences, either through an explicit model,27,29 a successor representation,ll or perhaps a form of f3-model. 26 One way of arranging this would use a VTE-like30 mechanism for proposing actions (perhaps using working memory in prefrontal cortex), in order to test their advantages. Figure 4D shows the consequence of using a learned successor representation underlying the advantage A1T(XO, a) shown in figure 4B. The dashed line shows the component of A1T(XO, a) dependent on a learned successor representation, and the prior bias about the value of the reward, and which is therefore sensitive to re-exposure (when the value accorded to the reward is decreased); the dotted line shows the remaining component of A 1T(XO, a), learned in the standard way. Re-exposure sensitivity (ie incentive learning) will exist over roughly iterations 25 - 75 . • SR models also force consideration of the repertoire of possible actions or responses available at a given state (figure 3B;C). We assume that both corticocortical and cortico-(dorsal) striatal plasticity sculpt this collection, using 8A (t) directly, and maybe also correlational learning rules. The details of the model are not experimentally fully determined, although its general scheme is based quite straightforwardly from the experimental evidence referred to (and many other experiments), and by consistency with the activity of dopamine cells (recordings of which have so far used only a single motivational state). 4 Discussion Experiments pose a critical challenge to our understanding of the psychological and neural implementation of reinforcement learning, 12,13 suggesting the importance of two different sorts of motivation in controlling behavior. With both empirical and theoretical bases, we have put these two aspects together through the medium of advantages. The most critical addition is a hard-wired, stimulus-substitution sensitive, route for the evaluation of stimuli and states, which competes with a plastic route through the amygdala and the oFC. This hard-wired route has the property of intrinsic sensitivity to various sorts of devaluation, and this leads to motivationally appropriate behavior. The computational basis of the new aspects of the model focus on motivational control of SR links (via VTT ), to add to motivational control of instrumental actions (via ATT). We also showed the potential decomposition of the advantages into a component based on the successor representation and therefore sensitive to re-exposure as in incentive learning, and a standard, learned, component. The model is obviously incomplete, and requires testing in richer environments. In particular, we have yet to explore how habits get created from actions as the maximal advantage goes to o. Acknowledgements I am very grateful to Christian Balkenius, Bernard Balleine, Tony Dickinson, Sham Kakade, Emmet Spier and Angela Yu for discussions. Funding was from the Gatsby Charitable Foundation. References [1] Adams, CD (1982) Variations in the sensitivity of instrumental responding to reinforcer devaluation. QJEP 34B:77-98. [2] Baird, LC (1993) Advantage Updating. Technical report WL-TR-93-1146, Wright-Patterson Air Force Base. [3] Balkenius, C (1995) Natural Intelligence in Artificial Creatures. PhD Thesis, Department of Cognitive Science, Lund University, Sweden. [4] Balleine, BW & Dickinson, A (1998) Goal-directed instrumental action: Contingency and incentive learning and their cortical substrates. Neuropharmacology 37:407-419. [5] Balleine, BW, Garner, C, Gonzalez, F & Dickinson, A (1995) Motivational control of heterogeneous instrumental chains. Journal of Experimental Psychology: Animal Behavior Processes 21:203-217. [6] Barto, AG, Sutton, RS & Anderson, CW (1983) Neuronlike elements that can solve difficult learning problems. IEEE SMC 13:834-846. [7] Berridge, KC (2000) Reward learning: Reinforcement, incentives, and expectations. In DL Medin, editor, The Psychology of Learning and Motivation 40:223-278. [8] Berridge, KC & Schulkin, J (1989) Palatability shift of a salt-associated incentive during sodium depletion. Quarterly Journal of Experimental Psychology: Comparative & Physiological Psychology 41:121-138. [9] Braver, TS, Barch, DM & Cohen, JD (1999) Cognition and control in schizophrenia: A computational model of dopamine and prefrontal function. Biological Psychiatry 46:312-328. [10] Corbit, LH, Muir, JL & Balleine, BW (200l) The role of the nucleus accumbens in instrumental conditioning: Evidence of a functional dissociation between accumbens core and shell. Journal of Neuroscience 21:3251-3260. [11] Dayan, P (1993) Improving generalisation for temporal difference learning: The successor representation. Neural Computation 5:6l3-624. [12] Dickinson, A & Balleine, B (1994) Motivational control of goal-directed action. Animal Learning & Behavior 22:1-18. [l3] Dickinson, A & Balleine, B (200l) The role of learning in motivation. In CR Gallistel, editor, Learning, Motivation and Emotion, Volume 3 of Steven's Handbook of Experimental Psychology, Third Edition. New York, NY: Wiley. [14] Dickinson, A, Balleine, B, Watt, A, Gonzalez, F & Boakes, RA (1995) Motivational control after extended instrumental training. Animal Learning & Behavior 23:197-206. [15] Estes, WK (1943). Discriminative conditioning. I. A discriminative property of conditioned anticipation. JEP 32:150-155. [16] Holland, PC & Gallagher, M (1999) Amygdala circuitry in attentional and representational processes. Trends in Cognitive Sciences 3:65-73. [17] Holland, PC & Rescorla, RA (1975) The effect of two ways of devaluing the unconditioned stimulus after first- and second-order appetitive conditioning. Journal of Experimental Psychology: Animal Behavior Processes 1:355-363. [18] Konorski, J (1948) Conditioned Reflexes and Neuron Organization. Cambridge, England: Cambridge University Press. [19] Konorski, J (1967) Integrative Activity of the Brain: An Interdisciplinary Approach. Chicago, 11: University of Chicago Press. [20] Mackintosh, NJ (1974) The Psychology of Animal Learning. New York, NY: Academic Press. [21] Montague, PR, Dayan, P, Person, C & Sejnowski TJ (1995) Bee foraging in uncertain environments using predictive hebbian learning. Nature 377:725-728. [22] Montague, PR, Dayan, P & Sejnowski, TJ (1996) A framework for mesencephalic dopamine systems based on predictive Hebbian learning. Journal of Neuroscience 16: 1936-1947. [23] Schoenbaum, G, Chiba, AA & Gallagher, M (1999) Neural encoding in orbitofrontal cortex and basolateral amygdala during olfactory discrimination learning. Journal of Neuroscience 19:1876-1884. [24] Schultz, W, Dayan, P & Montague, PR (1997) A neural substrate of prediction and reward. Science 275:1593-1599. [25] Spier, E (1997) From Reactive Behaviour to Adaptive Behaviour. PhD Thesis, Balliol College, Oxford. [26] Sutton, RS (1995) TD models: modeling the world at a mixture of time scales. In A Prieditis & S Russell, editors, Proceedings of the Twelfth International Conference on Machine Learning. San Francisco, CA: Morgan Kaufmann, 531-539. [27] Sutton, RS & Barto, AG (1981) An adaptive network that constructs and uses an internal model of its world. Cognition and Brain Theory 4:217246. [28] Sutton, RS & Barto, AG (1998) Reinforcement Learning. Cambridge, MA: MIT Press. [29] Sutton, RS & Pinette, B (1985) The learning of world models by connectionist networks. Proceedings of the Seventh Annual Conference of the Cognitive Science Society. Irvine, CA: Lawrence Erlbaum, 54-64. [30] Tolman, EC (1938) The determiners of behavior at a choice point. Psychological Review 45:1-41. [31] Watkins, CJCH (1989) Learning from Delayed Rewards. PhD Thesis, University of Cambridge, Cambridge, UK.
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Exact differential equation population dynamics for Integrate-and-Fire neurons Julian Eggert * HONDA R&D Europe (Deutschland) GmbH Future Technology Research Carl-Legien-StraBe 30 63073 Offenbach/Main, Germany julian. eggert@hre-ftr.f.rd.honda.co.jp Berthold Bauml Institut fur Robotik und Mechatronik Deutsches Zentrum fur Luft und Raumfahrt (DLR) o berpfaffenhofen Berthold.Baeuml@dlr.de Abstract Mesoscopical, mathematical descriptions of dynamics of populations of spiking neurons are getting increasingly important for the understanding of large-scale processes in the brain using simulations. In our previous work, integral equation formulations for population dynamics have been derived for a special type of spiking neurons. For Integrate- and- Fire type neurons, these formulations were only approximately correct. Here, we derive a mathematically compact, exact population dynamics formulation for Integrate- and- Fire type neurons. It can be shown quantitatively in simulations that the numerical correspondence with microscopically modeled neuronal populations is excellent. 1 Introduction and motivation The goal of the population dynamics approach is to model the time course of the collective activity of entire populations of functionally and dynamically similar neurons in a compact way, using a higher descriptionallevel than that of single neurons and spikes. The usual observable at the level of neuronal populations is the populationaveraged instantaneous firing rate A(t), with A(t)6.t being the number of neurons in the population that release a spike in an interval [t, t+6.t). Population dynamics are formulated in such a way, that they match quantitatively the time course of a given A(t), either gained experimentally or by microscopical, detailed simulation. At least three main reasons can be formulated which underline the importance of the population dynamics approach for computational neuroscience. First, it enables the simulation of extensive networks involving a massive number of neurons and connections, which is typically the case when dealing with biologically realistic functional models that go beyond the single neuron level. Second, it increases the analytical understanding of large-scale neuronal dynamics, opening the way towards better control and predictive capabilities when dealing with large networks. Third, it enables a systematic embedding of the numerous neuronal models operating at different descriptional scales into a generalized theoretic framework, explaining the relationships, dependencies and derivations of the respective models. Early efforts on population dynamics approaches date back as early as 1972, to the work of Wilson and Cowan [8] and Knight [4], which laid the basis for all current population-averaged graded-response models (see e.g. [6] for modeling work using these models). More recently, population-based approaches for spiking neurons were developed, mainly by Gerstner [3, 2] and Knight [5]. In our own previous work [1], we have developed a theoretical framework which enables to systematize and simulate a wide range of models for population-based dynamics. It was shown that the equations of the framework produce results that agree quantitatively well with detailed simulations using spiking neurons, so that they can be used for realistic simulations involving networks with large numbers of spiking neurons. Nevertheless, for neuronal populations composed of Integrate-and-Fire (I&F) neurons, this framework was only correct in an approximation. In this paper, we derive the exact population dynamics formulation for I&F neurons. This is achieved by reducing the I&F population dynamics to a point process and by taking advantage of the particular properties of I&F neurons. 2 Background: Integrate-and-Fire dynamics 2.1 Differential form We start with the standard Integrate- and- Fire (I&F) model in form of the wellknown differential equation [7] (1) which describes the dynamics of the membrane potential Vi of a neuron i that is modeled as a single compartment with RC circuit characteristics. The membrane relaxation time is in this case T = RC with R being the membrane resistance and C the membrane capacitance. The resting potential v R est is the stationary potential that is approached in the no-input case. The input arriving from other neurons is described in form of a current ji. In addition to eq. (1), which describes the integrate part of the I&F model, the neuronal dynamics are completed by a nonlinear step. Every time the membrane potential Vi reaches a fixed threshold () from below, Vi is lowered by a fixed amount Ll > 0, and from the new value of the membrane potential integration according to eq. (1) starts again. if Vi(t) = () (from below) . (2) At the same time, it is said that the release of a spike occurred (i.e., the neuron fired), and the time ti = t of this singular event is stored. Here ti indicates the time of the most recent spike. Storing all the last firing times, we gain the sequence of spikes {t{} (spike ordering index j, neuronal index i). 2.2 Integral form Now we look at the single neuron in a neuronal compound. We assume that the input current contribution ji from presynaptic spiking neurons can be described using the presynaptic spike times tf, a response-function ~ and a connection weight W· . ',J ji(t) = l: Wi,j l: ~(t - tf) (3) j f Integrating the I&F equation (1) beginning at the last spiking time tT, which determines the initial condition by Vi(ti) = vi(ti - 0) - 6., where vi(ti - 0) is the membrane potential just before the neuron spikes, we get 1 Vi(t) = vRest + fj(t - t:) + l: Wi,j l: a(t - t:; t - tf) , (4) j f with the refractory function fj(s) = - (vRest - Vi(t:)) e- S / T (5) and the alpha-function Sf a(s; s') = r ds" e-[sf -S"J/T ~(s") . JSI_S (6) If we start the integration at the time ti* of the spike before the last spike, then for ti* :::; t < ti the membrane potential is given by an expression like eq. (4), where ti is replaced by t:i* . Especially we can now express v( ti - 0) and therefore the initial condition for an integration starting at tT in terms of ti* and v(ti* - 0). In this way, we can proceed repetitively and move back into the past. After some simple algebra this results in Vi(t) = vRest + l:ry(t-t{)+ l:Wi,j l:a(oo;t - tf) , f j f ~ ~-------y~------~ vfef(t) v~yn(t) with a refractory function wich differs in the scale factor from that in eq. (5) ry(s) = -6. e- S / T • (7) (8) The components vref(t) and v?n(t) to the membrane potential indicate refractory and synaptic components to the neuron i, respectively, as normally used in the Spike- Response- Model (SRM) notation [2]. Both equations (4) and (7) formulate the neuronal dynamics using a refractory component, which depends on the own spike releases of a neuron, and a synaptic component, which comprises the integrated input contribution to the membrane potential by arrival of spikes from other neurons 2. The synaptic component is based on the alpha-function characteristic of isolated arriving spikes, with an increase of the membrane potential after spike arrival and a subsequent exponential decrease. 1 Strictly speaking, the constants vRest, T, () and ,6, and the function 1]( s) may vary for each neuron, so that they should be written with a subindex i [similarly for n(s; s') , which may vary for each connection j -+ i so that we should write it with subindices i, j]. For the sake of clarity, we omit these indices here. 2S0 the I&F model can be formulated as a special case of the Spike- Response- Model, which defines the neuronal dynamics in the integral formulation. The comparison of the equivalent expressions eq. (4) and eq. (7) reveals an interesting property of the I&F model. They look very similar, but in eq. (4), the refractory component depends only on the time elapsed since the last spike (thus reflecting a renewal property, sometimes also called a short term memory for refractory properties), whereas in eq. (7), it depends on a sum of the contributions of all past spikes. The simpler form of the refractory contribution in eq. (4) is achieved at the cost of an alpha-function that now depends on the time t - ti elapsed since the last own spike in addition to the times t - tf elapsed since the release of spikes at the presynaptic neurons j that provide input to i. In eq. (7), we have a more complex refractory contribution, but an alpha-function that does not depend on the last own spike time any more. 2.3 Probabilistic spike release Probabilistic firing is introduced into the I&F model eq. (4) resp. (7) by using threshold noise. The spike release of each neuron is controlled by a hazard function >.(v), so that >.(v)dt = Prob. that a neuron with membrane potential v spikes in [t, t + dt) (9) When a neuron spikes, we proceed as usual: The membrane potential is reset by a fixed amount 6. and the I&F dynamics continues. 3 Population dynamics 3.1 Density description Descriptions of neuronal populations usually assume a neuronal density function p(t; X) which depends on the state variables X of the neurons. The density quantifies the likelihood that a neuron picked out of the population will be found in the vicinity of the point X in state space, p(t; X) dX = Portion of neurons at time t with state in [X, X + dX) (10) If we know p(t; X) , the population activity A(t) can be easily calculated. Using the hazard function >'(t; X), the instantaneous population activity (spikes per time) can be calculated by computing the spike release averaged over the population, A(t) = J dX >.(t; X) p(t; X) (11) The population dynamics is then given by the time course of the neuronal density function p(t; X), which changes because each neuron evolves according to its own internal dynamics, e.g. after a spike release and the subsequent reset of the membrane potential. The main challenge for the formulation of a population dynamics resides in selecting a low-dimensional state space [for an easy calculation of A(t)] and a suitable form for gtp(t; X). As an example, for the population dynamics for I&F neurons it would be straightforward to use the membrane potential v from eq. (1) as the state variable X. But this leads to a complicated density dynamics, because the dynamics for v(t) consist of a continuous (differential equation (1)) and a discrete part (spike generation). Therefore, here we concentrate on an alternative description that allows a compact formulation of the desired I&F density dynamics. 3.2 Exact population dynamics for I&F neurons Which is the best state space for a population dynamics of I&F neurons? For the formulation of a population dynamics, it is usually assumed that the synaptic contributions to the membrane potential are identical for all neurons. This is the case if we group all neurons of the same dynamical type and with identical connectivity patterns into one population. That is, we say that neurons i and i' belong to the same population if Wi,j = Wi',j for all j (for simulations of realistic networks of spiking neurons, this will of course never be exactly the case, but it is reasonable to assume that a grouping of neurons into populations can be achieved to a good approximation) . In our case, looking at eq. (4), we see that, since o:(s, s') depends on s = t - ti and therefore on the own last spike time, the synaptic contribution to the membrane potential differs according to the state of the neuron. Thus we regard eq. (7). Here, we see that for identical connectivity patterns Wi,j, the synaptic contributions are the same for all neurons, because 0:(00, s') does not depend on the own spike time any more. Which are then the state variables of eq. (1) for the density description? We see that, for a fixed synaptic contribution, the membrane potential Vi is fully determined by the set of the own past spiking times {tf}. But this would mean an infinite-dimensional density for the state description of a population, and, accordingly, a computationally overly expensive calculation of the population activity A(t) according to eq. (11). To avoid this we take advantage of a particular property of the I&F model. According to eq. (8), the single spike refractory contributions 'TJ(s) are exponential. Since any sum of exponential functions with common relaxation constant T can be again expressed as as an exponential function with the same T , we can write instead of vrf(t) from eq. (7) (12) Now the membrane potential Vi(t) only depends on the time of the last own spike ti and the refractory contribution amplitude modulation factor at the last spike 'TJi . That is, we have transferred the effect of all spikes previous to the last one into 'TJi. In addition, we have to care about updating of ti and 'TJi when a neuron spikes so that we get 3 'TJi --+ 'TJi = 1 + 'TJie-(t-tn!T , ti --+ ti = t . (13) The effect of taking into account more than the most recent spike ti in the refractory component vief(t) leads to a modulation factor 'TJi greater than 1, in particular if spikes come in a rapid succession so that refractory contributions can accumulate. Instead of using a modulation factor 'TJi the effect of previous spikes can also be taken into account by introducing an effective last spiking time ii. vi"f(t) = 'TJ(t - in = 'TJi'TJ(t - tn , where ii and 'TJi are connected by i; = t; + TIn'TJi (14) (15) The effect of i* is sort of funny. Because of 'TJi ::::: 1 it holds for the effective last spiking time ii ::::: ti. This means, that, while at a given time t it is allways ti :::; t, it happens that ii ::::: t, meaning the neurons behave as if they would spike in the future. 3Here, the order of reemplacement matters; first we have to reemplace 1]:, then ti. For the membrane potential we get now instead of eq. (7) Vi(t) = vRest + ry(t - tn + 2..: Wi,j 2..: 0:(00; t - t;) (16) j f and for the update rule for the effective last spiking time t; follows tA* tA* f (t tA*) i-+i= 'i' (17) with (18) Therefore we can regard the dimensionality of the state space of the I&F dynamics as 1-dimensional in the description of eq. (16). The dynamics of the single I&F neurons now turns out to be very simple: Calculate the membrane potential Vi(t) using eq. (16) together with the state variable t;, and check if Vi(t) exceeds the threshold. If not, move forward in time and calculate again. If the membrane potential exceeds threshold, update t; according to eq. (17) and then proceed with the calculation of Vi(t) as normal. Using this single neuron dynamics , we can now proceed to gain a population dynamics using a density p(t; t*). The time t is here the explicit time dependence, whereas t* denote the state variable of the population. By fixing t* and the synaptic contribution vsyn(t) to the membrane potential, the state of a neuron is fully determined and the hazard function can be written as ,X[vsyn(t); t*]. The dynamics of the density p(t; t*) is then calculated as follows. Changes of p(t; t*) occur when neurons spike and t* is updated according to eq. (17). The hazard function controls the spike release, and, therefore, the change of p(t; t*). For chosen state variables, p(t; t*) decreases due to spiking of the neurons with the fixed t*, and increases because neurons with other t'* spike and get updated in just that way that after updating their state variable falls around t*. This occurs according to the reemplacement rule eq. (17) when f(t, t'*) = t* . (19) Taking all together the dynamics of the density p(t; t*) is given by decrease due to same state t* spiking A -ftp(t;t*) = '-,X[vsyn(t); t*]p(t; t*)' 1 + 00 + -00 dt'* 8[J(t, t'*) - t*] ,X[vsyn(t); t'*] p(t; t'*) increase due to spiking of neurons with other states t'* (20) The population activity can then be calculated using the density according to eq. (11) as follows 1 +00 A(t) = - 00 dt* ,X[vsyn(t); t*] p(t; t*) (21) Remark that the expression for the density dynamics (eq. 20) automatically conserves the norm of the density, so that 1 +00 - 00 dt* p( t; t*) = const , (22) which is a necessary condition because the number of neurons participating in the dynamics must remain constant. 4 Simulations The dynamics of a population of I&F neurons, represented by the time course of their joint activity, can now be easily calculated in terms of the differential equation (20), if the neuronal state density of the neuronal population p(t; i*) and the synaptic input vsyn(t) are known. This means that all we have to store is the density p(t; i*) for past and future effective last spiking times i* 4 . Favorably for numerical simulations, only a limited time window of i* around the actual time t is needed for the dynamics. The activity A(t) only appears as an auxiliary variable that is calculated with the help of the neuronal density. In figure 1 the simulation results for populations of of spiking neurons are shown. The neurons are uncoupled and a hazard function A(V) = ~ e2,B(v-e) , TO (23) with spike rate at threshold liTO = 1.0ms-1 , a kind of inverse temperature (3 = 2.0, which controls the noise level, and the threshold e = 1.0. The other parameters of the model in eq. (1) are: resting potential vRest = 0, jump in membrane potential after spike release ~ = 1 and time constant T = 20ms. This parameters are chosen to be biologicaly plausible. r------vsyr'i-' __ 1_00 __ 15_0 __ 20_0 __ 2_50 __ 30_0 ----,1 (ms) o~ I c) 100 150 200 250 300 I (ms) A (spikes/ms) 0.14 : b) = 0.12 II " " 0.1 n " " 0.08 :: :: I 0.06 !l 1l 1\ : ~ *:\ 0.04 ! \_, .. ----: ! \_ .. ----2 '-1 0.02 : ! ....... j ! I:, _______ J ~,' ~ .. ' r------Figure 1: Activity A(t) of simulated populations of neurons. The neurons are uncoupled and to each neuron the same synaptic field vsyn(t), ploted in c) and d), is applied. a) shows the activity A(t) for a population of I&F neurons simulated on the one hand as N = 10000 single neurons (solid line) using eq. (7) and on the other hand using the density dynamics eq. (20) (dashed line). In b) the activity A(t) of a population ofI&F neurons (dashed line) and a population of SRM neurons with renewal (solid line) are compared. For all simulations the same parameters as specified in the text were used. The simulations show that the density dynamics eq. (20) reproduces the activity A( t) of a population of single I&F neurons almost perfect, with the exception of the noise in the single neuron simulations due to the finite size effects. This holds even for the peaks occuring at the steps of the applied synaptic field vsyn (t), although the density dynamics is entirely based on differential equations and one would therefore not expect such an excellent match for fast changes in activity. 4 VSYll(t) only appears as a scalar in the dynamics, so that no integration over time takes place here. The simulations also show that there can be a big difference between I&F and SRM neurons with renewal. Because of the accumulation of the refractory effects of all former spikes in the case of I&F neurons the activity A(t) is generaly lower than for the SRM neurons with renewal and the higher the absolute actitvity level the bigger is the difference between both. 5 Conclusions In this paper we derived an exact differential equation density dynamics for a population of I&F neurons starting from the microscopical equations for a single neuron. This density dynamics allows a compuationaly efficient simulation of a whole population of neurons. In future work we want to simulate a network of connected neuronal populations. In such a network of populations (indexed e.g. by x) , a self-consistent system of differential equations based on the population's p(x, t; i*) and A(x, t) emerges if we constrain ourselves to neuronal populations connected synaptically according to the constraints given by the pool definition [2]. In this case, two neurons i and j belong to pools x and y, if Wi,j = W(x, y). This allows us to write for the synaptic component of the membrane potential vsyn(x,t) = 2: W (x , y) 1 00 ds'a(oo;s')A(y,t-s') y 0 (24) Using the alpha-function a(oo; s') as introduced in (6), and a "nice" responsefunction ~ for the input current time course after a spike, we can write eq. (24) using differential equations that use A(y, t) as input. This results in a system that is based entirely on differential equations and is very cheap to compute. References [1] J. Eggert and J.L. van Hemmen. Modeling neuronal assemblies: Theory and implementation. Neural Computation, 13(9):1923- 1974, 200l. [2] W. Gerstner. Population dynamics of spiking neurons: Fast transients, asynchronous states and locking. Neural Computation, 12:43- 89, 2000. [3] W . Gerstner and J . L. van Hemmen. Associative memory in a network of 'spiking' neurons. Network, 3:139- 164, 1992. [4] B. W. Knight. Dynamics of encoding in a populations of neurons. J. Gen. Physiology, 59:734- 766, 1972. [5] B. W. Knight. Dynamics of Encoding in Neuron Populations: Some General Mathematical Features. Neural Comput., 12:473- 518, 2000. [6] Z. Li. A neural model of contour integration in the primary visual cortex. Neural Comput. , 10(4):903- 940, 1998. [7] H. C. Tuckwell. Introduction to Theoretical Neurobiology. Cambridge University Press, Cambridge, 1988. [8] H. R. Wilson and J. D. Cowan. Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J., 12:1- 24, 1972.
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Improvisation and Learning Judy A. Franklin Computer Science Department Smith College Northampton, MA 01063 jfranklin@cs.smith.edu Abstract This article presents a 2-phase computational learning model and application. As a demonstration, a system has been built, called CHIME for Computer Human Interacting Musical Entity. In phase 1 of training, recurrentback-propagationtrains the machine to reproduce3 jazz melodies. The recurrent network is expanded and is further trained in phase 2 with a reinforcementlearning algorithm and a critique producedby a set of basic rules for jazz improvisation. After each phase CHIME can interactively improvise with a human in real time. 1 Foundations Jazz improvisation is the creation of a jazz melody in real time. Charlie Parker, Dizzy Gillespie, Miles Davis, John Coltrane, Charles Mingus, Thelonious Monk, and Sonny Rollins et al. were the founders of bebop and post bop jazz [9] where drummers, bassists, and pianists keep the beat and maintain harmonic structure. Other players improvise over this structure and even take turns improvising for 4 bars at a time. This is called trading fours. Meanwhile, artificial neural networks have been used in computer music [4, 12]. In particular, the work of (Todd [11]) is the basis for phase 1 of CHIME, a novice machine improvisor that learns to trade fours. Firstly, a recurrent network is trained with back-propagation to play three jazz melodies by Sonny Rollins [1], as described in Section 2. Phase 2 uses actor-critic reinforcement learning and is described in Section 3. This section is on jazz basics. 1.1 Basics: Chords, the ii-V-I Chord Progression and Scales The harmonic structure mentioned above is a series of chords that may be reprated and that are often grouped into standard subsequences. A chord is a group of notes played simultaneously. In the chromatic scale, C-Db-D-Eb-E-F-Gb-G-Ab-A-Bb-B-C, notes are separated by a half step. A flat (b) note is a half step below the original note; a sharp (#) is a half above. Two half steps are a whole step. Two whole steps are a major third. Three half steps are a minor third. A major triad (chord) is the first or tonic note, then the note a major third up, then the note a minor third up. When F is the tonic, F major triad is F-A-C. A minor triad (chord)is the tonic www.cs.smith.edu/˜jfrankli then a minor third, then a major third. F minor triad is F-Ab-C. The diminished triad is the tonic, then a minor third, then a minor third. F diminished triad is F-Ab-Cb. An augmented triad is the tonic, then a major third, then a major third. The F augmented triad is F-A-Db. A third added to the top of a triad forms a seventh chord. A major triad plus a major third is the major seventh chord. F-A-C-E is the F major seventh chord (Fmaj7). A minor triad plus a minor third is a minor seventh chord. For F it is F-Ab-C-Eb (Fm7). A major triad plus a minor third is a dominant seventh chord. For F it is F-A-C-Eb (F7). These three types of chords are used heavily in jazz harmony. Notice that each note in the chromatic scales can be the tonic note for any of these types of chords. A scale, a subset of the chromatic scale, is characterized by note intervals. Let W be a whole step and H be a half. The chromatic scale is HHHHHHHHHHHH. The major scale or ionian mode is WWHWWWH. F major scale is F-G-A-Bb-C-D-E-F. The notes in a scale are degrees; E is the seventh degree of F major. The first, third, fifth, and seventh notes of a major scale are the major seventh chord. The first, third, fifth, and seventh notes of other modes produce the minor seventh and dominant seventh chords. Roman numerals represent scale degrees and their seventh chords. Upper case implies major or dominant seventh and lower case implies minor seventh [9]. The major seventh chord starting at the scale tonic is the I (one) chord. G is the second degree of F major, and G-Bb-D-F is Gm7, the ii chord, with respect to F. The ii-V-I progression is prevalent in jazz [9], and for F it is Gm7-C7-Fmaj7. The minor ii-V-i progression is obtained using diminished and augmented triads, their seventh chords, and the aeolian mode. Seventh chords can be extended by adding major or minor thirds, e.g. Fmaj9, Fmaj11, Fmaj13, Gm9, Gm11, and Gm13. Any extension can be raised or lowered by 1 step [9] to obtain, e.g. Fmaj7#11, C7#9, C7b9, C7#11. Most jazz compositions are either the 12 bar blues or sectional forms (e.g. ABAB, ABAC, or AABA) [8]. The 3 Rollins songs are 12 bar blues. “Blue 7” has a simple blues form. In “Solid” and “Tenor Madness”, Rollins adds bebop variations to the blues form [1]. ii-V-I and VI-II-V-I progressions are added and G7+9 substitutes for the VI and F7+9 for the V (see section 1.2 below); the II-V in the last bar providesthe turnaroundto the I of the first bar to foster smooth repetition of the form. The result is at left and in Roman numeral notation at right: Bb7 Bb7 Bb7 Bb7 Eb7 Eb7 Bb7 G7+9 Cm7 F7 Bb7 G7+9 C7 F7+9 I I I I IV IV I VI ii V I VI II V 1.2 Scale Substitutions and Rules for Reinforcement Learning First note that the theory and rules derived in this subsection are used in Phase 2, to be described in Section 3. They are presented here since they derive from the jazz basics immediately preceding. One way a novice improvisor can play is to associate one scale with each chord and choose notes from that scale when the chord is presented in the musical score. Therefore, Rule 1 is that an improvisor may choose notes from a “standard” scale associated with a chord. Next, the 4th degree of the scale is often avoided on a major or dominant seventh chord (Rule 3), unless the player can resolve its dissonance. The major 7th is an avoid note on a dominant seventh chord (Rule 4) since a dominant seventh chord and its scale contain the flat 7th, not the major 7th. Rule 2 contains many notes that can be added. A brief rationale is given next. The C7 in Gm7-C7-Fmaj7 may be replaced by a C7#11, a C7+ chord, or a C7b9b5 or C7alt chord [9]. The scales for C7+ and C7#11 make available the raised fourth (flat 5), and flat 6 (flat 13) for improvising. The C7b9b5 and C7alt (C7+9) chords and their scales make available the flat9, raised 9, flat5 and raised 5 [1]. These substitutions provide the notes of Rule 2. These rules (used in phase 2) are stated below, using for reinforcement values very bad (-1.0), bad (-0.5), a little bad (-0.25), ok (0.25), good (0.5), and very good (1.0). The rules are discussed further in Section 4. The Rule Set: 1) Any note in the scale associated with the chord is ok (except as noted in rule 3). 2) On a dominant seventh, hip notes 9, flat9, #9, #11, 13 or flat13 are very good. One hip note 2 times in a row is a little bad. 2 hip notes more than 2 times in a row is a little bad. 3) If the chord is a dominant seventh chord, a natural 4th note is bad. 4) If the chord is a dominant seventh chord, a natural 7th is very bad. 5) A rest is good unless it is held for more than 2 16th notes and then it is very bad. 6) Any note played longer than 1 beat (4 16th notes) is very bad. 7) If two consecutive notes match the human’s, that is good. 2 CHIME Phase 1 In Phase 1, supervised learning is used to train a recurrent network to reproduce the three Sonny Rollins melodies. 2.1 Network Details and Training The recurrent network’s output units are linear. The hidden units are nonlinear (logistic function). Todd [11] used a Jordan recurrent network [6] for classical melody learning and generation. In CHIME, a Jordan net is also used, with the addition of the chord as input (Figure 1. 24 of the 26 outputs are notes (2 chromatic octaves), the 25th is a rest, and the 26th indicates a new note. The output with the highest value above a threshold is the next note, including the rest output. The new note output indicates if this is a new note, or if it is the same note being held for another time step ( note resolution). The 12 chord inputs (12 notes in a chromatic scale), are 1 or 0. A chord is represented as its first, third, fifth, and seventh notes and it “wraps around” within the 12 inputs. E.g., the Fm7 chord F-Ab-C-Eb is represented as C, Eb, F, Ab or 100101001000. One plan input per song enables distinguishingbetween songs. The 26 context inputs use eligibility traces, giving the hidden units a decaying history of notes played. CHIME (as did Todd) uses teacher forcing [13], wherein the target outputs for the previous step are used as inputs (so erroneous outputs are not used as inputs). Todd used from 8 to 15 hidden units; CHIME uses 50. The learning rate is 0.075 (Todd used 0.05). The eligibility rate is 0.9 (Todd used 0.8). Differences in values perhaps reflect contrasting styles of the songs and available computing power. Todd used 15 output units and assumed a rest when all note units are “turned off.” CHIME uses 24 output note units (2 octaves). Long rests in the Rollins tunes require a dedicated output unit for a rest. Without it, the note outputs learned to turn off all the time. Below are results of four representative experiments. In all experiments, 15,000 presentations of the songs were made. Each song has 192 16th note events. All songs are played at a fixed tempo. Weights are initialized to small random values. The squared error is the average squared error over one complete presentation of the song. “Finessing” the network may improve these values. The songs are easily recognized however, and an exact match could impair the network’s ability to improvise. Figure 2 shows the results for “Solid.” Experiment 1. Song: Blue Seven. Squared error starts at 185, decreases to 2.67. Experiment 2. Song: Tenor Madness. Squared error starts at 218, decreases to 1.05. Experiment 3. Song: Solid. Squared error starts at 184, decreases to 3.77. Experiment 4. Song: All three songs: Squared error starts at 185, decreases to 36. Figure 1: Jordan recurrent net with addition of chord input 2.2 Phase 1 Human Computer Interaction in Real Time In trading fours with the trained network, human note events are brought in via the MIDI interface [7]. Four bars of human notes are recorded then given, one note event at a time to the context inputs (replacing the recurrent inputs). The plan inputs are all 1. The chord inputs follow the “Solid” form. The machine generates its four bars and they are played in real time. Then the human plays again, etc. An accompaniment (drums, bass, and piano), produced by Band-in-a-Box software (PG Music), keeps the beat and provides chords for the human. Figure 3 shows an interaction. The machine’s improvisations are in the second and fourth lines. In bar 5 the flat 9 of the Eb7 appears; the E. This note is used on the Eb7 and Bb7 chords by Rollins in “Blue 7”, as a “passing tone.” D is played in bar 5 on the Eb7. D is the natural 7 over Eb7 (with its flat 7) but is a note that Rollins uses heavily in all three songs, and once over the Eb7. It may be a response to the rest and the Bb played by the human in bar 1. D follows both a rest and a Bb in many places in “Tenor Madness” and “Solid.” In bar 6, the long G and the Ab (the third then fourth of Eb7) figure prominently in “Solid.” At the beginning of bar 7 is the 2-note sequence Ab-E that appears in exactly the same place in the song “Blue 7.” The focus of bars 7 and 8 is jumping between the 3rd and 4th of Bb7. At the end of bar 8 the machine plays the flat 9 (Ab) then the flat 3 (Bb), of G7+9. In bars 13-16 the tones are longer, as are the human’s in bars 9-12. The tones are the 5th, the root, the 3rd, the root, the flat 7, the 3rd, the 7th, and the raised fourth. Except for the last 2, these are chord tones. 3 CHIME Phase 2 In Phase 2, the network is expanded and trained by reinforcement learning to improvise according to the rules of Section 1.2 and using its knowledge of the Sonny Rollins songs. 3.1 The Expanded Network Figure 4 shows the phase 2 network. The same inputs plus 26 human inputs brings the total to 68. The weights obtained in phase 1 initialize this network. The plan and chord weights Figure 2: At left “Solid” played by a human; at right the song reproduced by the ANN. are the same. The weights connecting context units to the hidden layer are halved. The same weights, halved, connect the 26 human inputs to the hidden layer. Each output unit gets the 100 hidden units’ outputs as input. The original 50 weights are halved and used as initial values of the two sets of 50 hidden unit weights to the output unit. 3.2 SSR and Critic Algorithms Using actor-critic reinforcement learning ([2, 10, 13]), the actor chooses the next note to play. The critic receives a “raw” reinforcement signal from the critique made by the rules of Section 1.2. For output j, the SSR (actor) computes mean
. A Gaussian distribution with mean and standard deviation chooses the output . is generated, the critic modifies and produces . is further modified by a self-scaling algorithm that tracks, via moving average, the maximum and minimum reinforcement and uses them to scale the signal to produce ! #"$ %'&(*)+", - %/.102"$ 3) . %2&4 !56 )78%'&49 %'&4 -5:)3;: < %/.10= -56 )= %2.>0+9 %/.10= !5:)?;: 3< %2&4 !56 )7A@B %2&4 !56 )6A ",@C)D %/.10= -56 )=E@B %2.10= !5F6 )6A ",@C)D The goal is to make small gains in reinforcement more noticeable and to scale the values between -1 and 1. If G %'&4 , then "A ! %2.>0H" %2&4*)I "AJ and if 8 %2.10 , then 8 ! %2.>02"I %'&4K)+" J7" . If JMLNLHO , the extremes of -1 and 1 are approached. The weight and standard deviation updates use : P !5F6 )=Q 4 !5:)F6SRT !M"UK )DVKKWXVK -5F6 )=8%'&49 CY[Z3\;]%/.10+9_^ !5:)6 "I^)`a ! %'&4b"U %2.10F)3;cBY < < ;cOdLH^IL If the difference between the max and min reinforcement stays large, over time will increase (to a max of Y[Z3\ ) and allow more exploration. When rmax-rmin is small, over time e will shrink (to a min of Y ). The actor’s hidden units are updated using backpropagation as before, using f`T ! "g ) as “error.” See [3, 5, 13] for more details on the SSR algorithm and its precursors. The critic inputs are the outputs of the hidden layer of the actor network; it “piggy-backs”on the actor and uses its learned features (see Figure 5). This also alleviates the computational burden so it can run in real time. There are delays in reward, e.g. in that a note played too many times in a row may result in punishment, and if 2 notes in a row coincide with the human’s it is rewarded. If !;:5:) is the prediction of future reward [10] for state x at time t, Figure 3: Phase1 trading 4 bars: 4 human, 4 machine, 4 human, 4 machine -56 )= e !5:)F6,^ ` ! -5F6 )3;:5:) "2 - -5:)3;:5:) for O LH^ L . The critic is a linear function of its inputs: ! -5:)3;]5:)T Y
!5:)D !5:) The weights are updated incrementally using the value of : -5F6 )=Q -5:)F6 V4 *WVK is in effect an error signal, a difference between consecutive predicted rewards [10]. The critic also uses eligibility traces of the inputs, so !5:) is actually !5:)a !5:)76 \ !5=" ) where \ -5:) !5:) . While this is all experimental, initial results show that the system with both the self-scalar and the critic performs better than with just one or without either one. A more systematic study is planned. 4 Results and Comments Recall the rules of Section 1.2. Rules 1-4 are based on discussions with John Payne, a professional jazz musician and instructor of 25 years1. The rules by no means encompass all of jazz theory or practice but are a starting point. The notes in rule 2 were cast as good in a “hip” situation. The notion of hip requires human sophistication so for now these notes are reinforced if played sporadically on the dominant seventh. Rule 5 was added to discourage not playing any notes. Rules 5 and 6 focus on not allowing an output of one note for too long. Each chord is assigned a scale for rule 1. C is limited, O O $L egL O , providing stability, and deliberate action uncertainty so different notes are played, for the same network state. Generally the goal of reinforcement learning is to find the best action for a given state, with uncertainty used for further exploration. Here, reinforcementlearning finds the best set of actions for a given state. In a typical example using the phase 1 network prior to phase 2 improvement, the average reinforcement value according to the rule set is -.37 (on a scale from -1 to 1). After Phase 2, the average reinforcement value is .28 after 30-100 off-line presentations of the human solo of 1800 note events. Figure 6 shows 12 bars of a human solo and 12 bars of a machine solo. The note durations 1The rules are not meant to represent John Payne Figure 4: Recurrent reinforcement learning network with human input used in phase 2. Figure 5: Phase 2 network with critic “piggy-backing” on hidden layer. are shortened, reflecting the rules to preventsettling onto one note. The machineplays chord tones, such as Bb and D in bars 1 and 2. The high G is the 13 of Bb7, a hip note. In bars 3 and 4 it plays C sharp, a hip note (the #9 of Bb7) and high G. These notes are played in bars 9 and 11 on Bb7. In bars 5 and 6 the 9 and 13 (F and C) are played on Eb7. The natural 7 (D) reflects its heavy use in Rollins’ melodies. Hip notes show up in bar 9 on Cm7: the 13 (G) and the 9 (D). In bar 11 G is played on G7+9 as is the hip flat 9 (the Ab). In bar 12, the Eb (flat 7 chord tone) is played on the F7+9. In bars 2, 4, 7, 9, and 10 the machine starts at the G at the top of the staff and descends through several chord tones, producing a recurring motif, an artifact of a “good” jazz solo. The phase 2 network has been used to interact with a human in real time while still learning. It keeps its recurrencesince the human has a separate set of inputs. A limitation to be addressed for CHIME is to move beyond one chord at a time. To achieve this, it must use more context, over more time. There are plenty of improvisation rules for chordprogressions[8]. Because CHIME employsreinforcementlearning, it has a stochastic element that allows it to play “outside the chord changes.” A research topic is to understand how to enable it to do this more pointedly. Figure 6: At left, 12 bars of human solo. At right, 12 bars the machine plays in response. References [1] J. Aebersold. You can play Sonny Rollins. A New Approach to Jazz Improvisation Vol 8. Jamey Aebersold, New Albany, IND., 1976. [2] A. G. Barto, R. S. Sutton, and C. W. Anderson. Neuronlike adaptive element that can solve difficult learning control problems. IEEE Transactions on Systems, Man, and Cybernetics, SMC13:834–846, 1983. [3] H. Benbrahim and J. Franklin. Biped walking using reinforcement learning. Robotics and Autonomous Systems, 22:283–302, 1997. [4] N. Griffith and P. Todd. Musical Networks: Parallel Distributed Perception and Performance. MIT Press, Cambridge MA, 1999. [5] V. Gullapalli, J. Franklin, and H. Benbrahim. Acquiring robot skills via reinforcement learning. IEEE Control Systems Magazine, 1994. [6] M. Jordan. Attractor dynamics and parallelism in a connectionist sequential machine. In Proceedings of the Eighth Annual Conference of the Cognitive Science Society, 1986. [7] P. Messick. Maximum MIDI. Manning Publications, Greenwich, CT, 1988. [8] S. Reeves. Creative Jazz Improvisation. 2nd Ed. Prentice Hall, Upper Saddle River NJ, 1995. [9] M. A. Sabatella. Whole Approach to Jazz Improvisation. A.D.G. Productions, Lawndale CA, 1996. [10] R. Sutton. Learning to predict by the methods of temporal differences. Machine Learning, 3:9– 44, 1988. [11] P. M. Todd. A connectionist approach to algorithmic composition. In P. M. Todd and e. D. Loy, editors, Music and Connectionism. MIT Press, Cambridge MA, 1991. [12] P. M. Todd and e. D. Loy. Music and Connectionism. MIT Press, Cambridge, MA, 1991. [13] R. J. Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 8:229–256, 1992.
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Geometrical Singularities in the Neuromanifold of Multilayer Perceptrons Shun-ichi Amari, Hyeyoung Park, and Tomoko Ozeki Brain Science Institute, RIKEN Hirosawa 2-1, Wako, Saitama, 351-0198, Japan {amari, hypark, tomoko} @brain.riken.go.jp Abstract Singularities are ubiquitous in the parameter space of hierarchical models such as multilayer perceptrons. At singularities, the Fisher information matrix degenerates, and the Cramer-Rao paradigm does no more hold, implying that the classical model selection theory such as AIC and MDL cannot be applied. It is important to study the relation between the generalization error and the training error at singularities. The present paper demonstrates a method of analyzing these errors both for the maximum likelihood estimator and the Bayesian predictive distribution in terms of Gaussian random fields, by using simple models. 1 Introduction A neural network is specified by a number of parameters which are synaptic weights and biases. Learning takes place by modifying these parameters from observed input-output examples. Let us denote these parameters by a vector () = (01 , .. . , On). Then, a network is represented by a point in the parameter space S, where () plays the role of a coordinate system. The parameter space S is called a neuromanifold. A learning process is represented by a trajectory in the neuromanifold. The dynamical behavior of learning is known to be very slow, because of the plateau phenomenon. The statistical physical method [1] has made it clear that plateaus are ubiquitous in a large-scale perceptron. In order to improve the dynamics of learning, the natural gradient learning method has been introduced by taking the Riemannian geometrical structure of the neuromanifold into account [2, 3]. Its adaptive version, where the inverse of the Fisher information matrix is estimated adaptively, is shown to have excellent behaviors by computer simulations [4, 5]. Because of the symmetry in the architecture of the multilayer perceptrons, the parameter space of the MLP admits an equivalence relation [6, 7]. The residue class divided by the equivalence relation gives rise to singularities in the neuromanifold, and plateaus exist at such singularities [8]. The Fisher information matrix becomes singular at singularities, so that the neuromanifold is strongly curved like the spacetime including black holes. In the neighborhood of singularities, the Fisher-Cramer-Rao paradigm does not hold, and the estimator is no more subject to the Gaussian distribution even asymptotically. This is essential in neural learning and model selection. The AlC and MDL criteria of model selection use the Gaussian paradigm, so that it is not appropriate. The problem was first pointed out by Hagiwara et al. [9]. Watanabe [10] applied algebraic geometry to elucidate the behavior of the Bayesian predictive estimator in MLP, showing sharp difference in regular cases and singular cases. Fukumizu [11] gives a general analysis of the maximum likelihood estimators in singular statistical models including the multilayer perceptrons. The present paper is a first step to elucidate effects of singularities in the neuromanifold of multilayer perceptrons. We use a simple cone model to elucidate how different the behaviors of the maximum likelihood estimator and the Bayes predictive distribution are from the regular case. To this end, we introduce the Gaussian random field [11, 12, 13], and analyze the generalization error and training error for both the mle (maximum likelihood estimator) and the Bayes estimator. 2 Topology of neuromanifold Let us consider MLP with h hidden units and one output unit, h Y = L Vi<{J (Wi· x) + n. (1) i = l where y is output, x is input and n is Gaussian noise. Let us summarize all the parameters in a single parameter vector () = (Wl , ···, Wh; Vl , ···, Vh) and write h f(x; ()) = L Vi<{J (Wi· x). (2) i=l Then, () is a coordinate system of the neuromanifold. Because of the noise, the input-output relation is stochastic, given by the conditional probability distribution 1 {I 2} p(ylx,()) = J2 exp -2(y-f(x;())) , (3) where we normalized the scale of noise equal to 1. Each point in the neuromanifold represents a neural network or its probability distribution. It is known that the behavior of MLP is invariant under 1) permutations of hidden units, and 2) sign change of both Wi and Vi at the same time. Two networks are equivalent when they are mapped by any of the above operations which form a group. Hence, it is natural to treat the residual space SI ::::J, where ::::J is the equivalence relation. There are some points which are invariant under a some nontrivial isotropy subgroup, on which singularities occurs. When Vi = 0, vi<{J (Wi· x) = 0 so that all the points on the sub manifold Vi = 0 are equivalent whatever Wi is. We do not need this hidden unit. Hence, in M = SI ::::J, all of these points are reduced to one and the same point. When Wi = Wj hold, these two units may be merged into one, and when Vi +Vj is the same, the two points are equivalent even when they differ in Vi - Vj. Hence, the dimension reduction takes place in the subspace satisfying Wi = Wj. Such singularities occur on the critical submanifolds of the two types (4) 3 Simple toy models Given training data, the parameters of the neural network are estimated or trained by learning. It is important to elucidate the effects of singularities on learning or estimation. We use simple toy models to attack this problem. One is a very simple multilayer percept ron having only one hidden unit. The other is a simple cone model: Let x be Gaussian random variable x E R d+2 , with mean p, and identity covariance matrix I , (5) and let 5 = {p,Ip, E Rd+2 } be the parameter space. The cone model M is a subset of 5, embedded as M : p, (6) where c is a constant, IIa2 11 = 1, W E 5 d and 5 d is a d-dimensional unit sphere. When d = 1, 51 is a circle so that W is replaced by angle B, and we have p, = ~ ccos B . ( 1 ) VI + c2 csinB (7) See Figure 1. The M is a cone, having (~, w) as coordinates, where the apex ~ = 0 is the singular point. , , Figure 1: One-dimensional cone model The input-output relation of a simple multilayer perceptron is given by y = v<p(w . x) + n (8) When v = 0, the behavior is the same whatever w is. Let us put w = (3w, where (3 = Iwl and W E 5 d , and ~ = vlwl, 'l/J(x;(3,w) = <p{(3(w· x)} /(3. We then have y = ~'l/J(x;(3,w) + n (9) which shows the cone structure with apex at ~ = O. In this paper, we assume that (3 is knwon and does not need to be estimateed. 4 Asymptotic statistical inference: generalization error and training error Let D = {Xl,···, XT} be T independent observations from the true distribution Po(x) which is specified by ~ = 0, that is, at the singular point. In the case of neural networks, the training set D is T input-output pairs (Xt, Yt), from the conditional probability distributions p(Ylx;~, w) and the true one is at ~ = O. The discussions go in parallel, so that we show here only the cone model. We study the characteristics of both the mle and the Bayesian predictive estimator. Let p(x) be the estimated distribution from data D . In the case of mle, it is given by p(x; 0) where 0 is the mle given by the maximizer of the log likelihood. For the Bayes estimator, it is given by the Bayes predictive distribution p(xID). We evaluate the estimator by the generalization error defined by the KL-divergence from Po(x) to p(x), Eg en = ED [K[po : pll, K[Po: p] = Epo [log ~(~i] . (10) Similarly, the training error is defined by using the empirical expectation, (11) In order to evaluate the estimator p, one uses E gen , but it is not computable. Instead, one uses the Etrain which is computable. Hence, it is important to see the difference between Egen and Etrain- This is used as a principle of model selection. When the statistical model M is regular, or the true distribution Po (x) is at a regular point, the mle-based p(x, 0) and the Bayes predictive distribution are asymptotically equivalent, and are Fisher efficient under reasonable regularity conditions, d Egen ~ 2T ' d Eg en ~ Etrain + T' (12) where d is the dimension number of parameter vector (j. All of these good relations do not hold in the singular case. The mle is no more asymptotically Gaussian, the mle and the Bayes estimators have different asymptotic characteristics, although liT consistency is guaranteed. The relation between the generalization and training error is different, so that we need a different model selection criterion to determine the number of hidden units. 5 Gaussian random fields and mle Here, we introduce the Gaussian random field [11, 12, 13] in the case of the cone model. The log likelihood of data D is written as 1 T L(D,~,w) = -"2l: Ilxt ~a(w)112. (13) t=l Following Hartigan [13] (see also [11] for details), we first fix wand search for the ~ that maximizes L. This is easy since L is a quadratic function of ( The maximum t is given by (14) Y(w) = a(w) · X, (15) By the central limit theorem, Y (w) = a( w) . x is a Gaussian random field defined on Sd = {w}. By substituting t(w) in (14) the log likelihood function becomes T , I", 2 1 2 L(w) = -2 ~ IIXtl1 + 2Y (w). t=l Therefore, the mle w is given by the maximizer of L(w), w = argmaxwy2(w). Theorem 1. In the case of the cone model, the mle satisfies Egen 2~ED h~p y 2(w)] , Etrain = - 2~ED h:p y 2 (w)] . Corollary 1. When d is large, the mle satisfies Egen c2d :::::: 2T(1 + c2 ) ' Etrain c2d :::::: 2T(1 + c2 )· (16) (17) (18) (19) (20) It should be remarked that the generalization and training errors depend on the shape parameter c as well as the dimension number d. 6 Bayesian predictive distribution The Bayes paradigm uses the posterior probability of the parameters based on the set of observations D. The posterior probability density is written as, T p(~,wID) = c(D)1f(~,w) rrp(xtl~ , w) , (21) t = l where c(D) is the normalization factor depending only on data D, 1f(~ , w) is a prior distribution on the parameter space. The Bayesian predictive distribution p(xID) is obtained by averaging p(xl~, w) with respect to the posterior distribution p(~, wiD), and can be written as p(xID) = J p(xl~ , w)p(~ , wID)d~dw. (22) The Bayes predictive distribution depends on the prior distribution 1f( ~, w) . As long as the prior is a smooth function, the first order asymptotic properties are the same for the mle and Bayes estimators in the regular case. However, at singularities, the situation is different. Here, we assume a uniform prior for w. For C we assume two different priors, the uniform prior and the Jeffreys prior. We show here a sketch of calculations in the case of Jeffreys prior, 7f(~,w) ex 1~ld . By introducing Id(u) = ~ J Iz + uldexp {_~Z2} dz, (23) after lengthy calculations, we obtain (24) where XT+! = ~(x + VTx) , Pd(x) = J Id(Y(w)) exp {~Y2(W)} dw. (25) Here Y(w) has the same form defined in (15), and Pd(x) is the function of the sufficient statistics x. By using the Edgeworth expansion, we have p(xID) (26) where \7 is the gradient and H2 (x) is the Hermite polynomial. We thus have the following theorem. Theorem 2. Under the Jeffreys prior for ~, the generalization error and the training error of the predictive distribution are given by Egen Etrain (27) (28) Under the uniform prior, the above results hold by replacing Id(Y) in the definition of Pd(X) by 1. In addition, From (24), we can easily obtain Egen = (d + 1)/2T for the Jeffreys prior, and Egen = 1/2T for the uniform prior. The theorem shows rather surprising results: Under the uniform prior, the generalization error is constant and does not depend on d. This is completely different from the regular case. However, this striking result is given rise to by the uniform prior on f The uniform prior puts strong emphasis on the singularity, showing that one should be very careful for choosing a prior when the model includes singularities. In the case of Jeffreys prior, the generalization error increases in proportion to d, which is the same result as the regular case. In addition, the symmetric duality between Egen and E train does not hold for both of the uniform prior and the Jeffreys prior. 7 Gaussian random field of MLP In the case of MLP with one hidden unit, the log likelihood is written as 1 T 2 L(D;~ , w)=-22:{Yt-~CPi9(w.Xt)} . (29) t=l Let us define a Gaussian random field depending on D and w, 1 T Y(w) = 1m LYt<P,6 (w· Xt) '"" N(O,A(w,w')) yT t = l where A(w, w') = Ex [<p,6(w . x)<p,6(w' . x)]. Theorem 3. For the mle, we have Egen Etrain where A(w) = A(w, w). In order to analyze the Bayes predictive distribution, we define 1 ( Y (W)) { 1 y2 (w) } Sd(D,w) = d+1 Id JA(W) exp --A() . J A(w) A(w) 2 w (30) (31) (32) (33) (34) We then have the Edgeworth expansion of the predictive distribution of the form, _1_ exp {_ y2 } {I + -'!L EW[V'Sd(D, w)<p,6(w . x)] p(Ylx, D) ~ f(L 1m [( )] (35) y27f 2 yT Ew Sd D,w ~ EW[V'V'Sd(D,w)A(w)] H ( )} + 2T EW[Sd(D,w)] 2 Y , where V' is the gradient with respect to Y(w). We thus have the following theorem. Theorem 4. Under the Jeffreys prior for ~ , the generalization error and the training error of the predictive distribution are given by Egen Etrain = (36) Under the uniform prior, the above results hold by redefining (37) We can also obtain Egen = (d + 1)/2T for the Jeffreys prior, and Egen = 1/2T for the uniform prior. There is a nice correspondence between the cone model and MLP. However, there is no sufficient statistics in the MLP case, while all the data are summarized in the sufficient statistics x in the cone model. 8 Conclusions and discussions We have analyzed the asymptotic behaviors of the MLE and Bayes estimators in terms of the generalization error and the training error by using simple statistical models (cone model and simple MLP), when the true parameter is at singularity. Since the classic paradigm of statistical inference based on the Cramer-Rao theorem does not hold in such a singular case, we need a new theory. The Gaussian random field has played a fundamental role. We can compare the estimation accuracy of the maximum likelihood estimator and the Bayesian predictive distribution from the results of analysis. Under the proposed framework, the various estimation methods can be studied and compared to each other. References [1] Saad, D. and Solla, S. A. (1995). Physical Review E, 52,4225-4243. [2] Amari, S. (1998). Neural Computation, 10,251-276. [3] Amari S. and Nagaoka, H. (2000). Methods of Information Geometry, AMS. [4] Amari, S., Park, H., and Fukumizu, F. (2000). Neural Computation, 12, 13991409. [5] Park, H., Amari, S. and Fukumizu, F. (2000). Neural Networks, 13, 755-764. [6] Chen, A. M., Lu, H., and Hecht-Nielsen, R. (1993). Neural Computations, 5, 910-927. [7] Riigger, S. M. and Ossen, A. (1997). Neural Processing Letters, 5, 63-72. [8] Fukumizu, K. and Amari, S. (2000) Neural Networks, 13 317-327. [9] Hagiwara, K., Hayasaka, K. , Toda, N., Usui, S., and Kuno, K. (2001). Neural Networks, 14 1419-1430. [10] Watanabe, S. (2001). Neural Computation, 13, 899-933. [11] Fukumizu, K. (2001). Research Memorandum, 780, lnst. of Statistical Mathematics. [12] Dacunha-Castelle, D. and Gassiat, E. (1997). Probability and Statistics, 1,285317. [13] Hartigan, J. A. (1985). Proceedings of Berkeley Conference in Honor of J. Neyman and J. Kiefer, 2, 807-810.
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A Sequence Kernel and its Application to Speaker Recognition William M. Campbell Motorola Human Interface Lab 7700 S. River Parkway Tempe, AZ 85284 Bill.Campbell@motorola.com Abstract A novel approach for comparing sequences of observations using an explicit-expansion kernel is demonstrated. The kernel is derived using the assumption of the independence of the sequence of observations and a mean-squared error training criterion. The use of an explicit expansion kernel reduces classifier model size and computation dramatically, resulting in model sizes and computation one-hundred times smaller in our application. The explicit expansion also preserves the computational advantages of an earlier architecture based on mean-squared error training. Training using standard support vector machine methodology gives accuracy that significantly exceeds the performance of state-of-the-art mean-squared error training for a speaker recognition task. 1 Introduction Comparison of sequences of observations is a natural and necessary operation in speech applications. Several recent approaches using support vector machines (SVM’s) have been proposed in the literature. The first set of approaches attempts to model emission probabilities for hidden Markov models [1, 2]. This approach has been moderately successful in reducing error rates, but suffers from several problems. First, large training sets result in long training times for support vector methods. Second, the emission probabilities must be approximated [3], since the output of the support vector machine is not a probability. A more recent method for comparing sequences is based on the Fisher kernel proposed by Jaakkola and Haussler [4]. This approach has been explored for speech recognition in [5]. The application to speaker recognition is detailed in [6]. We propose an alternative kernel based upon polynomial classifiers and the associated mean-squared error (MSE) training criterion [7]. The advantage of this kernel is that it preserves the structure of the classifier in [7] which is both computationally and memory efficient. We consider the application of text-independent speaker recognition; i.e., determining or verifying the identity of an individual through voice characteristics. Text-independent recognition implies that knowledge of the text of the speech data is not used. Traditional methods for text-independent speaker recognition are vector quantization [8], Gaussian mixture models [9], and artificial neural networks [8]. A state-of-the-art approach based on polynomial classifiers was presented in [7]. The polynomial approach has several advantages over traditional methods–1) it is extremely computationally-efficient for identification, 2) the classifier is discriminative which eliminates the need for a background or cohort model [10], and 3) the method generates small classifier models. In Section 2, we describe polynomial classifiers and the associated scoring process. In Section 3, we review the process for mean-squared error training. Section 4 introduces the new kernel. Section 5 compares the new kernel approach to the standard mean-squared error training approach. 2 Polynomial classifiers for sequence data We start by considering the problem of speaker verification–a two-class problem. In this case, the goal is to determine the correctness of an identity claim (e.g., a user id was entered in the system) from a voice input. If is the class, then the decision to be made is if the claim is valid, , or if an impostor is trying to break into the system,
. We motivate the classification process from a probabilistic viewpoint. For the verification application, a decision is made from a sequence of observations extracted from the speech input. We decide based on the output of a discriminant function using a polynomial classifier. A polynomial classifier of the form "!$# where is the vector of classifier parameters (model) and # is an expansion of the input space into the vector of monomials of degree % or less is used. For example, if % & and (' ) )+*-, ! , then # /. 01) ) * ) * ) ) * ) * * )+2 ) * ) * ) ) * * )32 *4 ! (1) Note that we do not use a nonlinear activation function as is common in higher-order neural networks; this allows us to find a closed form solution for training. Also, note that we use a bold # to avoid confusion with probabilities. If the polynomial classifier is trained with a mean-squared error training criterion and target values of 0 for (5 and 6 for 7(
, then will approximate the a posteriori probability 8 9:5; [11]. We can then find the probability of the entire sequence, 8< - ; =75 , as follows. Assuming independence of the observations [12] gives 8 > ; ? @BA 8C @ ; ? @BA 8C D; @ 8< @ 8 (2) For the purposes of classification, we can discard 8< @ . We take the logarithm of both sides to get the discriminant function EGF ; H @ A IBJLK M 8 D; @ 8 (N (3) where we have used the shorthand to denote the sequence C- . We use two terms of the Taylor series, IBJLK ) PO )"QR0 , to approximate the discriminant function and also normalize by the number of frames to obtain the final discriminant function E < ; 0 S H @ A 8C D; @ 8 (4) Note that we have discarded the QT0 in this discriminant function since this will not affect the classification decision. The key reason for using the Taylor approximation is that it reduces computation without significantly affecting classifier accuracy. Now assume we have a polynomial function < O 8 7 5; ; we call the vector the speaker model. Substituting in the polynomial function < gives E < ; =R5 0 S H @ A !$# @ 8C R5 0 S 8 =R5 ! H @ A # @ 0 8 =R5 ! # (5) where we have defined the mapping # as 0 S H @ A # @ (6) We summarize the scoring method. For a sequence of input vectors and a speaker model, , we construct # using (6). We then score using the speaker model, J
! # . Since we are performing verification, if J
is above a threshold then we declare the identity claim valid; otherwise, the claim is rejected as an impostor attempt. More details on this probabilistic scoring method can be found in [13]. Extending the sequence scoring framework to the case of identification (i.e., identifying the speaker from a list of speakers by voice) is straightforward. In this case, we construct speaker models for each speaker @ and then choose the speaker which maximizes ! @ # (assuming equal prior probability of each speaker). Note that identification has low computational complexity, since we must only compute one inner product to determine the speaker’s score. 3 Mean-squared error training We next review how to train the polynomial classifier to approximate the probability 8 D; ; this process will help us set notation for the following sections. Let be the desired speaker model and the ideal output; i.e., 5 0 and B 6 . The resulting problem is K
! # Q * (7) where denotes expectation. This criterion can be approximated using the training set as ! K
" #%$'&(*) H @ A ,+ + ! # < @ Q0 + + *.$'/ 0 ( H @BA 1+ + ! # 32 @ + + *54 (8) Here, the speaker’s training data is $ &(*) , and the anti-speaker data is 2 -62 $ / 0 ( . (Anti-speakers are designed to have the same statistical characteristics as the impostor set.) The training method can be written in matrix form. First, define 7983:<; as the matrix whose rows are the polynomial expansion of the speaker’s data; i.e., 7=83:<; ?>@ @A # ! # * ! ... # $B&(*) ! CED D F (9) Define a similar matrix for the impostor data, 7 : . Define 7 # 7=8 : ; 7 : 4 (10) The problem (8) then becomes D K 7 Q * (11) where is the vector consisting of 8 : ; ones followed by : zeros (i.e., the ideal output). The problem (11) can be solved using the method of normal equations, 7 ! 7 7 ! (12) We rearrange (12) to 7 ! 79 7 ! 8 : ; (13) where is the vector of all ones. If we define 7 ! 7 and solve for , then (13) becomes 7 ! 83:<; (14) 4 The naive a posteriori sequence kernel We can now combine the methods from Sections 2 and 3 to obtain a novel sequence comparison kernel in a straightforward manner. Combine the speaker model from (14) with the scoring equation from (5) to obtain the classifier score J
0 8C = 5 # ! 0 8 5 # ! 7 ! 8 : ; (15) Now 8C =75 83: ;
5 : 83:<; O 8 : ;
: (because of the large anti-speaker population), so that (15) becomes J
# ! # ! (16) where # is 0
83: ; *7 ! 83: ; (note that this exactly the same as mapping the training data using (6)), and is 0
:> . The scoring method in (16) is the basis of our sequence kernel. Given two sequences of speech feature vectors, and 2 , we compare them by mapping # and 2 # and then computing :<8- 2 # ! # (17) We call :<8 the naive a posteriori sequence kernel since scoring assumes independence of observations and training approximates the a posteriori probabilities. The value : 8- 62 can be interpreted as scoring using a polynomial classifier on the sequence 2 , see (5), with the MSE model trained from feature vectors (or vice-versa because of symmetry). Several observations should be made about the NAPS kernel. First, scoring complexity can be reduced dramatically in training by the following trick. We factor ! ! using the Cholesky decomposition. Then :<8- 2" # ! # . I.e., if we transform all the sequence data by ## before training, the sequence kernel is a simple inner product. For our application in Section 5, this reduces training time from $ hours per speaker down to %G6 seconds on a Sun Ultra &G6 , & &L6 MHz. Second, since the NAPS kernel explicitly performs the expansion to “feature space”, we can simplify the output of the support vector machine. Suppose ' # is the (soft) output of the SVM, ' # ( H @BA ) @ @ :<8- # @ # -+* (18) We can simplify this to ' # ( H @ A ) @ @ # @ - ! # (19) where ' * 6 6 , ! . That is, once we train the support vector machine, we can collapse all the support vectors down into a single model , where is the quantity in parenthesis in (19). Third, although the NAPS kernel is reminiscent of the Mahalanobis distance, it is distinct. No assumption of equal covariance matrices for different classes (speakers) is made for the new kernel–the kernel covariance matrix is a mixture of the individual class covariances. Also, the kernel is not a distance measure–no subtraction of means occurs as in the Mahalanobis distance. 5 Results 5.1 Setup The NAPS kernel was tested on the standard speaker recognition database YOHO [14] collected from 138 speakers. Utterances in the database consist of combination lock phrases of fixed length; e.g., “23-45-56.” Enrollment and verification session were recorded at distinct times. (Enrollment is the process of collecting data for training and generating a speaker model. Verification is the process of testing the system; i.e., the user makes an identity claim and then this hypothesis is verified.) For each speaker, enrollment consisted of four sessions each containing twenty-four utterances. Verification consisted of ten separate sessions with four utterances per session (again per speaker). Thus, there are 40 tests of the speaker’s identity and 40*137=5480 possible impostor attempts on a speaker. For clarity, we emphasize that enrollment and verification session data is completely separate. To extract features for each of the utterances, we used standard speech processing. Each utterance was broken up into frames of & 6 ms each with a frame rate of 0 6L6 frames/sec. The mean was removed from each frame, and the frame was preemphasized with the filter 0 Q 65 . A Hamming window was applied and then 0 linear prediction coefficients were found. The resulting coefficients were transformed to 0 cepstral coefficients. Endpointing was performed to eliminate non-speech frames. This typically resulted in approximately 6G6 observations per utterance. For verification, we measure performance in terms of the pooled and average equal error rates (EER). The average EER is found by averaging the individual EER for each speaker. The individual EER is the threshold at which the false accept rate (FAR) equals the false reject rate (FRR). The pooled EER is found by setting a constant threshold across the entire population. When the FAR equals the FRR for the entire population this is termed the pooled EER. For identification, the misclassification error rate is used. To eliminate bias in verification, we trained the first & speakers against the first & and the second & against the second & (as in [7]). We then performed verification using the second & as impostors to the first & speakers models and vice versa. This insures that the impostors are unknown. For identification, we trained all 0 & speakers against each other. 5.2 Experiments We trained support vector machines for each speaker using the software tool SVMTorch [15] and the NAPS kernel (17). The 0 cepstral features were mapped to a dimension % $ $ vector using a & rd degree polynomial classifier. Single utterances (i.e., “2345-56”) were converted to single vectors using the mapping (6) and then transformed with the Cholesky factor to reduce computation. We cross-validated using the first & enrollment sessions as training and the % th enrollment session as a test to determine the best tradeoff between margin and error; the best performing value of 65 0 was used with the final SVMTorch training. Using the identical set of features and the same methodology, classifier models were also trained using the mean-squared error criterion using the method in [7]. For final testing, all % enrollment session were used for training, and all verification sessions were used for testing. Results for verification and identification are shown in Table 1. The new kernel method reduces error rates considerably–the average EER is reduced by & , the pooled EER is reduced by % , and the identification error rate is reduced by % . The average number of support vectors was 0 & which resulted in a model size of about $ & $ %G6 bytes (in single precision floating point); using the model size reduction method in Section 4 resulted in a model size of 0 6 bytes–over a hundred times reduction in size. Table 1: Comparison of structural risk minimization and MSE training MSE NAPS SVM Average EER 1.63% 1.01% Pooled EER 2.76% 1.45% ID error rate 4.71% 2.72% We also plotted scores for all speakers versus a threshold, see Figure 1. We normalized the scores for the MSE and SVM approaches to the same range for comparison. One can easily see the reduction in pooled EER from the graph. Note also the dramatic shifting of the FRR curve to the right for the SVM training, resulting in substantially better error rates than the MSE training. For instance, when FAR is 65 0 , the MSE training method gives an FRR of % $ ; whereas, the SVM training method gives an FRR of 0 –a reduction by a factor of & $ in error. −4 −2 0 2 4 6 10 −1 10 0 10 1 10 2 Threshold Percent FAR(%) MSE FRR(%) MSE FAR(%) SVM FRR(%) SVM Figure 1: FAR/FRR rates for the entire population versus a threshold for the SVM and MSE training methods 6 Conclusions and future work A novel kernel for comparing sequences in speech applications was derived, the NAPS kernel. This data-dependent kernel was motivated by using a probabilistic scoring method and mean-squared error training. Experiments showed that incorporating this kernel in an SVM training architecture yielded performance superior to that of the MSE training criterion. Reduction in error rates of up to & $ times were observed while retaining the efficiency of the original MSE classifier architecture. The new kernel method is also applicable to more general situations. Potential applications include–using the approach with radial basis functions, application to automatic speech recognition, and extending to an SVM/HMM architecture. References [1] Vincent Wan and William M. Campbell, “Support vector machines for verification and identification,” in Neural Networks for Signal Processing X, Proceedings of the 2000 IEEE Signal Processing Workshop, 2000, pp. 775–784. [2] Aravind Ganapathiraju and Joseph Picone, “Hybrid SVM/HMM architectures for speech recognition,” in Speech Transcription Workshop, 2000. [3] John C. Platt, “Probabilities for SV machines,” in Advances in Large Margin Classifiers, Alexander J. Smola, Peter L. Bartlett, Bernhard Sch¨olkopf, and Dale Schuurmans, Eds., pp. 61–74. The MIT Press, 2000. [4] Tommi S. Jaakkola and David Haussler, “Exploiting generative models in discriminative classifiers,” in Advances in Neural Information Processing 11, M. S. Kearns, S. A. Solla, and D. A. Cohn, Eds. 1998, pp. 487–493, The MIT Press. [5] Nathan Smith, Mark Gales, and Mahesan Niranjan, “Data-dependent kernels in SVM classification of speech patterns,” Tech. Rep. CUED/F-INFENG/TR.387, Cambridge University Engineering Department, 2001. [6] Shai Fine, Jiˇr´i Navr´atil, and Ramesh A. Gopinath, “A hybrid GMM/SVM approach to speaker recognition,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, 2001. [7] William M. Campbell and Khaled T. Assaleh, “Polynomial classifier techniques for speaker verification,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, 1999, pp. 321–324. [8] Kevin R. Farrell, Richard J. Mammone, and Khaled T. Assaleh, “Speaker recognition using neural networks and conventional classifiers,” IEEE Trans. on Speech and Audio Processing, vol. 2, no. 1, pp. 194–205, Jan. 1994. [9] Douglas A. Reynolds, “Automatic speaker recognition using Gaussian mixture speaker models,” The Lincoln Laboratory Journal, vol. 8, no. 2, pp. 173–192, 1995. [10] Michael J. Carey, Eluned S. Parris, and John S. Bridle, “A speaker verification system using alpha-nets,” in Proceedings of the International Conference on Acoustics Speech and Signal Processing, 1991, pp. 397–400. [11] J¨urgen Sch¨urmann, Pattern Classification, John Wiley and Sons, Inc., 1996. [12] Lawrence Rabiner and Biing-Hwang Juang, Fundamentals of Speech Recognition, PrenticeHall, 1993. [13] William M. Campbell and C. C. Broun, “A computationally scalable speaker recognition system,” in Proceedings of EUSIPCO, 2000, pp. 457–460. [14] Joseph P. Campbell, Jr., “Testing with the YOHO CD-ROM voice verification corpus,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, 1995, pp. 341–344. [15] Ronan Collobert and Samy Bengio, “Support vector machines for large-scale regression problems,” Tech. Rep. IDIAP-RR 00-17, IDIAP, 2000.
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Convergence of Optimistic and Incremental Q-Learning Eyal Even-Dar* Abstract Yishay Mansourt Vie sho,v the convergence of tV/O deterministic variants of Qlearning. The first is the widely used optimistic Q-learning, which initializes the Q-values to large initial values and then follows a greedy policy with respect to the Q-values. We show that setting the initial value sufficiently large guarantees the converges to an Eoptimal policy. The second is a new and novel algorithm incremental Q-learning, which gradually promotes the values of actions that are not taken. We show that incremental Q-learning converges, in the limit, to the optimal policy. Our incremental Q-learning algorithm can be viewed as derandomization of the E-greedy Q-learning. 1 Introduction One of the challenges of Reinforcement Learning is learning in an unknown environment. The environment is modeled by an MDP and we can only observe the trajectory of states, actions and rewards generated by the agent wandering in the MDP. There are two basic conceptual approaches to the learning problem. The first is model base, where we first reconstruct a model of the MDP, and then find an optimal policy for the approximate model. Recently polynomial time algorithms have been developed for this approach, initially in [7] and latter extended in [3]. The second are direct methods that update their estimated policy after each step. The most popular of the direct methods is Q-learning [13]. Q-learning uses the information observed to approximate the optimal value function, from which one can construct an optimal policy. There are various proofs that Qlearning converges, in the limit, to the optimal value function, under very mild conditions [1, 11, 12, 8,6, 2]. In a recent result the convergence rates of Q-learning are computed and an interesting dependence on the learning rates is exhibited [4]. Q-learning is an off-policy that can be run on top of any strategy. ·Although, it is an off policy algorithm, in many cases its estimated value function is used to guide the selection of actions. Being always greedy with respect to the value function may result in poor performance, due to the lack of exploration, and often randomization is used guarantee proper exploration. We show the convergence of two deterministic strategies. The first is optimistic *School of Computer Science, Tel-Aviv University, Tel-Aviv, Israel. evend@cs.tau.ac.il tSchool of Computer Science, Tel-Aviv University, Israel. mansDur@cs.tau.ac.il Q-learning, that initializes the estimates to large values and then follows a greedy policy. Optimistic Q-Iearning is widely used in applications and has been recognized as having good convergence in practice [10]. We prove that optimistic Q-Iearning, with the right setting of initial values, converge to a near optimal policy. This is not the first theoretical result showing that optimism helps in reinforcement learning, however previous results where concern with model based methods [7,3]. We show the convergence of the widely used optimistic Q-Iearning, thus explaining and supporting the results observed in practice. Our second result is a new and novel deterministic algorithm incremental Qlearning, which gradually promotes the values of actions that are not taken. We show that the frequency of sub-optimal actions vanishes, in the limit, and that the strategy defined by incremental Q-Iearning converges, in the limit, to the optimal policy (rather than a near optimal policy). Another view of incremental Q-Iearning is as a derandomization of the E-greedy Q-Iearning. The E-greedy Qlearning performs the sub optimal action every liE times in expectation, while the incremental Q-learning performs sub optimal action every (Q(s, a(s)) - Q(s, b))jE times. Furthermore, by taking the appropriate values it can be similar to the Boltzman machine. 2 The Model We define a Markov Decision process (MDP) as follows Definition 2.1 A Markov Decision process (MDP) M is a 4-tuple (8, A, P, R), where S is a set of the states, A is a set of actions, P/:J(a) is the transition probability from state i to state j when performing action a E A in state i, and RM(S, a) is the reward received when performing action a in state s. A strategy for an MDP assigns, at each time t, for each state S a probability for performing action a E A, given a history Ft - 1 == {sl,al,rl, ...,St-l,at-l,rt-l} which includes the states, actions and rewards observed until time t - 1. While executing a strategy 1f we perform at time t action at in state St and observe a reward rt (distributed according to RM(S, a)), and the next state St+l distributed according to P:!,St+l (at). We combine the sequence of rewards to a single value called return, and our goal is to maximize the return. In this work we focus on discounted return, which has a parameter, E (0,1), and the discounted return of policy 1f is VM== L:o ,trt, where Tt is the reward observed at time t. We assume that RM(S, a) is non-negative and bounded by Rmax, i.e, "Is, a: 0:::; RM(S, a) :::; Rmax. This implies that the discounted return is bounded by Vmax == RrnalZ' 1-, . We define a value function for ~ach state s, under policy 1f, as VM(s) == E[L:o Ti,i] , where the expectation is over a run of policy 1f starting at state s, and a state-action value function Q:M(s, a) == E[RM(S, a)] +,LSI P:!sl (a)ViI(s/). Let 1f* be an optimal policy which maximizes the return from any start state. This implies that for any policy 1f and any state S we have VM* (s) ~ ViI (s), and 1f*(s) == argmaxa(E[RM(S, a)] +,(LsI P:!sl (a)V*(sl)). We use ViI and QM- for VM* and Q', respectively. We say that a policy 1f is an E-optimal if IIVM- Vull oo :::; €. Given a trajectory let Ts,a be the.set of times in which we perform action a in state s, TS == UaTs,a be the times when state s is visited, Ts,not(a) == TS \ Ts,a be the set of times where in state s an action a' =1= a is performed, and Tnot(s) == UsJ=I=sTsJ be the set of times in which a state s' =1= s is visited. Also, [#(8, a, t)] is the number of times action a is performed in state 8 up to time t, Le., ITs,a n [1, t]l. Finally, throughout the paper we assume that the MDP is a uni-chain (see [9]), namely that from every state we can reach any other state. 3 Q-Learning The Q-Learning algorithm [13] estimates the state-action value function (for discounted return) as follows: Qt+1 (s, a) == (1 - at(s, a))Qt (s, a) + at (8, a) (rt(8, a) + ,Vi(s')) where Sl is the state reached from state s when performing action a at time t, and Vi(s) == maxa Qt(s, a). We assume that at(sl, a') == 0 for t fj. TsJ,aJ. A learning rate at is well-behaved if for every state action pair (s, a): (1) 2::1 at(8, a) == 00 and (2) 2::1o'.;(s, a) < 00. If the learning rate is well-behaved and every state action pair is performed infinitely often then Q-Learning converges to Q* with probability 1 (see [1, 11, 12, 8, 6]). The convergence of Q-Iearning holds using any exploration policy, and only requires that each state action pair is executed infinitely often. The greedy policy with respect to the Q-values tries to exploit continuously, however, since it does not explore properly, it might result in poor return. At the other extreme random policy continuously explores, but its actual return may be very poor. An interesting compromise between the two extremes is the E-greedy policy, which is widely used in practice [10]. This policy executes the greedy policy with probability 1 E and the random policy with probability E. This balance between exploration and exploitation both guarantees convergence and often good performance. Common to many of the exploration techniques, is the use of randomization, which is also a very natural choice. In this work we explore strategies which perform exploration but avoids randomization and uses deterministic strategies. 4 Optimistic Q-Learning Optimistic Q-learning is a simple greedy algorithm with respect to the Q-values, where the initial Q-values are set to large values, larger than their optimal values. We show that optimistic Q-Iearning converges to an E-optimal policy if the initial Q-values are set sufficiently large. Let fiT == rr;=l (1 - ai). We set the initial conditions of the Q-values as follows: 1 Vs, a: Qo(s, a) = flT Vma:t, where T == T (E, 8, S, A, a.) will be specified later. Let T}i,T == ai rrj=i+1 (1 - aj) == ai(3T/fii. Note that T T Qt+l(s, a).== (l-at)Qt(s, a)+at(rt+,Vi(s/)) == (3TQO(S, a)+L T}i,Tri(S, a)+,L T}i,T"Vti (Si), i=l i=l where T == [#(s, a, t)] and Si is the next state arrived at time ti when action a is performed for the ith time in state s. First we show that as long as T == [#(s, a, t)] :::; T actions a are performed in state s, we have Qt(s, a) ~ Vmax ' Latter we will use this to show that action a is performed at least T times in state s. Lemma 4.1 In optimistic Q-learning for any state s, action a and time t, such that T == [#(s, a, t)] :::; T we have Qt(s, a) ~ Vmax ~ Q*(s, a). Lemma 4.1 follows from the following observation: . r r f' Qt(s, a) = f3rQO(s, a) + ~)7i,rri(s,a) + 'Y2: 17i,rVt;(Si) 2:: / Vmax 2:: V*(s). i=l i=l T Now we bound T as a function of the algorithm parameters (Le., E,8, lSI, IAI) and the learning rate. We need to set T large enough to guarantee that with probability 1 - 8, for any t >-T updates, using the given learning rate, the deviation from the true value is at most E. Formally, given a sequence X t of i.i.d. random variables with zero mean and bounded by Vmax , and a learning rate at == (l/[#(s, a, t)])W let Zt+1 == (1 - D:t)Zt + D:tXt. A time T(E, 8) is an initialization time if Prl'v't ~ T : Zt :::; E] ~ 1 - 8. The following lemma bounds the initialization time as a function of the parameter w of the learning rate. Lemma 4.2 The initialization time for X and a is at most T(E, 8) c ( (V~r (In(1/8) + In(Vmaxj€)))t), for some constant c. We define a modified process, in which we update using the optimal value function, rather than our current estimate. For t ~ 1 we have, where Sl is the next state. The following lemma bounds the difference between Q* and Qt. Lemma 4.3 Consider optimistic Q-learning and let T == T(E, 8) be the initialization time. Then with probability 1 - 8, for any t > T, we have Q*(s, a) - Q(s, a) :::; E. Proof: Let T == [#(s, a, t)]. By definition we have r r Qt(s, a) == f3rQO(s, a) +2: 'T/i,rri + 'Y2: 'T/i,rV*(Si)' i=l i=l This implies that, Q*(s, a) - Q(s, a) == -f3rQO(s, a) + error_r[s, a, t] + error_v[s, a, t] where error_r[s, a, t] E[R(s, a)] 2:;=1 'T/i,rri, and error_v[s, a, t] E[V*(SI)ls, a] - 2:;=1 'T/i,rV*(Si)' We bound both error_r[s, a, t] and error_v[s, a, t] using Lemma 4.2. Therefore, with probability 1- 8, we have Q*(s, a) - Q(s, a) :::; E, for any t ~ T. Q.E.D. Next we bound the difference between our estimate Vi(s) and V*(s). Lemma 4.4 Consider optimistic Q-learning and let T == T((I-I')E,8/ISIIAI) be the initialization time. With probability at least 1- 8 for any state s and time t, we have V*(s) - vt(s) :::; E. Proof: By Lemma 4.3 we have that with probability 1- {) for every state s, action a and time t we have Q* (s, a) - Qt(s, a) :::; (1-I')E. We show by induction on t that V*(s) - vt(s) :::; E, for every state s. For t == 0 we have Vo(s) > Vmax and hence the claim holds. For the inductive step assume it holds up to time t and show that it hold for time t + 1. Let (8, a) be the state action pair executed in time t + 1. If. [#(s, a, t + 1)] :::; T then by Lemma 4.1, vt(s) ~ Vmax ~ V*(s), and the induction claim holds. Otherwise, let a* be the optimal action at state s, then, V*(s) - vt+l(S) < Q*(s,a*) - Qt+l(s,a*) Q*(s,a*) - Qt+l(s,a*) + Qt+l(s,a*) - Qt+l(s,a*) ". < (1 -I')E + I'L 1Ji,,,.(V*(Si) - vti (Si)), i==l where T == [#(s, a, t)], ti is the time when the i-th time the action a is performed in state 8, and state Si is the next state. Since ti :::; t, by the inductive hypothesis we have that V* (Si) - vti (Si) :::; E, and therefore, V* (s) - vt+l (s) :::; (1 - I')E + I'E == E. Q.E.D. Lemma 4.5 Consider optimistic Q-learning and let T == T((I-I')E,{)/\SIIAI) be the initialization time. With probability at least 1 {) any state action pair (s, a) that is executed infinitely often is E-optimal, i.e., V*(s) - Q*(s, a) :::; E. Proof: Given a trajectory let U' be the set of state action pairs that are executed infinitely often, and let M' be the original MDP M restricted to U'. For M' we can use the classical convergence proofs, and claim that vt(s) converges to ViII (s) and Qt(s, a), for (s, a) E U', converges to QMI (s, a), both with probability 1. Since (8, a) E U' is performed infinitely often it implies that Qt(s, a) converges to vt(s) == VM,(s) and therefore QM' (s, a) == ViII (s). By Lemma 4.4 with probability 1- {) we have that VM(s) - yt(s) :::; E, therefore ViI(s) - QM(s, a) :::; ViI(s) - QM' (s, a) :::; E. Q.E.D. A simple corollary is that if we set E small enough, e.g., E < min(s,a){V*(s) Q*(s,a)IV*(s) f:. Q*(s,a)}, then optimistic Q-Iearning converges to the optimal policy. Another simple corollary is the following theorem. Theorem 4.6 Consider optimistic Q-learning and let T == T((I-I')E, {)/ISIIAI) be the initialization time. For any constant ~, with probability at least 1 {) there is a time T~ > T such that at any time t > T~ the strategy defined by the optimistic Q-learning is (E + ~)/(1 - ,)-optimal. 5 Incremental Q-Iearning In this section we describe a new algorithm that we call incremental Q-learning. The main idea of the algorithm is to achieve a deterministic tradeoff between exploration and exploitation. Incremental Q-Iearning is a greedy policy with respect to the estimated Q-values plus a promotion term. The promotion term of a state-action pair (s, a) is promoted each time the action a is not executed in state s, and zeroed each time action a is executed. We show that in incremental Q-Iearning every state-action pair is taken infinitely often, which implies standard convergence of the estimates. We show that the fraction of time in which sub-optimal actions are executed vanishes in the limit. This implies that the strategy defined by incremental Q-Iearning converges, in the limit, to the optimal policy. Incremental Q-Iearning estimates the Q-function as in Q-Iearning: Qt+l(S, a) == (1 - (It(s, a))Qt(s, a) + (It(s, a)(rt(s, a) + IVi(s/)) where Sl is the next state reached when performing action a in state s at time t. The promotion term At is define as follows: At+1(s, a) == 0: t E Ts,a At+1(s, a) == At(s, a) + "p([#(s, a, t)]): t E Ts,not(a) At+1(s, a) == At(s, a): t E Tnot(s) , where "p(i) is a promotion junction which in our case depends only on the number of times we performed (s, a'), al :j:. a, since the last time we performed (s, a). We say that a promotion function "p is well-behaved if: (1) The function "p converges to zero, Le., limi-+oo 'ljJ(i) == 0, and (2) "p(1) == 1 and 'ljJ(k) > "p(k+ 1) > o. For example "p(i) == t is well behaved promotion function. Incremental Q-Iearning is a greedy policy with respect to St(s, a) == Qt(s, a) + At(s, a). First we show that Qt, in incremental Q-Iearning, converges to Q*. Lemma 5.1 Consider incremental Q-Iearning using a well-behaved learning rate and a well-behaved promotion function. Then Qt converges to Q* with probability 1. Proof: Since the learning rate is well-behaved, we need only to show that each state action pair is performed infinitely often. We show that each state that is visited infinitely often, all of its actions are performed infinitely often. Since the MDP is uni-chain this will imply that with probability 1 we reach all states infinitely often, which completes the proof. Assume that state s is visited infinitely often. Since s is visited infinitely often, there has to be a non-empty subset of the actions AI which are performed infinitely often in s. The proof is by contradiction, namely assume that AI :j:. A. Let tl be the last time that an action not in A' is performed in state s. Since"p is well behaved we have that 'ljJ(tl) is constant for a fixed tl , it implies that At(s, a) diverges for a fj. AI. Therefore, eventually we reach a time t2 > tl such that At2 (s, a) > Vmax , for every a fj. AI. Since the actions in AI are performed infinitely often there is a time t3 > t2 such that each action al E AI is performed at least once in [t2, ts]. This implies that Ata (s, a) > Vmax + Ata (s, al ) for any al E AI and a fj. AI. Therefore, some action in a E A \ AI will be performed after t1 , contradicting our assumption. Q.E.D. The following lemma shows that the frequency of sub-optimal actions vanishes. Lemma 5.2 Consider incremental Q-learning using a well behaved learning rate and a well behaved promotion function. Let It(s, a) == ITs,al/ITsl and (s, a) be any sub-optimal state-action pair. Then limt-+oo It(s, a) == 0, with probability 1. The intuition behind Lemma 5.2 is the following. Let a* be an optimal action in state s and a be a sub-optimal action. By Lemma 5.1, with probability 1 both ~ lii> ~ 0.8 a.o Ql -5 " .... .... .... .... 0.2 J I I 100 200 300 I I I I I I 400 500 600 700 800 900 1000 Number of steps 103 Figure 1: Example of 50 states MDP, where the discount factor, {, is 0.9. The leaning rate of both Incremantal and epsilon greedy Q-Iearning is set to 0.8. The dashed line represents the epsilon greedy Q-Iearning. Qt(s, a*) converges to Q*(s, a*) == V*(s) and Qt(s, a) converges to Q*(s, a). This implies, intuitively, that At(s, a) has to be at least V*(s) - Q*(s, a) == h > 0 for (s, a) to be executed. Since the promotion function is well behaved, the number of time steps required until At(s, a) changes from 0 to h increases after each time we perform (s,a). Since the inter-time between executions of (s,a) diverges, the frequency ft(s, a) vanishes. The following corollary gives a quantitative bound. Corollary 5.3 Consider incremental Q-learning with learning rate at(s, a) == l/[#(s, a, t)] and '¢(k) == l/ek . Let (s, a) be a sub-optimal state-action pair. The number of times (s, a) is performed in the first n visits to state s is 8( V'(s~~~(s,a»)' for sufficiently large n. Furthermore, the return obtained by incremental Q-Iearning converges to the optimal return. Corollary 5.4 Consider incremental Q-learning using a well behaved learning rate and a well behaved promotion function. For every € there exists a time T f such that for any t > T f we have that the strategY.1T defined by incremental Q-Iearning is €-optimal with probability 1. 6 Experiments In this section we show some experimental results, comparing Incremental QLearning and epsilon-greedy Q-Learning. One can consider incremental Q-Iearning .as a derandomization of €t-greedy Q-Learning, where the promotion function satisfies 'l/Jt == €t·· The experiment was made on MDP, which includes 50 states and two actions per state. Each state action pair immediate reward is randomly chosen in the interval [0, 10]. For each state and action (s, a) the next state transition is random,i.e., for every state Sl we have a random variable X:: a E [0, 1] and PS~SI = E:::;.a. For the €t-greedy Q-Iearning, we have €t == 10000/t at time t, while for the incremental we have 'l/Jt == 10000/t. Each result in the experiment is an average of ten different runs. In Figure 1, we observe similar behavior of the two algorithms. This experiment demonstrates the strong experimental connection between these methods. We plan to further investigate the theoretical connection between €-greedy, Boltzman machine and incremental Q-Learning. 7 Acknowledgements This research was supported in part by a grant from the Israel Science Foundation. References [1] D. P. Bertsekas and J. N. Tsitsiklis. Neuro-Dynamic Programming. Athena Scientific, Belmont, MA, 1996. [2] V.S. Borkar and S.P. Meyn. The O.D.E. method for convergence of stochastic approximation and reinforcement learning. Siam J. control, 38 (2):447-69, 2000. [3] R. I. Brafman and M. Tennenholtz. R-max - a general polynomial time algorithm for near-optimal reinforcement learning. m IJCAI, 2001. [4] E. Even-Dar and Y. Mansour. Learning rates for Q-Iearning. m COLT, 2001. [5] J. C. Gittins and D. M. Jones. A dynamic allocation index for the sequential design of experiments. Progress in Statistics, pages 241 -266, 1974. [6] T. Jaakkola, M. I. Jordan, and S. P. Singh. On the convergence of stochastic iterative dynamic programming algorithms. Neural Computation, 6, 1994. [7] M. Kearns and S. Singh. Efficient reinforcement learning: theoretical framework and algorithms. In fCML, 1998. [8] M. Littman and Cs. Szepesvari. A generalized reinforcement learning model: convergence and applications. m ICML, 1996. [9] M.L Puterman. Markov Decision Processes - Discrete Stochastic Dynamic Programming. John Wiley & Sons. mc., New York, NY, 1994. [10] R. S. Sutton and A. G. Bato. Reinforcement Learning. MIT press, 1998. [11] J. N. Tsitsiklis. Asynchronous stochastic approximation and Q-Iearning. Machine Learning, 16:185-202, 1994. [12] C. Watkins and P. Dyan. Q-Iearning. Machine Learning, 8(3/4):279 -292, 1992. [13] C. Watkins. Learning from Delayed Rewards. PhD thesis, Cambridge University, 1989.
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. Rao-Blackwellised Particle Filtering Data Augmentation VIa Christophe Andrieu Statistics Group University of Bristol University Walk Bristol BS8 1TW, UK C.Andrieu@bristol.ac.uk N ando de Freitas Computer Science UC Berkeley 387 Soda Hall, Berkeley CA 94720-1776, USA jfgf@cs.berkeley.edu Abstract Arnaud Doucet EE Engineering University of Melbourne Parkville, Victoria 3052 Australia doucet@ee.mu.oz.au In this paper, we extend the Rao-Blackwellised particle filtering method to more complex hybrid models consisting of Gaussian latent variables and discrete observations. This is accomplished by augmenting the models with artificial variables that enable us to apply Rao-Blackwellisation. Other improvements include the design of an optimal importance proposal distribution and being able to swap the sampling an selection steps to handle outliers. We focus on sequential binary classifiers that consist of linear combinations of basis functions, whose coefficients evolve according to a Gaussian smoothness prior. Our results show significant improvements. 1 Introduction Sequential Monte Carlo (SMC) particle methods go back to the first publically available paper in the modern field of Monte Carlo simulation (Metropolis and Ulam 1949); see (Doucet, de Freitas and Gordon 2001) for a comprehensive review. SMC is often referred to as particle filtering (PF) in the context of computing filtering distributions for statistical inference and learning. It is known that the performance of PF often deteriorates in high-dimensional state spaces. In the past, we have shown that if a model admits partial analytical tractability, it is possible to combine PF with exact algorithms (Kalman filters, HMM filters , junction tree algorithm) to obtain efficient high dimensional filters (Doucet, de Freitas, Murphy and Russell 2000, Doucet, Godsill and Andrieu 2000). In particular, we exploited a marginalisation technique known as Rao-Blackwellisation (RB). Here, we attack a more complex model that does not admit immediate analytical tractability. This probabilistic model consists of Gaussian latent variables and binary observations. We show that by augmenting the model with artificial variables, it becomes possible to apply Rao-Blackwellisation and optimal sampling strategies. We focus on the problem of sequential binary classification (that is, when the data arrives one-at-a-time) using generic classifiers that consist of linear combinations of basis functions, whose coefficients evolve according to a Gaussian smoothness prior (Kitagawa and Gersch 1996). We have previously addressed this problem in the context of sequential fault detection in marine diesel engines (H0jen-S0rensen, de Freitas and Fog 2000). This application is of great importance as early detection of incipient faults can improve safety and efficiency, as well as, help to reduce downtime and plant maintenance in many industrial and transportation environments. 2 Model Specification and Estimation Objectives Let us consider the following binary classification model. Given at time t = 1,2, .. . an input Xt we observe Zt E {O, I} such that Pr( Zt = llxt ,.8t) = CP(f(xl, .8t}), (1) where CP (u) = vk J::oo exp (_a2 /2) da is the cumulative function of the standard normal distribution. This is the so-called pro bit link. By convention, researchers tend to adopt a logistic (sigmoidal) link function 'P (u) = (1 + exp (_U)) -1 . However, from a Bayesian computational point of view, the probit link has many advantages and is equally valid. The unknown function is modeled as K !(Xt, .8t) = L .8t,k\[ldxt) = \[IT (Xt).8t , k=1 where we have assumed that the basis functions \[I (Xt) £ (\[11 (Xt) , ... , \[I K (Xt)/ do not depend on unknown parameters; see (Andrieu, de Freitas and Doucet 1999) for the more general case . .8t £ (.8t,1,' .. ,.8t,K) T E ~K is a set of unknown time-varying regression coefficients. To complete the model, we assume that they satisfy .8t = At.8t-1 + BtVt, .80'" N (rna, Po) (2) where Vt i·~.:f N (0, In.) and A and B control model correlations and smoothing (regularisation). Typically K is rather large, say 10 or 100, and the bases \[Ik (.) are multivariate splines, wavelets or radial basis functions (Holmes and Mallick 1998). 2.1 Augmented Statistical Model We augment the probabilistic model artificially to obtain more efficient sampling algorithms, as will be detailed in the next section. In particular, we introduce the set of independent variables Yt , such that Yt =! (Xt,.8t) + nt, (3) h i.i.d. N (0 1) d d fi {I if Yt > 0, were nt '" "an e ne Zt = 0 otherwise. It is then easy to check that one has Pr (Zt = 11 Xt, .8t) = CP (f (Xl, .8t)) . This data augmentation strategy was first introduced in econometrics by economics Nobel laureate Daniel McFadden (McFadden 1989). In the MCMC context, it has been used to design efficient samplers (Albert and Chib 1993). Here, we will show how to take advantage of it in an SMC setting. 2.2 Estimation objectives Given, at time t, the observations Ol:t £ (Xl:t, Zl:t), any Bayesian inference is based on the posterior distribution1 P (d.8o:tl Ol:t)' We are, therefore, interested in estimating sequentially in time this distribution and some of its features, such as IFor any B, we use P (dBo,tl au) to denote the distribution and p (Bo,tl au) to denote the density, where P (dBo,tl au) = p (Bo,tl au) dBo,t. Also, Bo,t ~ {Bo, BI, ... , Bd. lE ( f (xt, ,Bt) I Ol:t) or the marginal predictive distribution at time t for new input data Xt+1, that is Pr (Zt+1 = 11 01:t, xHd. The posterior density satisfies a time recursion according to Bayes rule, but it does not admit an analytical expression and, consequently, we need to resort to numerical methods to approximate it. 3 Sequential Bayesian Estimation via Particle Filtering A straightforward application of SMC methods to the model (1)-(2) would focus on sampling from the high-dimensional distribution P (d,Bo:t I 01:t) (H0jen-S0rensen et al. 2000). A substantially more efficient strategy is to exploit the augmentation of the model to sample only from the low-dimensional distribution P ( dY1:t I 01:t). The low-dimensional samples allow us then to compute the remaining estimates analytically, as shown in the following subsection. 3.1 Augmentation and Rao-Blackwellisation Consider the extended model defined by equations (1)-(2)-(3). One has p(,Bo:tlo1:t) = J p( ,Bo:tl x 1:t,Y1:t)p(Y1:tl o1:t)dY1:t· Thus if we have a Monte Carlo approximation of P (dY1:t I ol:d of the form then P (,Bo:tl 01:t) can be approximated via N PN (,Bo:tl 01:t) = L w~i)p ( ,Bo:tl x1:t,yi:O ' i=l that is a mixture of Gaussians. From this approximation, one can estimate lE(,Btlxl:t,Yl:t) and lE(,Bt-Llxl:t,Yl:t). For example, an estimate of the predictive distribution is given by PrN(Zt+1 = Il ol:t,XH1) = J Pr( Zt+1 = lIYH1)PN(dYl:t+1 lol:t, xt+1) (4) N ,, (i) ( (i) ) = ~ Wt ][(0,+00) YHl , i=l where Y~21 ~ P ( dYHll Xl:t+1, Yi~~). This shows that we can restrict ourselves to the estimation of P (Y1:t1 Ol:t) for inference purposes. In the SMC framework, we must estimate the "target" density P (Y1:t1 Ol:t) pointwise up to a normalizing constant. By standard factorisation, one has t p(Yl:tlol:t) IX IT Pr( zk IYk)p(Ykl xl:k,Yl:k-l), wherep(YIIY1:0,Xl:0) ,@,p(Yll xd· k=l Since Pr (Zk I Yk) is known, we only need to estimate P (Yk I Xl:k, Yl:k-d up to a normalizing constant. This predictive density can be computed using the Kalman filter. Given (Xl:k' Yl:k-l), the Kalman filter equations are the following. Set ,Bolo = mo and ~o l o = ~o, then for t = 1, ... , k - 1 compute ,Btlt-1 = At,Bt- 1It- 1 ~t l t-1 = At~t-1 I t-1AI + BtBI St = \[IT (xt} ~tlt- 1 \[I (Xt) + 1 Yt lt-1 = \[IT (Xt) ,Btlt-1 ,Btlt = ,Bt lt -1 + ~tlt- 1 \[I (Xt) St- 1 (Yt Yt lt - t) ~tit = ~t l t-1 ~t l t-1 \[I (xt} St- 1\[lT (Xt) ~t l t-1' (5) where ,Btlt - 1 ~ 1E(,BtIXl:t-1,Yl:t-d, ,Btlt ~ 1E(,Btlxl:t,Yl:t), Ytlt- 1 IE(Ytlxl:t,Yl:t- d, ~t l t-1 ~ cov(,BtIXl:t- 1,Y1:t- 1), ~t l t ~ cov(,Btlxl:t,Yl:t) and St ~ cov (Ytl Xl:t,Y1:t-1). One obtains P (Yk I X1:k, Y1:k-d = N (Yk;Y klk- 1' Sk) . (6) 3.2 Sampling Algorithm In this section, we briefly outline the PF algorithm for generating samples from p(dYl:tlol:t). (For details, please refer to our extended technical report at http://www . cs. berkeley. edu/ '" jfgf /publications . html.) Assume that at time t - 1 we have N particles {Yi~Ld~l distributed according to P (dYl:t- 11 ol:t- d from which one can get the following empirical distribution approximation 1 N PN (dYl:t-11 ol:t-d = N L JYi~;_l (dYl:t-d . i= l Various SMC methods can be used to obtain N new paths {Yi~~}~l distributed approximately according to P (dYl:t1 Ol:t)' The most successful of these methods typically combine importance sampling and a selection scheme. Their asymptotic convergence (N --t 00) is satisfied under mild conditions (Crisan and Doucet 2000). Since the selection step is standard (Doucet et al. 2001), we shall concentrate on describing the importance sampling step. To obtain samples from P( dYl:t IOl:t), we can sample from a proposal distribution Q(dYl:t) and weight the samples appropriately. Typically, researchers use the transition prior as proposal distribution (Isard and Blake 1996). Here, we implement an optimal proposal distribution, that is one that minimizes the variance of the importance weights W (Yl:t) conditional upon not modifying the path Y1:t-1' In our case, we have ( I ) {p(YtIXl:t ,Y1:t-dlI[o,+ oo) (Yt) if Zt = 1 P Yt X1:t,Yl:t-1,Zt ex: p(Ytlxl:t ,Yl:t-dlI(- oo,o) (Yt) if Zt = 0 ' which is a truncated Gaussian version of (6) of and consequently W (Yl:t) ex: Pr (Zt I Xl:t, Y1:t- d = (1 _ <I> ( _ Y$,l ) ) z, <I> ( _ Y$,l ) 1-z, (7) The algorithm is shown in Figure 1. (Please refer to our technical report for convergence details.) Remark 1 When we adopt the optimal proposal distribution, the importance weight Wt ex: Pr (Zt I X1:t, Y1:t- d does not depend on Yt. It is thus possible to carry out the selection step before the sampling step. The algorithm is then similar to the auxiliary variable particle filter of (Pitt and Shephard 1999). This modification to the original algorithm has important implications. It enables us to search for more Sequential importance sampling step . -(i) h. (i) :::{i) ( (i) ) • For t = 1, ... , N, (3t lt-1 = (3t lt-1 and sample Yt ~ P dYtl Xl:t, Yl:t-1 ' Zt . • For i = 1, ... , N, evaluate the importance weights using (7). Selection step • Multiply/Discard particles {~i ),,B~i~ _l}~l with respect to high/low impor. h (i) b' N . I { (i) (3(i) }N · tance welg ts W t to 0 tam partlc es Yt , t lt- 1 i=l ' Updatmg step • Compute ~t+1 I t given ~t l t - 1' • For i = 1, ... , N, use one step of the Kalman recursion (5) to compute {,B~i~ l l t } C) -C) given {y/ ,(3 ti t-1 } and ~t l t-1' Figure 1: RBPF for semiparametric binary classification. likely regions of the posterior at time t-1 using the information at time t to generate better samples at time t. In practice, this increases the robustness of the algorithm to outliers and allows us to apply it in situations where the distributions are very peaked (e.g., econometrics and almost deterministic sensors and actuators). Remark 2 The covariance updates of the Kalman jilter are outside the loop over particles. This results in substantial computational savings. 4 Simulations To compare our model, using the RBPF algorithm, to standard logistic and probit classification with PF, we generated data from clusters that change with time as shown in Figure 2. This data set captures the characteristics of a fault detection problem that we are currently studying. (For some results of applying PF to fault detection in marine diesel engines, please refer to (H0jen-S0rensen et al. 2000). More results will become available once permission is granted.) This data cannot be easily separated with an algorithm based on a time-invariant model. For the results presented here, we set the initial distributions to: (30 '" N(O , 51) and Yo '" N(O, 51). The process matrices were set to A = I and B = JI, where 82 = 0.1 is a smoothing parameter. The number of bases (cubic splines with random locations) was set to 10. (It is of course possible, when we have some data already, to initialise the bases locations so that they correspond to the input data. This trick for efficient classification in high dimensional input spaces is used in the support vector machines setting (Vapnik 1995).) The experiment was repeated with the number of particles varying between 10 and 400. Figure 3 shows the "value for money" summary plot. The new algorithm has a lower computational cost and shows a significant reduction in estimation variance. Note that the computation of the RBPF stays consistently low even for small numbers of particles. This has enabled us to apply the technique to large models consisting of hundreds of Bases using a suitable regulariser. Another advantage of PF algorithms for classification is that they yield entire probability estimates of class membership as shown in Figure 4. -:5'----o:------::5 Data from t=1 to t=100 - 5'---------5 0 5 Data from t=200 to t=300 0'<>0 -:5:---'-'---0:------::5 Data from t=1 00 to t=200 - 5'-------"-----5 0 5 Data from t=1 to t=300 Figure 2: Time-varying data. 35 ! ..... 'j r,. I I ' . ~ , ........... I i T ... I 1''''' L -n. ... "',i ... T L~ ~ ... ~ .......... i .. ,- -,... . • 50'----~--L-~--~8-~,0~-~,2--,~ 4-~,6-~,8 Computation (flops) ,10' Figure 3: Number of classification errors as the number of particles varies between 10 and 400 (different computational costs). The algorithm with the augmentation trick (RBPF) is more efficient than standard PF algorithms. 5 Conclusions In this paper, we proposed a dynamic Bayesian model for time-varying binary classification and an efficient particle filtering algorithm to perform the required computations. The efficiency of our algorithm is a result of data augmentation, RaoBlackwellisation, adopting the optimal importance distribution, being able to swap the sampling and selection steps and only needing to update the Kalman filter means in the particles loop. This extends the realm of efficient particle filtering to the ubiquitous setting of Gaussian latent variables and binary observations. Extensions to n-ary observations, different link functions and estimation of the hyper-parameters can be carried out in the same framework. 50 - 1 Figure 4: Predictive density. References Albert, J. and Chib, S. (1993) . Bayesian analysis of binary and polychotomous response data, Journal of the American Statistical Association 88(422): 669- 679. Andrieu, C., de Freitas, N. and Doucet, A. (1999) . Sequential Bayesian estimation and model selection applied to neural networks, Technical Report CUED/F-INFENG/TR 341, Cambridge University Engineering Department. Crisan, D. and Doucet, A. (2000). Convergence of sequential Monte Carlo methods, Technical Report CUED/F-INFENG/TR 381, Cambridge University Engineering Department. Doucet, A., de Freitas, N. and Gordon, N. J. (eds) (2001). Sequential Monte Carlo Methods in Practice, Springer-Verlag. Doucet, A., de Freitas, N., Murphy, K. and Russell, S. (2000). Rao blackwellised particle filtering for dynamic Bayesian networks, in C. Boutilier and M. Godszmidt (eds), Uncertainty in Artificial Intelligence, Morgan Kaufmann Publishers, pp. 176- 183. Doucet, A., Godsill, S. and Andrieu, C. (2000). On sequential Monte Carlo sampling methods for Bayesian filtering, Statistics and Computing 10(3): 197- 208. H0jen-S0rensen, P. , de Freitas, N. and Fog, T. (2000). On-line probabilistic classification with particle filters, IEEE Neural Networks for Signal Processing, Sydney, Australia. Holmes, C. C. and Mallick, B. K. (1998). Bayesian radial basis functions of variable dimension, Neural Computation 10(5): 1217- 1233. Isard, M. and Blake, A. (1996) . Contour tracking by stochastic propagation of conditional density, European Conference on Computer Vision, Cambridge, UK, pp. 343- 356. Kitagawa, G. and Gersch, W. (1996). Smoothness Priors Analysis of Time Series, Vol. 116 of Lecture Notes In Statistics, Springer-Verlag. McFadden, D. (1989). A method of simulated momemts for estimation of discrete response models without numerical integration, Econometrica 57: 995- 1026. Metropolis, N. and Uiam, S. (1949). The Monte Carlo method, Journal of the American Statistical Association 44(247): 335- 341. Pitt, M. K. and Shephard, N. (1999). Filtering via simulation: Auxiliary particle filters, Journal of the American Statistical Association 94(446): 590- 599. Vapnik, V. (1995). The Nature of Statistical Learning Theory, Springer-Verlag, New York.
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Boosting and Maximum Likelihood for Exponential Models Guy Lebanon School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 lebanon@cs.cmu.edu John Lafferty School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 lafferty@cs.cmu.edu Abstract We derive an equivalence between AdaBoost and the dual of a convex optimization problem, showing that the only difference between minimizing the exponential loss used by AdaBoost and maximum likelihood for exponential models is that the latter requires the model to be normalized to form a conditional probability distribution over labels. In addition to establishing a simple and easily understood connection between the two methods, this framework enables us to derive new regularization procedures for boosting that directly correspond to penalized maximum likelihood. Experiments on UCI datasets support our theoretical analysis and give additional insight into the relationship between boosting and logistic regression. 1 Introduction Several recent papers in statistics and machine learning have been devoted to the relationship between boosting and more standard statistical procedures such as logistic regression. In spite of this activity, an easy-to-understand and clean connection between these different techniques has not emerged. Friedman, Hastie and Tibshirani [7] note the similarity between boosting and stepwise logistic regression procedures, and suggest a least-squares alternative, but view the loss functions of the two problems as different, leaving the precise relationship between boosting and maximum likelihood unresolved. Kivinen and Warmuth [8] note that boosting is a form of “entropy projection,” and Lafferty [9] suggests the use of Bregman distances to approximate the exponential loss. Mason et al. [10] consider boosting algorithms as functional gradient descent and Duffy and Helmbold [5] study various loss functions with respect to the PAC boosting property. More recently, Collins, Schapire and Singer [2] show how different Bregman distances precisely account for boosting and logistic regression, and use this framework to give the first convergence proof of AdaBoost. However, in this work the two methods are viewed as minimizing different loss functions. Moreover, the optimization problems are formulated in terms of a reference distribution consisting of the zero vector, rather than the empirical distribution of the data, making the interpretation of this use of Bregman distances problematic from a statistical point of view. In this paper we present a very basic connection between boosting and maximum likelihood for exponential models through a simple convex optimization problem. In this setting, it is seen that the only difference between AdaBoost and maximum likelihood for exponential models, in particular logistic regression, is that the latter requires the model to be normalized to form a probability distribution. The two methods minimize the same extended Kullback-Leibler divergence objective function subject to the same feature constraints. Using information geometry, we show that projecting the exponential loss model onto the simplex of conditional probability distributions gives precisely the maximum likelihood exponential model with the specified sufficient statistics. In many cases of practical interest, the resulting models will be identical; in particular, as the number of features increases to fit the training data the two methods will give the same classifiers. We note that throughout the paper we view boosting as a procedure for minimizing the exponential loss, using either parallel or sequential update algorithms as in [2], rather than as a forward stepwise procedure as presented in [7] or [6]. Given the recent interest in these techniques, it is striking that this connection has gone unobserved until now. However in general, there may be many ways of writing the constraints for a convex optimization problem, and many different settings of the Lagrange multipliers (or Kuhn-Tucker vectors) that represent identical solutions. The key to the connection we present here lies in the use of a particular non-standard presentation of the constraints. When viewed in this way, there is no need for special-purpose Bregman distances to give a unified account of boosting and maximum likelihood, as we only make use of the standard Kullback-Leibler divergence. But our analysis gives more than a formal framework for understanding old algorithms; it also leads to new algorithms for regularizing AdaBoost, which is required when the training data is noisy. In particular, we derive a regularization procedure for AdaBoost that directly corresponds to penalized maximum likelihood using a Gaussian prior. Experiments on UCI data support our theoretical analysis, demonstrate the effectiveness of the new regularization method, and give further insight into the relationship between boosting and maximum likelihood exponential models. 2 Notation Let and be finite sets. We denote by
the set of nonnegative measures on , and by the set of conditional probability distributions, ! #"%$&('*),+.-0/&1%+ for each )2 . For 3 , we will overload the notation ('*),+.-0/ and ('*-&45)6/ ; the latter will be suggestive of a conditional probability distribution, but in general it need not be normalized. Let 798 :
, ;<=1>+@??@?6+. , be given functions, which we will refer to as features. These will correspond to the weak learners in boosting, and to the sufficient statistics in an exponential model. Suppose that we have data A'B)DC.+E-%CF/GH CJI,K with empirical distribution LMN'B)O+.-0/ and marginal L MP'*)D/ ; thus, L MN'B)O+.-0/Q K H H CRI,KTS '*)UCV+E)D/ S 'B-%C.+E-0/G? We assume, without loss of generality, that L MN'*)6/XWZY for all ) . Throughout the paper, we assume (for notational convenience) that the training data has the following property. Consistent Data Assumption. For each )[ with L MN'*)6/QWY , there is a unique -\] for which L MN'*-&45)6/PW^Y . This - will be denoted L -T'*)6/ . For most data sets of interest, each ) appears only once, so that the assumption trivially holds. However, if ) appears more than once, we require that it is labeled consistently. We make this assumption mainly to correspond with the conventions used to present boosting algorithms; it is not essential to what follows. Given 7 8 , we define the exponential model _#`D'*-&45)D/ , for a33cb , by _`d'B-4e)D/f K g<hUikjRlm nVoqprm h.sRt9u>v `%w xzyJ{zw r|F} where ~FaO+V7c'*),+.-0/.^ b 8 I,K a 8 7 8 '*),+.-0/ . The maximum likelihood estimation problem is to determine parameters a that maximize the conditional loglikelihood %'FaD/ {zw LMN'B)O+.-0/N_`U'*-&45)6/ or minimize the log loss %' aU/ . The objective function to be minimized in the multi-label boosting algorithm AdaBoost.M2 [2] is the exponential loss given by M2 'FaD/] H CJI,K Z I
ucv `%w x%yR{ w r| x%yR{ w
*|B} ? As has been often noted, the log loss and the exponential loss are qualitatively different. The exponential loss grows exponentially with increasing negative “margin,” while the log loss grows linearly. 3 Correspondence Between AdaBoost and Maximum Likelihood Since we are working with unnormalized models we make use of the extended conditional Kullback-Leibler divergence or -divergence, given by ',+E_z/ def { L MQ'*)D/ ,'B45)6/ '*-45)D/ _ '*4e)D/ '*-&45)D/ _ 'B-4e)D/ defined on (possibly taking on the value ). Note that if ,' #45)6/ and _ ' #45)6/ then this becomes the more familiar KL divergence for probabilities. Let features 78 and a fixed default distribution _ be given. We define ]'*L MP+V7T/N as '*L MN+k7T/ ![ { L MN'B)D/ '*-45)D/ 'F7 8 'B)O+.-0/"$#% &(' 7 8 45)*) / Y0+ all ;d>? (1) Since L M + , this set is non-empty. Note that under the consistent data assumption, we have that #,% &(' 7 45)-)X 7c'B)O+VL -6'B)D/E/ . Consider now the following two convex optimization problems, labeled . K and .0/ . '1. K@/ minimize 'O+V_ / subject to [2 ]' L MN+k7T/ '1. / / minimize '3O+V_ / subject to ]4 '*L M +V7T/ Thus, problem . / differs from . K only in that the solution is required to be normalized. As we’ll show, the dual problem .65 K corresponds to AdaBoost, and the dual problem .75 / corresponds to maximum likelihood for exponential models. This presentation of the constraints is the key to making the correspondence between AdaBoost and maximum likelihood. Note that the constraint { L MQ'*)D/ Z ,'B4e)D/07c'B)O+.-0/& #% &0' 78) , which is the usual presentation of the constraints for maximum likelihood (as dual to maximum entropy), doesn’t make sense for unnormalized models, since the two sides of the equation may not be “on the same scale.” Note further that attempting to rescale by dividing by the mass of to get { L MP'B)D/ ,'B-4e)D/07c'B)O+E/ Z 9,'*-&45)6/ :#% &0' 78) would yield nonlinear constraints. We now derive the dual problems formally; the following section gives a precise statement of the duality result. To derive the dual problem .75 K , we calculate the Lagrangian as ; K '3O+kaU/ { LMN'*)6/ ,'B4e)D/ < ,'B-4e)D/ _ '*-&45)D/ 1,^~FaO+V7c'B)O+.-0/"$#% &0' 7<45)*) ? For a b , the connecting equation _#` def arg =?>@BA "C ; K9'3O+VaD/ is given by _`D'*-45)6/X _ '*-&45)6/ uAv `%w x%yR{zw @|9D8E FHG x"I*{KJ } ? Thus, the dual function L M6K9' aU/N ; K%'B_`U+VaD/ is given by L M6K9'FaD/ { L MP'*)6/ _ '*-45)D/ u v `%w xzyJ{zw r|9D8E FHG x"IB{KJ } ? (2) The dual problem is to determine a8N POQRS=TOVU ` L MO' aU/ . To derive the dual for . / , we simply add additional Lagrange multipliers W,{ for the constraints 8,'B-4e)D/ =1 . 3.1 Special cases It is now straightforward to derive various boosting and logistic regression problems as special cases of the above optimization problems. Case 1: AdaBoost.M2. Take _ A'B-4e)D/ 1 . Then the dual problem =TO U `L M K 'FaD/ is equivalent to computing a*N OVQ! =?><@ ` C I
uAv `zw x%yR{ w r| x%yR{ w
*| which is the optimization problem of AdaBoost.M2. Case 2: Binary AdaBoost. In addition to the assumptions for the previous case, now assume that 2 &1%+
1% , and take 7 8 'B)O+E/ K / -N7 8 'B)D/ . Then the dual problem is given by a N OQR =7>@ ` C u 6
v `%w x%yR{ B|B} which is the optimization problem of binary AdaBoost. Case 3: Maximum Likelihood for Exponential Models. In this case we take the same setup as for AdaBoost.M2 but add the additional normalization constraints: 8,'B-4e) C / 1%+ 1>+@?@??T+ ? If these constraints are satisfied, then the other constraints take the form { L MN'*)6/
'*-&45)6/.7 8 'B)O+.-0/ {zw L MN'B)O+.-0/E7 8 '*),+.-0/ and the connecting equation becomes _ ` '*-&45)6/ K p _K >'B-4e)D/ uAv `zw x%yR{%w r|B} were { is the normalizing term { Z c_ >'*- 4 )D/ u>v `%w xzyJ{zw r|F} , which corresponds to setting the Lagrange multiplier W { to the appropriate value. In this case, after a simple calculation the dual problem is seen to be L M / ' aU/< { L MQ'*)O+E-0/< _`D'B-4e)D/ which corresponds to maximum likelihood for a conditional exponential model with sufficient statistics 798>'*)O+E-0/ . Case 4: Logistic Regression. Returning to the case of binary AdaBoost, we see that when we add normalization constraints as above, the model is equivalent to binary logistic regression, since _`D'e1 4e)D/& K K i jRlm nEo p sJt ? We note that it is not necessary to scale the features by a constant factor here, as in [7]; the correspondence between logistic regression and boosting is direct. 3.2 Duality Let K and / be defined as the following exponential families: K 'F_K %+k7T/ #_ [ &_ '*-&45)6/ _K A'*-&45)6/ u v `%w x%yR{zw @| x%yR{%w yR{ |B|B} +Ua][ b /%'F_K %+k7T/ #_ &_ '*-&45)6/ _K A'*-&45)D/ u v `%w xzyJ{zw r|F} +Ua][ b A? Thus K is unnormalized while / is normalized. We now define the boosting solution _ Nboost and maximum likelihood solution _ Nml as _ Nboost OQR =7>@
#" T { L MN'*)6/d
c_ '*-&45)D/ _ Nml :OVQ!S=TO U
#" O { L MP'*)D/B< _ ' -&45)6/ where denotes the closure of the set . The following theorem corresponds to Proposition 4 of [3] for the usual KL divergence; in [4] the duality theorem is proved for a general class of Bregman distances, including the extended KL divergence as a special case. Note that we do not work with divergences such as ' Y0+V_z/ as in [2], but rather '*L MN+E_z/ , which is more natural and interpretable from a statistical point-of-view. Theorem. Suppose that ' L MN+V_ / . Then _ Nboost and _ Nml exist, are unique, and satisfy _ Nboost OVQ! =?><@ A " '3O+V_ / OVQ!S=7>@ " '*L MN+E_z/ _ Nml OVQ! =?><@ A " ! '3O+V_K / OVQ!S=7>@ " '*L MN+#" /6? Moreover, _ Nml is computed in terms of _ Nboost as _ Nml :OVQ! =?>@ & " ! 'O+V_ Nboost / . PSfrag replacements _ Nboost ml K PSfrag replacements _ Nboost _ Nml K Figure 1: Geometric view of the duality theorem. Minimizing the exponential loss finds the member of that intersects with the feasible set of measures satisfying the moment constraints (left). When we impose the additional constraint that each conditional distribution
must be normalized, we introduce a Lagrange multiplier for each training example , giving a higher-dimensional family . By the duality theorem, projecting the exponential loss solution onto the intersection of the feasible set with the simplex gives the maximum likelihood solution. This result has a simple geometric interpretation. The unnormalized exponential family K intersects the feasible set of measures satisfying the constraints (1) at a single point. The algorithms presented in [2] determine this point, which is the exponential loss solution _ Nboost :OVQ!S=?><@ " '*L M +E_z/ (see Figure 1, left). On the other hand, maximum conditional likelihood estimation for an exponential model with the same features is equivalent to the problem _VNml OQRS=?>@ " ' L M +V_z/ where K is the exponential family with additional Lagrange multipliers, one for each normalization constraint. The feasible set for this problem is . Since , by the Pythagorean equality we have that _ Nml OQR =7>@ A " ! '3O+V_ Nboost / (see Figure 1, right). 4 Regularization Minimizing the exponential loss or the log loss on real data often fails to produce finite parameters. Specifically, this happens when for some feature 7 8 7 8 '*),+.-0/"7 8 '*),+GL - '*)6/./^Y for all - and ) with L MP'B)D/PWY (3) or 7 8 '*),+.-0/"7 8 '*),+GL - '*)6/./^Y for all - and ) with L MP'B)D/PWY0? This is especially harmful since often the features for which (3) holds are the most important for the purpose of discrimination. Of course, even when (3) does not hold, models trained by maximum likelihood or the exponential loss can overfit the training data. A standard regularization technique in the case of maximum likelihood employs parameter priors in a Bayesian framework. See [11] for non-Bayesian alternatives in the context of boosting. In terms of convex duality, parameter priors for the dual problem correspond to “potentials” on the constraint values in the primal problem. The case of a Gaussian prior on a , for example, corresponds to a quadratic potential on the constraint values in the primal problem. We now consider primal problems over '3O+ / where ^^ and \ b is a parameter vector that relaxes the original constraints. Define '*L MN+k7D+ @/N^ as ]' L M +k7D+ @/ [ { L MQ'*)D/ '*-&45)6/'F78>'B)O+.-0/"$#% &0' 78 45)*) / V8 (4) and consider the primal problem . Kkw reg given by ' .cKGw reg / minimize ',+E_ / ' @/ subject to [4 '*L M +V7D+ @/ where f% b is a convex function whose minimum is at Y . To derive the dual problem, the Lagrangian is calculated as ; ',+ 9+kaU/& ; ',+VaD/S ' @/ and the dual function is M Kkw reg 'FaD/ M K 'FaD/( 5z'FaD/ where 5%'FaD/ is the convex conjugate of . For a quadratic penalty ' @/ 8 K /
/ 8 / 8 , we have 5%'FaD/X 8 K /
/ 8 a / 8 and the dual function becomes M6KGw reg ' aU/ { LMN'B)D/ _ '*-&45)6/ u v `%w x%yR{%w r| x%yR{%w yR{ |B} 8 a / 8
/ 8 ? (5) A sequential update rule for (5) incurs the small additional cost of solving a nonlinear equation by Newton-Raphson every iteration. See [1] for a discussion of this technique in the context of exponential models in statistical language modeling. 5 Experiments We performed experiments on some of the UC Irvine datasets in order to investigate the relationship between boosting and maximum likelihood empirically. The weak learner was the decision stump FindAttrTest as described in [6], and the training set consisted of a randomly chosen 90% of the data. Table 1 shows experiments with regularized boosting. Two boosting models are compared. The first model _zK was trained for 10 features generated by FindAttrTest, excluding features satisfying condition (3). Training was carried out using the parallel update method described in [2]. The second model, _ / , was trained using the exponential loss with quadratic regularization. The performance was measured using the conditional log-likelihood of the (normalized) models over the training and test set, denoted train and test, as well as using the test error rate test. The table entries were averaged by 10-fold cross validation. For the weak learner FindAttrTest, only the Iris dataset produced features that satisfy (3). On average, 4 out of the 10 features were removed. As the flexibility of the weak learner is increased, (3) is expected to hold more often. On this dataset regularization improves both the test set log-likelihood and error rate considerably. In datasets where _K shows significant overfitting, regularization improves both the log-likelihood measure and the error rate. In cases of little overfitting (according to the log-likelihood measure), regularization only improves the test set log-likelihood at the expense of the training set log-likelihood, however without affecting test set error. Next we performed a set of experiments to test how much _Nboost differs from _ Nml , where the boosting model is normalized (after training) to form a conditional probability distribution. For different experiments, FindAttrTest generated a different number of features (10–100), and the training set was selected randomly. The top row in Figure 2 shows for the Sonar dataset the relationship between train 'F_ Nml / and train 'F_ Nboost / as well as between train 'F_ Nml / and train 'B_ Nml +V_ Nboost / . As the number of features increases so that the training Unregularized Regularized Data train 'F_K/ test 'B_#K@/ test 'F_K/ train 'B_ / / test 'F_ / / test 'B_ / / Promoters -0.29 -0.60 0.28 -0.32 -0.50 0.26 Iris -0.29 -1.16 0.21 -0.10 -0.20 0.09 Sonar -0.22 -0.58 0.25 -0.26 -0.48 0.19 Glass -0.82 -0.90 0.36 -0.84 -0.90 0.36 Ionosphere -0.18 -0.36 0.13 -0.21 -0.28 0.10 Hepatitis -0.28 -0.42 0.19 -0.28 -0.39 0.19 Breast -0.12 -0.14 0.04 -0.12 -0.14 0.04 Pima -0.48 -0.53 0.26 -0.48 -0.52 0.25 Table 1: Comparison of unregularized to regularized boosting. For both the regularized and unregularized cases, the first column shows training log-likelihood, the second column shows test loglikelihood, and the third column shows test error rate. Regularization reduces error rate in some cases while it consistently improves the test set log-likelihood measure on all datasets. All entries were averaged using 10-fold cross validation. data is more closely fit ( train 'B_ ml /7U Y ), the boosting and maximum likelihood models become more similar, as measured by the KL divergence. This result does not hold when the model is unidentifiable and the two models diverge in arbitrary directions. The bottom row in Figure 2 shows the relationship between the test set log-likelihoods, test 'B_ Nml / and test 'F_ Nboost / , together with the test set error rates test 'B_ Nml / and test 'F_ Nboost / . In these figures the testing set was chosen to be 50% of the total data. In order to indicate the number of points at each error rate, each circle was shifted by a small random value to avoid points falling on top of each other. While the plots in the bottom row of Figure 2 indicate that train 'F_ Nml / W train 'B_ Nboost / , as expected, on the test data the linear trend is reversed, so that test 'B_ Nml / test 'F_ Nboost / . Identical experiments on Hepatitis, Glass and Promoters resulted in similar results and are omitted due to lack of space. The duality result suggests a possible explanation for the higher performance of boosting with respect to test. The boosting model is less constrained due to the lack of normalization constraints, and therefore has a smaller -divergence to the uniform model. This may be interpreted as a higher extended entropy, or less concentrated conditional model. However, as %'F_ Nml /D Y , the two models come to agree (up to identifiability). It is easy to show that for any exponential model _#`\ / + train 'B_ Nml +V_` / %'B_ Nml /" %'B_`>/r? By taking _`( _ Nboost it is seen that as the difference between %'F_VNml / and %'B_ Nboost / gets smaller, the divergence between the two models also gets smaller. The empirical results are consistent with the theoretical analysis. As the number of features is increased so that the training data is fit more closely, the model matches the empirical distribution L M and the normalizing term ` 'B)D/ becomes a constant. In this case, normalizing the boosting model " Nboost does not violate the constraints, and results in the maximum likelihood model. Acknowledgments We thank Michael Collins, Michael Jordan, Andrew Ng, Fernando Pereira, Rob Schapire, and Yair Weiss for helpful comments on an early version of this paper. Part of this work was carried out while the second author was visiting the Department of Statistics, University of California at Berkeley. −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 PSfrag replacements train ml train boost −0.25 −0.2 −0.15 −0.1 −0.05 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 PSfrag replacements train
ml train ml boost −25 −20 −15 −10 −5 −25 −20 −15 −10 −5 PSfrag replacements test
ml test boost 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.1 0.15 0.2 0.25 0.3 0.35 0.4 PSfrag replacements test ml test boost Figure 2: Comparison of AdaBoost and maximum likelihood for Sonar dataset. The top row compares train
ml to train
boost (left) and train
ml to train
ml boost (right). The bottom row shows the relationship between test
ml and test
boost (left) and test
ml and test
boost (right). The experimental results for other UCI datasets were very similar. References [1] S. Chen and R. Rosenfeld. A survey of smoothing techniques for ME models. IEEE Transactions on Speech and Audio Processing, 8(1), 2000. [2] M. Collins, R. E. Schapire, and Y. Singer. Logistic regression, AdaBoost and Bregman distances. Machine Learning, to appear. [3] S. Della Pietra, V. Della Pietra, and J. Lafferty. Inducing features of random fields. IEEE Transactions on Pattern Analysis and Machine Intelligence, 19(4), 1997. [4] S. Della Pietra, V. Della Pietra, and J. Lafferty. Duality and auxiliary functions for Bregman distances. Technical Report CMU-CS-01-109, Carnegie Mellon University, 2001. [5] N. Duffy and D. Helmbold. Potential boosters? In Neural Information Processing Systems, 2000. [6] Y. Freund and R. E. Schapire. Experiments with a new boosting algorithm. In International Conference on Machine Learning, 1996. [7] J. Friedman, T. Hastie, and R. Tibshirani. Additive logistic regression: A statistical view of boosting. The Annals of Statistics, 28(2), 2000. [8] J. Kivinen and M. K. Warmuth. Boosting as entropy projection. In Computational Learning Theory, 1999. [9] J. Lafferty. Additive models, boosting, and inference for generalized divergences. In Computational Learning Theory, 1999. [10] L. Mason, J. Baxter, P. Bartlett, and M. Frean. Functional gradient techniques for combining hypotheses. In A. Smola, P. Bartlett, B. Sch¨olkopf, and D. Schuurmans, editors, Advances in Large Margin Classifiers, 1999. [11] G. R¨atsch, T. Onoda, and K.-R. M¨uller. Soft margins for AdaBoost. Machine Learning, 2001.
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Iterative Double Clustering for Unsupervised and Semi-Supervised Learning Ran El-Yaniv Oren Souroujon Computer Science Department Technion - Israel Institute of Technology (rani,orenso)@cs.technion.ac.il Abstract We present a powerful meta-clustering technique called Iterative Double Clustering (IDC). The IDC method is a natural extension of the recent Double Clustering (DC) method of Slonim and Tishby that exhibited impressive performance on text categorization tasks [12]. Using synthetically generated data we empirically find that whenever the DC procedure is successful in recovering some of the structure hidden in the data, the extended IDC procedure can incrementally compute a significantly more accurate classification. IDC is especially advantageous when the data exhibits high attribute noise. Our simulation results also show the effectiveness of IDC in text categorization problems. Surprisingly, this unsupervised procedure can be competitive with a (supervised) SVM trained with a small training set. Finally, we propose a simple and natural extension of IDC for semi-supervised and transductive learning where we are given both labeled and unlabeled examples. 1 Introduction Data clustering is a fundamental and challenging routine in information processing and pattern recognition. Informally, when we cluster a set of elements we attempt to partition it into subsets such that points in the same subset are more “similar” to each other than to points in other subsets. Typical clustering algorithms depend on a choice of a similarity measure between data points [6], and a “correct” clustering result depends on an appropriate choice of a similarity measure. However, the choice of a “correct” measure is an ill-defined task without a particular application at hand. For instance, consider a hypothetical data set containing articles by each of two authors such that half of the articles authored by each author discusses one topic, and the other half discusses another topic. There are two possible dichotomies of the data which could yield two different bi-partitions: one according to topic, and another, according to writing style. When asked to cluster this set into two sub-clusters, one cannot successfully achieve the task without knowing the goal: Are we interested in clusters that reflect writing style or semantics? Therefore, without a suitable target at hand and a principled method for choosing a similarity measure suitable for the target, it can be meaningless to interpret clustering results. The information bottleneck (IB) method of Tishby, Pereira and Bialek [8] is a recent framework that can sometimes provide an elegant solution to this problematic “metric selection” aspect of data clustering (see Section 2). The original IB method generates a soft clustering assignments for the data. In [10], Slonim and Tishby developed a simplified “hard” variant of the IB clustering, where there is a hard assignment of points to their clusters. Employing this hard IB clustering, the same authors introduced an effective two-stage clustering procedure called Double Clustering (DC) [12]. An experimental study of DC on text categorization tasks [12] showed a consistent advantage of DC over other clustering methods. A striking finding in [12] is that DC sometimes even attained results close to those of supervised learning.1 In this paper we present a powerful extension of the DC procedure which we term Iterative Double Clustering (IDC). IDC performs iterations of DC and whenever the first DC iteration succeeds in extracting a meaningful structure of the data, a number of the next consecutive iterations can continually improve the clustering quality. This continual improvement achieved by IDC is due to generation of progressively less noisy data representations which reduce variance. Using synthetically generated data, we study some properties of IDC. Not only that IDC can dramatically outperform DC whenever the data is noisy, our experiments indicate that IDC attains impressive categorization results on text categorization tasks. In particular, we show that our unsupervised IDC procedure is competitive with an SVM (and Naive Bayes) trained over a small sized training set. We also propose a natural extension of IDC for transductive semi-supervised transductive. Our preliminary empirical results indicate that our transductive IDC can yield effective text categorization. 2 Information Bottleneck and Double Clustering We consider a data set X of elements, each of which is a d-dimensional vector over a set F of features. We focus on the case where feature values are non-negative real numbers. For every element x = (f1, . . . , fd) ∈X we consider the empirical conditional distribution {p(fi|x)} of features given x, where p(fi|x) = fi/ Pd i=1 fi. For instance, X can be a set of documents, each of which is represented as a vector of word-features where fi is the frequency of the ith word (in some fixed word enumeration). Thus, we represent each element as a distribution over its features, and are interested in a partition of the data based on these feature conditional distributions. Given a predetermined number of clusters, a straightforward approach to cluster the data using the above “distributional representation” would be to choose some (dis)similarity measure for distributions (e.g. based on some Lp norm or some statistical measure such as the KL-divergence) and employ some “plug-in” clustering algorithm based on this measure (e.g. agglomerative algorithms). Perhaps due to feature noise, this simplistic approach can result in mediocre results (see e.g. [12]). Suppose that our data is given via observations of a random variable S. In the information bottleneck (IB) method of Tishby et al. [8] we attempt to extract the essence of the data S using co-occurrence observations of S together with a target variable T. The goal is to extract a compressed representation ˜S of S with minimum compromise of information content with respect to T. This way, T can direct us to extract meaningful clustering from S where the meaning is determined by the target T. Let I(S, T) = P s∈S,t∈T p(s, t) log p(s,t) p(s)p(t), the mutual information between S and T [3]. 1Specifically, the DC method obtained in some cases accuracy close to that obtained by a naive Bayes classifier trained over a small sized sample [12]. The IB method attempts to compute p(˜s|s), a “soft” assignment of a data point s to clusters ˜s, so as to minimize I(S, ˜S)−βI( ˜S, T), given the Markov condition T →S →˜S (i.e., T and ˜S are conditionally independent given S). Here, β is a Lagrange multiplier that controls a constraint on I( ˜S, T) and thus, the tradeoffbetween the desired compression level and the predictive power of ˜S with respect to T. As shown in [8], this minimization yields a system of coupled equations for the clustering mapping p(˜s|s) in terms of the cluster representations p(t|˜s) and the cluster weights p(˜s). The paper [8] also presents an algorithm similar to deterministic annealing [9] for recovering a solution for the coupled equations. Slonim and Tishby [10] proposed a simplified IB approach for the computation of “hard” cluster assignments. In this hard IB variant, each data point s, represented by {p(t|s)}t, is associated with one centroid ˜s. They also devised a greedy agglomerative clustering algorithm that starts with the trivial clustering, where each data point s is a single cluster; then, at each step, the algorithm merges the two clusters that minimize the loss of mutual information I( ˜S, T). The reduction in I( ˜S, T) due to a merge of two clusters ˜si and ˜sj is shown to be (p(˜si) + p(˜sj))DJS[p(t|˜si), p(t|˜sj)], (1) where, for any two distributions p(x) and q(x), with priors λp and λq, λp + λq = 1, DJS[p(x), q(x)] is the Jensen-Shannon divergence (see [7, 4]), DJS[p(x), q(x)] = λpDKL(p||p + q 2 ) + λqDKL(q||p + q 2 ). Here, p+q 2 denotes the distribution (p(x)+q(x))/2 and DKL(·||·) is the Kullbak-Leibler divergence [3]. This agglomerative algorithm is of course only locally optimal, since at each step it greedily merges the two most similar clusters. Another disadvantage of this algorithm is its time complexity of O(n2) for a data set of n elements (see [12] for details). The IB method can be viewed as a meta-clustering procedure that, given observations of the variables S and T (via their empirical co-occurrence samples p(s, t)), attempts to cluster s-elements represented as distributions over t-elements. Using the merging cost of equation (1) one can approximate IB clustering based on other “plug-in” vectorial clustering routines applied within the simplex containing the s-elements distributional representations. DC [12] is a two-stage procedure where during the first stage we IB-cluster features represented as distributions over elements, thus generating feature clusters. During the second stage we IB-cluster elements represented as distributions over the feature clusters (a more formal description follows). For instance, considering a document clustering domain, in the first stage we cluster words as distributions over documents to obtain word clusters. Then in the second stage we cluster documents as distributions over word clusters, to obtain document clusters. Intuitively, the first stage in DC generates more coarse pseudo features (i.e. feature centroids), which can reduce noise and sparseness that might be exhibited in the original feature values. Then, in the second stage, elements are clustered as distributions over the “distilled” pseudo features, and therefore can generate more accurate element clusters. As reported in [12], this DC two-stage procedure outperforms various other clustering approaches as well as DC variants applied with other dissimilarity measures (such as the variational distance) different from the optimal JS-divergence of Equation (1). It is most striking that in some cases, the accuracy achieved by DC was close to that achieved by a supervised Naive Bayes classifier. 3 Iterative Double Clustering (IDC) Denote by IBN(T|S) the clustering result, into N clusters, of the IB hard clustering procedure when the data is S and the target variable is T (see Section 2). For instance, if T represents documents and S represents words, the application of IBN(T = documents|S = words) will cluster the words, represented as distributions over the documents, into N clusters. Using the notation of our problem setup, with X denoting the data and F denoting the features, Figure 1 provides a pseudo-code of the IDC meta-clustering algorithm, which clusters X into N ˜ X clusters. Note that the DC procedure is simply an application of IDC with k = 1. The code of Figure 1 requires to specify k, the number of IDC iterations to run, N ˜ X, the number of element clusters (e.g. the desired number of of document clusters) and N ˜ F , the number of feature clusters to use during each iteration. In the experiments reported below we always assumed that we know the correct N ˜ X. Our experiments show that the algorithm is not too sensitive to an overestimate of N ˜ F . Note that the choice of these parameters is the usual model order selection problem. Perhaps the first question regarding k (number of iterations) to ask is whether or not IDC converges to a steady state (e.g. where two consecutive iterations generate identical partitions). Unfortunately, a theoretical understanding of this convergence issue is left open in this paper. In most of our experiments IDC converged after a small number of iterations. In all the experiments reported below we used a fixed k = 15. Input: X (input data) N ˜ X (number of element clusters) N ˜ F (number of feature clusters to use) k (number of iterations) Initialize: S ←F, T ←X, loop {k times} N ←N ˜ F ˜F ←IBN(T|S) N ←N ˜ X, S ←X, T ←˜F ˜X ←IBN(T|S) S ←F, T ←˜X end loop Output ˜X Figure 1: Pseudo-code for IDC The “hard” IB-clustering originally presented by [12] uses an agglomerative procedure as its underlying clustering algorithm (see Section 2). The “soft” IB [8] applies a deterministic annealing clustering [9] as its underlying procedure. As already discussed, the IB method can be viewed as meta-clustering which can employ many vectorial clustering routines. We implemented IDC using several routines including agglomerative clustering and deterministic annealing. Since both these algorithms are computationally intensive, we also implemented IDC using a simple fast algorithm called Add-C proposed by Guedalia et al. [5]. Add-C is an online greedy clustering algorithm with linear running time and can be viewed as a simple online approximation of k-means. For this reason, all the results reported below were computed using Add-C (whose description is omitted, for lack of space, see [5] for details). For obtaining a better approximation to the IB method we of course used the JS-divergence of (1) as our cost measure. Following [12] we chose to evaluate the performance of IDC with respect to a labeled data set. Specifically, we count the number of classification errors made by IDC as obtained from labeled data. In order to better understand the properties of IDC, we first examined it within a controlled setup of synthetically generated data points whose feature values were generated by d-dimensional Gaussian distributions (for d features) of the form N(µ, Σ), where Σ = σ2I, with σ constant. In order to simulate different sources, we assigned different µ values (from a given constant range) to each combination of source and feature. Specifically, for data simulating m classes and |F| features, |F| × m different distributions were selected. We introduced feature noise by distorting each entry with value v by adding a random sample from N(0, (α · v)2), where α is the “noise amplitude” (resulting negative values were rounded to zero). In figure 2(a), we plot the average accuracy of 10 runs of IDC. As can be seen, at low level noise amplitudes IDC attains perfect accuracy. When the noise amplitude increases, both IDC and DC deteriorate but the multiple rounds of IDC can better resist the extra noise. After observing the large accuracy gain between DC and IDC at a specific interval of noise amplitude within the feature noise setup, we set the noise amplitude to values in that interval and examined the behavior of the IDC run in more detail. Figure 2(b) shows a typical trace of the accuracy obtained at each of the 20 iterations of an IDC run over noisy data. This learning curve shows a quick improvement in accuracy during the first few rounds, and then reaches a plateau. Following [12] we used the 20 Newsgroups (NG20) [1] data set to evaluate IDC on real, labeled data. We chose several subsets of NG20 with various degrees of difficulty. In the first set of experiments we used the following four newsgroups (denoted as NG4), two of which deal with sports subjects: ‘rec.sport.baseball’, ‘rec.sport.hockey’, ‘alt.atheism’ and ‘sci.med’. In these experiments we tested some basic properties of IDC. In all the experiments reported in this section we performed the following preprocessing: We lowered the case of all letters, filtered out low frequency words which appeared up to (and including) 3 times in the entire set and filtered out numerical and non-alphabetical characters. Of course we also stripped offnewsgroup headers which contain the class labels. In Figure 2(c) we display accuracy vs. number of feature clusters (N ˜ F ). The accuracy deteriorates when N ˜ F is too small and we see a slight negative trend when it increases. We performed an additional experiment which tested the performance using very large numbers of feature clusters. Indeed, these results indicate that after a plateau in the range of 10-20 there is a minor negative trend in the accuracy level. Thus, with respect to this data set, the IDC algorithm is not too sensitive to an overestimation of the number N ˜ F of feature clusters. Other experiments over the NG4 data set confirmed the results of [12] that the JSdivergence dissimilarity measure of Equation (1) outperforms other measures, such as the variational distance (L1 norm), the KL-divergence, the square-Euclidean distance and the ‘cosine’ distance. Details of all these experiments will be presented in the full version of the paper. In the next set of experiments we tested IDC’s performance on the same newsgroup subsets used in [12]. Table 1(a) compares the accuracy achieved by DC to the the last (15th) round of IDC with respect to all data sets described in [12]. Results of DC were taken from [12] where DC is implemented using the agglomerative routine. Table 1(b) displays a preliminary comparison of IDC with the results of a Naive Bayes (NB) classifier (reported in [11]) and a support vector machine (SVM). In each of the 5 experiments the supervised classifiers were trained using 25 documents per class and tested on 475 documents per class. The input for the unsupervised IDC was 500 unlabeled documents per class. As can be seen, IDC outperforms in this setting both the naive Bayes learner and the SVM. 4 Learning from Labeled and Unlabeled Examples In this section, we present a natural extension of IDC for semi-supervised transductive learning that can utilize both labeled and unlabeled data. In transductive learning, the testing is done on the unlabeled examples in the training data, while in semi-supervised Newsgroup DC IDC-15 Binary1 0.70 0.85 Binary2 0.68 0.83 Binary3 0.75 0.80 Multi51 0.59 0.86 Multi52 0.58 0.88 Multi53 0.53 0.86 Multi101 0.35 0.56 Multi102 0.35 0.49 Multi103 0.35 0.55 Average 0.54 0.74 Data Set NB SVM IDC-15 IDC-1 COMP (5) 0.50 0.51 0.50 0.34 SCIENCE (4) 0.73 0.68 0.79 0.44 POLITICS (3) 0.67 0.76 0.78 0.42 RELIGION (3) 0.55 0.78 0.60 0.38 SPORT (2) 0.75 0.78 0.89 0.76 Average 0.64 0.70 0.71 0.47 Table 1: Left: Accuracy of DC vs. IDC on most of the data sets described in [12]. DC results are taken from [12]; Right: Accuracy of Naive Bayes (NB) and SVM classifiers vs. IDC on some of the data sets described in [11]. The IDC-15 column shows final accuracy achieved at iteration 15 of IDC; the IDC-1 column shows first iteration accuracy. The NB results are taken from [11]. The SVM results were produced using the LibSVM package [2] with its default parameters. In all cases the SVM was trained and tested using the same training/test set sizes as described in [11] (25 documents per newsgroup for training and 475 for testing; the number of unlabeled documents fed to IDC was 500 per newsgroup). The number of newsgroups in each hyper-category is specified in parenthesis (e.g. COMP contains 5 newsgroups). inductive learning it is done on previously unseen data. Here we only deal with the transductive case. In the full version of the paper we will present a semi-supervised inductive learning version of IDC. For motivating the transductive IDC, consider a data set X that has emerged from a statistical mixture which includes several sources (classes). Let C be a random variable indicating the class of a random point. During the first iteration of a standard IDC we cluster the features F so as to preserve I(F, X). Typically, X contains predictive information about the classes C. In cases where I(X, C) is sufficiently large, we expect that the feature clusters ˜F will preserve some information about C as well. Having available some labeled data points, we may attempt to generate feature clusters ˜F which preserve more information about class labels. This leads to the following straightforward idea. During the first IB-stage of the IDC first iteration, we cluster the features F as distributions over class labels (given by the labeled data). This phase results in feature clusters ˜F. Then we continue as usual; that is, in the second IB-phase of the first IDC iteration we cluster X, represented as distributions over ˜F. Subsequent IDC iterations use all the unlabeled data. In Figure 2(d) we show the accuracy obtained by DC and IDC in categorizing 5 newsgroups as a function of the training (labeled) set size. For instance, we see that when the algorithm has 10 documents available from each class it can categorize the entire unlabeled set, containing 90 unlabeled documents in each of the classes, with accuracy of about 80%. The benchmark accuracy of IDC with no labeled examples obtained about 73%. In Figure 2(e) we see the accuracy obtained by DC and transductive IDC trained with a constant set of 50 labeled documents, on different unlabeled (test) sample sizes. The graph shows that the accuracy of DC significantly degrades, while IDC manages to sustain an almost constant high accuracy. 5 Concluding Remarks Our contribution is threefold. First, we present a natural extension of the successful double clustering algorithm of [12]. Empirical evidence indicates that our new iterative DC algorithm has distinct advantages over DC, especially in noisy settings. Second, we applied the unsupervised IDC on text categorization problems which are typically dealt with by supervised learning algorithms. Our results indicate that it is possible to achieve performance competitive to supervised classifiers that were trained over small samples. Finally, we present a natural extension of IDC that allows for transductive learning. Our preliminary empirical evaluation of this scheme over text categorization appears to be promising. A number of interesting questions are left for future research. First, it would be of interest to gain better theoretical understanding of several issues: the generalization properties of DC and IDC, the convergence of IDC to a steady state and precise conditions on attribute noise settings within which IDC is advantageous. Second, it would be important to test the empirical performance of IDC with respect to different problem domains. Finally, we believe it would be of great interest to better understand and characterize the performance of transductive IDC in settings having both labeled and unlabeled data. Acknowledgements We thank Naftali Tishby and Noam Slonim for helpful discussions and for providing us with the detailed descriptions of the NG20 data sets used in their experiments. We also thank Ron Meir, Yiftach Ravid and the anonymous referees for their constructive comments. This research was supported by the Israeli Ministry of Science References [1] 20 newsgroup data set. http://www.ai.mit.edu/˜jrennie/20 newsgroups/. [2] Libsvm. http://www.csie.ntu.edu.tw/˜cjlin/libsvm. [3] T.M. Cover and J.A. Thomas. Elements of Information Theory. John Wiley & Sons, Inc., 1991. [4] R. El-Yaniv, S. Fine, and N. Tishby. Agnostic classification of markovian sequences. In NIPS97, 1997. [5] I.D. Guedalia, M. London, and M. Werman. A method for on-line clustering of nonstationary data. Neural Computation, 11:521–540, 1999. [6] A.K. Jain and R.C. Dubes. Algorithms for Clustering Data. Prentice-Hall, New Jersey, 1988. [7] J. Lin. Divergence measures based on the shannon entropy. IEEE Transactions on Information Theory, 37(1):145–151, 1991. [8] F.C. Pereira N. Tishby and W. Bialek. Information bottleneck method. In 37-th Allerton Conference on Communication and Computation, 1999. [9] K. Rose. Deterministic annealing for clustering, compression, classification, regression and related optimization problems. Proceedings of the IEEE, 86(11):2210–2238, 1998. [10] N. Slonim and N. Tishby. Agglomerative information bottleneck. In NIPS99, 1999. [11] N. Slonim and N. Tishby. The power of word clustering for text classification. To appear in the European Colloquium on IR Research, ECIR, 2001. [12] Noam Slonim and Naftali Tishby. Document clustering using word clusters via the information bottleneck method. In ACM SIGIR 2000, 2000. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 10 20 30 40 50 60 70 80 90 100 Feature Noise Amplitude Accuracy First Iteration Accuracy Last Iteration Accuracy 0 2 4 6 8 10 12 14 16 18 20 30 40 50 60 70 80 90 100 IDC Iteration Accuracy (a) (b) 2 4 6 8 10 12 14 16 18 20 0 10 20 30 40 50 60 70 80 90 100 Number Of Feature Clusters Accuracy First Iteration Accuracy Last Iteration Accuracy 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 80 90 100 Training Set Size Accuracy First Iteration Accuracy Last Iteration Accuracy (c) (d) 100 150 200 250 300 350 400 450 500 0 10 20 30 40 50 60 70 80 90 100 Test Set Size Accuracy First Iteration Accuracy Last Iteration Accuracy (e) Figure 2: (a) Average accuracy over 10 trials for different amplitudes of proportional feature noise. Data set: A synthetically generated sample of 200 500-dimensional elements in 4 classes. (b) A trace of a single IDC run. The x-axis is the number of IDC iterations and the y-axis is accuracy achieved in each iteration. Data set: Synthetically generated sample of 500, 400-dimensional elements in 5 classes; Noise: Proportional feature noise with α = 1.0; (c) Average accuracy (10 trials) for different numbers of feature clusters. Data set: NG4. (d) Average accuracy of (10 trials of) transductive categorization of 5 newsgroups. Sample size: 80 documents per class, X-axis is training set size. Upper curve shows trans. IDC-15 and lower curve is trans. IDC-1. (e) Average accuracy of (10 trials of) transductive categorization of 5 newsgroups. Sample size: constant training set size of 50 documents from each class. The x-axis counts the number of unlabeled samples to be categorized. Upper curve is trans. IDC-15 and lower curve is trans. IDC-1. Each error bar (in all graphs) specifies one std.
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Multiagent Planning with Factored MDPs Carlos Guestrin Computer Science Dept Stanford University guestrin@cs.stanford.edu Daphne Koller Computer Science Dept Stanford University koller@cs.stanford.edu Ronald Parr Computer Science Dept Duke University parr@cs.duke.edu Abstract We present a principled and efficient planning algorithm for cooperative multiagent dynamic systems. A striking feature of our method is that the coordination and communication between the agents is not imposed, but derived directly from the system dynamics and function approximation architecture. We view the entire multiagent system as a single, large Markov decision process (MDP), which we assume can be represented in a factored way using a dynamic Bayesian network (DBN). The action space of the resulting MDP is the joint action space of the entire set of agents. Our approach is based on the use of factored linear value functions as an approximation to the joint value function. This factorization of the value function allows the agents to coordinate their actions at runtime using a natural message passing scheme. We provide a simple and efficient method for computing such an approximate value function by solving a single linear program, whose size is determined by the interaction between the value function structure and the DBN. We thereby avoid the exponential blowup in the state and action space. We show that our approach compares favorably with approaches based on reward sharing. We also show that our algorithm is an efficient alternative to more complicated algorithms even in the single agent case. 1 Introduction Consider a system where multiple agents, each with its own set of possible actions and its own observations, must coordinate in order to achieve a common goal. We want to find a mechanism for coordinating the agents’ actions so as to maximize their joint utility. One obvious approach to this problem is to represent the system as a Markov decision process (MDP), where the “action” is a joint action for all of the agents and the reward is the total reward for all of the agents. Unfortunately, the action space is exponential in the number of agents, rendering this approach impractical in most cases. Alternative approaches to this problem have used local optimization for the different agents, either via reward/value sharing [11, 13] or direct policy search [10]. We present a novel approach based on approximating the joint value function as a linear combination of local value functions, each of which relates only to the parts of the system controlled by a small number of agents. We show how such factored value functions allow the agents to find a globally optimal joint action using a very natural message passing scheme. We provide a very efficient algorithm for computing such a factored approximation to the true value function using a linear programming approach. This approach is of independent interest, as it is significantly faster and compares very favorably to previous approximate algorithms for single agent MDPs. We also compare our multiagent algorithm to the multiagent reward and value sharing algorithms of Schneider et al. [11], showing that our algorithm achieves superior performance which in fact is close to the achievable optimum for this class of problems. A1 A4 A2 A3 1 Q 2 Q 4 Q 3 Q A3 A3 A4 A4 A2 A2 A1 A1 X1 X3 X4 X2 X3’ X3’ X4’ X2’ X1’ (a) (b) Figure 1: (a) Coordination graph for a 4-agent problem. (b) A DBN for a 4-agent MDP. 2 Cooperative Action Selection We begin by considering a simpler problem of selecting a globally optimal joint action in order to maximize the joint immediate value achieved by a set of agents. Suppose we have a collection of agents, where each agent chooses an action . We use to denote
. Each agent has a local Q function , which represents its local contribution to the total utility function. The agents are jointly trying to maximize . An agent’s local function might be influenced by its action and those of some other agents; we define the scope "!$#&%('*) + $,.-/ to be the set of agents whose action influences . Each 0 may be further decomposed as a linear combination of functions that involve fewer agents; in this case, the complexity of the algorithm may be further reduced. Our task is to select a joint action 1 that maximizes *2 1+3 . The fact that the depend on the actions of multiple agents forces the agents to coordinate their action choices. We can represent the coordination requirements of the system using a coordination graph, where there is a node for each agent and an edge between two agents if they must directly coordinate their actions to optimize some particular 54 . Fig. 1(a) shows the coordination graph for an example where 6 7 82:9; 9=< 3?>@ <*2A9< 9B 3?>CED 2:9; 9 DF3.>C B&2:9 D* 9B 3 . A graph structure suggests the use of a cost network [5], which can be solved using non-serial dynamic programming [1] or a variable elimination algorithm which is virtually identical to variable elimination in a Bayesian network. The key idea is that, rather than summing all functions and then doing the maximization, we maximize over variables one at a time. Specifically, when maximizing over 9HG , only summands involving 9 G participate in the maximization. Let us begin our optimization with agent 4. To optimize B , functions I and D are irrelevant. Hence, we obtain: JLKNM OPRQ O$STQ OTU I 2:9 N 9 <3.>V D 2:9 9 D 3.> JWK8M ORX ) E< 2:9 <8 9 BF3.>VB 2A9 D 9 B3H,H We see that to optimally choose B , the agent must know the values of 0< and D . In effect, it is computing a conditional strategy, with a (possibly) different action choice for each action choice of agents 2 and 3. Agent 4 can summarize the value that it brings to the system in the different circumstances using a new function Y*B 2 Z<&[ D 3 whose value at the point 9 <8 9 D is the value of the internal JLKNM expression. Note that Y8B introduces a new induced communication dependency between agents < and D , the dashed line in Fig. 1(a). Our problem now reduces to computing JLKNM OPRQ OTS$Q OTU 82:9; 9< 3\>]ED 2:9; 9 D3L> Y B&2A9< 9 D3 , having one fewer agent. Next, agent 3 makes its decision, giving: JLKNM O P Q O S ^ 2A9 9 <F3_>`Y D 2A9 N 9 <3 , where Y D 2A9 9 <3E JLKNM O U ) D 2:9 9 D 3_>`Y8 2A9 <* 9 D 3H, . Agent 2 now makes its decision, giving Y*< 2:9 35 JWK8M O S ) ^ 2A9 8 9 <F3a>bY D 2:9 9 <F3H, , and agent 1 can now simply choose the action 9c that maximizes Y JLKNM OP Y <*2A9" 3 . We can recover the maximizing set of actions by performing the entire process in reverse: The maximizing choice for Y selects the action 9+d for agent 1. To fulfill its commitment to agent 1, agent 2 must choose the value 9cd < which maximizes YN< 2A9;d 3 . This, in turn, forces agent 3 and then agent 4 to select their actions appropriately. In general, the algorithm maintains a set of functions, which initially contains
IF?R* . The algorithm then repeats the following steps: (1) Select an uneliminated agent G . (2) Take all Y $[Y whose scope contains G . (3) Define a new function Y0 JLKNM O Y and introduce it into ; the scope of Y is "!$#&%(') Y ,
G . As above, the maximizing action choices are recovered by sending messages in the reverse direction. The cost of this algorithm is linear in the number of new “function values” introduced, or exponential in the induced width of the coordination graph [5]. Furthermore, each agent does not need to communicate directly with every other agent, instead the necessary communication bandwidth will also be the induced width of the graph, further simplifying the coordination mechanism. We note that this algorithm is essentially a special case of the algorithm used to solve influence diagrams with multiple parallel decisions [7] (as is the one in the next section). However, to our knowledge, these ideas have not been applied to the problem of coordinating the decision making process of multiple collaborating agents. 3 One-Step Lookahead We now consider two elaborations to the action selection problem of the previous section. First, we assume that the agents are acting in a space described by a set of discrete state variables,
, where each 4 takes on values in some finite domain # J 2 4 3 . A state defines a value 4Z# J 2 43 for each variable 4 . The scope of the local functions that comprise the value can include both action choices and state variables. We assume that the agents have full observability of the relevant state variables, so by itself, this extension is fairly trivial: The functions define a conditional cost network. Given a particular state , the agents instantiate the state variables and then solve the cost network as in the previous section. However, we note that the agents do not actually need to have access to all of the state variables: agent only needs to observe the variables that are in the scope of its local function, thereby decreasing considerably the amount of information each agent needs to observe. The second extension is somewhat more complicated: We assume that the agents are trying to maximize the sum of an immediate reward and a value that they expect to receive one step in the future. We describe the dynamics of the system using a dynamic decision network (DDN) [4]. Let 4 denote the variable 4 at the current time and 4 the variable at the next step. The transition graph of a DDN is a two-layer directed acyclic graph whose nodes are
N[ Z ? , and where only nodes in
have parents. We denote the parents of 4 in the graph by Parents 2 4 3 . For simplicity of exposition, we assume that Parents 2 4 3
! . (This assumption can be relaxed, but our algorithm becomes somewhat more complex.) Each node " 4 is associated with a conditional probability distribution (CPD) # 2 " 4%$ Parents 2 4 3[3 . The transition probability # 2 $ [1(3 is then defined to be & 4 # 2 4 $(' 4 3 , where ' 4 is the value in [1 of the variables in Parents 2 4 3 . The immediate rewards are a set of functions ) ?) , and the next-step values are a set of functions *?N+*+ . Here, we assume that both ) 4 ’s and * 4 ’s are functions that depend only on a small set of variables. Fig. 1(b) shows a DDN for a simple four-agent problem, where the ovals represent the variables 4 and the rectangles the agent actions. The diamond nodes in the first time step represent the immediate reward, while the * nodes in the second time step represent the future value associated with a subset of the state variables. For any setting of the state variables, , the agents aim to maximize , 2 .3 JLKNM O P Q.-.-.Q O/ ) )$ 2 1+3&> 1032 # 2 $ [1(3*" 2 3H, , i.e., the immediate reward plus the expected value of the next state. The expectation is a summation over an exponential number of future states. As shown in [8], this can be simplified substantially. For example, if we consider the function * 82 3 in Fig. 1(b), we can see that its expected value is a function only of 0N[ Z< . More generally, we define 4 2 [1(3a 0 2 # 2 $ [1(3*" 2 3 . Recall our assumption that the scope of each *+ is only a small subset of variables 5I . Then, the scope of 4N is 2 5E F3 2 2 Parents 2 4 3 . Specifically, 4 2 [1+3 2 # 2
$ 3 2
3 , where is a value of 2 5 3 . Note that the cost of the computation depends linearly on $ Z# J 2 2 5 3[3 $ , which depends on 5 (the scope of * ) and on the complexity of the process dynamics. By replacing the expectation with the backprojection, we can once again generate a set of local functions )T ?> 4 , and apply our coordination graph algorithm unchanged. 4 Markov Decision Processes We now turn our attention to the substantially more complex case where the agents are acting in a dynamic environment, and are jointly trying to maximize their expected longterm return. The Markov Decision Process (MDP) framework formalizes this problem. An MDP is defined as a 4-tuple 2
L5 # 3 where:
is a finite set of $
$ states; is a set of actions; is a reward function
IR, such that 2 1+3 represents the reward obtained in state after taking action 1 ; and # is a Markovian transition model where # 2 $ [1+3 represents the probability of going from state to state with action 1 . We assume that the MDP has an infinite horizon and that future rewards are discounted exponentially with a discount factor ) +! 3 . The optimal value function , d is defined so that the value of a state must be the maximal value achievable by any action at that state. More precisely, we define #" 2 [1+3W 2 1+3 >$ E 032 # 2 $ 1+3 , 2 3 , and the Bellman operator % d to be % d , 2 .3V JLKNM'& "_2 [1+3 . The optimal value function , d is the fixed point of % d : , d (% d , d . A stationary policy ) for an MDP is a mapping )*
( , where ) 2 .3 is the action the agent takes at state . For any value function , , we can define the policy obtained by acting greedily relative to , : Greedy 2 , 3 2 .3a K,+-_JLKNM.& " 2 1+3 . The greedy policy relative to the optimal value function , d is the optimal policy ) d Greedy 2 , d 3 . There are several algorithms for computing the optimal policy. One is via linear programming. Numbering the states in
as 8 #/ , our variables are 0N?01/ , where 0 4 represents , 2 4 3 . The LP is: Minimize: 432 2 4 30 454 Subject to: 0 476 2 4 [1(3.>8 E G # 2 G $ 4 1+390;:=< 4
1 >L The state relevance weights 2 are any convex weights, with 2 2 .3@?A and 0 2 2 .3a B . In our setting, the state space is exponentially large, with one state for each assignment to
$ . We use the common approach of restricting attention to value functions that are compactly represented as a linear combination of basis functions C
* ? * : . A linear value function over C is a function , that can be written as , 2 .3a : ED * &2 .3 for some coefficients F 2 D D : 3 . The LP approach can be adapted to use this value function representation [12]: Variables: D 8 D : 4 Minimize: G2 D 4 Subject to: : HD *" 2 4 3 6 2 4 1+3[> E 0 2 # 2 G $ ?4 1+3+ : HD * &2 G 3 4 < ?4
1 I\ Where 2 0 2 2 ?4H3 * &2 ?4 3 . This transformation has the effect of reducing the number of free variables in the LP to J but the number of constraints remains $
$ $ $ . There is, in general, no guarantee as to the quality of the approximation : ;D * , but recent work of de Farias and Van Roy [3] provides some analysis of the error relative to that of the best possible approximation in the subspace, and some guidance as to selecting the 2 ’s so as to improve the quality of the approximation. 5 Factored MDPs Factored MDPs [2] allow the representation of large structured MDPs by using a dynamic Bayesian network to represent the transition model. Our representation of the one-step transition dynamics in Section 3 is precisely a factored MDP, where we factor not only the states but also the actions. In [8], we proposed the use of factored linear value functions to approximate the value function in a factored MDP. These value functions are a weighted linear combination of basis functions, as above, but each basis function is restricted to depend only on a small subset of state variables. The * functions in Fig. 1(b) are an example. If we had a value function , represented in this way, then we could use our algorithm of Section 3 to implement Greedy 2 , 3 by having the agents use our message passing coordination algorithm at each step. (Here we have only one function * per agent, but our approach extends trivially to the case of multiple * functions.) In previous work [9, 6], we presented algorithms for computing approximate value functions of this form for factored MDPs. These algorithms can circumvent the exponential blowup in the number of state variables, but explicitly enumerate the action space of the MDP, making them unsuitable for the exponentially large action space in multiagent MDPs. We now provide a novel algorithm based on the LP of the previous section. In particular, we show how we can solve this LP exactly in closed form, without explicitly enumerating the exponentially many constraints. Our first task is to compute the coefficients 2 in the objective function. Note that, 2 0 2 2 .3* 2 .3I 2 2 39 2 3 , as basis * has scope restricted to 5 . Here, 2 2 3 represents the marginal of the state relevance weights 2 over 5I . Thus, the coefficients 2 can be pre-computed efficiently if 2 is represented compactly by its marginals 2 2 5? F3 . Our experiments used uniform weights 2 2 .3a , thus, 2 2
3_ . We must now deal with the exponentially large constraint set. Using the backprojection from Section 3, we can rewrite our constraints as: : D *" 2 .3 6 2 1+3.> : D 4 2 1+3 4 <
[1 I where 4 &2 1+3 032 # 2 $ 1+3* 2 3 . Note that this exponentially large set of linear constraints can be replaced by a single, equivalent, non-linear constraint: 6 JLK8M 0=Q & 2 1+3.> : D ) 4 82 [1+3 * *2 .3 , In a factored MDP, the reward function is represented as the sum of local rewards 4 ) 4 . Furthermore, the basis *+ and its backprojection 48 are also functions that depend only on a small set of variables. Thus, the right side of the constraint can be viewed as the sum of restricted scope functions parameterized by F . For a fixed F , we can compute the maximum over
[1 using a cost network, as in Section 2. If F is not specified, the maximization induces a family of cost networks parameterized by F . As we showed in [6], we can turn this cost network into a compact set of LP constraints on the free variable F . More generally, suppose we wish to enforce the constraint 6 JLKNM 2
3 , where 2 3 2
3 such that each has a restricted scope. Here, the superscript F indicates that each might be multiplied by a weight D , but this dependency is linear. Consider the cost network used to maximize ; let Y by any function used in the network, including the original ’s, and let be its scope. For any assignment to , we introduce a variable , whose value represents Y 2 3 , into the linear program. For the initial functions , we include the constraint that 2 3 . As is linear in F , this constraint is linear in the LP variables. Now, consider a new function Y introduced into by eliminating a variable (G . Let Y8Y be the functions extracted from , with scope 8 Offline: 1. Select a set of restricted scope basis functions
. 2. Apply efficient LP-based approximation algorithm offline (Section 5) to compute coefficients of the approximate value function . 3. Use the one-step lookahead planning algorithm (Section 3) with as a value function estimate to compute local functions for each agent. Online: At state : 1. Each agent instantiates with values of state variables in scope of . 2. Agents apply coordination graph algorithm (Section 2) with local functions to coordinate approximately optimal global action. Figure 2: Algorithm for multiagent planning with factored MDPs respectively. As in the cost network, we want that JWK8M ) , where is the value of in the instantiation 2 &G:3 . We enforce this by introducing a set of constraints into our LP: 6 < G . The last function generated in the elimination, Y! , has an empty domain. We introduce the additional constraint 6 " , which is equivalent to the global constraint 6 JWK8M 2 3 . In the case of cooperative multiagent MDPs, the actions of the individual agents become variables in the cost network, so that the set # is simply
. The functions are simply the local functions corresponding to the rewards ) , the bases * and their backprojections 4N . We can thus write down constraints that enforce : ;D *= 2 .3 6 2 1+3 >* E 032 # 2 $ [1(3+ : HD * &2 3 over the entire exponential state space and joint action space using a number of constraints which is only exponential in the induced tree width of the cost network, rather than exponential in the number of actions and state variables in the problem. A traditional single agent is, of course, a special case of the multiagent case. The LP approach described in this section provides an attractive alternative to the methods described in [9] and [6]. In particular, our approach requires that we solve a single LP, whose size is essentially the size of the cost network. The approach of [6] (which is substantially more efficient than that of [9]) requires that we solve an LP for each step in policy iteration, and each LP contains constraints corresponding to multiple cost networks (whose number depends on the complexity of the policy representation). Furthermore, the LP approach eliminates the restrictions on the action model made in [9, 6]. Our overall algorithm for multiagent planning with factored MDPs in shown in Fig. 2. 6 Experimental results We first validate our approximate LP approach by comparing the quality of the solution to the approximate policy iteration (PI) approach of [6]. As the approximate PI algorithm is not able to deal with the exponentially large action spaces of multiagent problems, we compare these two approaches on the single agent SysAdmin problem presented in [6], on a unidirectional ring network of up to 32 machines (over 4 billion states). As shown in Fig. 3(b), our new approximate LP algorithm for factored MDPs is significantly faster than the approximate PI algorithm. In fact, approximate PI with single-variable basis functions variables is more costly than the LP approach using basis functions over consecutive triples of variables. As shown in Fig. 3(c), for singleton basis functions, the approximate PI policy obtains slightly better performance for some problem sizes. However, as we increase the number of basis functions for the approximate LP formulation, the value of the resulting policy is much better. Thus, in this problem, our new approximate linear programming formulation allows us to use more basis functions and to obtain a resulting policy of higher value, while still maintaining a faster running time. We constructed a multiagent version of the SysAdmin problem, applied to various netUnidirectional Ring Server Star Ring of Rings 0 20 40 60 80 100 120 140 160 180 200 0 5 10 15 20 25 30 35 number of machines Total running time (minutes) PI single basis LP single basis LP pair basis LP triple basis 0 100 200 300 400 0 10 20 30 40 number of machines Discounted reward of final policy (averaged over 50 trials of 100 steps) PI single basis LP single basis LP pair basis LP triple basis Figure 3: (a) Network topologies used in our experiments. Graphs: Approximate LP versus approximate PI on single agent SysAdmin on unidirectional ring: (b) running time; (c) estimated value of policy. 0 20 40 60 80 100 120 140 160 180 2 4 6 8 10 12 14 16 Number of agents Total running time (seconds) Star: LP Single basis Star: LP Pair basis Ring of Rings: LP Single basis 3.4 3.6 3.8 4 4.2 4.4 2 4 6 8 10 12 14 16 Number of agents Estimated value per agent (100 runs) LP Single basis LP Pair basis Distributed value function Distributed reward Utopic maximum value 3.2 3.4 3.6 3.8 4 4.2 4.4 5 10 15 20 25 30 Number of agents Estimate value per agent (100 runs) LP Single basis Distributed reward Distributed value function Utopic maximum value Figure 4: (a) Running time for approximate LP for increasing number of agents. Policy performance of approximate LP and DR/DRF: (b) on “star”; (c) on “ring of rings”. work architectures shown in Fig. 3(a). Each machine is associated with an agent 4 and two variables: Status .4
good, faulty, dead , and Load 4
idle, loaded, process successful . A dead machine increases the probability that its neighbors will become faulty and die. The system receives a reward of 1 if a process terminates successfully. If the Status is faulty, processes take longer to terminate. If the machine dies, the process is lost. Each agent 4 must decide whether machine should be rebooted, in which case the Status becomes good and any running process is lost. For a network of machines, the number of states in the MDP is and the joint action space contains possible actions, e.g., a problem with agents has over < states and a billion possible actions. We implemented the factored approximate linear programming and the message passing coordination algorithms in Matlab, using CPLEX as the LP solver. We experimented with two types of basis functions: “single”, which contains an indicator basis function for each value of each 4 and 4 ; and “pair” which, in addition, contains indicators over joint assignments of the Status variables of neighboring agents. We use ; . As shown in Fig. 4(a), the total running time of the algorithm grows linearly in the number of agents, for each fixed network and basis type. This is the expected asymptotic behavior, as each problem has a fixed induced tree width of the cost network. (The induced tree width for pair basis on the “ring of rings” problem was too large.) For comparison, we also implemented the distributed reward (DR) and distributed value function (DRF) algorithms of Schneider et al. [11]. Here we used 10000 learning iterations, with learning and exploration rates starting at ; and * respectively and a decaying schedule after 5000 iterations; the observations for each agent were the status and load of its machine. The results of the comparison are shown in Fig. 4(b) and (c). We also computed a utopic upper bound on the value of the optimal policy by removing the (negative) effect of the neighbors on the status of the machines. This is a loose upper bound, as a dead neighbor increases the probability of a machine dying by about , . For both network topologies tested, the estimated value of the approximate LP solution using single basis was significantly higher than that of the DR and DRF algorithms. Note that the single basis solution requires no coordination when acting, so this is a “fair” comparison to DR and DRF which also do not communicate while acting. If we allow for pair bases, which implies agent communication, we achieve a further improvement in terms of estimated value. The policies obtained tended to be intuitive: e.g., for the “star” topology with pair basis, if the server becomes faulty, it is rebooted even if loaded. but for the clients, the agent waits until the process terminates or the machine dies before rebooting. 7 Conclusion We have provided principled and efficient approach to planning in multiagent domains. Rather than placing a priori restrictions on the communication structure between agents, we first choose the form of an approximate value function and derive the optimal communication structure given the value function architecture. This approach provides a unified view of value function approximation and agent communication. We use a novel linear programming technique to find an approximately optimal value function. The inter-agent communication and the LP avoid the exponential blowup in the state and action spaces, having computational complexity dependent, instead, upon the induced tree width of the coordination graph used by the agents to negotiate their action selection. By exploiting structure in both the state and action spaces, we can deal with considerably larger MDPs than those described in previous work. In a family of multiagent network administration problems with over < states and over a billion actions, we have demonstrated near optimal performance which is superior to a priori reward or value sharing schemes. We believe the methods described herein significantly further extend the efficiency, applicability and general usability of factored value functions and models for the control of dynamic systems. Acknowledgments: This work was supported by ONR under MURI “Decision Making Under Uncertainty”, the Sloan Foundation, and the first author was also supported by a Siebel scholarship. References [1] U. Bertele and F. Brioschi. Nonserial Dynamic Programming. Academic Press, 1972. [2] C. Boutilier, T. Dean, and S. Hanks. Decision theoretic planning: Structural assumptions and computational leverage. Journal of Artificial Intelligence Research, 11:1 – 94, 1999. [3] D.P. de Farias and B. Van Roy. The linear programming approach to approximate dynamic programming. submitted to the IEEE Transactions on Automatic Control, January 2001. [4] T. Dean and K. Kanazawa. A model for reasoning about persistence and causation. Computational Intelligence, 5(3):142–150, 1989. [5] R. Dechter. Bucket elimination: A unifying framework for reasoning. Artificial Intelligence, 113(1–2):41–85, 1999. [6] C. Guestrin, D. Koller, and R. Parr. Max-norm projections for factored MDPs. In Proc. 17th IJCAI, 2001. [7] F. Jensen, F. Jensen, and S. Dittmer. From influence diagrams to junction trees. In Uncertainty in Artificial Intelligence: Proceedings of the Tenth Conference, pages 367–373, Seattle, Washington, July 1994. Morgan Kaufmann. [8] D. Koller and R. Parr. Computing factored value functions for policies in structured MDPs. In Proceedings of the Sixteenth International Joint Conference on Artificial Intelligence (IJCAI99). Morgan Kaufmann, 1999. [9] D. Koller and R. Parr. Policy iteration for factored MDPs. In Proc. 16th UAI, 2000. [10] L. Peshkin, N. Meuleau, K. Kim, and L. Kaelbling. Learning to cooperate via policy search. In Proc. 16th UAI, 2000. [11] J. Schneider, W. Wong, A. Moore, and M. Riedmiller. Distributed value functions. In Proc. 16th ICML, 1999. [12] P. Schweitzer and A. Seidmann. Generalized polynomial approximations in Markovian decision processes. Journal of Mathematical Analysis and Applications, 110:568 – 582, 1985. [13] D. Wolpert, K. Wheller, and K. Tumer. General principles of learning-based multi-agent systems. 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ACh, Uncertainty, and Cortical Inference Peter Dayan Angela Yu Gatsby Computational Neuroscience Unit 17 Queen Square, London, England, WC1N 3AR. dayan@gatsby.ucl.ac.uk feraina@gatsby.ucl.ac.uk Abstract Acetylcholine (ACh) has been implicated in a wide variety of tasks involving attentional processes and plasticity. Following extensive animal studies, it has previously been suggested that ACh reports on uncertainty and controls hippocampal, cortical and cortico-amygdalar plasticity. We extend this view and consider its effects on cortical representational inference, arguing that ACh controls the balance between bottom-up inference, influenced by input stimuli, and top-down inference, influenced by contextual information. We illustrate our proposal using a hierarchical hidden Markov model. 1 Introduction The individual and joint computational roles of neuromodulators such as dopamine, serotonin, norepinephrine and acetylcholine are currently the focus of intensive study.5,7,9–11,16,27 A rich understanding of the effects of neuromodulators on the dynamics of networks has come about through work in invertebrate systems.21 Further, some general computational ideas have been advanced, such as that they change the signal to noise ratios of cells. However, more recent studies, particularly those focusing on dopamine,26 have concentrated on specific computational tasks. ACh was one of the first neuromodulators to be attributed a specific role. Hasselmo and colleagues,10,11 in their seminal work, proposed that cholinergic (and, in their later work, also GABAergic12) modulation controls read-in to and readout from recurrent, attractor-like memories, such as area CA3 of the hippocampus. Such memories fail in a characteristic manner if the recurrent connections are operational during storage, thus forcing new input patterns to be mapped to existing memories. Not only would these new patterns lose their specific identity, but, worse, through standard synaptic plasticity, the size of the basin of attraction of the offending memory would actually be increased, making similar problems more likely. Hasselmo et al thus suggested, and collected theoretical and experimental evidence in favor of, the notion that ACh (from the septum) should control the suppression and plasticity of specific sets of inputs to CA3 neurons. During read-in, high levels of ACh would suppress the recurrent synapses, but make them readily plastic, so that new memories would be stored without being pattern-completed. Then, during read-out, low levels of ACh would boost the impact of the recurrent weights (and reduce their plasticity), allowing auto-association to occur. The ACh signal to the hippocampus can be characterized as reporting the unfamiliarity of the input with which its release is associated. This is analogous to its characterization as reporting the uncertainty associated with predictions in theories of attentional influences over learning in classical conditioning.4 In an extensive series of investigations in rats, Holland and his colleagues14,15 have shown that a cholinergic projection from the nucleus basalis to the (parietal) cortex is important when animals have to devote more learning (which, in conditioning, is essentially synonymous with paying incremental attention) to stimuli about whose consequences the animal is uncertain.20 We have4 interpreted this in the statistical terms of a Kalman filter, arguing that the ACh signal reported this uncertainty, thus changing plasticity appropriately. Note, however, that unlike the case of the hippocampus, the mechanism of action of ACh in conditioning is not well understood. In this paper, we take the idea that ACh reports on uncertainty one step farther. There is a wealth of analysis-by-synthesis unsupervised learning models of cortical processing.1,3,8,13,17,19,23 In these, top-down connections instantiate a generative model of sensory input; and bottom-up connections instantiate a recognition model, which is the statistical inverse of the generative model, and maps inputs into categories established in the generative model. These models, at least in principle, permit stimuli to be processed according both to bottom-up input and top-down expectations, the latter being formed based on temporal context or information from other modalities. Top-down expectations can resolve bottom-up ambiguities, permitting better processing. However, in the face of contextual uncertainty, top-down information is useless. We propose that ACh reports on top-down uncertainty, and, as in the case of area CA3, differentially modulates the strength of synaptic connections: comparatively weakening those associated with the top-down generative model, and enhancing those associated with bottom-up, stimulus-bound information.2 Note that this interpretation is broadly consistent with existent electrophysiology data, and documented effects on stimulus processing of drugs that either enhance (eg cholinesterase inhibitors) or suppress (eg scopolamine) the action of ACh.6,25,28 There is one further wrinkle. In exact bottom-up, top-down, inference using a generative model, top-down contextual uncertainty does not play a simple role. Rather, all possible contexts are treated simultaneously according to the individual posterior probabilities that they currently pertain. Given the neurobiologically likely scenario in which one set of units has to be used to represent all possible contexts, this exact inferential solution is not possible. Rather, we propose that a single context is represented in the activities of high level (presumably pre-frontal) cortical units, and uncertainty associated with this context is represented by ACh. This cholinergic signal then controls the balance between bottom-up and top-down influences over inference. In the next section, we describe the simple hierarchical generative model that we use to illustrate our proposal. The ACh-based recognition model is introduced in section 3 and discussed in section 4. 2 Generative and Recognition Models Figure 1A shows a very simple case of a hierarchical generative model. The generative model is a form of hidden Markov model (HMM), with a discrete hidden state , which will capture the idea of a persistent temporal context, and a twodimensional, real-valued, output . Crucially, there is an extra layer, between and . The state is stochastically determined from , and controls which of a set of 2d Gaussians (centered at the corners of the unit square) is used to generate . In this austere case, is the model’s representation of , and the key inference −1 0 1 2 −1 0 1 2 1 2 3 4 0 200 400 1 2 3 4 −1 0 1 2 −1 0 1 2
! Figure 1: Generative model. A) Three-layer model "$#&%('*),+.-/102#3%4'*)5+4-6/178#,9;: with dynamics ( < ) in the " layer ( =8> "?A@B"C?ED FHG.@BI.J K(L ), a probabilistic mapping ( M ) from ";NO0 ( =8> 0(? @P"C?RQ "?SGT@UI.J L V ), and a Gaussian model WX> 7YQ 04G with means at the corners of the unit square and standard deviation IZJ V in each direction. The model is rotationally invariant; only some of the links are shown for convenience. B) Sample sequence showing the slow dynamics in " ; the stochastic mapping into 0 and the substantial overlap in 7 (different symbols show samples from the different Gaussians shown in A). problem will be to determine the distribution over which generated , given the past experience [ ]\^`_ba ^4cdd d c ]\ ^fe and itself. Figure 1B shows an example of a sequence of gihih steps generated from the model. The state in the layer stays the same for an average of about jZh timesteps; and then switches to one of the other states, chosen equally at random. The transition matrix is kTlnmSoqpRlRm . The state in the layer is more weakly determined by the state in the layer, with a probability of only j*r(g that !_ . The stochastic transition from to is governed by the transition matrix s l mHtum . Finally, is generated as a Gaussian about a mean specified by . The standard deviation of these Gaussians ( h dv in each direction) is sufficiently large that the densities overlap substantially. The naive solution to inferring is to use only the likelihood term (ie only the probabilities wyx Cz H{ ). The performance of this is likely to be poor, since the Gaussians in for the different values of overlap so much. However, and this is why it is a paradigmatic case for our proposal, contextual information, in this case past experience, can help to determine . We show how the putative effect of ACh in controlling the balance between bottom-up and top-down inference in this model can be used to build a good approximate inference model. In order to evaluate our approximate model, we need to understand optimal inference in this case. Figure 2A shows the standard HMM inference model, which calculates the exact posteriors wyx Cz [ H{ and wyx Cz [ H{ . This is equivalent to just the forward part of the forwards-backwards algorithm22 (since we are not presently interested in learning the parameters of the model). The adaptation to include the layer is straightforward. Figures 3A;D;E show various aspects of exact inference for a particular run. The histograms in figure 3A show that wyx z [ { captures quite well the actual states !| that generated the data. The upper plot shows the posterior probabilities of the actual states in the sequence – these should be, and are, usually high; the lower histogram the posterior probability of the other possible states; these should be, and are, usually low. Figure 3D shows the actual state sequence | ; figure 3E shows the states that are individually most likely at each time step (note that this is not the maximum likelihood state sequence, as found by the Viterbi algorithm, for instance).
"!# $ % $ & ' ( ) * Figure 2: Recognition models. A) Exact recognition model. =8> " ?ED F Q + ?ED F G is propagated to provide the prior =8> "? Q +`?ED F G (shown by the lengths of the thick vertical bars) and thus the prior =8> 0(? Q +`?ED F G . This is combined with the likelihood term from the data 7? to give the true =8> 0 ? Q + ? G . B) Bottom-recognition model uses only a generic prior over 0 ? (which conveys no information), and so the likelihood term dominates. C) ACh model. A single estimated state , "?ED F is used, in conjunction with its certainty -6? D F , reported by cholinergic activity, to produce an approximate prior , =8>., "C?nQ/, "C?ED F]G over "C? (which is a mixture of a delta function and a uniform), and thus an approximate prior over 0 ? . This is combined with the likelihood to give an approximate , =8> 0f? Q +`? G , and a new cholinergic signal -? is calculated. 0 1 0 0.25 0 1 0 0.8 0 1 0 1 0 1 0 1 0 400 1 2 3 4 0 400 1 2 3 4 0 400 1 2 3 4 021436587 9;: 021=< 3 5 7 9;: > ? @ ABC?ED=> 5 ?EF&G
HI?J 021 > 7 9K: L > M N O P Q R M N O P Q R L 0213S7 9K: 0UTE13S7 9K: V W X Y Z [ A \]HI^ A'\]H_^ A'\]H_^ Figure 3: Exact and approximation recognition. A) Histograms of the exact posterior distribution =8> 0!Q +`G over the actual state 0a` ? (upper) and the other possible states 0Kb @ 0c` ? (lower, written =8>'d 0 ` G ). This shows the quality of exact representational inference. B;C) Comparison of the purely bottom up =e4> 0f? Q 7 ?SG (B) and the ACh-based approximation , =8> 0(?nQ + G (C) with the true =8> 0(? Q +`G across all values of 0 . The ACh-based approximation is substantially more accurate. D) Actual " ? . E) Highest probability " state from the exact posterior distribution. F) Single , " state in the ACh model. Figure 2B shows a purely bottom up model that only uses the likelihood terms to infer the distribution over . This has wKfix z ]{ _hg x Cz H{ rji where i is a normalization factor. Figure 3B shows the representational performance of this model, through a scatter-plot of wkfZx z { against the exact posterior wyx z [ { . If bottom-up inference was correct, then all the points would lie on the line of equality – the bow-shape shows that purely bottom-up inference is relatively poor. Figure 4C shows this in a different way, indicating the difference between the average summed log probabilities of the actual states under the bottom up model and those under the true posterior. The larger and more negative the difference, the worse the approximate inference. Averaging over l hZhih runs, the difference is m_n4h log units (compared with a total log likelihood under the exact model of mpoql h ). 3 ACh Inference Model Figure 2C shows the ACh-based approximate inference model. The information about the context comes in the form of two quantities: ]\^ , the approximated contextual state having seen [ ]\^ , and ]\ ^ , which is the measure of uncertainty in that contextual state. The idea is that ]\ ^ is reported by ACh, and is used to control (indicated by the filled-in ellipse) the extent to which top-down information based on ]\^ is used to influence inference about . If we were given the full exact posterior distribution wyx ]\^ c ]\^ z [ ]\ ^ { , then one natural definition for this ACh signal would be the uncertainty in the most likely contextual state ]\^,_ lm l wyx ]\ ^`_ Xz [ ]\ ^ { (1) Figure 4A shows the resulting ACh signal for the case of figure 3. As expected, ACh is generally high at times when the true state | is changing, and decreases during the periods that | is constant. During times of change, top-down information is confusing or potentially incorrect, and so bottom-up information should dominate. This is just the putative inferential effect of ACh. However, the ACh signal of figure 4A was calculated assuming knowledge of the true posterior, which is unreasonable. The model of figure 2C includes the key approximation that the only other information from [ ]\ ^ about the state of is in the single choice of context variable ]\ ^ . The full approximate inference algorithm becomes wyx ]\ ^ ]\^u{_ ]\ ^ r
tlm ]\ ^ l m oqp [ ]\^ approximation (2) w x Cz ]\ ^ ]\^u{_ l wyx ]\^!_ ]\^ { k lul m prior over (3) wyx uc z ]\ ^ ]\^u{_ wyx Cz ]\ ^ ]\^u{ s l m tum propagation to (4) wyx c z [ { wyx c z ]\^ ]\ ^ { wyx z { conditioning (5) wyx z [ {_ l wyx c _ z [ { marginalization (6) wyx z [ {_ t w x _ c z [ { marginalization (7) _ argmax l wyx _ Xz [ { contextual inference (8) _ lm !"Al wyx _ Xz [ { ACh level (9) where uc ]\^4c ]\^ are used as approximate sufficient statistics for [ , the number of states is
t (here
t _ g ), $#&% is the Kronecker delta, and the constant of proportionality in equation 5 normalizes the full conditional distribution. The last two lines show the information that is propagated to the next time step; equation 6 shows the representational answer from the model, the distribution over given [ . These computations are all local and straightforward, except for the representation and normalization of the joint distribution over and , a point to which we return later. Crucially, ACh exerts its influence through equation 2. If ]\ ^ is high, then the input stimulus controlled, likelihood term dominates in the conditioning process (equation 5); if ]\^ is low, then temporal context ( ]\^ ) and the likelihood terms balance. One potentially dangerous aspect of this inference procedure is that it might get unreasonably committed to a single state ]\^ _ _Uddd because it does not represent explicitly the probability accorded to the other possible values of ]\^ given [ ]\^ . A natural way to avoid this is to bound the ACh level from below by a constant, ' , making approximate inference slightly more stimulus-bound than exact inference. This approximation should add robustness. In practice, rather than use equation 9, we use $_ '( lm' lm l wyx ._ z [ {) (10) 0 0.5 1 0 50 100 150 200 250 300 350 400 0 0.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −70 −50 −30 −10
! #"%$'&( )+* -,.,./10 23 45 67 89 & :23;'23 8<= >
: ? @BA 0CED #"GFIH.*KJ @BA 0CL M-"GFNH'*KJ Figure 4: ACh model. A) ACh level from the exact posterior for one run. B) ACh level ? in the approximate model in the same run. Note the coarse similarity between A and B. C) Solid: the mean extra representational cost for the true state 0 ` ? over that in the exact posterior using the ACh model as a function of the minimum allowed ACh level O . Dashed: the same quantity for the pure bottom-up model (which is equivalent to the approximate model for O`@ ' ). Errorbars (which are almost invisible) show standard errors of the means over ' I IfI trials. Figure 4B shows the approximate ACh level for the same case as figure 4A, using ' _ h d l . Although the detailed value of this signal is clearly different from that arising from an exact knowledge of the posterior probabilities (in figure 4A), the gross movements are quite similar. Note the effect of ' in preventing the ACh level from dropping to h . Figure 3C shows that the ACh-based approximate posterior values w x z [ { are much closer to the true values than for the purely bottom-up model, particularly for values of wyx z [ ]{ near h and l , where most data lie. Figure 3F shows that inference about is noisy, but the pattern of true values | is certainly visible. Figure 4C shows the effect of changing ' on the quality of inference about the true states | . This shows differences between approximate and exact log probabilities of the true states | , averaged over l hihZh cases. If ' _ l , then inference is completely stimulus-bound, just like the purely bottom-up model; values of ' less than h d o appear to do well for this and other settings of the parameters of the problem. An upper bound on the performance of approximate inference can be calculated in three steps by: i) using the exact posterior to work out and , ii) using these values to approximate wyx { as in equation 2, and iii) using this approximate distribution in equation 4 and the remaining equations. The average resulting cost (ie the average resulting difference from the log probability under exact inference) is m j d v log units. Thus, the ACh-based approximation performs well, and much better than purely bottom-up inference. 4 Discussion We have suggested that one of the roles of ACh in cortical processing is to report contextual uncertainty in order to control the balance between stimulus-bound, bottom-up, processing, and contextually-bound, top-down processing. We used the example of a hierarchical HMM in which representational inference for a middle layer should correctly reflect such a balance, and showed that a simple model of the drive and effects of ACh leads to competent inference. This model is clearly overly simple. In particular, it uses a localist representation for the state , and so exact inference would be feasible. In a more realistic case, distributed representations would be used at multiple levels in the hierarchy, and so only one single context could be entertained at once. Then, it would also not be possible to represent the degree of uncertainty using the level of activities of the units representing the context, at least given a population-coded representation. It would also be necessary to modify the steps in equations 4 and 5, since it would be hard to represent the joint uncertainty over representations at multiple levels in the hierarchy. Nevertheless, our model shows the feasibility of using an ACh signal in helping propagate and use approximate information over time. Since it is straightforward to administer cholinergic agonists and antagonists, there are many ways to test aspects of this proposal. We plan to start by using the paradigm of Ress et al,24 which uses fMRI techniques to study bottom-up and top-down influences on the detection of simple visual targets. Preliminary simulation studies indicate that a hidden Markov model under controllable cholinergic modulation can capture several aspects of existent data on animal signal detection tasks.18 Acknowledgements We are very grateful to Michael Hasselmo, David Heeger, Sham Kakade and Szabolcs K´ali for helpful discussions. Funding was from the Gatsby Charitable Foundation and the NSF. Reference [28] is an extended version of this paper. References [1] Carpenter, GA & Grossberg, S, editors (1991) Pattern Recognition by SelfOrganizing Neural Networks. Cambridge, MA: MIT Press. [2] Dayan, P (1999). Recurrent sampling models for the Helmholtz machine. Neural Computation, 11:653-677. [3] Dayan, P, Hinton, GE, Neal, RM & Zemel, RS (1995) The Helmholtz machine. Neural Computation 7:889-904. [4] Dayan, P, Kakade, S & Montague, PR (2000). Learning and selective attention. Nature Neuroscience, 3:1218-1223. [5] Doya, K (1999) Metalearning, neuromodulation and emotion. The 13th Toyota Conference on Affective Minds, 46-47. [6] Everitt, BJ & Robbins, TW (1997) Central cholinergic systems and cognition. Annual Review of Psychology 48:649-684. [7] Fellous, J-M, Linster, C (1998) Computational models of neuromodulation. Neural Computation 10:771-805. [8] Grenander, U (1976-1981) Lectures in Pattern Theory I, II and III: Pattern Analysis, Pattern Synthesis and Regular Structures. Berlin:Springer-Verlag. [9] Hasselmo, ME (1995) Neuromodulation and cortical function: Modeling the physiological basis of behavior. Behavioural Brain Research 67:1-27. [10] Hasselmo, M (1999) Neuromodulation: acetylcholine and memory consolidation. Trends in Cognitive Sciences 3:351-359. [11] Hasselmo, ME & Bower, JM (1993) Acetylcholine and memory. Trends in Neurosciences 16:218-222. [12] Hasselmo, ME, Wyble, BP & Wallenstein, GV (1996) Encoding and retrieval of episodic memories: Role of cholinergic and GABAergic modulation in the hippocampus. Hippocampus 6:693-708. [13] Hinton, GE, & Ghahramani, Z (1997) Generative models for discovering sparse distributed representations. Philosophical Transactions of the Royal Society of London. B352:1177-1190. [14] Holland, PC (1997) Brain mechanisms for changes in processing of conditioned stimuli in Pavlovian conditioning: Implications for behavior theory. Animal Learning & Behavior 25:373-399. [15] Holland, PC & Gallagher, M (1999) Amygdala circuitry in attentional and representational processes. Trends In Cognitive Sciences 3:65-73. [16] Kakade, S & Dayan, P (2000). Dopamine bonuses. In TK Leen, TG Dietterich & V Tresp, editors, NIPS 2000. [17] MacKay, DM (1956) The epistemological problem for automata. In CE Shannon & J McCarthy, editors, Automata Studies. Princeton, NJ: Princeton University Press, 235-251. [18] McGaughy, J, Kaiser, T, & Sarter, M. (1996). Behavioral vigilance following infusions of 192 IgG=saporin into the basal forebrain: selectivity of the behavioral impairment and relation to cortical AChE-positive fiber density. Behavioral Neuroscience 110: 247-265. [19] Mumford, D (1994) Neuronal architectures for pattern-theoretic problems. In C Koch & J Davis, editors, Large-Scale Theories of the Cortex. Cambridge, MA:MIT Press, 125-152. [20] Pearce, JM & Hall, G (1980) A model for Pavlovian learning: Variation in the effectiveness of conditioned but not unconditioned stimuli. Psychological Review 87:532-552. [21] Pfluger, HJ (1999) Neuromodulation during motor development and behavior. Current Opinion in Neurobiology 9:683-689. [22] Rabiner, LR (1989) A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE 77:257-286. [23] Rao, RPN & Ballard, DH (1997) Dynamic model of visual recognition predicts neural response properties in the visual cortex. Neural Computation 9:721-763. [24] Ress, D, Backus, BT & Heeger, DJ (2000) Activity in primary visual cortex predicts performance in a visual detection task. Nature Neuroscience 3:940-945. [25] Sarter, M, Bruno, JP (1997) Cognitive functions of cortical acetylcholine: Toward a unifying hypothesis. Brain Research Reviews 23:28-46. [26] Schultz, W (1998) Predictive reward signal of dopamine neurons. Journal of Neurophysiology 80:1–27. [27] Schultz, W, Dayan, P & Montague, PR (1997). A neural substrate of prediction and reward. Science, 275, 1593-1599. [28] Yu, A & Dayan, P (2002). Acetylcholine in cortical inference. Submitted to Neural Networks.
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Categorization by Learning and Combining Object Parts Bernd Heisele Thomas Serre Massimiliano Pontil Thomas Vetter Tomaso Poggio Center for Biological and Computational Learning, M.I.T., Cambridge, MA, USA Honda R&D Americas, Inc., Boston, MA, USA Department of Information Engineering, University of Siena, Siena, Italy Computer Graphics Research Group, University of Freiburg, Freiburg, Germany heisele,serre,tp @ai.mit.edu pontil@ing.unisi.it vetter@informatik.uni-freiburg.de Abstract We describe an algorithm for automatically learning discriminative components of objects with SVM classifiers. It is based on growing image parts by minimizing theoretical bounds on the error probability of an SVM. Component-based face classifiers are then combined in a second stage to yield a hierarchical SVM classifier. Experimental results in face classification show considerable robustness against rotations in depth and suggest performance at significantly better level than other face detection systems. Novel aspects of our approach are: a) an algorithm to learn component-based classification experts and their combination, b) the use of 3-D morphable models for training, and c) a maximum operation on the output of each component classifier which may be relevant for biological models of visual recognition. 1 Introduction We study the problem of automatically synthesizing hierarchical classifiers by learning discriminative object parts in images. Our motivation is that most object classes (e.g. faces, cars) seem to be naturally described by a few characteristic parts or components and their geometrical relation. Greater invariance to viewpoint changes and robustness against partial occlusions are the two main potential advantages of component-based approaches compared to a global approach. The first challenge in developing component-based systems is how to choose automatically a set of discriminative object components. Instead of manually selecting the components, it is desirable to learn the components from a set of examples based on their discriminative power and their robustness against pose and illumination changes. The second challenge is to combine the component-based experts to perform the final classification. 2 Background Global approaches in which the whole pattern of an object is used as input to a single classifier were successfully applied to tasks where the pose of the object was fixed. In [6] Haar wavelet features are used to detect frontal and back views of pedestrians with an SVM classifier. Learning-based systems for detecting frontal faces based on a gray value features are described in [14, 13, 10, 2]. Component-based techniques promise to provide more invariance since the individual components vary less under pose changes than the whole object. Variations induced by pose changes occur mainly in the locations of the components. A component-based method for detecting faces based on the empirical probabilities of overlapping rectangular image parts is proposed in [11]. Another probabilistic approach which detects small parts of faces is proposed in [4]. It uses local feature extractors to detect the eyes, the corner of the mouth, and the tip of the nose. The geometrical configuration of these features is matched with a model configuration by conditional search. A related method using statistical models is published in [9]. Local features are extracted by applying multi-scale and multi-orientation filters to the input image. The responses of the filters on the training set are modeled as Gaussian distributions. In [5] pedestrian detection is performed by a set of SVM classifiers each of which was trained to detect a specific part of the human body. In this paper we present a technique for learning relevant object components. The technique starts with a set of small seed regions which are gradually grown by minimizing a bound on the expected error probability of an SVM. Once the components have been determined, we train a system consisting of a two-level hierarchy of SVM classifiers. First, component classifiers independently detect facial components. Second, a combination classifier learns the geometrical relation between the components and performs the final detection of the object. 3 Learning Components with Support Vector Machines 3.1 Linear Support Vector Machines Linear SVMs [15] perform pattern recognition for two-class problems by determining the separating hyperplane with maximum distance to the closest points in the training set. These points are called support vectors. The decision function of the SVM has the form:
(1) where is the number of data points and ! #"%$ $ is the class label of the data point & . The coefficients are the solution of a quadratic programming problem. The margin ' is the distance of the support vectors to the hyperplane, it is given by: ' $ ( ) * (2) The margin is an indicator of the separability of the data. In fact, the expected error probability of the SVM, +-,.0/1/ , satisfies the following bound [15]: +2,3.4/5/%6 $ +87:9<; ' ;>= (3) where 9 is the diameter of the smallest sphere containing all data points in the training set. 3.2 Learning Components Our method automatically determines rectangular components from a set of object images. The algorithm starts with a small rectangular component located around a pre-selected point in the object image (e.g. for faces this could be the center of the left eye). The component is extracted from each object image to build a training set of positive examples. We also generate a training set of background patterns that have the same rectangular shape as the component. After training an SVM on the component data we estimate the performance of the SVM based on the upper bound on the error probability. According to Eq. (3) we calculate: 9<; ' ; * (4) As shown in [15] this quantity can be computed by solving a quadratic programming problem. After determining we enlarge the component by expanding the rectangle by one pixel into one of the four directions (up, down, left, right). Again, we generate training data, train an SVM and determine . We do this for expansions into all four directions and finally keep the expansion which decreases the most. This process is continued until the expansions into all four directions lead to an increase of . In order to learn a set of components this process can be applied to different seed regions. 4 Learning Facial Components Extracting face patterns is usually a tedious and time-consuming work that has to be done manually. Taking the component-based approach we would have to manually extract each single component from all images in the training set. This procedure would only be feasible for a small number of components. For this reason we used textured 3-D head models [16] to generate the training data. By rendering the 3-D head models we could automatically generate large numbers of faces in arbitrary poses and with arbitrary illumination. In addition to the 3-D information we also knew the 3-D correspondences for a set of reference points shown in Fig. 1a). These correspondences allowed us to automatically extract facial components located around the reference points. Originally we had seven textured head models acquired by a 3-D scanner. Additional head models were generated by 3-D morphing between all pairs of the original head models. The heads were rotated between " and in depth. The faces were illuminated by ambient light and a single directional light pointing towards the center of the face. The position of the light varied between " and in azimuth and between and in elevation. Overall, we generated 2,457 face images of size 58 58. Some examples of synthetic face images used for training are shown in Fig. 1b). The negative training set initially consisted of 10,209 58 58 non-face patterns randomly extracted from 502 non-face images. We then applied bootstrapping to enlarge the training data by non-face patterns that look similar to faces. To do so we trained a single linear SVM classifier and applied it to the previously used set of 502 non-face images. The false positives (FPs) were added to the non-face training data to build the final training set of size 13,654. We started with fourteen manually selected seed regions of size 5 5. The resulting components were located around the eyes (17 17 pixels), the nose (15 20 pixels), the mouth (31 15 pixels), the cheeks (21 20 pixels), the lip (13 16 pixels), the nostrils (
$ pixels), the corners of the mouth (18 11 pixels), the eyebrows (19 15 pixels), and the bridge of the nose (18 16 pixels). a) b) Figure 1: a) Reference points on the head models which were used for 3-D morphing and automatic extraction of facial components. b) Examples of synthetic faces. 5 Combining Components An overview of our two-level component-based classifier is shown in Fig. 2. On the first level the component classifiers independently detect components of the face. Each classifier was trained on a set of facial components and on a set of non-face patterns generated from the training set described in Section 4. On the second level the combination classifier performs the detection of the face based on the outputs of the component classifiers. The maximum real-valued outputs of each component classifier within rectangular search regions around the expected positions of the components are used as inputs to the combination classifier. The size of the search regions was estimated from the mean and the standard deviation of the locations of the components in the training images. The maximum operation is performed both during training and at run-time. Interestingly it turns out to be similar to the key pooling mechanism postulated in a recent model of object recognition in the visual cortex [8]. We also provide the combination classifier with the precise positions of the detected components relative to the upper left corner of the 58 58 window. Overall we have three values per component classifier that are propagated to the combination classifier: the maximum output of the component classifier and the & image coordinates of the maximum. 3. For each component k, determine its maximum output within a search region and its location: Combination classifier: Linear SVM Combination classifier: Linear SVM Left Eye expert: Linear SVM Left Eye expert: Linear SVM ... 1. Shift 58x58 window over input image *Outputs of component experts: bright intensities indicate high confidence. 2. Shift component experts over 58x58 window 4. Final decision: face / background ) , , ( 14 14 14 Y X O ) , , ,..., , , ( 14 14 14 1 1 1 Y X O Y X O Nose expert: Linear SVM Nose expert: Linear SVM Mouth expert: Linear SVM Mouth expert: Linear SVM ... * * * ) , , ( 1 1 1 Y X O ) , , ( k k k Y X O ) , , ( k k k Y X O Figure 2: System overview of the component-based classifier. 6 Experiments In our experiments we compared the component-based system to global classifiers. The component system consisted of fourteen linear SVM classifiers for detecting the components and a single linear SVM as combination classifier. The global classifiers were a single linear SVM and a single second-degree polynomial SVM both trained on the gray values of the whole face pattern. The training data for these three classifiers consisted of 2,457 synthetic gray face images and 13,654 non-face gray images of size 58 58. The positive test set consisted of 1,834 faces rotated between about " and in depth. The faces were manually extracted from the CMU PIE database [12]. The negative test set consisted of 24,464 difficult non-face patterns that were collected by a fast face detector [3] from web images. The FP rate was calculated relative to the number of non-face test images. Because of the resolution required by the component-based system, a direct comparison with other published systems on the standard MIT-CMU test set [10] was impossible. For an indirect comparison, we used a second-degree polynomial SVM [2] which was trained on a large set of 19 19 real face images. This classifier performed amongst the best face detection systems on the MIT-CMU test set. The ROC curves in Fig. 3 show that the component-based classifier is significantly better than the three global classifiers. Some detection results generated by the component system are shown in Fig. 4. Figure 3: Comparison between global classifiers and the component-based classifier. Figure 4: Faces detected by the component-based classifier. A natural question that arises is about the role of geometrical information. To answer this question–which has relevant implications for models of cortex–we tested another system in which the combination classifier receives as inputs only the output of each component classifier but not the position of its maximum. As shown in Fig. 5 this system still outperforms the whole face systems but it is worse than the system with position information. Figure 5: Comparison between a component-based classifier trained with position information and a component-based classifier without position information. 7 Open Questions An extension under way of the component-based approach to face identification is already showing good performances [1]. Another natural generalization of the work described here involves the application of our system to various classes of objects such as cars, animals, and people. Still another extension regards the question of view-invariant object detection. As suggested by [7] in a biological context and demonstrated recently by [11] in machine vision, full pose invariance in recognition tasks can be achieved by combining view-dependent classifiers. It is interesting to ask whether the approach described here could also be used to learn which views are most discriminative and how to combine them optimally. Finally, the role of geometry and in particular how to compute and represent position information in biologically plausible networks, is an important open question at the interface between machine and biological vision. References [1] B. Heisele, P. Ho, and T. Poggio. Face recognition with support vector machines: global versus component-based approach. In Proc. 8th International Conference on Computer Vision, Vancouver, 2001. [2] B. Heisele, T. Poggio, and M. Pontil. Face detection in still gray images. A.I. memo 1687, Center for Biological and Computational Learning, MIT, Cambridge, MA, 2000. [3] B. Heisele, T. Serre, S. Mukherjee, and T. Poggio. Feature reduction and hierarchy of classifiers for fast object detection in video images. In Proc. IEEE Conference on Computer Vision and Pattern Recognition, Hawaii, 2001. [4] T. K. Leung, M. C. Burl, and P. Perona. Finding faces in cluttered scenes using random labeled graph matching. In Proc. International Conference on Computer Vision, pages 637–644, Cambridge, MA, 1995. [5] A. Mohan, C. Papageorgiou, and T. Poggio. Example-based object detection in images by components. In IEEE Transactions on Pattern Analysis and Machine Intelligence, volume 23, pages 349–361, April 2001. [6] C. Papageorgiou and T. Poggio. A trainable system for object detection. In International Journal of Computer Vision, volume 38, 1, pages 15–33, 2000. [7] T. Poggio and S. Edelman. A network that learns to recognize 3-D objects. Nature, 343:163–266, 1990. [8] M. Riesenhuber and T. Poggio. Hierarchical models of object recognition in cortex. Nature Neuroscience, 2(11):1019–1025, 1999. [9] T. D. Rikert, M. J. Jones, and P. Viola. A cluster-based statistical model for object detection. In Proc. IEEE Conference on Computer Vision and Pattern Recognition, volume 2, pages 1046–1053, Fort Collins, 1999. [10] H. A. Rowley, S. Baluja, and T. Kanade. Rotation invariant neural network-based face detection. Computer Science Technical Report CMU-CS-97-201, CMU, Pittsburgh, 1997. [11] H. Schneiderman and T. Kanade. A statistical method for 3d object detection applied to faces and cars. In Proc. IEEE Conference on Computer Vision and Pattern Recognition, pages 746–751, 2000. [12] T. Sim, S. Baker, and M. Bsat. The CMU pose, illumination, and expression (PIE) database of human faces. Computer Science Technical Report 01-02, CMU, 2001. [13] K.-K. Sung. Learning and Example Selection for Object and Pattern Recognition. PhD thesis, MIT, Artificial Intelligence Laboratory and Center for Biological and Computational Learning, Cambridge, MA, 1996. [14] R. Vaillant, C. Monrocq, and Y. Le Cun. An original approach for the localisation of objects in images. In International Conference on Artificial Neural Networks, pages 26–30, 1993. [15] V. Vapnik. Statistical learning theory. John Wiley and Sons, New York, 1998. [16] T. Vetter. Synthesis of novel views from a single face. International Journal of Computer Vision, 28(2):103–116, 1998.
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On Discriminative vs. Generative classifiers: A comparison of logistic regression and naive Bayes Andrew Y. Ng Computer Science Division University of California, Berkeley Berkeley, CA 94720 Michael I. Jordan C.S. Div. & Dept. of Stat. University of California, Berkeley Berkeley, CA 94720 Abstract We compare discriminative and generative learning as typified by logistic regression and naive Bayes. We show, contrary to a widelyheld belief that discriminative classifiers are almost always to be preferred, that there can often be two distinct regimes of performance as the training set size is increased, one in which each algorithm does better. This stems from the observation- which is borne out in repeated experiments- that while discriminative learning has lower asymptotic error, a generative classifier may also approach its (higher) asymptotic error much faster. 1 Introduction Generative classifiers learn a model of the joint probability, p( x, y), of the inputs x and the label y, and make their predictions by using Bayes rules to calculate p(ylx), and then picking the most likely label y. Discriminative classifiers model the posterior p(ylx) directly, or learn a direct map from inputs x to the class labels. There are several compelling reasons for using discriminative rather than generative classifiers, one of which, succinctly articulated by Vapnik [6], is that "one should solve the [classification] problem directly and never solve a more general problem as an intermediate step [such as modeling p(xly)]." Indeed, leaving aside computational issues and matters such as handling missing data, the prevailing consensus seems to be that discriminative classifiers are almost always to be preferred to generative ones. Another piece of prevailing folk wisdom is that the number of examples needed to fit a model is often roughly linear in the number of free parameters of a model. This has its theoretical basis in the observation that for "many" models, the VC dimension is roughly linear or at most some low-order polynomial in the number of parameters (see, e.g., [1, 3]), and it is known that sample complexity in the discriminative setting is linear in the VC dimension [6]. In this paper, we study empirically and theoretically the extent to which these beliefs are true. A parametric family of probabilistic models p(x, y) can be fit either to optimize the joint likelihood of the inputs and the labels, or fit to optimize the conditional likelihood p(ylx), or even fit to minimize the 0-1 training error obtained by thresholding p(ylx) to make predictions. Given a classifier hGen fit according to the first criterion, and a model hDis fit according to either the second or the third criterion (using the same parametric family of models) , we call hGen and hDis a Generative-Discriminative pair. For example, if p(xly) is Gaussian and p(y) is multinomial, then the corresponding Generative-Discriminative pair is Normal Discriminant Analysis and logistic regression. Similarly, for the case of discrete inputs it is also well known that the naive Bayes classifier and logistic regression form a Generative-Discriminative pair [4, 5]. To compare generative and discriminative learning, it seems natural to focus on such pairs. In this paper, we consider the naive Bayes model (for both discrete and continuous inputs) and its discriminative analog, logistic regression/linear classification, and show: (a) The generative model does indeed have a higher asymptotic error (as the number of training examples becomes large) than the discriminative model, but (b) The generative model may also approach its asymptotic error much faster than the discriminative model- possibly with a number of training examples that is only logarithmic, rather than linear, in the number of parameters. This suggests-and our empirical results strongly support-that, as the number of training examples is increased, there can be two distinct regimes of performance, the first in which the generative model has already approached its asymptotic error and is thus doing better, and the second in which the discriminative model approaches its lower asymptotic error and does better. 2 Preliminaries We consider a binary classification task, and begin with the case of discrete data. Let X = {O, l}n be the n-dimensional input space, where we have assumed binary inputs for simplicity (the generalization offering no difficulties). Let the output labels be Y = {T, F}, and let there be a joint distribution V over X x Y from which a training set S = {x(i) , y(i) }~1 of m iid examples is drawn. The generative naive Bayes classifier uses S to calculate estimates p(xiIY) and p(y) of the probabilities p(xi IY) and p(y), as follows: P' (x- = 11Y = b) = #s{xi=l ,y=b}+1 (1) , #s{y-b}+21 (and similarly for p(y = b),) where #s{-} counts the number of occurrences of an event in the training set S. Here, setting l = ° corresponds to taking the empirical estimates of the probabilities, and l is more traditionally set to a positive value such as 1, which corresponds to using Laplace smoothing of the probabilities. To classify a test example x, the naive Bayes classifier hGen : X r-+ Y predicts hGen(x) = T if and only if the following quantity is positive: (rr~-d) (x i ly = T))p(y = T) ~ p(xilY = T) p(y = T) IGen(x ) = log (rrn '( _I _ F)) '( _ F) = L..,log ' ( _I _ F) + log ' ( _ F)' (2) i=1 P X, Y P Y i=1 P X, Y P Y In the case of continuous inputs, almost everything remains the same, except that we now assume X = [O,l]n, and let p(xilY = b) be parameterized as a univariate Gaussian distribution with parameters {tily=b and if; (note that the j1's, but not the if's, depend on y). The parameters are fit via maximum likelihood, so for example {tily=b is the empirical mean of the i-th coordinate of all the examples in the training set with label y = b. Note that this method is also equivalent to Normal Discriminant Analysis assuming diagonal covariance matrices. In the sequel, we also let J.tily=b = E[Xi IY = b] and a; = Ey[Var(xily)] be the "true" means and variances (regardless of whether the data are Gaussian or not). In both the discrete and the continuous cases, it is well known that the discriminative analog of naive Bayes is logistic regression. This model has parameters [,8, OJ, and posits that p(y = Tlx; ,8, O) = 1/(1 +exp(-,8Tx - 0)). Given a test example x, the discriminative logistic regression classifier hois : X I-t Y predicts hOis (x) = T if and only if the linear discriminant function lDis(x) = L~=l (3ixi + () (3) is positive. Being a discriminative model, the parameters [(3, ()] can be fit either to maximize the conditionallikelikood on the training set L~= llogp(y(i) Ix(i); (3, ()), or to minimize 0-1 training error L~= ll{hois(x(i)) 1- y(i)}, where 1{-} is the indicator function (I{True} = 1, I{False} = 0). Insofar as the error metric is 0-1 classification error, we view the latter alternative as being more truly in the "spirit" of discriminative learning, though the former is also frequently used as a computationally efficient approximation to the latter. In this paper, we will largely ignore the difference between these two versions of discriminative learning and, with some abuse of terminology, will loosely use the term "logistic regression" to refer to either, though our formal analyses will focus on the latter method. Finally, let 1i be the family of all linear classifiers (maps from X to Y); and given a classifier h : X I-t y, define its generalization error to be c(h) = Pr(x,y)~v [h(x) 1- y]. 3 Analysis of algorithms When V is such that the two classes are far from linearly separable, neither logistic regression nor naive Bayes can possibly do well, since both are linear classifiers. Thus, to obtain non-trivial results, it is most interesting to compare the performance of these algorithms to their asymptotic errors (cf. the agnostic learning setting). More precisely, let hGen,oo be the population version of the naive Bayes classifier; i.e. hGen,oo is the naive Bayes classifier with parameters p(xly) = p(xly),p(y) = p(y). Similarly, let hOis,oo be the population version of logistic regression. The following two propositions are then completely straightforward. Proposition 1 Let hGen and hDis be any generative-discriminative pair of classifiers, and hGen,oo and hois,oo be their asymptotic/population versions. Thenl c(hDis,oo) :S c(hGen,oo). Proposition 2 Let hDis be logistic regression in n-dimensions. Then with high probability c(hois) :S c(hois,oo) + 0 (J ~ log ~) Thus, for c(hOis) :S c(hOis,oo) + EO to hold with high probability (here, EO > 0 is some fixed constant), it suffices to pick m = O(n). Proposition 1 states that aymptotically, the error of the discriminative logistic regression is smaller than that of the generative naive Bayes. This is easily shown by observing that, since c(hDis) converges to infhE1-l c(h) (where 1i is the class of all linear classifiers), it must therefore be asymptotically no worse than the linear classifier picked by naive Bayes. This proposition also provides a basis for what seems to be the widely held belief that discriminative classifiers are better than generative ones. Proposition 2 is another standard result, and is a straightforward application of Vapnik's uniform convergence bounds to logistic regression, and using the fact that 1i has VC dimension n. The second part of the proposition states that the sample complexity of discriminative learning- that is, the number of examples needed to approach the asymptotic error- is at most on the order of n. Note that the worst case sample complexity is also lower-bounded by order n [6]. lUnder a technical assumption (that is true for most classifiers, including logistic regression) that the family of possible classifiers hOis (in the case of logistic regression, this is 1l) has finite VC dimension. The picture for discriminative learning is thus fairly well-understood: The error converges to that of the best linear classifier, and convergence occurs after on the order of n examples. How about generative learning, specifically the case of the naive Bayes classifier? We begin with the following lemma. Lemma 3 Let any 101,8 > ° and any l 2: ° be fixed. Assume that for some fixed Po > 0, we have that Po :s: p(y = T) :s: 1 - Po. Let m = 0 ((l/Ei) log(n/8)). Then with probability at least 1 - 8: 1. In case of discrete inputs, IjJ(XiIY = b) - p(xilY = b)1 :s: 101 and IjJ(y = b) - p(y = b) I :s: 101, for all i = 1, ... ,n and bEY. 2. In the case of continuous inputs, IPily=b - f-lily=b I :s: 101, laT - O"T I :s: 101, and IjJ(y = b) - p(y = b) I :s: 101 for all i = 1, ... ,n and bEY. Proof (sketch). Consider the discrete case, and let l = ° for now. Let 101 :s: po/2. By the Chernoff bound, with probability at least 1 - 81 = 1- 2exp(-2Eim), the fraction of positive examples will be within 101 of p(y = T) , which implies IjJ(y = b) - p(y = b)1 :s: 101, and we have at least 1m positive and 1m negative examples, where I = Po 101 = 0(1). So by the Chernoff bound again, for specific i, b, the chance that IjJ(XiIY = b) - p(xilY = b)1 > 101 is at most 82 = 2exp(-2Ehm). Since there are 2n such probabilities, the overall chance of error, by the Union bound, is at most 81 + 2n82 . Substituting in 81 and 8/s definitions, we see that to guarantee 81 + 2n82 :s: 8, it suffices that m is as stated. Lastly, smoothing (l > 0) adds at most a small, O(l/m) perturbation to these probabilities, and using the same argument as above with (say) 101/2 instead of 101, and arguing that this O(l/m) perturbation is at most 101/2 (which it is as m is at least order l/Ei) , again gives the result. The result for the continuous case is proved similarly using a Chernoff-bounds based argument (and the assumption that Xi E [0,1]). D Thus, with a number of samples that is only logarithmic, rather than linear, in n, the parameters of the generative classifier hGen are uniformly close to their asymptotic values in hGen,oo. Is is tempting to conclude therefore that c(hGen), the error of the generative naive Bayes classifier, also converges to its asymptotic value of c(hGen,oo) after this many examples, implying only 0 (log n) examples are required to fit a naive Bayes model. We will shortly establish some simple conditions under which this intuition is indeed correct. Note that this implies that, even though naive Bayes converges to a higher asymptotic error of c(hGen,oo) compared to logistic regression's c:(hDis,oo), it may also approach it significantly faster-after O(log n), rather than O(n), training examples. One way of showing c(hGen) approaches c(hGen,oo) is by showing that the parameters' convergence implies that hGen is very likely to make the same predictions as hGen,oo. Recall hGen makes its predictions by thresholding the discriminant function lGen defined in (2). Let lGen,oo be the corresponding discriminant function used by hGen,oo. On every example on which both lGen and lGen,oo fall on the same side of zero, hGen and hGen,oo will make the same prediction. Moreover, as long as lGen,oo (x) is, with fairly high probability, far from zero, then lGen (x), being a small perturbation of lGen,oo(x), will also be usually on the same side ofzero as lGen,oo(x). Theorem 4 Define G(T) = Pr(x,y)~v[(lGen ,oo(x ) E [O,Tn] A y = T) V (lGen,oo(X) E [-Tn, O]A Y = F)]. Assume that for some fixed Po > 0, we have Po :s: p(y = T) :s: 1 - Po, and that either Po :s: P(Xi = 11Y = b) :s: 1 - Po for all i, b (in the case of discrete inputs), or O"T 2: Po (in the continuous case). Then with high probability, c:(hGen) :s: c:(hGen,oo) + G (0 (J ~ logn)) . (4) Proof (sketch). c(hGen) - c(hGen,oo) is upperbounded by the chance that hGen,oo correctly classifies a randomly chosen example, but hGen misclassifies it. Lemma 3 ensures that, with high probability, all the parameters of hGen are within O( j(log n)/m) of those of hGen,oo. This in turn implies that everyone of the n + 1 terms in the sum in lGen (as in Equation 2) is within O( j(1ogn)/m) of the corresponding term in lGen,oo, and hence that IlGen(x) -lGen,oo(x)1 :S O(nj(1ogn)/m). Letting T = O( j(logn)/m), we therefore see that it is possible for hGen,oo to be correct and hGen to be wrong on an example (x , y) only if y = T and lGen,oo(X) E [0, Tn] (so that it is possible that lGen,oo(X) ::::: 0, lGen(x) :S 0), or if y = F and lGen,oo(X) E [-Tn, 0]. The probability of this is exactly G(T), which therefore upperbounds c(hGen) - c(hGen,oo). D The key quantity in the Theorem is the G(T) , which must be small when T is small in order for the bound to be non-trivial. Note G(T) is upper-bounded by Prx[lGen,oo(x) E [-Tn, Tn]]-the chance that lGen,oo(X) (a random variable whose distribution is induced by x ""' V) falls near zero. To gain intuition about the scaling of these random variables, consider the following: Proposition 5 Suppose that, for at least an 0(1) fraction of the features i (i = 1, ... ,n), it holds true that IP(Xi = 11Y = T) - P(Xi = 11Y = F)I ::::: 'Y for some fixed'Y > 0 (or IJLily=T JLily=FI ::::: 'Y in the case of continuous inputs). Then E[lGen,oo(x)ly = T] = O(n), and -E[lGen,oo(x)ly = F] = O(n). Thus, as long as the class label gives information about an 0(1) fraction of the features (or less formally, as long as most of the features are "relevant" to the class label), the expected value of IlGen,oo(X) I will be O(n). The proposition is easily proved by showing that, conditioned on (say) the event y = T, each of the terms in the summation in lGen,oo(x) (as in Equation (2), but with fi's replaced by p's) has non-negative expectation (by non-negativity of KL-divergence), and moreover an 0(1) fraction of them have expectation bounded away from zero. Proposition 5 guarantees that IlGen,oo (x)1 has large expectation, though what we want in order to bound G is actually slightly stronger, namely that the random variable IlGen,oo (x)1 further be large/far from zero with high probability. There are several ways of deriving sufficient conditions for ensuring that G is small. One way of obtaining a loose bound is via the Chebyshev inequality. For the rest of this discussion, let us for simplicity implicitly condition on the event that a test example x has label T. The Chebyshev inequality implies that Pr[lGen,oo(x) :S E[lGen,oo(X)] - t] :S Var(lGen,oo(x))/t2 . Now, lGen,oo (X) is the sum of n random variables (ignoring the term involving the priors p(y)). If (still conditioned on y), these n random variables are independent (i.e. if the "naive Bayes assumption," that the xi's are conditionally independent given y, holds), then its variance is O(n); even if the n random variables were not completely independent, the variance may still be not much larger than 0 (n) (and may even be smaller, depending on the signs of the correlations), and is at most O(n2). So, if E[lGen,oo(x)ly = T] = an (as would be guaranteed by Proposition 5) for some a > 0, by setting t = (a - T)n, Chebyshev's inequality gives Pr[lGen,oo(x) :S Tn] :S O(l/(a - T)2n1/) (T < a), where 1} = 0 in the worst case, and 1} = 1 in the independent case. This thus gives a bound for G(T), but note that it will frequently be very loose. Indeed, in the unrealistic case in which the naive Bayes assumption really holds, we can obtain the much stronger (via the Chernoff bound) G(T):S exp(-O((a - T)2n)) , which is exponentially small in n. In the continuous case, if lGen,oo(x) has a density that, within some small interval [-m,mJ, is uniformly bounded by O(l/n), then we also have G(T) = O(T). In any case, we also have the following Corollary to Theorem 4. Corollary 6 Let the conditions of Theorem 4 hold, and suppose that G(T) :S Eo/2+ F(T) for some function F(T) (independent of n) that satisfies F(T) -+ 0 as T -+ 0, and some fixed EO > O. Then for €(hGen) :S c(hGen,oo) + EO to hold with high pima (continuous) adult (continuous) 0.5 0.5 0.45 " 0.45 0.4 ~-2~~~, ____ 0.4 e gO.35 ;;; ~ 0.35 0.3 0.3 '--, 0.25 0.250 20 40 60 0"0 10 20 30 optdigits (O's and 1 's, continuous) optdigits (2's and 3's, continuous) o.4,-------------, 0.4", -------------. 0.3 0.3 ~0.2 0.1 01 ~ 50 100 150 200 liver disorders (continuous) 0.5,-----------------, 0.45 ~ ~ 0.4 ~ 0.35 0.4 0.3 0.350 20 40 60 0.250 20 40 60 80 100 120 promoters (discrete) lymphography (discrete) 0.5 0.5 ~:: ~ ......•...•..••.•.•..•.•..•.•.•.•.•...•...•. •.. _. ~ 0.4 ~'\~.~:::.:~ •...• ~ ,~0 . 3 "'" 0.2 ..... 0.1 0.2 ..... ".""""",,. %L--~20~-~ 40---'-6~0---'---'8~0-~ 100 0.10'------,5~0---.,.10~0---.---,J 150 lenses (predict hard vs. soft, discrete) sick (discrete) 0.5 0.8,-------------, 0.4 0'6\=~_ gOA ~ 0.2 ..... ------------ -- --- --- --- 0.2 0.10'---~-~10c--~15--2~0-~25 %~----,5~0---~10~0~--,,! 150 boston (predict it > median price, continuous) 0.45,-------------, 0.4 I::: \ ... ~~ __ 02Q---""2"'0---4"'0-----"cJ60 ionosphere (continuous) 0.5,-----------------, 0.2 adult (discrete) 0.7,-----------------, 0.6 100 200 300 400 breast cancer (discrete) 0.45 ~ 0.4 ~ 0.35 0.3 0.250 100 200 300 voting records (discrete) 0.4 0.3 gO.2 ~ 0.1 \--. 00 20 40 60 80 Figure 1: Results of 15 experiments on datasets from the VCI Machine Learning repository. Plots are of generalization error vs. m (averaged over 1000 random train/test splits). Dashed line is logistic regression; solid line is naive Bayes. probability, it suffices to pick m = O(log n). Note that the previous discussion implies that the preconditions of the Corollary do indeed hold in the case that the naive Bayes (and Proposition 5's) assumption holds, for any constant fa so long as n is large enough that fa ::::: exp( -O(o:2n)) (and similarly for the bounded Var(lGen,oo(x)) case, with the more restrictive fa ::::: O(I/(o:2n17))). This also means that either ofthese (the latter also requiring T) > 0) is a sufficient condition for the asymptotic sample complexity to be 0 (log n). 4 Experiments The results of the previous section imply that even though the discriminative logistic regression algorithm has a lower asymptotic error, the generative naive Bayes classifier may also converge more quickly to its (higher) asymptotic error. Thus, as the number of training examples m is increased, one would expect generative naive Bayes to initially do better, but for discriminative logistic regression to eventually catch up to, and quite likely overtake, the performance of naive Bayes. To test these predictions, we performed experiments on 15 datasets, 8 with continuous inputs, 7 with discrete inputs, from the VCI Machine Learning repository.2 The results ofthese experiments are shown in Figure 1. We find that the theoretical predictions are borne out surprisingly well. There are a few cases in which logistic regression's performance did not catch up to that of naive Bayes, but this is observed primarily in particularly small datasets in which m presumably cannot grow large enough for us to observe the expected dominance of logistic regression in the large m limit. 5 Discussion Efron [2] also analyzed logistic regression and Normal Discriminant Analysis (for continuous inputs), and concluded that the former was only asymptotically very slightly (1/3- 1/2 times) less statistically efficient. This is in marked contrast to our results, and one key difference is that, rather than assuming P(xly) is Gaussian with a diagonal covariance matrix (as we did), Efron considered the case where P(xly) is modeled as Gaussian with a full convariance matrix. In this setting, the estimated covariance matrix is singular if we have fewer than linear in n training examples, so it is no surprise that Normal Discriminant Analysis cannot learn much faster than logistic regression here. A second important difference is that Efron considered only the special case in which the P(xly) is truly Gaussian. Such an asymptotic comparison is not very useful in the general case, since the only possible conclusion, if €(hDis,oo) < €(hGen,oo), is that logistic regression is the superior algorithm. In contrast, as we saw previously, it is in the non-asymptotic case that the most interesting "two-regime" behavior is observed. Practical classification algorithms generally involve some form of regularization- in particular logistic regression can often be improved upon in practice by techniques 2To maximize the consistency with the theoretical discussion, these experiments avoided discrete/continuous hybrids by considering only the discrete or only the continuous-valued inputs for a dataset where necessary. Train/test splits were random subject to there being at least one example of each class in the training set, and continuous-valued inputs were also rescaled to [0, 1] if necessary. In the case of linearly separable datasets, logistic regression makes no distinction between the many possible separating planes. In this setting we used an MCMC sampler to pick a classifier randomly from them (i.e., so the errors reported are empirical averages over the separating hyperplanes) . Our implementation of Normal Discriminant Analysis also used the (standard) trick of adding € to the diagonal of the covariance matrix to ensure invertibility, and for naive Bayes we used I = 1. such as shrinking the parameters via an L1 constraint, imposing a margin constraint in the separable case, or various forms of averaging. Such regularization techniques can be viewed as changing the model family, however, and as such they are largely orthogonal to the analysis in this paper, which is based on examining particularly clear cases of Generative-Discriminative model pairings. By developing a clearer understanding of the conditions under which pure generative and discriminative approaches are most successful, we should be better able to design hybrid classifiers that enjoy the best properties of either across a wider range of conditions. Finally, while our discussion has focused on naive Bayes and logistic regression, it is straightforward to extend the analyses to several other models, including generativediscriminative pairs generated by using a fixed-structure, bounded fan-in Bayesian network model for P(xly) (of which naive Bayes is a special case). Acknowledgments We thank Andrew McCallum for helpful conversations. A. Ng is supported by a Microsoft Research fellowship. This work was also supported by a grant from Intel Corporation, NSF grant IIS-9988642, and ONR MURI N00014-00-1-0637. References [1] M. Anthony and P. Bartlett. Neural Network Learning: Theoretical Foundations. Cambridge University Press, 1999. [2] B. Efron. The efficiency of logistic regression compared to Normal Discriminant Analysis. Journ. of the Amer. Statist. Assoc., 70:892- 898, 1975. [3] P. Goldberg and M. Jerrum. Bounding the VC dimension of concept classes parameterized by real numbers. Machine Learning, 18:131-148, 1995. [4] G.J. McLachlan. Discriminant Analysis and Statistical Pattern Recognition. Wiley, New York, 1992. [5] Y. D. Rubinstein and T. Hastie. Discriminative vs. informative learning. In Proceedings of the Third International Conference on Knowledge Discovery and Data Mining, pages 49- 53. AAAI Press, 1997. [6] V. N. Vapnik. Statistical Learning Theory. John Wiley & Sons, 1998.
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Risk Sensitive Particle Filters Sebastian Thrun, John Langford, Vandi Verma School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 thrun,jcl,vandi @cs.cmu.edu Abstract We propose a new particle filter that incorporates a model of costs when generating particles. The approach is motivated by the observation that the costs of accidentally not tracking hypotheses might be significant in some areas of state space, and next to irrelevant in others. By incorporating a cost model into particle filtering, states that are more critical to the system performance are more likely to be tracked. Automatic calculation of the cost model is implemented using an MDP value function calculation that estimates the value of tracking a particular state. Experiments in two mobile robot domains illustrate the appropriateness of the approach. 1 Introduction In recent years, particle filters [3, 7, 8] have found widespread application in domains with noisy sensors, such as computer vision and robotics [2, 5]. Particle filters are powerful tools for Bayesian state estimation in non-linear systems. The key idea of particle filters is to approximate a posterior distribution over unknown state variables by a set of particles, drawn from this distribution. This paper addresses a primary deficiency of particle filters: Particle filters are insensitive to costs that might arise from the approximate nature of the particle representation. Their only criterion for generating a particle is the posterior likelihood of a state. To illustrate this point, consider the example of a Space Shuttle. Failures of the engine system are extremely unlikely, even in the presence of evidence to the contrary. Should we therefore not track the possibility of such failures, just because they are unlikely? If failure to track such lowlikelihood events may incur high costs—such as a mission failure—these variables should be tracked even when their posterior probability is low. This observation suggests that costs should be taken into consideration when generating particles in the filtering process. This paper proposes a particle filter that generates particles according to a distribution that combines the posterior probability with a risk function. The risk function measures the importance of a state location on future cumulative costs. We obtain this risk function via an MDP that calculates the approximate future risk of decisions made in a particular state. Experimental results in two robotic domains illustrate that our approach yields significantly better results than a particle filter insensitive to costs. 2 The “Classical” Particle Filter Particle filters are a popular means of estimating the state of partially observable controllable Markov chains [3], sometimes referred to as dynamical systems [1]. To do so, particle filters require two types of information: data, and a probabilistic generative model of the system. The data generally comes in two flavors: measurements (e.g., camera images) and controls (e.g., robot motion commands). The measurement at time will be denoted , and denotes the control asserted in the time interval
. Thus, the data is given by and
Following common notation in the controls literature, we use the subscript to refer to an event at time and the superscript to denote all events leading up to time . Particle filters, like any member of the family of Bayes filters such as Kalman filters and HMMs, estimate the posterior distribution of the state of the dynamical system conditioned on the data, !#" $
&% . They do so via the following recursive formula '#" $ % ( ' $ " %*) '#" $ " +, % !#" + $ +
+, %" +, (1) where ( is a normalization constant. To calculate this posterior, three probability distributions are required, which together are commonly referred as the probabilistic model of the dynamical system: (1) A measurement model ! $ " % , which describes the probability of measuring when the system is in state " . (2) A control model !#",$ . "*+, % , which characterizes the effect of controls ! on the system state by specifying the probability that the system is in state " after executing control in state " +, . (3) An initial state distribution '#" / % , which specifies the user’s knowledge about the initial system state. See [2, 5] for examples of such models in practical applications. Eqn. 1 is easily derived under the common assumption that the system is Markov: '#" 0$ % 132547658 ( * !0$ " 9
+
% !#"*$ +
% : 2;#<=5> ( * !0$ " % '"*$ +, % ( ! $ " %*) !#" $ +
" + % '" +, $ +, %" + : 2;#<=5> ( * !0$ " % ) !#" $ .
"*+, % !#"*+, $ +
+, %?"*+, (2) Notice that this filter, in the general form stated here, is commonly known as a Bayes filter. Approximations to Bayes filters includes the Kalman filter, the hidden Markov model, binary filters, and of course particle filters. In many applications, the key concern in implementing this probabilistic filter is the continuous nature of the states " , controls , and measurements . Even in discrete applications, the state space is often too large to compute the entire posterior in reasonable time. The particle filter addresses these concerns by approximating the posterior using sets of state samples (particles): @ ",A BDC BFE 9GIHIHIH G J (3) The set @ consists of K particles "'A BLC , for some large number K (e.g, K M
N NMN ). Together, these particles approximates the posterior !#" $ &% . @ is calculated recursively. Initially, at time N , the particles "A BLC / are generated from the initial state distribution '" / % . The -th particle set @ is then calculated recursively from @ +, as follows: 1 set @ @ 2 2 for to K do 3 pick the -th sample "?A &C + @ + 4 draw "'A &C '"*$ " A &C + % 5 set A &C !7$ ",A &C % 6 add
#"'A &C A &C to @ 2 7 endfor 8 for to K do 9 draw " A BDC from @ 2 with probability proportional to A BDC 10 add "!A BDC to @ 11 endfor Lines 2 through 7 generates a new set of particles that incorporates the control . Lines 8 through 11 apply a technique known as importance-weighted resampling [11] to account for the measurement . It is a well-known fact that (for large K ) the resulting weighted particles are asymptotically distributed according to the desired posterior [12] !#" $ 5% In recent years, researchers have actively developed various extensions of the basic particle filter, capable of coping with degenerate situations that are often relevant in practice [3, 7, 8]. The common aim of this rich body of literature, however, is to generate samples from the posterior '" $ &% . If different controls at different states infer drastically different costs, generating samples according to the posterior runs the risk of not capturing important events that warrant action. Overcoming this deficiency is the very aim of this paper. 3 Risk Sensitive Particle Filters This section describes a modified particle filter that is sensitive to the risk arising from the approximate nature of the particle representation. To arrive at a notion of risk, our approach requires a cost function #"
% (4) This function assigns real-valued costs to states and control. From a decision theoretic point of view, the goal of risk sensitive sampling is to generate particles that minimize the cumulative increase in cost due to the particle approximation. To translate this into a practical algorithm, we extend the basic paradigm in two ways. First, we modify the basic particle filters so that particles are generated in a risk-sensitive way, where the risk is a function of . Second, an appropriate risk function is defined that approximates the cumulative expected costs relative to tracking individual states. This risk function is calculated using value iteration. 3.1 Risk-Sensitive Sampling Risk-sensitive sampling generates particles factoring in a risk function, #" % . Formally, all we have to ask of a risk function is that it be positive and finite almost everywhere. Not all risk functions will be equally useful, however, so deriving the “right” risk function is important. Decision theory gives us a framework for deciding what the “right” action is in any given state. By considering approximation errors due to monte carlo sampling in decision theory and making a sequence of rough approximations, we can arrive at the choice of #" % , which is discussed further below. The full derivation is omitted for lack of space. For now, let us simply assume are given a suitable risk function. Risk sensitive particle filters generate samples that are distributed according to "* % !#" $
% (5) Here #" % '"!$ % "* +, is a normalization constant that ensures that the term in (5) is indeed a probability distribution. Thus, the probability that a state sample " A BLC is part of @ is not only a function of its posterior probability, but also of the risk #" A BLC % associated with that sample. Sampling from (5) is easily achieved by the following two modifications of the basic particle filter algorithm. First, the initial set of particles " A BLC / is generated from the distribution / #" / % '#" / % (6) Second, Line 5 of the particle filter algorithm is replaced by the following assignment: set A &C #" A &C % " A &C +, % + ! $ " A &C % (7) We conjecture that this simple modification results in a particle filter with samples distributed according to #" % !#" $
&% . Our conjecture is obviously true for the base case N , since the risk function was explicitly incorporated in the construction of @ / (see eqn. 6). By induction, let us assume that the particles in @ + are distributed according to + "*+ % !#"*+, $ +
+,0% . Then Line 3 of the modified algorithm generates "A &C + + "*+ % !#"*+, $ +
+,9% . Line 4 gives us "A &C + #" + % '"*$ 7
"*+ % '"*+, $ +, +% . Samples generated in Line 9 are distributed according to A &C + "*+ % !#"*$ .
"*+, % !#"*+,$ +,
+, % (8) Substituting in the modified weight (eqn. 7) we find the final sample distribution: #" % #" + % +, ' 0$ "* % + #" + % '"*$ 7
"*+ % '"*+M$ +, + % +, "* % '9$ " % '"*$ 7 " + % '"*+M$ +, + % (9) This term is, up to the normalization constant ( +, +, , equivalent to the desired distribution (5) (see also eqn. 1), which proves our conjecture. Thus, the risk sensitive particle filter successfully generates samples from a distribution that factors in the risk . 3.2 The Risk Function The remaining question is: What is an appropriate risk function ? How important is it to track a state " ? Our approach rests on the assumption that there are two possible situations, one in which the state is tracked well, and one in which the state is tracked poorly. In the first situation, we assume that any controller will basically chose the right control, whereas in the second situation, it is reasonable to assume that controls are selected anywhere between random and in the worst possible way. To complete this model, we assume that with small probability, the state estimator might move from “well-tracked” to “lost track” and vice versa. These assumptions are sufficient to formulate an MDP that models the effect of tracking accuracy on the expected costs. The MDP is defined over an augmented state space
" (see also [10]), where N3 is a binary state variable that models the event that the estimator tracks the state with sufficient ( ) or insufficient ( N ) accuracy. The various probabilities of the MDP are easily obtained from the known probability distributions via the natural assumption that the variable is conditionally independent of the system state " : '
" $
" +, + % '#" $ " +, % ! $ + % !0$
"*9 9 % '0$ "* % !
" / / % '#" / % ! / %
"
% #"
% (10) The expressions on the left hand side define all necessary components of the augmented model. The only unspecified terms on the right hand side are the initial tracking probability ' 9/ % and the transition probabilities for the state estimator ' $ +, % . The former must be set in accordance to the initial knowledge state (e.g., 1 if the initial system state is known, 0 if it is unknown). For the latter, we adopt a model where with high likelihood the tracking state is retained ( ! 9+ % N3 ) and with low likelihood it changes ( ' 9+, % N3 N ). The MDP is solved via value iteration. To model the effect of poor tracking on the control policy, our approach uses the following value iteration rule (stated here without discounting for simplicity), in which denotes the value function, and is an auxiliary variable:
#"' %
#"
% if
"
% & M % )
"
%* if N
" % #"
% E / )
" % ' $ % !#" 5$ '
" %" (11) This value iteration rule considers two cases: When , i.e., the state is estimated sufficiently accurately, it is assumed that the controller acts by minimizing costs. If N , however, the controller adopts a mixture of picking the worst possible control , and a random control. These two options are traded off by the gain factor , which controls the “pessimism” of the approach. suggests that poor state estimation leads to the worst possible control. N is more optimistic, in that control is assumed to be random. Our experiments have yielded somewhat indifferent results relative to the choice of , and we use N3 for all experiments reported here. Finally, the risk is defined as the difference between the value function that arises from accurate versus inaccurate state estimation: " % " N % " % (12) Under mild assumptions, #" % can be shown to be strictly positive. 4 Experimental Results We have applied our approach to two complimentary real-world robotic domains: robot localization, and mobile robot diagnostics. Both yield superior results using our new risk sensitive approach when compared to the standard particle filter. 4.1 Mobile Robot Localization Our first evaluation domain involves the problem of localizing a mobile robot from sensor data [2]. In our experiments, we focused on the most difficult of all localization problems: (a) A B C (b) Figure 1: (a) Robot Pearl, as it interacts with elderly people at an assisted living facility in Oakmont, PA. (b) Occupancy grid map. Shown here are also three testing locations labeled A, B, and C, and regions of high costs (black contours). (a) (b) Figure 2: (a) Risk function : the darker a location, the higher the risk. This function, which is used in the proposal distribution, is derived from the immediate risk function shown in Figure 1b. (b) Sample of a uniform distribution, taking into consideration the risk function. standard filter risk sensitive filter steps to re-localize when ported to A 120 13.7 89.3 12.3 steps to re-localize when ported to B 301 35.2 203 37.6 steps to re-localize when ported to C 63.2 6.2 53.2 7.7 number of violations after global kidnapping 96.1 14.1 57.4 10.3 Table 1: Localization results for the kidnapped robot problem, which emulates a total localization failure. Our new approach requires consistently fewer steps for re-localization in high-cost areas, and therefore incurs less cost. The kidnapped robot problem [4]. Here a well-localized robot is “tele-ported” to some unknown location and has to recover from this event. This problem plays an important role in evaluating the robustness of a localization algorithm. Figure 1a shows the robot Pearl, which has recently been deployed in an assisted living facility as an assistant to the elderly and cognitively frail. Our study is motivated by the fact that some of the robot’s operational area is a densely cluttered dining room, where the robot is not allowed to cross certain boundaries due to the danger of physically harming people. These boundaries are illustrated by the black contours shown in Figure 1b, which also depicts an occupancy grid map of the facility. Beyond the boundaries, the robot’s sensor are somewhat insufficient to avoid collisions, since they can only sense obstacles at one specific height (34 cm). Figure 2a shows the risk function , projected into 2D. The darker a location, the higher the risk. A sample set drawn from this risk function is shown in Figure 2b. This sample set represents a uniform posterior. Since risk sensitive particle filters incorporate the risk (a) B L Ry Rx α Sy Sx W1 W2 W4 W3 v1 v2 v3 v4 (b) −4 −3 −2 −1 0 1 2 3 4 0 1 2 3 4 5 6 Rover position at time step 1, 10, 22 and 35 x −> y −> (c) Figure 3: (a) The Hyperion rover, a mobile robot being developed at CMU. (b) Kinematic model. (c) Rover position at time step 1, 10, 22 and 35. 0 20 40 0 5 10 1000 samples 0 20 40 0 2 4 6 8 0 20 40 0 0.5 1 Time step −> 0 20 40 0 5 10 Most Likely State 100 samples 0 20 40 0 2 4 6 8 Sample Variance 0 20 40 0 0.5 1 Error using 1−0 loss Time step −> 0 20 40 0 5 10 10,000 samples 0 20 40 0 2 4 6 8 0 20 40 0 0.5 1 Time step −> 0 20 40 0 5 10 100,000 samples 0 20 40 0 2 4 6 8 0 20 40 0 0.5 1 Time step −> (a) 10 20 30 40 0 5 10 Most likely state 100 samples 0 20 40 0 5 10 15 Avg. sample variance 10 20 30 40 0 0.5 1 Median error (1−0 loss) 10 20 30 40 −0.1 0 0.1 Time step −> Error variance 0 20 40 0 5 10 1000 samples 0 20 40 0 5 10 15 0 20 40 −1 0 1 0 20 40 −1 0 1 Time step −> 0 20 40 0 5 10 10000 samples 0 20 40 0 5 10 15 0 20 40 −1 0 1 0 20 40 −1 0 1 Time step −> (b) Figure 4: Tracking curves obtained with (a) plain particle filters, and (b) our new risk sensitive filter. The bottom curves show the error, which is much smaller for our new approach. function into the sampling process, however, the density of samples is proportional to the risk function . Numerical results are summarized in Table 1, using data collected in the facility at dinner time. We ran two types of experiments: First, we kidnapped the robot to any of the locations marked A, B, and C in Figure 1, and measured the number of sensor readings required to recover from this global failure. All three locations are within the high-risk area so the recovery time is significantly shorter than with plain particle filters. Second, we measured the number of times a simple-minded planner that always looks at the most likely pose would violate the safety constraint. Here we find that our approach is almost twice as safe as the conventional particle filter, at virtually the same computational expense. All experiments were repeated 20 times, and rely on real-world data and operating conditions. 4.2 Mobile Robot Diagnosis In some domains, particle filters simply cannot be applied in real time because of a large number of high loss and low probability events. One example is the fault detection domain illustrated in Figure 3. Our evaluation involves a data set where a rover is driven with a variety of different control inputs in the normal operation mode. At the time step, wheel #3 becomes stuck and locked against a rock. The wheel is then driven in the backward direction, fixing the problem. The rover returns to the normal operation mode and continues to operate normally until the gear on wheel #4 breaks at the N time step. This fault is not recoverable and the controller just alters its input based on this state. Notice that both failures lead to very similar sensor measurement, despite the fact that they are caused by quite different events. Tracking results in Figure 4 show that our approach yields superior results to the standard particle filter. Even though failures are very unlikely, our approach successfully identifies them due to the high risk associated with such a failure while the plain particle filter essentially fails to do so. The estimation error is shown in the bottom row of Figure 4, which is practically zero for our approach when 1,000 or more samples are used. Vanialle particle filters exhibit non-zero error even with 100,000 samples. However, it is important to notice that these results were obtained using simulated data and a hand-tuned loss function approach. 5 Discussion We have proposed a particle filter algorithm that considers a cost model when generating samples. The key idea is that particles are generated in proportion to their posterior likelihood and to the risk that arises relative to a control goal. An MDP algorithm was developed that computes the risk function as a differential cumulative cost. Experimental results in two robotic domains show the superior performance of our new approach. An alternative approach for solving the problem addressed in this paper would be to analyze the estimation process as a partially observable Markov decision process (POMDP) [6]. Bounds on the performance loss due to the approximate nature of particle filters can be found in [9]. Pursuing the problem of risk-sensitive particle generation within the POMDP framework might be a promising future line of research. Acknowledgment The authors thank Dieter Fox and Wolfram Burgard, who generously provided some the localization software on which this research is built. Financial support by DARPA (TMR, MARS, CoABS and MICA programs) and NSF (ITR, Robotics, and CAREER programs) is gratefully acknowledged. References [1] X. Boyen and D. Koller. Tractable inference for complex stochastic processes. In Proc. UAI-98. [2] F. Dellaert, D. Fox, W. Burgard, and S. Thrun. Monte carlo localization for mobile robots. In Proc. ICRA-99. [3] A. Doucet, J.F.G. de Freitas, and N.J. Gordon, editors. Sequential Monte Carlo Methods In Practice. Springer, 2001. [4] S. Engelson. Passive Map Learning and Visual Place Recognition. PhD thesis, Computer Science Department, Yale University, 1994. [5] M. Isard and A. Blake. CONDENSATION: conditional density propagation for visual tracking. International Journal of Computer Vision, 29(1):5–28, 1998. [6] L.P. Kaelbling, M.L. Littman, and A.R. Cassandra. Planning and acting in partially observable stochastic domains. Artificial Intelligence, 101(1-2):99–134, 1998. [7] J. Liu and R. Chen. Sequential monte carlo methods for dynamic systems. Journal of the American Statistical Association, 93:1032–1044, 1998. [8] M. Pitt and N. Shephard. Filtering via simulation: auxiliary particle filter. Journal of the American Statistical Association, 94:590–599, 1999. [9] P. Poupart, L.E. Ortiz, and C. Boutilier. Value-directed sampling methods for monitoring POMDPs. In Proc. UAI-2001. [10] N. Roy and S. Thrun. Coastal navigation with mobile robot. In Proc. NIPS-99. [11] D.B. Rubin. Using the SIR algorithm to simulate posterior distributions. In Bayesian Statistics 3. Oxford Univ. Press, 1988. [12] M.A. Tanner. Tools for Statistical Inference. Springer, 1996.
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Characterizing neural gain control using spike-triggered covariance Odelia Schwartz Center for Neural Science New York University odelia@cns.nyu.edu E. J. Chichilnisky Systems Neurobiology The Salk Institute ej@salk.edu Eero P. Simoncelli Howard Hughes Medical Inst. Center for Neural Science New York University eero.simoncelli@nyu.edu Abstract Spike-triggered averaging techniques are effective for linear characterization of neural responses. But neurons exhibit important nonlinear behaviors, such as gain control, that are not captured by such analyses. We describe a spike-triggered covariance method for retrieving suppressive components of the gain control signal in a neuron. We demonstrate the method in simulation and on retinal ganglion cell data. Analysis of physiological data reveals significant suppressive axes and explains neural nonlinearities. This method should be applicable to other sensory areas and modalities. White noise analysis has emerged as a powerful technique for characterizing response properties of spiking neurons. A sequence of stimuli are drawn randomly from an ensemble and presented in rapid succession, and one examines the subset that elicit action potentials. This “spike-triggered” stimulus ensemble can provide information about the neuron’s response characteristics. In the most widely used form of this analysis, one estimates an excitatory linear kernel by computing the spike-triggered average (STA); that is, the mean stimulus that elicited a spike [e.g., 1, 2]. Under the assumption that spikes are generated by a Poisson process with instantaneous rate determined by linear projection onto a kernel followed by a static nonlinearity, the STA provides an unbiased estimate of this kernel [3]. Recently, a number of authors have developed interesting extensions of white noise analysis. Some have examined spike-triggered averages in a reduced linear subspace of input stimuli [e.g., 4]. Others have recovered excitatory subspaces, by computing the spiketriggered covariance (STC), followed by an eigenvector analysis to determine the subspace axes [e.g., 5, 6]. Sensory neurons exhibit striking nonlinear behaviors that are not explained by fundamentally linear mechanisms. For example, the response of a neuron typically saturates for large amplitude stimuli; the response to the optimal stimulus is often suppressed by the presence of a non-optimal mask [e.g., 7]; and the kernel recovered from STA analysis may change shape as a function of stimulus amplitude [e.g., 8, 9]. A variety of these nonlinear behaviors can be attributed to gain control [e.g., 8, 10, 11, 12, 13, 14], in which neural responses are suppressively modulated by a gain signal derived from the stimulus. Although the underlying mechanisms and time scales associated with such gain control are current topics of research, the basic functional properties appear to be ubiquitous, occurring throughout the nervous system. k 0 0 0 a b Figure 1: Geometric depiction of spike-triggered analyses. a, Spike-triggered averaging with two-dimensional stimuli. Black points indicate raw stimuli. White points indicate stimuli eliciting a spike, and the STA (black vector), which provides an estimate of , corresponds to their center of mass. b, Spike-triggered covariance analysis of suppressive axes. Shown are a set of stimuli lying on a plane perpendicular to the excitatory kernel, . Within the plane, stimuli eliciting a spike are concentrated in an elliptical region. The minor axis of the ellipse corresponds to a suppressive stimulus direction: stimuli with a significant component along this axis are less likely to elicit spikes. The stimulus component along the major axis of the ellipse has no influence on spiking. Here we develop a white noise methodology for characterizing a neuron with gain control. We show that a set of suppressive kernels may be recovered by finding the eigenvectors of the spike-triggered covariance matrix associated with smallest variance. We apply the technique to electrophysiological data obtained from ganglion cells in salamander and macaque retina, and recover a set of axes that are shown to reduce responses in the neuron. Moreover, when we fit a gain control model to the data using a maximum likelihood procedure within this subspace, the model accounts for changes in the STA as a function of contrast. 1 Characterizing suppressive axes As in all white noise approaches, we assume that stimuli correspond to vectors, , in some finite-dimensional space (e.g., a neighborhood of pixels or an interval of time samples). We assume a gain control model in which the probability of a stimulus eliciting a spike grows monotonically with the halfwave-rectified projection onto an excitatory linear kernel,
, and is suppressively modulated by the fullwave-rectified projection onto a set of linear kernels, . First, we recover the excitatory kernel, . This is achieved by presenting spherically symmetric input stimuli (e.g., Gaussian white noise) to the neuron and computing the STA (Fig. 1a). STA correctly recovers the excitatory kernel, under the assumption that each of the gain control kernels are orthogonal (or equal) to the excitatory kernel. The proof is essentially the same as that given for recovering the kernel of a linear model followed by a monotonic nonlinearity [3]. In particular, any stimulus can be decomposed into a component in the direction of the excitatory kernel, and a component in a perpendicular direction. This can be paired with another stimulus that is identical, except that its component in the perpendicular direction is negated. The two stimuli are equally likely to occur in a spherically Gaussian stimulus set (since they are equidistant from the origin), and they are equally likely to elicit a spike (since their excitatory components are equal, and their rectified perpendicular components are equal). Their vector average lies in the direction of the excitatory kernel. Thus, the STA (which is an average over all such stimuli, or all such stimulus pairs) must also lie in that direction. In a subsequent section we explain how to Arbitrary Excitatory: Model: Weights Retrieved: Eigenvalues: Axis number 1 3{ Variance (eigenvalue) 2.5{ 2{ 1.5{ 1{ 0 350 Suppressive: Excitatory: Suppressive: Figure 2: Estimation of kernels from a simulated model (equation 2). Left: Model kernels. Right: Sorted eigenvalues of covariance matrix of stimuli eliciting spikes (STC). Five eigenvalues fall significantly below the others. Middle: STA (excitatory kernel) and eigenvectors (suppressive kernels) associated with the lowest eigenvalues. recover the excitatory kernel when it is not orthogonal to the suppressive kernels. Next, we recover the suppressive subspace, assuming the excitatory kernel is known. Consider the stimuli lying on a plane perpendicular to this kernel. These stimuli all elicit the same response in the excitatory kernel, but they may produce different amounts of suppression. Figure 1b illustrates the behavior in a three-dimensional stimulus space, in which one axis is assumed to be suppressive. The distribution of raw stimuli on the plane is spherically symmetric about the origin. But the distribution of stimuli eliciting a spike is narrower along the suppressive direction: these stimuli have a component along the suppressive axis and are therefore less likely to elicit a spike. This behavior is easily generalized from this plane to the entire stimulus space. If we assume that the suppressive axes are fixed, then we expect to see reductions in variance in the same directions for any level of numerator excitation. Given this behavior of the spike-triggered stimulus ensemble, we can recover the suppressive subspace using principal component analysis. We construct the sample covariance matrix of the stimuli eliciting a spike:
(1) where is the number of spikes. To ensure the estimated suppressive subspace is orthogonal to the estimated (as in Figure 1b), the stimuli are first projected onto the subspace perpendicular to the estimated . The principal axes (eigenvectors) of that are associated with small variance (eigenvalues) correspond to directions in which the response of the neuron is modulated suppressively. We illustrate the technique on simulated data for a neuron with a spatio-temporal receptive field. The kernels are a set of orthogonal bandpass filters. The stimulus vectors of this input sequence are defined over a 18-sample spatial region and a 18-sample time window (i.e., a ! #" -dimensional space). Spikes are generated using a Poisson process with mean rate determined by a specific form of gain control [14]: $&% ')( +* -, . / 10 .1243 .65 (2) The goal of simulation is to recover excitatory kernel , the suppressive subspace spanned by , weights 0 , and constant 3 . 0 Arbitrary Retrieved kernels: Eigenvalues: Excitatory: Suppressive: Axis number Variance (eigenvalue) actual 1 95 % confidence 0 26 Figure 3: Left: Retrieved kernels from STA and STC analysis of ganglion cell data from a salamander retina (cell 1999-11-12-B6A). Right: sorted eigenvalues of the spike-triggered covariance matrix, with corresponding eigenvectors. Low eigenvalues correspond to suppressive directions, while other eigenvalues correspond to arbitrary (ignored) directions. Raw stimulus ensemble was sphered (whitened) prior to analysis and low-variance axes underrepresented in stimulus set were discarded. Figure 2 shows the original and estimated kernels for a model simulation with 600K input samples and 36.98K spikes. First, we note that STA recovers an accurate estimate of the excitatory kernel. Next, consider the sorted eigenvalues of , as plotted in Figure 2. The majority of the eigenvalues descend gradually (the covariance matrix of the white noise source should have constant eigenvalues, but remember that those in Figure 2 are computed from a finite set of samples). The last five eigenvalues are significantly below the values one would obtain with randomly selected stimulus subsets. The eigenvectors associated with these lowest eigenvalues span approximately the same subspace as the suppressive kernels. Note that some eigenvectors correspond to mixtures of the original suppressive kernels, due to non-uniqueness of the eigenvector decomposition. In contrast, eigenvectors corresponding to eigenvalues in the gradually-descending region appear arbitrary in their structure. Finally, we can recover the scalar parameters of this specific model ( 0 and 3 ) by selecting them to maximize the likelihood of the spike data according to equation (2). Note that a direct maximum likelihood solution on the raw data would have been impractical due to the high dimensionality of the stimulus space. 2 Suppressive Axes in Retinal Ganglion Cells Retinal ganglion cells exhibit rapid [8, 15] as well as slow [9, 16, 17] gain control. We now demonstrate that we can recover a rapid gain control signal by applying the method to data from salamander retina [9]. The input sequence consists of 80K time samples of full-field 33Hz flickering binary white noise (contrast 8.5%). The stimulus vectors of this sequence are defined over a 60-segment time window. Since stimuli are finite in number and binary, they are not spherically distributed. To correct for this, we discard low-variance axes and whiten the stimuli within the remaining axes. Figure 3 depicts the kernels estimated from the 623 stimulus vectors eliciting spikes. Similar to the model simulation, the eigenvalues gradually fall off, but four of the eigenvalues appear to drop significantly below the rest. To make this more concrete, we test the hypothesis that the majority of the eigenvalues are consistent with those of randomly selected stimulus vectors, but that the last " eigenvalues fall significantly below this range. Specifically, we perform a Monte Carlo simulation, drawing (with replacement) random subsets of 623 stimuli from the full set of raw stimuli. We also randomly select " (orthogonal) projection onto arbitrary kernel projection onto excitatory kernel 0 0. 5 0 projection onto arbitrary kernel projection onto suppressive kernel a b -0. 5 0. 5 -0. 5 0 0. 5 0 -0. 5 0. 5 -0. 5 Figure 4: Scatter plots from salamander ganglion cell data (cell 1999-11-12-B6A). Black points indicate the raw stimulus set. White points indicate stimuli eliciting a spike. a, Projection of stimuli onto estimated excitatory kernel vs. arbitrary kernel. b, Projection of stimuli onto an estimated suppressive kernel vs. arbitrary kernel. axes, representing a suppressive subspace, and project this subspace out of the set of randomly chosen stimuli. We then compute the eigenvalues of the sample covariance matrix of these stimuli. We repeat this times, and estimate a 95 percent confidence interval for each of the eigenvalues. The figure shows that the first eigenvalues lie within the confidence interval. In practice, we repeat this process in a nested fashion, assuming initially no directions are significantly suppressive, then one direction, and so on up to four directions. These low eigenvalues correspond to eigenvectors that are concentrated in recent time (as is the estimated excitatory kernel). The remaining eigenvectors appear to be arbitrary, spanning the full temporal window. We emphasize that these kernels should not be interpreted to correspond to receptive fields of individual neurons underlying the suppressive signal, but merely provide an orthogonal basis for a suppressive subspace. We can now verify that the recovered STA axis is in fact excitatory, and the kernels corresponding to the lowest eigenvalues are suppressive. Figure 4a shows a scatter plot of the stimuli projected onto the excitatory axis vs. an arbitrary axis. Spikes are seen to occur only when the component along the excitatory axis is high, as expected. Figure 4b is a scatter plot of the stimuli projected onto one of the suppressive axes vs. an arbitrary (ignored) axis. The spiking stimuli lie within an ellipse, with the minor axis corresponding to the suppressive kernel. This is exactly what we would expect in a suppressive gain control system (see Figure 1b). Figure 5 illustrates recovery of a two-dimensional suppressive subspace for a macaque retinal ganglion cell. The subspace was computed from the 36.43K stimulus vectors eliciting spikes out of a total of 284.74K vectors. The data are qualitatively similar to those of the salamander cell, although both the strength of suppression and specific shapes of the scatter plots differs. In addition to suppression, the method recovers facilitation (i.e., high-variance axes) in some cells (not shown here). 3 Correcting for Bias in Kernel Estimates The kernels in the previous section were all recovered from stimuli of a single contrast. However, when the STA is computed in a ganglion cell for low and high contrast stimuli, the low-contrast kernel shows a slower time course [9] (figure 7,a). This would appear inconsistent with the method we describe, in which the STA is meant to provide an estimate of a single excitatory kernel. This behavior can be explained by assuming a model of the form given in equation 2, and in addition dropping the constraint that the gain control kernels are orthogonal (or identical) to the excitatory kernel. projection onto arbitrary kernel projection onto excitatory kernel projection onto arbitrary kernel projection onto suppressive kernel 0. 5 0 0.5 -0. 5 0 0.5 0. 5 0 0.5 -0. 5 0 0.5 0 60 1 actual 95% confidence
Variance (eigenvalue) a b c ! " # $ %& ' ' " ()*) +($ Figure 5: a, Sorted eigenvalues of stimuli eliciting spikes from a macaque retina (cell 200109-29-E6A). b-c, Scatter plots of stimuli projected onto recovered axes. k0k STA estimate Gain kernel Figure 6: Demonstration of estimator bias. When a gain control kernel is not orthogonal to the excitatory kernel, the responses to one side of the excitatory kernel are suppressed more than those on the other side. The resulting STA estimate is thus biased away from the true excitatory kernel, . First we show that when the orthogonality constraint is dropped, the STA estimate of the excitatory kernel is biased by the gain control signal. Consider a situation in which a suppressive kernel contains a component in the direction of the excitatory kernel, . We write -, 2 /. , where /. is perpendicular to the excitatory kernel. Then, for example, a stimulus 210 2. , with 043 , produces a suppressive component along equal to , . 250 2. . , but the corresponding paired stimulus vector 0 2. produces a suppressive component of , . 0 /. . . Thus, the two stimuli are equally likely to occur but not equally likely to elicit a spike. As a result, the STA will be biased in the direction 2. . Figure 6 illustrates an example in which a non-orthogonal suppressive axis biases the estimate of the STA. Now consider the model in equation 2 in the presence of a non-orthogonal suppressive subspace. Note that the bias is stronger for larger amplitude stimuli because the constant term 3 . dominates the gain control signal for weak stimuli. Indeed, we have previously hypothesized that changes in receptive field tuning can arise from divisive gain control models that include an additive constant [14]. Even when the STA estimate is biased by the gain control signal, we can still obtain an (asymptotically) unbiased estimate of the excitatory kernel. Specifically, the true excitatory kernel lies within the subspace spanned by the estimated (biased) excitatory and suppressive kernels. So, assuming a particular gain control model, we can again maximize the likelihood of the data, but now allowing both the excitatory and suppressive kernels to move within the subspace spanned by the initial estimated kernels. The resulting suppresMn Excitatory: Suppressive: { a b Low contrast STA High contrast STA 0 -0.5 0 0.1 Time preceding spike (sec) -1.8 0 -0.5 0 0.1 Time preceding spike (sec) -1.8 Low contrast STA High contrast STA { { 0.99 0.97 0.87 { { 0.52 0.46 Weights c Figure 7: STA kernels estimated from low (8.5%) and high (34%) contrast salamander retinal ganglion cell data (cell 1999-11-12-B6A). Kernels are normalized to unit energy. a, STA kernels derived from ganglion cell spikes. b, STA kernels derived from simulated spikes using ML-estimated model. c, Kernels and corresponding weights of ML-estimated model. sive kernels need not be orthogonal to the excitatory kernel. We maximize the likelihood of the full two-contrast data set using a model that is a generalization of that given by equation (2): $ % ')( +* , % / 10 . , . 243 (3) The exponent ' is incorporated to allow for more realistic contrast-response functions. The excitatory axis is initially set to the STA and the suppressive axes are set to the low-eigenvalue eigenvectors of the STC, along with the STA (e.g., to allow for selfsuppression). The recoveredaxes and weights are shown in Figure 7b, and remaining model parameters are: ' 5 , 3 5 " . Whereas the axes recovered from the STA/STC analysis are orthogonal, the axes determined during the maximum likelihood stage need not be (and in the data example are not) orthogonal. Figure 7b also demonstrates that the fitted model accounts for the change in STA observed at different contrast levels. Specifically, we simulate responses of the model (equation (3) with Poisson spike generation) on each of the two contrast stimulus sets, and then compute the STA based on these simulated spike trains. Although it is based on a single fixed excitatory kernel, the model exhibits a change in STA shape as a function of contrast very much like the salamander neuron. 4 Discussion We have described a spike-triggered covariance method for characterizing a neuron with gain control, and demonstrated the plausibility of the technique through simulation and analysis of neural data. The suppressive axes recovered from retinal ganglion cell data appear to be significant because: (1) As in the model simulation, a small number of eigenvalues are significantly below the rest; (2) The eigenvectors associated with these axes are concentrated in a temporal region immediately preceding the spike, unlike the remaining axes; (3) Projection of the multi-dimensional stimulus vectors onto these axes reveal reductions of spike probability; (4) The full model, with parameters recovered through maximum likelihood, explains changes in STA as a function of contrast. Models of retinal processing often incorporate gain control [e.g., 8, 10, 15, 17, 18]. We have shown for the first time how one can use white noise analysis to recover a gain control subspace. The kernels defining this subspace correspond to relatively short timescales. Thus, it is interesting to compare the recovered subspace to models of rapid gain control. In particular, Victor [15] proposed a retinal gain model in which the gain signal consists of time-delayed copies of the excitatory kernel. In fact, for the cell shown in Figure 3, the recovered suppressive subspace lies within the space spanned by shifted copies of the excitatory kernel. The fact that we do not see evidence for slow gain control in the analysis might indicate that these signals do not lie within a low-dimensional stimulus subspace. In addition, the analysis is not capable of distinguishing between physiological mechanisms that could underlie gain control behaviors. Potential candidates may include internal biochemical adjustments, non-Poisson spike generation mechanisms, synaptic depression, and shunting inhibition due to other neurons. This technique should be applicable to a far wider range of neural data than has been shown here. Future work will incorporate analysis of data gathered using stimuli that vary in both time and space (as in the simulated example of Figure 2). We are also exploring applicability of the technique to other visual areas. Acknowledgments We thank Liam Paninski and Jonathan Pillow for helpful discussions and comments, and Divya Chander for data collection. References [1] E deBoer and P Kuyper. Triggered correlation. In IEEE Transact. Biomed. Eng., volume 15, pages 169–179, 1968. [2] J P Jones and L A Palmer. The two-dimensional spatial structure of simple receptive fields in the cat striate cortex. J Neurophysiology, 58:1187–11211, 1987. [3] E J Chichilnisky. A simple white noise analysis of neuronal light responses. Network: Computation in Neural Systems, 12(2):199–213, 2001. [4] D L Ringach, G Sapiro, and R Shapley. A subspace reverse-correlation technique for the study of visual neurons. Vision Research, 37:2455–2464, 1997. [5] R de Ruyter van Steveninck and W Bialek. Coding and information transfer in short spike sequences. In Proc.Soc. Lond. B. Biol. Sci., volume 234, pages 379–414, 1988. [6] B A Y Arcas, A L Fairhall, and W Bialek. What can a single neuron compute? In Advances in Neural Information Processing Systems, volume 13, pages 75–81, 2000. [7] M Carandini, D J Heeger, and J A Movshon. Linearity and normalization in simple cells of the macaque primary visual cortex. Journal of Neuroscience, 17:8621–8644, 1997. [8] R M Shapley and J D Victor. The effect of contrast on the transfer properties of cat retinal ganglion cells. J. Physiol. (Lond), 285:275–298, 1978. [9] D Chander and E J Chichilnisky. Adaptation to temporal contrast in primate and salamander retina. J Neurosci, 21(24):9904–9916, 2001. [10] R Shapley and C Enroth-Cugell. Visual adaptation and retinal gain control. Progress in Retinal Research, 3:263–346, 1984. [11] R F Lyon. Automatic gain control in cochlear mechanics. In P Dallos et al., editor, The Mechanics and Biophysics of Hearing, pages 395–420. Springer-Verlag, 1990. [12] W S Geisler and D G Albrecht. Cortical neurons: Isolation of contrast gain control. Vision Research, 8:1409–1410, 1992. [13] D J Heeger. Normalization of cell responses in cat striate cortex. Vis. Neuro., 9:181–198, 1992. [14] O Schwartz and E P Simoncelli. Natural signal statistics and sensory gain control. Nature Neuroscience, 4(8):819–825, August 2001. [15] J D Victor. The dynamics of the cat retinal X cell centre. J. Physiol., 386:219–246, 1987. [16] S M Smirnakis, M J Berry, David K Warland, W Bialek, and M Meister. Adaptation of retinal processing to image contrast and spatial scale. Nature, 386:69–73, March 1997. [17] K J Kim and F Rieke. Temporal contrast adaptation in the input and output signals of salamander retinal ganglion cells. J. Neurosci., 21(1):287–299, 2001. [18] M Meister and M J Berry. The neural code of the retina. Neuron, 22:435–450, 1999.
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Speech Recognition with Missing Data using Recurrent Neural Nets P.D. Green Speech and Hearing Research Group Department of Computer Science University of Sheffield Sheffield S14DP, UK p.green@dcs.shef.ac.uk S. Parveen Speech and Hearing Research Group Department of Computer Science University of Sheffield Sheffield S14DP, UK s.parveen@dcs.shef.ac.uk Abstract In the ‘missing data’ approach to improving the robustness of automatic speech recognition to added noise, an initial process identifies spectraltemporal regions which are dominated by the speech source. The remaining regions are considered to be ‘missing’. In this paper we develop a connectionist approach to the problem of adapting speech recognition to the missing data case, using Recurrent Neural Networks. In contrast to methods based on Hidden Markov Models, RNNs allow us to make use of long-term time constraints and to make the problems of classification with incomplete data and imputing missing values interact. We report encouraging results on an isolated digit recognition task. 1. Introduction Automatic Speech Recognition systems perform reasonably well in controlled and matched training and recognition conditions. However, performance deteriorates when there is a mismatch between training and testing conditions, caused for instance by additive noise (Lippmann, 1997). Conventional techniques for improving recognition robustness (reviewed by Furui 1997) seek to eliminate or reduce the mismatch, for instance by enhancement of the noisy speech, by adapting statistical models for speech units to the noise condition or simply by training in different noise conditions. Missing data techniques provide an alternative solution for speech corrupted by additive noise which make minimal assumptions about the nature of the noise. They are based on identifying uncorrupted, reliable regions in the frequency domain and adapting recognition algorithms so that classification is based on these regions. Present missing data techniques developed at Sheffield (Barker et al. 2000a, Barker et al. 2000b, Cooke et al., 2001) and elsewhere (Drygaglo et al., 1998, Raj et al., 2000) adapt the prevailing technique for ASR based on Continuous Density Hidden Markov Models. CDHMMs are generative models which do not give direct estimates of posterior probabilities of the classes given the acoustics. Neural Networks, unlike HMMs, are discriminative models which do give direct estimates of posterior probabilities and have been used with success in hybrid ANN/HMM speech recognition systems (Bourlard et al., 1998). In this paper, we adapt a recurrent neural network architecture introduced by (Gingras & Bengio, 1998) for robust ASR with missing data. 2. Missing data techniques for Robust ASR 2.1 Missing data masks Speech recognition with missing data is based on the assumption that some regions in time/frequency remain uncorrupted for speech with added noise. See (Cooke et al., 2001) for arguments to support this assumption. Initial processes, based on local signal-to-noise estimates, on auditory grouping cues, or a combination (Barker et al., 2001) define a binary ‘missing data mask’: ones in the mask indicate reliable (or ‘present’) features and zeros indicate unreliable (or ‘missing’) features. 2.2 Classification with missing data Techniques for classification with incomplete data can be divided into imputation and marginalisation. Imputation is a technique in which missing features are replaced by estimated values to allow the recognition process proceed in normal way. If the missing values are replaced by either zeros, random values or their means based on training data, the approach is called unconditional imputation. On the other hand in conditional imputation conditional statistics are used to estimate the missing values given the present values. In the marginalisation approach missing values are ignored (by integrating over their possible ranges) and recognition is performed with the reduced data vector which is considered reliable. For the multivariate mixture Gaussian distributions used in CDHMMs, marginalisation and conditional imputation can be formulated analytically (Cooke et al., 2001). For missing data ASR further improvements in both techniques follow from using the knowledge that for spectral energy features the unreliable data is bounded between zero and the energy in speech+noise mixture (Vizinho et al., 1999), (Josifovski et al., 1999). These techniques are referred to as bounded marginalisation and bounded imputation. Coupled with a ‘softening’ of the reliable/unreliable decision, missing data techniques produce good results on a standard connected-digits-in-noise recognition task: performance using models trained on clean data is comparable, and in severe noise superior, to conventional systems trained across different noise conditions (Barker et al., 2001). 2.3 Why recurrent neural nets for missing data robust ASR? Several neural net architectures have been proposed to deal with the missing data problem in general (Ahmed & Tresp, 1993), (Ghahramani & Jordan, 1994). The problem in using neural networks with missing data is to compute the output of a node/unit when some of its input values are unavailable. For marginalisation, this involves finding a way of integrating over the range of the missing values. A robust ASR system to deal with missing data using neural networks has recently been proposed by (Morris et al., 2000). This is basically a radial basis function neural network with the hidden units associated with a diagonal covariance gaussian. The marginal over the missing values can be computed in this case and hence the resulting system is equivalent to the HMM based missing data speech recognition system using marginalisation. Reported performance is also comparable to that of the HMM based speech recognition system. In this paper missing data is dealt with by imputation. We use recurrent neural networks to estimate missing values in the input vector. RNNs have the potential to capture long-term contextual effects over time, and hence to use temporal context to compensate for missing data which CDHMM based missing data techniques do not do. The only contextual information available in CDHMM decoding come from the addition of temporal derivatives to the feature vector. RNNs also allow a single net to perform both imputation and classification, with the potential of combining these processes to mutual benefit. The RNN architecture proposed by Gingras et al. (1998) is based on a fully-connected feedforward network with input, hidden and output layers using hyperbolic tangent activation functions. The output layer has one unit for each class and the network is trained with the correct classification as target. Recurrent links are added to the feedforward net with unit delay from output to the hidden units as in Jordan networks (Jordan, 1988). There are also recurrent links with unit delay from hidden units to missing input units to impute missing features. In addition, there are self delayed terms with a fixed weight for each unit which basically serve to stabilise RNN behaviour over time and help in imputation as well. Gingras et al. used this RNN both for a pattern classification task with static data (one input vector for each example) and sequential data (a sequence of input values for each example). Our aim is to adapt this architecture for robust ASR with missing data. Some preliminary static classification experiments were performed on vowel spectra (individual spectral slices excised from the TIMIT database). RNN performance on this task with missing data was better than standard MLP and gaussian classifiers. In the next section we show how the net can be adapted for dynamic classification of the spectral sequences constituting words. 3. RNN architecture for robust ASR with missing data Figure 1 illustrates our modified version of the Gingras and Bengio architecture. Instead of taking feedback from the output to the hidden layer we have chosen a fully connected or Elman RNN (Elman, 1990) where there are full recurrent links from the past hidden layer to the present hidden layer (figure 1). We have observed that these links produce faster convergence, in agreement with (Pedersen, 1997). The number of input units depends on the size of feature vector, i.e. the number of spectral channels. The number of hidden units is determined by experimentation. There is one output unit for each pattern class. In our case the classes are taken to be whole words, so in the isolated digit recognition experiments we report, there are eleven output units, for ‘1’ - ‘9’, ‘zero’ and ‘oh’. In training, missing inputs are initialised with their unconditional means. The RNN is then allowed to impute missing values for the next frame through the recurrent links, after a feedforward pass. Where is the missing feature at time t, is the learning rate, indicates recurrent links from a hidden unit to the missing input and is the activation of hidden unit j at time t-1. The average of the RNN output over all the frames of an example is taken after these frames have gone through a forward pass. The sum squared error between the correct targets and the RNN output for each frame is back-propagated through time and RNN X m t ( , ) 1 γ – ( )X m t 1 – ( , ) v jm f hid j t 1 – , ( ) ( ) j 1 = H ∑ + = X m t ( , ) γ v jm hid j t 1 – , ( ) weights are updated until a stopping criterion is reached. The recognition phase consists of a forward pass to produce RNN output for unseen data and imputation of missing features at each time step. The highest value in the averaged output vector is taken as the correct class. 4. Isolated word recognition experiments Continuous pattern classification experiments were performed using data from 30 male speakers in the isolated digits section of the TIDIGIT database (Leonard, 1984). There were two examples per speaker of each of the 11 words (i.e. 1-9, zero, oh). 220 examples were chosen from a subset of 10 speakers for training. Recognition was performed on 110 examples from the speakers not included in training. A validation set of 110 examples was used for early stopping. Features were extracted from hamming windowed speech with a window size of 25 msec and 50% overlap. Two types of feature vectors used for the experiments were total energies in the four frequency bands (115-629 Hz, 565-1370 Hz, 1262-2292 Hz and 22123769 Hz) and 20 mel scaled FFT filter bank energies. In the initial experiments we report, the missing data masks were formed by deleting Figure 1: RNN architecture for robust ASR with missing data technique. Solid arrows show full forward and recurrent connections between two layers. Shaded blocks in the input layer indicate missing inputs which keep changing at every time step. Missing inputs are fully connected (solid arrows) with the hidden layer with a unit delay in addition to delayed self-connection (thin arrows) with a fixed weight. -1 h i d d e n -1 one two three nine oh zero o u t p u t Reliable features -1 spectral energy features at random. This allows comparison with early results with HMMbased missing data recognition (Cooke et al. 1996) and close experimental control. For training 1/3rd of the training examples were clean, 1/3rd had 25% deletions and 1/3rd had 50% deletions. Recognition performance was evaluated with 0% to 80% missing features with an increment of 10%. 5. Results 5.1 RNN performance as a classifier An RNN with 20 inputs, 65 hidden and 11 output units was chosen for recognition and imputation with 20 features per time frame. Its performance on various amounts of missing features from 0% to 80%, shown in Figure 2 (the ‘RNN imputation’ curve), is much better than the standard Elman RNN trained on clean speech only for classification task and tested with the mean imputation. Use of the self delayed term in addition to the recurrent links for imputation of missing features contributes positively in case of sequential data. Results resemble those reported for HMMs in (Cooke et al. 1996). We also show that results are superior to ‘last reliable imputation’ in which the imputed value of a feature is the last reliable value for that feature. 5.2 RNN performance on pattern completion Imputation, or pattern completion, performance was observed for an RNN trained with 4 features per frame of the speech and is shown in Figure 3. The RNN for this task had 4 input, 45 hidden and 11 output units. In figure 3(a), solid curves show the true values of the feature in each frequency band at every frame for an example of a spoken ‘9’, the horizontal lines are mean feature values, and the circles are the missing values imputed by the RNN. Imputed values are encouragingly close to the true values. For this network, classification error for recognition was 10.7% at 0% missing and 46.4% at 80% missing. Figure 2: Comparison of RNN classification performance for different imputation methods. 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 90 % missing classification error % missing values replaced by means missing values replaced by "last reliable" values missing values imputed by RNN The top curve in figure 3(b) shows the average pattern completion error when an RNN with 20 input channels was trained on clean speech and during recognition missing features were imputed from their unconditional means. The bottom curve is the average pattern completion error with missing features imputed by the network. This demonstrates the clear advantage of using the RNN for both imputation and classification. 6. Conclusion & future work The experiments reported in section 5 constitute no more than a proof of concept. Our next step will be to extend this recognition system for the connected digits recognition task with missing data, following the Aurora standard for robust ASR (Pearce et al. 2000). This will provide a direct comparison with HMM-based missing data recognition (Barker et al., 2001). In this case we will need to introduce ‘silence’ as an additional recognition class, and the training targets will be obtained by forced-alignment on clean speech with an existing recogniser. We will use realistic missing data masks, rather than random deletions. This is known to be a more demanding condition (Cooke et al. 1996). When we are training using clean speech with added noise, another possibility is to use the true values of the corrupted features as training targets for imputation. Use of actual targets for missing values has been reported by (Seung, 1997) but the RNN architecture in the latter work supports only pattern completion. Acknowledgement This work is being supported by Nokia Mobile Phones, Denmark and the UK Overseas Research Studentship scheme. References Ahmed, S. & Tresp, V. (1993). Some solutions to the missing feature problem in vision. Advances in 10 20 30 40 50 60 70 80 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 % missing average imputation error mean imputation rnn imputation (b) (a) Figure 3: (a) Missing values for digit 9 imputed by an RNN (b) Average imputation errors for mean imputation and RNN imputation 0 5 10 15 20 25 30 35 40 −4 −2 0 2 energy band1 0 5 10 15 20 25 30 35 40 −4 −2 0 2 energy band2 0 5 10 15 20 25 30 35 40 −3 −2 −1 0 energy band3 0 5 10 15 20 25 30 35 40 −2.5 −2 −1.5 energy band4 frame number true values RNN imputation mean Neural Information Processing Systems 5 (S.J.Hanson, J.D.Cowan & C.L.Giles, eds.), Morgan Kaufmann, San Mateo, CA, 393-400. Barker, J., Green, P.D. and Cooke, M.P. (2001). Linking auditory scene analysis and robust ASR by missing data techniques. Workshop on Innovation in Speech Processing 2001, Stratford-upon-Avon, UK. Barker, J., Josifovski, L., Cooke, M.P. and Green, P.D. (2000a). Soft decisions in missing data techniques for robust automatic speech recognition. Accepted for ICSLP-2000, Beijing. Barker, J., Cooke, M.P. and Ellis, D.P.W. (2000b). Decoding speech in the presence of other sound sources. Accepted for ICSLP-2000, Beijing Bourlard, H. and N. Morgan (1998). Hybrid HMM/ANN systems for speech recognition: Overview and new research directions. In C. L.Giles and M. Gori (Eds.), Adaptive Processing of Sequences and Data Structures, Volume 1387 of Lecture Notes in Artificial Intelligence, pp. 389--417. Springer. Cooke, M., Green, P., Josifovski, L. and Vizinho, A. (2001). Robust automatic speech recognition with missing and unreliable acoustic data. submitted to Speech Communication, 24th June 1999. Cooke, M.P., Morris, A. & Green, P.D. (1996). Recognising occluded speech. ESCA Tutorial and Workshop on the Auditory Basis of Speech Perception, Keele University, July 15-19. Drygajlo, A. & El-Maliki, M. (1998). Speaker verification in noisy environment with combined spectral subtraction and missing data theory. Proc ICASSP-98, vol. I, pp121-124. Elman, J.L. (1990). Finding structure in time. Cognitive Science, 14, 179-211. Furui, S. (1997). Recent advances in robust speech recognition. Proc. ESCA-NATO Tutorial and Research Workshop on Robust Speech Recognition for Unknown Communication Channels, France, pp.11-20. Gingras, F. and Bengio, Y. (1998). Handling Asynchronous or Missing Data with Recurrent Networks. International Journal of Computational Intelligence and Organizations, vol. 1, no. 3, pp. 154-163. Ghahramani, Z. & Jordan, M.I. (1994). Supervised learning from incomplete data via an EM approach. Advances in Neural Information Processing Systems 6 (J.D. Cowan, G. Tesauro & J. Alspector, eds.), Morgan Kaufmann, San Mateo, CA, pp.120-129. Jordan, M. I. (1998). Supervised learning and systems with excess degrees of freedom. Technical Report COINS TR 88-27, Massachusetts Institute of Technology, 1988. Josifovski, L., Cooke, M., Green, P. and Vizinho, A. (1999). State based imputation of missing data for robust speech recognition and speech enhancement. Proc. Eurospeech’99, Budapest, Vol. 6, pp. 2837-2840. Leonard, R. G., (1984). A Database for Speaker-Independent Digit Recognition. Proc. ICASSP 84, Vol. 3, p. 42.11, 1984. Lippmann, R. P. (1997). Speech recognition by machines and humans. Speech Communication vol. 22 no. 1 pp. 1-15. Morris, A., Josifovski, L., Bourlard, H., Cooke, M.P. and Green, P.D. (2000). A neural network for classification with incomplete data: application to robust ASR. ICSLP 2000, Beijing China. Pearce, D. and Hirsch, H.--G. (2000). The aurora experimental framework for the performance evaluation of speech recognition systems under noisy conditions. In Proc. ICSLP 2000, IV, 29--32, Beijing, China. Pedersen, M. W. (1997). Optimization of Recurrent Neural Networks for Time Series Modeling. PhD thesis. Technical University of Denmark. Raj, B., Seltzer, M., & Stern, R. (2000). Reconstruction of damaged spectrographic features for robust speech recognition. ICSLP 2000. Seung, H. S. (1997). Learning continuous attractors in Recurrent Networks. Proc. NIPS’97 pp 654660. Vizinho, A., Green, P., Cooke, M. and Josifovski, L. (1999). Missing data theory, spectral subtraction and signal-to-noise estimation for robust ASR: An integrated study. Proc. Eurospeech’99, Budapest, Sep. 1999, Vol. 5, pp. 2407-2410.
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On Spectral Clustering: Analysis and an algorithm Andrew Y. Ng CS Division U.C. Berkeley ang@cs.berkeley.edu Michael I. Jordan CS Div. & Dept. of Stat. U.C. Berkeley jordan@cs.berkeley.edu Abstract Yair Weiss School of CS & Engr. The Hebrew Univ. yweiss@cs.huji.ac.il Despite many empirical successes of spectral clustering methodsalgorithms that cluster points using eigenvectors of matrices derived from the data- there are several unresolved issues. First, there are a wide variety of algorithms that use the eigenvectors in slightly different ways. Second, many of these algorithms have no proof that they will actually compute a reasonable clustering. In this paper, we present a simple spectral clustering algorithm that can be implemented using a few lines of Matlab. Using tools from matrix perturbation theory, we analyze the algorithm, and give conditions under which it can be expected to do well. We also show surprisingly good experimental results on a number of challenging clustering problems. 1 Introduction The task of finding good clusters has been the focus of considerable research in machine learning and pattern recognition. For clustering points in Rn-a main application focus of this paper- one standard approach is based on generative models, in which algorithms such as EM are used to learn a mixture density. These approaches suffer from several drawbacks. First, to use parametric density estimators, harsh simplifying assumptions usually need to be made (e.g., that the density of each cluster is Gaussian) . Second, the log likelihood can have many local minima and therefore multiple restarts are required to find a good solution using iterative algorithms. Algorithms such as K-means have similar problems. A promising alternative that has recently emerged in a number of fields is to use spectral methods for clustering. Here, one uses the top eigenvectors of a matrix derived from the distance between points. Such algorithms have been successfully used in many applications including computer vision and VLSI design [5, 1]. But despite their empirical successes, different authors still disagree on exactly which eigenvectors to use and how to derive clusters from them (see [11] for a review). Also, the analysis of these algorithms, which we briefly review below, has tended to focus on simplified algorithms that only use one eigenvector at a time. One line of analysis makes the link to spectral graph partitioning, in which the second eigenvector of a graph's Laplacian is used to define a semi-optimal cut. Here, the eigenvector is seen as a solving a relaxation of an NP-hard discrete graph partitioning problem [3], and it can be shown that cuts based on the second eigenvector give a guaranteed approximation to the optimal cut [9, 3]. This analysis can be extended to clustering by building a weighted graph in which nodes correspond to datapoints and edges are related to the distance between the points. Since the majority of analyses in spectral graph partitioning appear to deal with partitioning the graph into exactly two parts, these methods are then typically applied recursively to find k clusters (e.g. [9]). Experimentally it has been observed that using more eigenvectors and directly computing a k way partitioning is better (e.g. [5, I]). Here, we build upon the recent work of Weiss [11] and Meila and Shi [6], who analyzed algorithms that use k eigenvectors simultaneously in simple settings. We propose a particular manner to use the k eigenvectors simultaneously, and give conditions under which the algorithm can be expected to do well. 2 Algorithm Given a set of points S = {81' ... ,8n } in jRl that we want to cluster into k subsets: 1. Form the affinity matrix A E Rnx n defined by A ij = exp(-Ilsi - sjW/2(2 ) if i # j , and Aii = O. 2. Define D to be the diagonal matrix whose (i, i)-element is the sum of A's i-th row, and construct the matrix L = D-l / 2AD-l / 2 .1 3. Find Xl , X2 , ... , Xk , the k largest eigenvectors of L (chosen to be orthogonal to each other in the case of repeated eigenvalues), and form the matrix X = [XIX2 . . . Xk) E Rn xk by stacking the eigenvectors in columns. 4. Form the matrix Y from X by renormalizing each of X's rows to have unit length (i.e. Yij = X ij/CL.j X~)1 / 2). 5. Treating each row of Y as a point in Rk , cluster them into k clusters via K-means or any other algorithm (that attempts to minimize distortion). 6. Finally, assign the original point Si to cluster j if and only if row i of the matrix Y was assigned to cluster j. Here, the scaling parameter a 2 controls how rapidly the affinity Aij falls off with the distance between 8i and 8j, and we will later describe a method for choosing it automatically. We also note that this is only one of a large family of possible algorithms, and later discuss some related methods (e.g., [6]). At first sight, this algorithm seems to make little sense. Since we run K-means in step 5, why not just apply K-means directly to the data? Figure Ie shows an example. The natural clusters in jR2 do not correspond to convex regions, and Kmeans run directly finds the unsatisfactory clustering in Figure li. But once we map the points to jRk (Y's rows) , they form tight clusters (Figure lh) from which our method obtains the good clustering shown in Figure Ie. We note that the clusters in Figure lh lie at 90 0 to each other relative to the origin (cf. [8]). lReaders familiar with spectral graph theory [3) may be more familiar with the Laplacian 1- L. But as replacing L with 1- L would complicate our later discussion, and only changes the eigenvalues (from Ai to 1 - Ai) and not the eigenvectors, we instead use L. 3 Analysis of algorithm 3.1 Informal discussion: The "ideal" case To understand the algorithm, it is instructive to consider its behavior in the "ideal" case in which all points in different clusters are infinitely far apart. For the sake of discussion, suppose that k = 3, and that the three clusters of sizes n1, n2 and n3 are 8 1,82 , and 8 3 (8 = 8 1 U 82 U 8 3 , n = n1 +n2 + n3)' To simplify our exposition, also assume that the points in 8 = {Sl,'" ,Sn} are ordered according to which cluster they are in, so that the first n1 points are in cluster 8 1 , the next n2 in 82 , etc. We will also use "j E 8/' as a shorthand for s· E 8i . Moving the clusters "infinitely" far apart corresponds to zeroing all the efements Aij corresponding to points Si and Sj in different clusters. More precisely, define Aij = 0 if Xi and Xj are in different clusters, and Aij = Aij otherwise. Also let t , D , X and Y be defined as in the previous algorithm, but starting with A instead of A. Note that A and t are therefore block-diagonal: [ A(ll) 0 A. = 0 A(22) o 0 o 1 A [L(11) o ; L = 0 A~~ 0 o £(22) o (1) where we have adopted the convention of using parenthesized superscripts to index into subblocks of vectors/matrices, and Lrii) = (D(ii)) - 1/2A(ii) (D(ii)) - 1/2. Here, A(ii) = A(ii) E jRni xni is the matrix of "intra-cluster" affinities for cluster i. For future use, also define d(i) E jRni to be the vector containing D(ii) 's diagonal elements, and dE jRn to contain D's diagonal elements. To construct X, we find t's first k = 3 eigenvectors. Since t is block diagonal, its eigenvalues and eigenvectors are the union of the ei~envalues and eigenvectors of its blocks (the latter padded appropriately with zeros). It is straightforward to show that Lrii) has a strictly positive principal eigenvector xii) E jRni with eigenvalue 1. Also, since A)~) > 0 (j i:- k), the next eigenvalue is strictly less than 1. (See, e.g., [3]). Thus, stacking t 's eigenvectors in columns to obtain X, we have: [ xi1) 0 0 1 X = 0 xi2) 0 E jRnx3. o 0 xi3) (2) Actually, a subtlety needs to be addressed here. Since 1 is a repeated eigenvalue in t, we could just as easily have picked any other 3 orthogonal vectors spanning the same subspace as X's columns, and defined them to be our first 3 eigenvectors. That is, X could have been replaced by XR for any orthogonal matrix R E jR3X3 (RT R = RRT = 1). Note that this immediately suggests that one use considerable caution in attempting to interpret the individual eigenvectors of L, as the choice of X's columns is arbitrary up to a rotation, and can easily change due to small perturbations to A or even differences in the implementation of the eigensolvers. Instead, what we can reasonably hope to guarantee about the algorithm will be arrived at not by considering the (unstable) individual columns of X, but instead the subspace spanned by the columns of X, which can be considerably more stable. Next, when we renormalize each of X's rows to have unit length, we obtain: [ y(l) 1 [r 0 0 1 y = y(2) 0 r 0 R y(3) 0 0 r (3) where we have used y(i) E jRni xk to denote the i-th subblock of Y. Letting fiji) denote the j-th row of17(i) , we therefore see that fjY) is the i-th row ofthe orthogonal matrix R. This gives us the following proposition. Proposition 1 Let A's off-diagonal blocks A(ij ) , i =I- j, be zero. Also assume that each cluster Si is connected.2 Then there exist k orthogonal vectors 1'1, . .. ,1' k (1'; l' j = 1 if i = j, 0 otherwise) so that Y's rows satisfy , (i) ( ) ~ =G 4 for all i = 1, ... ,k, j = 1, ... ,ni. In other words, there are k mutually orthogonal points on the surface of the unit k-sphere around which Y 's rows will cluster. Moreover, these clusters correspond exactly to the true clustering of the original data. 3.2 The general case In the general case, A's off-diagonal blocks are non-zero, but we still hope to recover guarantees similar to Proposition 1. Viewing E = A - A as a perturbation to the "ideal" A that results in A = A+E, we ask: When can we expect the resulting rows of Y to cluster similarly to the rows of Y? Specifically, when will the eigenvectors of L, which we now view as a perturbed version of L, be "close" to those of L? Matrix perturbation theory [10] indicates that the stability of the eigenvectors of a matrix is determined by the eigengap. More precisely, the subspace spanned by L's first 3 eigenvectors will be stable to small changes to L if and only if the eigengap 8 = IA3 - A41, the difference between the 3rd and 4th eigenvalues of L, is large. As discussed previously, the eigenvalues of L is the union of the eigenvalues of D11), D22), and D33), and A3 = 1. Letting Ay) be the j-th largest eigenvalue of Dii), we therefore see that A4 = maxi A~i). Hence, the assumption that IA3 - A41 be large is exactly the assumption that maXi A~i ) be bounded away from 1. Assumption AI. There exists 8 > 0 so that, for all i = 1, ... ,k, A~i) :s: 1 - 8. Note that A~i) depends only on Dii), which in turn depends only on A(ii) = A(ii) , the matrix of intra-cluster similarities for cluster Si' The assumption on A~i) has a very natural interpretation in the context of clustering. Informally, it captures the idea that if we want an algorithm to find the clusters Sl, S2 and S3, then we require that each of these sets Si really look like a "tight" cluster. Consider an example in which Sl = S1.1 U S1.2 , where S1.1 and S1.2 are themselves two well separated clusters. Then S = S1.1 U S1.2 U S2 U S3 looks like (at least) four clusters, and it would be unreasonable to expect an algorithm to correctly guess what partition of the four clusters into three subsets we had in mind. This connection between the eigengap and the cohesiveness of the individual clusters can be formalized in a number of ways. Assumption ALl. Define the Cheeger constant [3] of the cluster Si to be _. ~lE I, kIi' I A;;.,') h(S.) - mmI . {~ '(.) ~ ,(.)}. (5) mm lEI d , kli'I dk where the outer minimum is over all index subsets I ~ {I, ... ,nd. Assume that there exists 8 > 0 so that (h(Si))2 /2 ~ 8 for all i. 2This condition is satisfied by A.j~) > 0 (j i- k) , which is true in our case. A standard result in spectral graph theory shows that Assumption Al.l implies Assumption Al. Recall that d)i) = 2:k A)~) characterizes how "well connected" or how "similar" point j is to the other points in the same cluster. The term in the minI{·} characterizes how well (I, I) partitions Si into two subsets, and the minimum over I picks out the best such partition. Specifically, if there is a partition of Si'S points so that the weight of the edges across the partition is small, and so that each of the partitions has moderately large "volume" (sum of dY) 's), then the Cheeger constant will be small. Thus, the assumption that the Cheeger constants h(Si) be large is exactly that the clusters Si be hard to split into two subsets. We can also relate the eigengap to the mixing time of a random walk (as in [6]) defined on the points of a cluster, in which the chance of transitioning from point i to j is proportional to Aij , so that we tend to jump to nearby-points. Assumption Al is equivalent to assuming that, for such a walk defined on the points of any one of the clusters Si , the corresponding transition matrix has second eigenvalue at most 1-8. The mixing time of a random walk is governed by the second eigenvalue; thus, this assumption is exactly that the walks mix rapidly. Intuitively, this will be true for tight (or at least fairly "well connected") clusters, and untrue if a cluster consists of two well-separated sets of points so that the random walk takes a long time to transition from one half of the cluster to the other. Assumption Al can also be related to the existence of multiple paths between any two points in the same cluster. Assumption A2. There is some fixed fl > 0, so that for every iI , i2 E {I, ... ,k}, il =j:. i2, we have that (6) To gain intuition about this, consider the case of two "dense" clusters il and i2 of size O(n) each. Since dj measures how "connected" point j is to other points in the same cluster, it will be dj = O(n) in this case, so the sum, which is over 0(n2 ) terms, is in turn divided by djdk = O(n2 ) . Thus, as long as the individual Ajk's are small, the sum will also be small, and the assumption will hold with small fl. Whereas dj measures how connected Sj E Si is to the rest of Si, 2:k:k'itSi Ajk measures how connected Sj is to points in other clusters. The next assumption is that all points must be more connected to points in the same cluster than to points in other clusters; specifically, that the ratio between these two quantities be small. Assumption A3. For some fixed f2 > 0, for every i = 1, ... ,k, j E Si, we have: (7) For intuition about this assumption, again consider the case of densely connected clusters (as we did previously). Here, the quantity in parentheses on the right hand side is 0(1), so this becomes equivalent to demanding that the following ratio be small: (2:k:k'itSi Ajk)/dj = (2:k:k'itSi Ajk)/(2:k:kESi Ajk ) = 0(f2) . Assumption A4. There is some constant C > ° so that for every i = 1, .. . ,k, . _ ' (i) ni ' (i ) J - 1, ... ,ni, we have dj ~ (2:k=l dk )/(Cni). This last assumption is a fairly benign one that no points in a cluster be "too much less" connected than other points in the same cluster. Theorem 2 Let assumptions Al, A2, A3 and A4 hold. Set f = Jk(k - l)fl + kE~. If 0 > (2 + V2}::, then there exist k orthogonal vectors rl, . .. ,rk (rF r j = I if i = j, o otherwise) so that Y's rows satisfy (8) Thus, the rows of Y will form tight clusters around k well-separated points (at 90 0 from each other) on the surface of the k-sphere according to their "true" cluster Si. 4 Experiments To test our algorithm, we applied it to seven clustering problems. Note that whereas (J2 was previously described as a human-specified parameter, the analysis also suggests a particularly simple way of choosing it automatically: For the right (J2, Theorem 2 predicts that the rows of Y will form k "tight" clusters on the surface of the k-sphere. Thus, we simply search over (J2 , and pick the value that, after clustering Y 's rows, gives the tightest (smallest distortion) clusters. K-means in Step 5 of the algorithm was also inexpensively initialized using the prior knowledge that the clusters are about 90 0 apart.3 The results of our algorithm are shown in Figure l a-g. Giving the algorithm only the coordinates of the points and k, the different clusters found are shown in the Figure via the different symbols (and colors, where available). The results are surprisingly good: Even for clusters that do not form convex regions or that are not cleanly separated (such as in Figure 19), the algorithm reliably finds clusterings consistent with what a human would have chosen. We note that there are other, related algorithms that can give good results on a subset of these problems, but we are aware of no equally simple algorithm that can give results comparable to these. For example, we noted earlier how K-means easily fails when clusters do not correspond to convex regions (Figure Ii). Another alternative may be a simple "connected components" algorithm that, for a threshold T, draws an edge between points Si and Sj whenever Iisi - sjl12 :s: T, and takes the resulting connected components to be the clusters. Here, T is a parameter that can (say) be optimized to obtain the desired number of clusters k. The result of this algorithm on the threecircles-j oined dataset with k = 3 is shown in Figure lj. One of the "clusters" it found consists of a singleton point at (1.5,2). It is clear that this method is very non-robust. We also compare our method to the algorithm of Meila and Shi [6] (see Figure lk). Their method is similar to ours, except for the seemingly cosmetic difference that they normalize A's rows to sum to I and use its eigenvectors instead of L's, and do not renormalize the rows of X to unit length. A refinement of our analysis suggests that this method might be susceptible to bad clusterings when the degree to which different clusters are connected (L: j d;il) varies substantially across clusters. 3 Briefiy, we let the first cluster centroid be a randomly chosen row of Y , and then repeatedly choose as the next centroid the row of Y that is closest to being 90° from all the centroids (formally, from the worst-case centroid) already picked. The resulting K-means was run only once (no restarts) to give the results presented. K-means with the more conventional random initialization and a small number of restarts also gave identical results. In contrast, our implementation of Meila and Shi's algorithm used 2000 restarts. flips,8clusten (a) squiggles, 4 clusteNl (d) Ihreecirdes_joined,3clusters (g) threecircles-joined, 3 clusters(conoecled """'l""'enlS) (j) (b) (e) RowsoJYO ittered , rarKlomlysubsa mpled) lorlW<lCirc~ (h) ~near.dballs , 3 dus\efs(Meila and Shi algor1lhm) o o q, o~ 00 0 o~ ~& 0 (k) o o 0 &~llO ~~ o 0 o o o (c) th reeci~es-joiJ\ed,2c1ust8fS (f) lWo circles, 2 cluSle<S (K_means) (i) flips, 6 cluste<s (Kannan elal,aigor;thm) N (I) Figure 1: Clustering examples, with clusters indicated by different symbols (and colors, where available). (a-g) Results from our algorithm, where the only parameter varied across runs was k. (h) Rows of Y (jittered, subsampled) for twocircles dataset. (i) K-means. (j) A "connected components" algorithm. (k) Meila and Shi algorithm. (1) Kannan et al. Spectral Algorithm I. (See text.) 5 Discussion There are some intriguing similarities between spectral clustering methods and Kernel peA, which has been empirically observed to perform clustering [7, 2]. The main difference between the first steps of our algorithm and Kernel PCA with a Gaussian kernel is the normalization of A (to form L) and X. These normalizations do improve the performance of the algorithm, but it is also straightforward to extend our analysis to prove conditions under which Kernel PCA will indeed give clustering. While different in detail, Kannan et al. [4] give an analysis of spectral clustering that also makes use of matrix perturbation theory, for the case of an affinity matrix with row sums equal to one. They also present a clustering algorithm based on k singular vectors, one that differs from ours in that it identifies clusters with individual singular vectors. In our experiments, that algorithm very frequently gave poor results (e.g., Figure 11). Acknowledgments We thank Marina Meila for helpful conversations about this work. We also thank Alice Zheng for helpful comments. A. Ng is supported by a Microsoft Research fellowship. This work was also supported by a grant from Intel Corporation, NSF grant IIS-9988642, and ONR MURI N00014-00-1-0637. References [1] C. Alpert, A. Kahng, and S. Yao. Spectral partitioning: The more eigenvectors, the better. Discrete Applied Math, 90:3- 26, 1999. [2] N. Christianini, J. Shawe-Taylor, and J. Kandola. Spectral kernel methods for clustering. In Neural Information Processing Systems 14, 2002. [3] F. Chung. Spectral Graph Theory. Number 92 in CBMS Regional Conference Series in Mathematics. American Mathematical Society, 1997. [4] R. Kannan, S. Vempala, and A. Yetta. On clusterings- good, bad and spectral. In Proceedings of the 41st Annual Symposium on Foundations of Computer Science, 2000. [5] J. Malik, S. Belongie, T. Leung, and J. Shi. Contour and texture analysis for image segmentation. In Perceptual Organization for Artificial Vision Systems. Kluwer, 2000. [6] M. Meila and J. Shi. Learning segmentation by random walks. In Neural Information Processing Systems 13, 200l. [7] B. Scholkopf, A. Smola, and K.-R Miiller. Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10:1299- 1319, 1998. [8] G. Scott and H. Longuet-Higgins. Feature grouping by relocalisation of eigenvectors of the proximity matrix. In Proc. British Machine Vision Conference, 1990. [9] D. Spielman and S. Teng. Spectral partitioning works: Planar graphs and finite element meshes. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science, 1996. [10] G. W. Stewart and J.-G. Sun. Matrix Perturbation Th eory. Academic Press, 1990. [11] Y. Weiss. Segmentation using eigenvectors: A unifying view. In International Conference on Computer Vision, 1999.
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Grouping with Bias Stella X. Yu Robotics Institute Carnegie Mellon University Center for the Neural Basis of Cognition Pittsburgh, PA 15213-3890 stella. yu@es. emu. edu Abstract Jianbo Shi Robotics Institute Carnegie Mellon University 5000 Forbes Ave Pittsburgh, PA 15213-3890 jshi@es.emu.edu With the optimization of pattern discrimination as a goal, graph partitioning approaches often lack the capability to integrate prior knowledge to guide grouping. In this paper, we consider priors from unitary generative models, partially labeled data and spatial attention. These priors are modelled as constraints in the solution space. By imposing uniformity condition on the constraints, we restrict the feasible space to one of smooth solutions. A subspace projection method is developed to solve this constrained eigenproblema We demonstrate that simple priors can greatly improve image segmentation results. 1 " Introduction Grouping is often thought of as the process of finding intrinsic clusters or group structures within a data set. In image segmentation, it means finding objects or object segments by clustering pixels and segregating them from background. It is often considered a bottom-up process. Although never explicitly stated, higher level of knowledge, such as familiar object shapes, is to be used only in a separate post-processing step. The need for the integration of prior knowledge arises in a number of applications. In computer vision, we would like image segmentation to correspond directly to object segmentation. In data clustering, if users provide a few examples of clusters, we would like a system to adapt the grouping process to achieve the desired properties. In this case, there is an intimate connection to learning classification with partially labeled data. We show in this paper thatit is possible to integrate both bottom-up and top-down information in a single grouping process. In the proposed method, the bottom-up grouping process is modelled as a graph partitioning [1, 4, 12, 11, 14, 15] problem, and the top-down knowledge is encoded as constraints on the solution space. Though we consider normalized cuts criteria in particular, similar derivation can be developed for other graph partitioning criteria as well. We show that it leads to a constrained eigenvalue problem, where the global optimal solution can be obtained by eigendecomposition. Our model is expanded in detail in Section 2. Results and conclusions are given in Section 3. 2 Model In graph theoretic methods for grouping, a relational graph GA == (V, E, W) is first constructed based on pairwise similarity between two elements. Associated with the graph edge between vertex i and j is weight Wij , characterizing their likelihood of belonging in the same group. For image segmentation, pixels are taken as graph nodes, and pairwise pixel similarity can be evaluated based on a number of low level grouping cues. Fig. Ic shows one possible defini~ion, where the weight b.etween two pixels is inversely proportional to the magnitude of the strongest intervening edge [9]. a)Image. d)NCuts. e)Segmentation. Figure 1: Segmentation by graph partitioning. a)200 x 129 image with a few pixels marked(+). b)Edge map extracted using quadrature filters.c)Local affinity fields of marked pixels superimposed together. For every marked pixel, we compute its affinity with its neighbours within a radius of 10. The value is determined by a Gaussian function of the maximum magnitude of edges crossing the straight line connecting the two pixels [9]. When there is a strong edge separating the two, the affinity is low. Darker intensities mean larger values. d)Solution by graph partitioning. It is the second eigenvector from normalized cuts [15] on the affinity matrix. It assigns a value to each pixel. Pixels of similar values belong to the same group. e)Segmentation by thresholding the eigenvector with o. This gives a bipartitioning of the image which corresponds to the best cuts that have maximum within-region coherence and between-region distinction. After an image is transcribed into a graph, image segmentation becomes a vertex partitioning problem. Consider segmenting an image into foreground and background. This corresponds to vertex bipartitioning (VI, V2) on graph G, where V = VI U V2 and VI nV2 = 0. A good segmentation seeks a partitioning such that nodes within partitions are tightly connected and nodes across partitions are loosely connected. A number of criteria have been proposed to achieve this goal. For normalized cuts [15], the solution is given by some eigenvector of weight matrix W (Fig. Id). Thresholding on it leads to a discrete segmentation (Fig. Ie). W.hile we will focus on normalized cuts criteria [15], most of the following discussions apply to other criteria as well. 2.1 Biased grouping as constrained optimization Knowledge other than the image itself can greatly change the segmentation we might obtain based on such low level cues. Rather than seeing boundaries between black and white regions, we see objects. The sources of priors we consider in this paper are: unitary generative models (Fig. 2a), which could arise from sensor models in MRF [5], partial grouping (Fig. 2b), which could arise from human computer interaction [8], and spatial attention (Fig. 2c). All of these provide additional, often long-range, binding information for grouping. We model such prior knowledge in the form of constraints on a valid grouping configuration. In particular, we see that all such prior knowledge defines a partial a)Bright foreground. b)Partial grouping. c)Spatial attention. Figure 2: Examples of priors considered in this paper. a)Local constraints from unitary generative models. In this case, pixels of light(dark) intensities are likely to be the foreground(background). This prior knowledge is helpful not only for identifying the tiger as the foreground, but also for perceiving the river as one piece. How can we incorporate these unitary constraints into a- graph that handles only pairwise relationships between pixels? b)Global configuration constraints from partial grouping a priori. In this case, we have manually selected two sets of pixels to be grouped together in foreground (+) and background (JJ.) respectively. They are distributed across the image and often have distinct local features. How can we force them to be in the same group and further bring similar pixels along and push dissimilar pixels apart? c)Global constraints from spatial attention. We move our eyes to the place of most interest and then devote our limited visual processing to it. The complicated scene structures in the periphery can thus be ignored while sparing the parts associated with the object at fovea. How can we use this information to facilitate figural popout in segmentation? grouping solution, indicating which set of pixels should belong to one partition. Let Hz, 1 == 1"" ,n, denote a partial grouping. Ht have pixels known to be in Vt , t == 1,2. These sets are derived as follows. Unitary generative models: H l and H 2 contains a set of pixels that satisfy the unitary generative models for foreground and background respectively. For example, in Fig. 2a, Hl (H2 ) contains pixels of brightest(darkest) intensities. Partial grouping: Each Hz, 1== 1, ... ,n, contains a set of pixels that users specify to belong together. The relationships between Hz, 1> 2 and Vt , t == 1,2 are indefinite. Spatial attention: H l == 0 and H 2 contains pixels randomly selected outside the visual fovea, since we want to maintain maximum discrimination at the-fovea but merging pixels far away from the fovea to be one group. To formulate these constraints induced on the graph partitioning, we introduce binary group indicators X == [Xl, X 2]. Let N == IVI be the number of nodes in the graph. For t == 1,2, Xt is an N x 1 vector where Xt(k) == 1 if vertex k E Vt and 0 otherwise. The constrained grouping problem can be formally written as: min €(Xl ,X2 ) s.t. Xt(i) == Xt(j), i, j E HE, 1== 1"" ,n, t == 1,2, Xt(i) =1= Xt(j), i E Hl , j E H2 , t == 1,2, where €(X1,X2 ) is some graph partitioning cost function, such as minimum cuts [6], average cuts [7], or normalized cuts [15]. The first set of constraints can be re-written in matrix form: UT X == 0 , where, e.g. for some column k, Uik == 1, Ujk == ~1. We search for the optimal solution only in the feasible set determined by all the constraints. 2.2 Conditions on grouping constraints The above formulation can be implemented by the maximum-flow algorithm for minimum cuts criteria [6, 13, 3], where two special nodes called source and sink are introduced,.with infinite weights set up between nodes in HI (H2 ) and source(sink). In the context of learning from labeled and unlabeled data, the biased mincuts are linked to minimizing leave-one-out cross validation [2]. In the normalize cuts formulation, this leads to a constrained eigenvalue problem, as soon to be seen. However, simply forcing a few nodes to be in the same group can produce some undesirable graph partitioning results, illustrated in Fig. 3. Without bias, the data points are naturally first organized into top and bottom groups, and then subdivided into left and right halves (Fig. 3a). When we assign points from top and bottom clusters to be together, we do not just want one of the groups to lose its labeled point to the other group (Fig. 3b), but rather we desire the biased grouping process to explore their neighbouring connections and change the organization to left and right division accordingly. Larger Cut Desired Cut a a a a a a a a a a a a a a a a a a I a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a 0 0 a 0 0 0 0 0 0 0 0 a a a a a a a a a a a 0 a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a .A. a a Min Perturbed Cut Min Cut a a a a a a a a a a a a a a II a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a 0 a a a a a a a a a a a a a a a a a a a a a a a 0 a a a a a a a a a a a a a a a a a a 0 a a a a a a a a a a a a a a a a a a a a 0 a a a a a a a a a a a a a a a a a a a)No bias. b)With bias. Figure 3: Undesired grouping caused by simple grouping constraints. a)Data points are distributed in four groups, with a larger spatial gap between top and bottom groups than that between left and right groups. Defining weights based on proximity, we find the top-bottom grouping as the optimal bisection. b)Introduce two pairs of filled nodes to be together. Each pair has one point from the top and the other from the bottom group. The desired partitioning should now be the left-right division. However, perturbation on the unconstrained optimal cut can lead to a partitioning that satisfies the constraints while producing the smallest cut cost. The desire of propagating partial grouping information on the constrained nodes is, however, not reflected in the constrained partitioning criterion itself. Often, a slightly perturbed version of the optimal unbiased cut becomes the legitimate optimum. One reason for such a solution being undesirable is that some of the "perturbed" nodes-are isolated from their close neighbours. To fix this problem, we introduce the notion of uniformity of a graph partitioning. Intuitively, if two labeled nodes, i and j, have similar connections to their neighbours, we desire a cut to treat them fairly so that if i gets grouped with i's friends, j also gets grouped with j's friends (Fig. 3b). This uniformity condition is one way to propagate prior grouping information from labeled nodes to their neighbours. For normalized cuts criteria, we define the normalized cuts of a single node to be NC t (·.X)- EXt(k)=I=Xt(i),YtWik u s ~,D.. . n This value is high for a node isolated from its close neighbours in partitioning X. We may not know in advance what this value is for the optimal partitioning, but we desire this value to be the same for any pair of nodes preassigned together: NCuts(i;X) == NCuts(j;X), \li,j E Hz, l == 1,··· ,no While this condition does not force NCuts(i; X) to be small for each labeled node, it is unlikely for all of them to have a large value while producing the minimum NCuts for the global bisection. Similar measures can be defined for other criteria. In Fig. 4, we show that the uniformity condition on the bias helps preserving the smoothness of solutions at every labeled point. Such smoothing is necessary especially when partially labeled data are scarce. I 300 0.5 :-....~ ....-.... _o5A~,S/ -1 [l...--V_-_l...--_-----'---_----'l -1[·..... o 300 0 100 0.5 -0.5 a)Point set data. b)Simple bias. c)Conditioned bias. 0.5 0.5 o 0 o -0.2 -0.4 -0.5 -1 -1 100 100 d)NCuts wlo bias. e)NCuts wi bias b). f)NCuts wi bias c). Figure 4: Condition constraints with uniformity. a)Data consist of three strips, with 100 points each, numbered from left toright. Two points from the side strips are randomly chosen to be pre-assigned together. b)Simple constraint UT X == 0 forces any feasible solution to have equal valuation on the two marked points. c)Conditioned constraint UTpX == o. Note that now we cannot tell which points are biased. We compute W using Gaussian function of distance with u == 3. d)Segmentation without bias gives three separate groups. e)Segmentation with simple bias not only fails to glue the two side strips into one, but also has two marked points isolated from their neighbours. f)Segmentaion with conditioned bias brings two side strips into one group. See the definition of P below. 2.3 COlllpntation: subspace projection To develop a computational solution for the c9nstrained optimization problem, we introduce some notations. Let the degree matrix D be a diagonal matrix, Dii == Ek Wik, \Ii. Let P == D-IW be the normalized weight matrix. It is a transition probability matrix for nonnegative weight matrix W [10]. Let a == xI~~l be the degree ratio of VI, where 1 is the vector of ones. We define a new variable x == (1 - a)XI - aX2 • We can show that for normalized cuts, the biased grouping with the uniformity condition is translated into: . xT(D-W)x T mIn E(X) == TD ' s.t. U Px == o. x x Note, we have dropped the constraint Xt(i) =1= Xt(j), i E HI, j E H 2 , t == 1,2. Using Lagrange multipliers, we find that the optimal solution x* satisfies: QPx* == AX*, E(X*) == 1 - A, where Q is a projector onto the feasible solution space: Q == I - D-1V(VTD-1V)-lVT, V == pTU. Here we assume that the conditioned constraint V is of full rank, thus VTD-1V is invertible. Since 1 is still the trivial 'Solution corresponding to the largest eigenvalue of 1, the second leading right eigenvector of the matrix QP is the solution we seek. To summarize, given weight matrix W, partial grouping in matrix form UT x == 0, we do the following to find the optimal bipartitioning: Step 1: Compute degree matrix D, Dii == E j Wij , Vi. Step 2: Compute normalized matrix P == D-1W. Step 3: Compute conditioned constraint V == pTU. Step 4: Compute projected weight matrix W == QP==p-n-1V(VTn-1V)-lVTp. Step 5: Compute the second largest eigenvector x*: Wx* == AX*. Step 6: Threshold x* to get a discrete segmentation. 3 Results and conclusions We apply our method to the images in Fig. 2. For all the examples, we compute pixel affinity W as in Fig. 1. All the segmentation results are obtained by thresholding the eigenvectors using their mean values. The results without bias, with simple bias UT x == 0 and conditioned bias UT Px == 0 are compared in Fig. 5, 6, 7. b)Prior. c)NCuts' on W. d)Seg. wlo bias. e)Simple bias. f)Seg. on e) g)Conditioned bias. h)Seg. w/ bias. Figure 5: Segmentation with bias from unitary generative models. a)Edge map of the 100 x 150 image. N = 15000. b)We randomly sample 17 brightest pixels for HI (+),48 darkest pixels for H2 (~), producing 63 constraints in total. c) and d) show the solution without bias. It picks up the optimal bisection based on intensity distribution. e) and f) show the solution with simple bias. The labeled nodes have an uneven influence on grouping. g) and h) show the solution with conditioned bias. It successfully breaks the image into tiger and river as our general impression of this image. The computation for the three cases takes 11, 9 and 91ms respectively. Prior knowledge is particularly useful in supplying long-range grouping information which often lacks in data grouping based on low level cues. With our model, the partial grouping prior can be integrated into the bottom-up grouping framework by seeking the optimal solution in a restricted domain. We show that the uniformity constraint is effective in eliminating spurious solutions resulting from simple perturbation on the optimal unbiased solutions. Segmentation from the discretization of the continuous eigenvectors also becomes trivial. e)Simple bias. f)Seg. on e) g)Conditioned bias. h)Seg. w/ bias. Figure 6: Segmentation with bias from hand-labeled partial grouping. a)Edge map of the 80 x 82 image. N = 6560. b)Hand-labeled partial grouping includes 21 pixels for HI (+), 31 pixels for H 2 (A), producing 50 constraints in total. c) and d) show the solution without bias. It favors a few largest nearby pieces of similar intensity. e) and f) show the solution with simple bias. Labeled pixels in cluttered contexts are poor at binding their segments together. g) and h) show the solution with conditioned bias. It successfully pops out the pumpkin made of many small intensity patches. The computation for the three cases takes 5, 5 and 71ms respectively. f)4th eig. b) g)6th eig. b) h)4th eig. d) i)6th eig. d) j)8th eig. d) Figure 7: Segmentation with bias from spatial attention. N = 25800. a)We randomly choose 86 pixels far away from the fovea (Fig. 2c) for H 2 (A), producing 85 constraints. b) and c) show the solution with simple bias. It is similar to the solution without bias (Fig. 1). d) and e) show the solution with conditioned bias. It ignores the variation in the background scene, which includes not only large pieces of constant intensity, but also many small segments of various intensities. The foreground .successfully clips out the human figure. f) and g) are two subsequent eigenvectors with simple bias. h), i) and j.) are those with conditioned bias. There is still a lot of structural organization in the former, but almost none in the latter. This greatly simplifies the task we face when seeking a segmentation from the continuous eigensolution. The computation for the three cases takes 16, 25 and 220ms respectively. All these benefits come at a computational cost that is 10 times that for the original unbiased grouping problem. We note that we can also impose both UT x == 0 and UT Px == 0, or even UT pBX == 0, S > 1. Little improvement is observed in our examples.' Since projected weight matrix W becomes denser, the computation slows down. We hope that this problem can be alleviated by using multi-scale techniques. It remains open for future research. Acknowledgelllents This research is supported by (DARPA HumanID) ONR NOOOI4-00-1-091~and NSF IRI-9817496. References [1] A. Amir· and M. Lindenbaum. Quantitative analysis of grouping process. In European Conference on Computer Vision, pages 371-84, 1996. [2] A. Blum and S. Chawla. Learning from labeled and unlabeled data using graph mincuts, 2001. [3] Y. Boykov, O. Veksler, and R. Zabih. Fast approximate energy minimization via graph cuts. In International Conference on Computer Vision, 1999. [4] Y. Gdalyahu, D. Weinshall, and M. Werman. A randpmized algorithm for pairwise clustering. ill Neural Information Processing Systems, pages 424-30, 1998. [5] S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6(6):721-41, 1984. [6] H. Ishikawa and D. Geiger. Segmentation by grouping junctions. In IEEE Conference on Computer Vision and Pattern Recognition, 1998. [7] I. H. Jermyn and H. Ishikawa. Globally optimal regions and boundaries. In International Conference on Computer Vision, 1999. [8] M. Kass, A. Witkin, and D. Terzopoulos. Snakes: Active contour models. International Journal of Computer Vision, pages 321-331, 1988. [9] J. Malik, S. Belongie, T. Leung, and J. Shi. Contour and texture analysis for image segmentation. International Journal of Computer Vision, 2001. [10] M. Meila and J. Shi. Learning segmentation with random walk. ill Neural Information Processing Systems, 2001. [11] P. Perona and W. Freeman. A factorization approach to grouping. In European Conference on Computer Vision, pages 655-70, 1998. [12] J. Puzicha, T. Hofmann, and J. Buhmann. Unsupervised texture segmentation in a deterministic annealing framework. IEEE Transactions on Pattern Analysis and Machine Intelligence, 20(8):803-18, 1998. [13] S. Roy and I. J. Cox. A maximum-flow formulation of then-camera stereo correspondence problem. In International Conference on Computer Vision, 1998. [14] E. Sharon, A. Brandt, and R. Basri. Fast multiscale image segmentation. In IEEE Conference on Computer Vision and Pattern Recognition, pages 70-7, 2000. [15] J. Shi and J. Malik. Normalized cuts and image segmentation. In IEEE Conference on Computer Vision and Pattern Recognition, pages 731-7, June 1997.
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Learning hierarchical structures with Linear Relational Embedding Alberto Paccanaro Geoffrey E. Hinton Gatsby Computational Neuroscience Unit UCL, 17 Queen Square, London, UK alberto,hinton @gatsby.ucl.ac.uk Abstract We present Linear Relational Embedding (LRE), a new method of learning a distributed representation of concepts from data consisting of instances of relations between given concepts. Its final goal is to be able to generalize, i.e. infer new instances of these relations among the concepts. On a task involving family relationships we show that LRE can generalize better than any previously published method. We then show how LRE can be used effectively to find compact distributed representations for variable-sized recursive data structures, such as trees and lists. 1 Linear Relational Embedding Our aim is to take a large set of facts about a domain expressed as tuples of arbitrary symbols in a simple and rigid syntactic format and to be able to infer other “common-sense” facts without having any prior knowledge about the domain. Let us imagine a situation in which we have a set of concepts and a set of relations among these concepts, and that our data consists of few instances of these relations that hold among the concepts. We want to be able to infer other instances of these relations. For example, if the concepts are the people in a certain family, the relations are kinship relations, and we are given the facts ”Alberto has-father Pietro” and ”Pietro has-brother Giovanni”, we would like to be able to infer ”Alberto has-uncle Giovanni”. Our approach is to learn appropriate distributed representations of the entities in the data, and then exploit the generalization properties of the distributed representations [2] to make the inferences. In this paper we present a method, which we have called Linear Relational Embedding (LRE), which learns a distributed representation for the concepts by embedding them in a space where the relations between concepts are linear transformations of their distributed representations. Let us consider the case in which all the relations are binary, i.e. involve two concepts. In this case our data consists of triplets
, and the problem we are trying to solve is to infer missing triplets when we are given only few of them. Inferring a triplet is equivalent to being able to complete it, that is to come up with one of its elements, given the other two. Here we shall always try to complete the third element of the triplets 1. LRE will then represent each concept in the data as a learned vector in a 1Methods analogous to the ones presented here that can be used to complete any element of a triplet can be found in [4]. Euclidean space and each relationship between the two concepts as a learned matrix that maps the first concept into an approximation to the second concept. Let us assume that our data consists of such triplets containing distinct concepts and binary relations. We shall call this set of triplets ;
will denote the set of -dimensional vectors corresponding to the concepts, and the set of matrices corresponding to the relations. Often we shall need to indicate the vectors and the matrix which correspond to the concepts and the relation in a certain triplet . In this case we shall denote the vector corresponding to the first concept with , the vector corresponding to the second concept with and the matrix corresponding to the relation with . We shall therefore write the triplet as
where and . The operation that relates a pair !" to a vector # is the matrix-vector multiplication, %$ , which produces an approximation to . If for every triplet ! we think of $ & as a noisy version of one of the concept vectors, then one way to learn an embedding is to maximize the probability that it is a noisy version of the correct completion, ' . We imagine that a concept has an average location in the space, but that each “observation” of the concept is a noisy realization of this average location. Assuming spherical Gaussian noise with a variance of ()+* on each dimension, the discriminative goodness function that corresponds to the log probability of getting the right completion, summed over all training triplets is: , . 0/ ( 1 24365 6798:8 ;
<>= ?<@7BAC<@8:8 D . E+F0GIH 798:8 ;
<>= ?<J7 E F 8:8 D (1) where 1 is the number of triplets in having the first two terms equal to the ones of , but differing in the third term 2. Learning based on maximizing , with respect to all the vector and matrix components has given good results, and has proved successful in generalization as well [5]. However, when we learn an embedding by maximizing , , we are not making use of exactly the information that we have in the triplets. For each triplet , we are making the vector representing the correct completion K more probable than any other concept vector given $ B , while the triplet states that $ & must be equal to L . The numerator of , does exactly this, but we also have the denominator, which is necessary in order to stay away from the trivial M solution 3. We noticed however that the denominator is critical at the beginning of the learning, but as the vectors and matrices differentiate we could gradually lift this burden, allowing N 0/ O $ &QPRS O to become the real goal of the learning. To do this we modify the discriminative function to include a parameter T , which is annealed from ( to U during learning 4: V . 0/ ( 1 24365 679W0;
<J= ?<J7BA%<XW0D Y . E+FZGIH 79W0;
<J= ?<>7 E F W0DX[]\ (2) 2We would like our system to assign equal probability to each of the correct completions. The discrete probability distribution that we want to approximate is therefore: ^_a`cb d N d e4f b6g6hji eBk l m where g is the discrete delta function and l ranges over the vectors in n . Our system implements the discrete probability distribution: o _ ` b prqJsIt h kQu>vxw@ykzl"u@{Xm where | is the normalization factor. The }X~ factor in eq.1 ensures that we are minimizing the Kullback-Leibler divergence between ^ and o . 3The obvious approach to find an embedding would be to minimize the sum of squared distances between v w y and i over all the triplets, with respect to all the vector and matrix components. Unfortunately this minimization (almost) always causes all of the vectors and matrices to collapse to the trivial solution. 4For one-to-many relations we must not decrease the value of all the way to , because this would cause some concept vectors to become coincident. This is because the only way to make v wJy equal to different vectors, is by collapsing them onto a unique vector. During learning this function V (for Goodness) is maximized with respect to all the vector and matrix components. This gives a much better generalization performance than the one obtained by just maximizing , . The results presented in the next sections were obtained by maximizing V using gradient ascent. All the vector and matrix components were updated simultaneously at each iteration. One effective method of performing the optimization is conjugate gradient. Learning was fast, usually requiring only a few hundred updates. It is worth pointing out that, in general, different initial configurations and optimization algorithms caused the system to arrive at different solutions, but these solutions were almost always very similar in terms of generalization performance. 2 LRE results Here we present the results obtained applying LRE to the Family Tree Problem [1]. In this problem, the data consists of people and relations among people belonging to two families, one Italian and one English, shown in fig.1 (left) 5. All the information in these trees can be represented in simple propositions of the form
. Using the relations father, mother, husband, wife, son, daughter, uncle, aunt, brother, sister, nephew, niece there are 112 such triplets in the two trees. Fig.1 (right) shows the embedding obtained after training with LRE. Notice how the Italians are linearly separable from the English people. From the Hinton diagram, we can see that each member of a family is symmetric to the corresponding member in the other family. The sign of the third component of the vectors is (almost) a feature for the nationality. When testing the generalization Grazia = Pierino Margaret = Arthur Victoria = James Jennifer = Charles Charlotte Colin Christopher = Penelope Andrew = Christine 1 7 2 8 11 5 3 9 6 12 Aurelio = Maria Giannina = Pietro Mariemma Alberto Doralice = Marcello Bortolo = Emma 23 13 19 14 20 21 15 17 24 18 22 16 10 4 −5 0 5 −5 0 5 −2 0 2 English Italians Figure 1: Left: Two isomorphic family trees. The symbol “=” means “married to”. Right Top: layout of the vectors representing the people obtained for the Family Tree Problem in 3D. Vectors end-points are indicated by *, the ones in the same family tree are connected to each other. All (6(* triplets were used for training. Right Bottom: Hinton diagrams of the 3D vectors shown above. The vector of each person is a column, ordered according to the numbering on the tree diagram on the left. performance, for each triplet in the test set , we chose as completion the concepts according to their probability, given $ & . The system was generally able to complete correctly all (6(* triplets even when * of them, picked at random, had been left out during training. These results on the Family Tree Problem are much better than the ones obtained using any other method on the same problem: Quinlan’s FOIL [7] could generalize on triplets, while Hinton (1986) and O’Reilly (1996) made one or more errors when only test cases were held out during training. 5The names of the Italian family have been altered from those originally used in Hinton (1986) to match those of one of the author’s family. For most problems there exist triplets which cannot be completed. This is the case, for example, of (Christopher, father, ?) in the Family Tree Problem. Therefore, here we argue that it is not sufficient to test generalization by merely testing the completion of those complete-able triplets which have not been used for training. The proper test for generalization is to see how the system completes any triplet of the kind where ranges over the concepts and R over the relations. We cannot assume to have knowledge of which triplets admit a completion, and which do not. To our knowledge this issue has never been analyzed before (even though FOIL handles this problem correctly). To do this the system needs a way to indicate when a triplet does not admit a completion. Therefore, once the maximization of V is terminated, we build a new probabilistic model around the solution which has been found. This new model is constituted, for each relation, of a mixture of identical spherical Gaussians, each centered on a concept vector, and a Uniform distribution. The Uniform distribution will take care of the “don’t know” answers, and will be competing with all the other Gaussians, each representing a concept vector. For each relation the Gaussians have different variances and the Uniform a different height. The parameters of this probabilistic model are, for each relation , the variances of the Gaussians ; and the relative density under the Uniform distribution, which we shall write as ZP ; )I* ; . These parameters are learned using a validation set, which will be the union of a set of complete-able (positive) triplets and a set of pairs which cannot be completed (negative); that is >
/ and c Z/ where indicates the fact that the result of applying relation to does not belong to . This is done by maximizing the following discriminative goodness function over the validation set : . / 24365 P D D P D D . E+F0GIH ZP O $ P O * ; . / ( 1 $ 2 3I5 P W0; = ? 7BA IW D D ZP D D ! . E+F GIH ZP O $ " P O * ; (3) with respect to the ; and ; parameters, while everything else is kept fixed. Having learned these parameters, in order to complete any triplet > we compute the probability distribution over each of the Gaussians and the Uniform distribution given $ . The system then chooses a vector or the “don’t know” answer according to those probabilities, as the completion to the triplet. We used this method on the Family Tree Problem using a train, test and validation sets built in the following way. The test set contained (* positive triplets chosen at random, but such that there was a triplet per relation. The validation set contained a group of (* positive and a group of (* negative triplets, chosen at random and such that each group had a triplet per relation. The train set contained the remaining positive triplets. After learning a distributed representation for the entities in the data by maximizing V over the training set, we learned the parameters of the probabilistic model by maximizing over the validation set. The resulting system was able to correctly complete all the * possible triplets > . Figure 2 shows the distribution of the probabilities when completing one complete-able and one uncomplete-able triplet in the test set. LRE seems to scale up well to problems of bigger size. We have used it on a much bigger version of the Family Tree Problem, where the family tree is a branch of the real family tree of one of the authors containing "# people over $ generations. Using the same set of (* relations used in the Family Tree Problem, there is a total of % positive triplets. After learning using a training set of $I* positive triplets, and a validation set constituted 1 2 3 4 5 6 7 8 9 10111213141516 171819202122 232425 0 0.2 0.4 0.6 0.8 1 Charlotte uncle Emma aunt 1 2 3 4 5 6 7 8 9 10111213141516 171819202122 232425 0 0.2 0.4 0.6 0.8 1 Figure 2: Distribution of the probabilities assigned to each concept for one complete-able (left) and one uncomplete-able (right) triplet written above each diagram. The completeable triplet has two correct completions but neither of the triplets had been used for training. Black bars from ( to * are the probabilities of the people ordered according to the numbering in fig.1. The last grey bar on the right, is the probability of the “don’t know” answer. by IU positive and IU negative triplets, the system is able to complete correctly almost all the possible triplets. When many completions are correct, a high probability is always assigned to each one of them. Only in few cases is a non-negligible probability assigned to some wrong completions. Almost all the generalization errors are of a specific form. The system appears to believe that ”brother/sister of” means ”son/daughter of parents of”. It fails to model the extra restriction that people cannot be their own brother/sister. On the other hand, nothing in the data specifies this restriction. 3 Using LRE to represent recursive data structures In this section, we shall show how LRE can be used effectively to find compact distributed representations for variable-sized recursive data structures, such as trees and lists. Here we discuss binary trees, but the same reasoning applies to trees of any valence. The approach is inspired by Pollack’s RAAM architecture [6]. A RAAM is an auto-encoder which is trained using backpropagation. Figure 3 shows the architecture of the network for binary trees. The system can be thought as being composed of two networks. The first one, called C2 C1 C1 R1 C1 C2 R2 C2 R1 R2 R2 R1 r 1 R C R 2 l r l r r~ ~ ~ Compressor Reconstructor l l~ Adjective Noun a b Noun Phrase Verb Noun c d Verb Phrase Figure 3: Left: the architecture of a RAAM for binary trees. The layers are fully connected. Adapted from [6]. Center: how LRE can be used to learn a representation for binary trees in a RAAM-like fashion. Right: the binary tree structure of the sentences used in the experiment. compressor encodes two fixed-width patterns into a single pattern of the same size. The second one, called reconstructor, decodes a compressed pattern into facsimiles of its parts, and determines when the parts should be further decoded. To encode a tree the network must learn as many auto-associations as the total number of non-terminal nodes in the tree. The codes for the terminal nodes are supplied, and the network learns suitable codes for the other nodes. The decoding procedure must decide whether a decoded vector represents a terminal node or an internal node which should be further decoded. This is done by using binary codes for the terminal symbols, and then fixing a threshold which is used for checking for “binary-ness” during decoding. The RAAM approach can be cast as an LRE problem, in which concepts are trees, subtrees or leaves, or pairs of trees, sub-trees or leaves, and there exist relationships: implementing the compressor, and and which jointly implement the reconstructor (see fig.3). We can then learn a representation for all the trees, and the matrices by maximizing V in eq.2. This formulation, which we have called Hierarchical LRE (HLRE), solves two problems encountered in RAAMs. First, one does not need to supply codes for the leaves of the trees, since LRE will learn an appropriate distributed representation for them. Secondly, one can also learn from the data when to stop the decoding process. In fact, the problem of recognizing whether a node needs to be further decoded, is similar to the problem of recognizing that a certain triplet does not admit a completion, that we solved in the previous section. While before we built an outlier model for the “don’t know” answers, now we shall build one for the non-terminal nodes. This can be done by learning appropriate values of and for relations and maximizing in eq.3. The set of triplets !B where is not a leaf of the tree, will play the role of the set which appears in eq.3. We have applied this method to the problem of encoding binary trees which correspond to sentences of words from a small vocabulary. Sentences had a fixed structure: a noun phrase, constituted of an adjective and a noun, followed by a verb phrase, made of a verb and a noun (see fig.3). Thus each sentence had a fixed grammatical structure, to which we added some extra semantic structure in the following way. Words of each grammatical category were divided into two disjoint sets. Nouns were in \ girl, woman, scientist or in dog, doctor, lawyer ; adjectives were in \ pretty, young or in ugly, old ; verbs were in \ help, love or in hurt, annoy . Our training set was constituted by (U sentences of the type:
\ \
\ \ and (U of the type
\ , where the suffix indicates the set to which each word type belongs. In this way, sentences of the kind “pretty girl annoy scientist” were not allowed in the training set, and there were ( possible sentences that satisfied the constraints which were implicit in the training set. We used HLRE to learn a distributed representation for all the nodes in the trees, maximizing V using the *IU sentences in the training set. In 7D, after having built the outlier model for the non-terminal symbols, given any root or internal node the system would reconstruct its children, and if they were non-terminal symbols would further decode each of them. The decoding process would always halt providing the correct reconstruction for all the *+U sentences in the training set. The top row of fig.4 shows the distributed representations found for each word in the vocabulary. Notice how the T and sets of adjectives and verbs are almost symmetric with respect to the origin; the difference between the T and sets is less evident for the nouns, due to the fact that while there exists a restriction on which nouns can be used in position
, there is no restriction on the nouns appearing in position in the training sentences (see fig.3, right). We tested how well this system could generalize beyond the training set using the same procedure used by Pollack to enumerate the set of trees that RAAMs are able to represent [6]: for every pair of patterns for trees, first we encoded them into a pattern for a new higher level tree, and then we decoded this tree back into the patterns of the two sub-trees. If the norm of the difference between the original and the reconstructed sub-trees was within a tolerance, which we set to U ( , then the tree could be considered to be well formed. The system shows impressive generalization performance: after training using the *+U sentences, the four-word sentences it generates are all the ( well formed sentences, and only those. It does not generate sentences which are either grammatically wrong, like “dog old girl annoy”, nor sentences which violate semantic constraints, like “pretty girl annoy scientist”. This is striking when compared to the poor generalization performance obtained by the RAAM on similar problems. As recognized by Adjectives Nouns Verbs C1 ×
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girl C1 ×
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girl C2 ×
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girl R1 ×
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Nouns Figure 4: For Hinton diagrams with multiple rows, each row relates to a word, in the following order - Adjectives: 1=pretty; 2=young; 3=ugly; 4=old ; Nouns: 1=girl; 2=woman; 3=scientist; 4=dog; 5=doctor; 6=lawyer ; Verbs: 1=help; 2=love; 3=hurt; 4=annoy ; . Black bars separate \ , \ , \ (higher), from , , (lower). Top row: The distributed representation of the words in the sentences found after learning. Center row: The different contributions given to the root of the tree by the word “girl” when placed in position ,
, and in the tree. Bottom row: The contribution of each leaf to the reconstruction of , when adjectives, nouns, verbs and nouns are applied in positions ,
, and respectively. Pollack [6], this was almost certainly due to the fact that for the RAAMs the representation for the leaves was too similar, a problem that the HLRE formulation solves, since it learns their distributed representations. Let us try to explain why HLRE can generalize so well. The matrix can be decomposed into two sub-matrices, and , such that for any two children of a given node, and , we have: $ Y [ $ R $ , where “;” denotes the concatenation operator. Therefore we have a pair of matrices, either or , associated to each link in the graph. Once the system has learned an embedding, finding a distributed representation for a given tree amounts to multiplying the representation of its leaves by all the matrices found on all the paths from the leaves to the root, and adding them up. Luckily matrix multiplication is non-commutative, and therefore every sequence of words on its leaves can generate a different representation at the root node. The second row of fig.4 makes this point clear showing the different contributions given to the root of the tree by the word “girl” , depending on its position in the sentence. A tree can be “unrolled” from the root to its leaves by multiplying its distributed representation using the matrices. We can now analyze how a particular leaf is reconstructed. Leaf , for example, is reconstructed as: $ $ $ $ $
$ $ $ $ $ The third row of fig.4 shows the contribution of each leaf to the reconstruction of , when adjectives, nouns, verbs and nouns are placed on leaves ,
, and respectively. We can see that the contributions from the adjectives, match very closely their actual distributed representations, while the contributions from the nouns in position are negligible. This means that any adjective placed on will tend to be reconstructed correctly, and that its reconstruction is independent of the noun we have in position . On the other hand, the contributions from nouns and verbs in positions
and are non-negligible, and notice how those given by words belonging to the T subsets are almost symmetric to those given by words in the subsets. In this way the system is able to enforce the semantic agreement between words in positions ,
and . Finally, the reconstruction of , when adjectives, nouns, verbs and nouns are not placed on leaves ,
, and respectively, assigns a very low probability to any word, and thus the system does not generate sentences which are not well formed. 4 Conclusions Linear Relational Embedding is a new method for learning distributed representations of concepts and relations from data consisting of instances of relations between given concepts. It finds a mapping from the concepts into a feature-space by imposing the constraint that relations in this feature-space are modeled by linear operations. LRE shows excellent generalization performance. The results on the Family Tree Problem are far better than those obtained by any previously published method. Results on other problems are similar. Moreover we have shown elsewhere [4] that, after learning a distributed representation for a set of concepts and relations, LRE can easily modify these representations to incorporate new concepts and relations and that it is possible extract logical rules from the solution and to couple LRE with FOIL [7]. Learning is fast and LRE rarely converges to solutions with poor generalization. We began introducing LRE for binary relations, and then we saw how these ideas can be easily extended to higher arity relation by simply concatenating concept vectors and using rectangular matrices for the relations. The compressor relation for binary trees is a ternary relation; for trees of higher valence the compressor relation will have higher arity. We have seen how HLRE can be used to find distributed representations for hierarchical structures, and its generalization performance is much better than the one obtained using RAAMs on similar problems. It is easy to prove that, when all the relations are binary, given a sufficient number of dimensions, there always exists an LRE-type of solution that satisfies any set of triplets [4]. However, due to its linearity, LRE cannot represent some relations of arity greater than * . This limitation can be overcome by adding an extra layer of non-linear units for representing the relations. This new method, called Non-Linear Relational Embedding (NLRE) [4], can represent any relation and has given good generalization results. References [1] Geoffrey E. Hinton. Learning distributed representations of concepts. In Proceedings of the Eighth Annual Conference of the Cognitive Science Society, pages 1–12. Erlbaum, NJ, 1986. [2] Geoffrey E. Hinton, James L. McClelland, and David E. Rumelhart. Distributed representations. In David E. Rumelhart, James L. McClelland, and the PDP research Group, editors, Parallel Distributed Processing, volume 1, pages 77–109. The MIT Press, 1986. [3] Randall C. O’Reilly. The LEABRA model of neural interactions and learning in the neocortex. PhD thesis, Department of Psychology, Carnegie Mellon University, 1996. [4] Alberto Paccanaro. Learning Distributed Representations of Relational Data using Linear Relational Embedding. PhD thesis, Computer Science Department, University of Toronto, 2002. [5] Alberto Paccanaro and Geoffrey E. Hinton. Learning distributed representations by mapping concepts and relations into a linear space. In Pat Langley, editor, Proceedings of ICML2000, pages 711–718. Morgan Kaufmann, Stanford University, 2000. [6] Jordan B. Pollack. Recursive distributed representations. Artificial Intelligence, 46:77–105, 1990. [7] J. R. Quinlan. Learning logical definitions from relations. Machine Learning, 5:239–266, 1990.
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Constructing Distributed Representations Using Additive Clustering Wheeler Ruml Division of Engineering and Applied Sciences Harvard University 33 Oxford Street, Cambridge, MA 02138 ruml@eecs.harvard.edu Abstract If the promise of computational modeling is to be fully realized in higherlevel cognitive domains such as language processing, principled methods must be developed to construct the semantic representations used in such models. In this paper, we propose the use of an established formalism from mathematical psychology, additive clustering, as a means of automatically constructing binary representations for objects using only pairwise similarity data. However, existing methods for the unsupervised learning of additive clustering models do not scale well to large problems. We present a new algorithm for additive clustering, based on a novel heuristic technique for combinatorial optimization. The algorithm is simpler than previous formulations and makes fewer independence assumptions. Extensive empirical tests on both human and synthetic data suggest that it is more effective than previous methods and that it also scales better to larger problems. By making additive clustering practical, we take a significant step toward scaling connectionist models beyond hand-coded examples. 1 Introduction Many cognitive models posit mental representations based on discrete substructures. Even connectionist models whose processing involves manipulation of real-valued activations typically represent objects as patterns of 0s and 1s across a set of units (Noelle, Cottrell, and Wilms, 1997). Often, individual units are taken to represent specific features of the objects and two representations will share features to the degree to which the two objects are similar. While this arrangement is intuitively appealing, it can be difficult to construct the features to be used in such a model. Using random feature assignments clouds the relationship between the model and the objects it is intended to represent, diminishing the model’s value. As Clouse and Cottrell (1996) point out, hand-crafted representations are tedious to construct and it can be difficult to precisely justify (or even articulate) the principles that guided their design. These difficulties effectively limit the number of objects that can be encoded, constraining modeling efforts to small examples. In this paper, we investigate methods for automatically synthesizing feature-based representations directly from the pairwise object similarities that the model is intended to respect. This automatic Table 1: An 8-feature model derived from consonant confusability data. With c = 0.024, the model accounts for 91.8% of the variance in the data. Wt. Objects with feature Interpretation .350 fθ front unvoiced fricatives .243 dg back voiced stops .197 p k unvoiced stops (without t) .182 b v∂ front voiced .162 ptk unvoiced stops .127 mn nasals .075 dgv∂z˘z voiced (without b) .049 ptkfθs˘s unvoiced approach eliminates the manual burden of selecting and assigning features while providing an explicit design criterion that objectively connects the representations to empirical data. After formalizing the problem, we will review existing algorithms that have been proposed for solving it. We will then investigate a new approach, based on combinatorial optimization. When using a novel heuristic search technique, we find that the new approach, despite its simplicity, performs better than previous algorithms and that, perhaps more important, it maintains its effectiveness on large problems. 1.1 Additive Clustering We will formalize the problem of constructing discrete features from similarity information using the additive clustering model of Shepard and Arabie (1979). In this framework, abbreviated ADCLUS, clusters represent arbitrarily overlapping discrete features. Each of the k features has a non-negative real-valued weight wk, and the similarity between two objects i and j is just the sum of the weights of the features they share. If fik is 1 if object i has feature k and 0 otherwise, and c is a real-valued constant, then the similarity of i and j is modeled as ˆsij = X k wkfikfjk + c . This class of models is very expressive, encompassing non-hierarchical as well as hierarchical arrangements of clusters. An example model, derived using the ewindclus-klb algorithm described below, is shown in Table 1. The representation of each object is simply the binary column specifying its membership or absence in each cluster. Additive clustering is asymmetric in the sense that only the shared features of two objects contribute to their similarity, not the ones they both lack. (This is the more general formulation, as an additional feature containing the set complement of the original feature could always be used to produce such an effect.) With a model formalism in hand, we can then phrase the problem of constructing feature assignments as simply finding the ADCLUS model that best matches the given similarity data using the desired number of features. The fit of a model (comprising F, W, and c) to a matrix, S, can be quantified by the variance accounted for (VAF), which compares the model’s accuracy to merely predicting using the mean similarity: VAF = 1 − P i,j(sij −ˆsij)2 P i,j(sij −¯s)2 A VAF of 0 can always be achieved by setting all wk to 0 and c to ¯s. 2 Previous Algorithms Additive clustering is a difficult 0-1 quadratic programming problem and only heuristic methods, which do not guarantee an optimal model, have been proposed. Many different approaches have been taken: Subsets: Shepard and Arabie (1979) proposed an early algorithm based on subset analysis that was clearly superseded by Arabie’s later work below. Hojo (1983) also proposed an algorithm along these lines. We will not consider these algorithms further. Non-discrete Approximation: Arabie and Carroll (1980) and Carroll and Arabie (1983) proposed the two-stage indclus algorithm. In the first stage, cluster memberships are treated as real values and optimized for each cluster in turn by gradient descent. At the same time, a penalty term for non-0-1 values is gradually increased. Afterwards, a combinatorial clean-up stage tries all possible changes to 1 or 2 cluster memberships. Experiments reported below use the original code, modified slightly to handle large instances. Random initial configurations were used. Asymmetric Approximation: In the sindclus algorithm, Chaturvedi and Carroll (1994) optimize an asymmetric model with two sets of cluster memberships, having the form ˆsij = P k wkfikgjk +c. By considering each cluster in turn, this formulation allows a fast method for determining each of F, G, and w given the other two. In practice, F and G often become identical, yielding an ADCLUS model. Experiments reported below use both a version of the original implementation that has been modified to handle large instances and a reimplemented version (resindclus) that differs in its behavior at boundary cases (handling 0 weights, empty clusters, ties). Models from runs in which F and G did not converge were each converted into several ADCLUS models by taking only F, only G, their intersection, or their union. The weights and constants of each model were optimized using constrained least-squares linear regression (Stark and Parker, 1995), ensuring non-negative cluster weights, and the one with the highest VAF was used. Alternating Clusters: Kiers (1997) proposed an element-wise simplified sindclus algorithm, which we abbreviate as ewindclus. Like sindclus, it considers each cluster in turn, alternating between the weights and the cluster memberships, although only one set of clusters is maintained. Weights are set by a simple regression and memberships are determined by a gradient function that assumes object independence and fixed weights. The experiments reported below use a new implementation, similar to the reimplementation of sindclus. Expectation Maximization: Tenenbaum (1996) reformulated ADCLUS fitting in probabilistic terms as a problem with multiple hidden factorial causes, and proposed a combination of the EM algorithm, Gibbs sampling, and simulated annealing to solve it. The experiments below use a modified version of the original implementation which we will notate as em-indclus. It terminates early if 10 iterations of EM pass without a change in the solution quality. (A comparison with the original code showed this modification to give equivalent results using less running time.) Unfortunately, it is not clear which of these approaches is the best. Most published comparisons of additive clustering algorithms use only a small number of test problems (or only artificial data) and report only the best solution found within an unspecified amount of time. Because the algorithms use random starting configurations and often return solutions of widely varying quality even when run repeatedly on the same problem, this leaves it unclear which algorithm gives the best results on a typical run. Furthermore, different Table 2: The performance of several previously proposed algorithms on data sets from psychological experiments. indclus sindclus re-sindclus ewindclus Name VAF IQR VAF IQR r VAF IQR r VAF IQR r animals-s 77 75–80 66 65–65 8 78 79 –80 12 64 60–69 4 numbers 83 81–86 84 82 –86 5 78 75–81 7 82 79–85 5 workers 83 82–85 81 79–83 9 84 82–85 7 67 63–72 2 consonants 89 89–90 88 87–89 6 81 80–82 5 51 44–57 1 animals 71 69–74 66 66–66 9 66 66–66 13 72 71 –73 26 letters 80 80–80 78 78–79 7 68 65–72 5 74 73–75 17 Table 3: The performance of indclus and em-indclus on the human data sets. indclus em-indclus Name n k VAF IQR r VAF IQR animals-s 10 3 80 80–80 23 80 80–80 numbers 10 8 91 90–91 157 90 89–90 workers 14 7 89 88–89 89 87 87–89 consonants 16 8 91 91–91 291 91 91–91 animals 26 12 71 69–74 1 N/A letters 30 5 82 82–83 486 82 82–83 algorithms require very different running times, and multiple runs of a fast algorithm with high variance in solution quality may produce a better result in the same time as a single run of a more predictable algorithm. The next section reports on a new empirical comparison that addresses these concerns. 2.1 Evaluation of Previous Algorithms We compared indclus, both implementations of sindclus, ewindclus, and emindclus on 3 sets of problems. The first set is a collection of 6 typical data sets from psychological experiments that have been used in previous additive clustering work (originally by Shepard and Arabie (1979), except for animals-s, Mechelen and Storms (1995), and animals, Chaturvedi and Carroll (1994)). The number of objects (n) and the number of features used (k) are listed for each instance as part of Table 3. The second set of problems contains noiseless synthetic data derived from ADCLUS models with 8, 16, 32, 64, and 128 objects. In a rough approximation of the human data, the number of clusters was set to 2 log2(n), and as in previous ADCLUS work, each object was inserted in each cluster with probability 0.5. A single similarity matrix was generated from each model using weights and constants uniformly distributed between 1 and 6. The third set of problems was derived from the second by adding gaussian noise with a variance of 10% of the variance of the similarity data and enforcing symmetry. Each algorithm was run at least 50 times on each data set. Runs that crashed or resulted in a VAF < 0 were ignored. To avoid biasing our conclusions in favor of methods requiring more computation time, those results were then used to derive the distribution of results that would be expected if all algorithms were run simultaneously and those that finished early were re-run repeatedly until the slowest algorithm finished its first run, with any re-runs in progress at that point discarded.1 1Depending as it does on running time, this comparison remains imprecise due to variations in the degree of code tuning and the quality of the compilers used, and the need to normalize timings between the multiple machines used in the tests. Summaries of the time-equated results produced by each algorithm on each of the human data sets are shown in Table 2. (em-indclus took much longer than the other algorithms and its performance is shown separately in Table 3.) The mean VAF for each algorithm is listed, along with the inter-quartile range (IQR) and the mean number of runs that were necessary to achieve time parity with the slowest algorithm on that data set (r). On most instances, there is remarkable variance in the VAF achieved by each algorithm.2 Overall, despite the variety of approaches that have been brought to bear over the years, the original indclus algorithm appears to be the best. (Results in which another algorithm was superior to indclus are marked with a box.) Animals-s is the only data set on which its median performance was not the best, and its overall distribution of results is consistently competitive. It is revealing to note the differences in performance between the original and reimplemented versions of sindclus. Small changes in the handling of boundary cases make a large difference in the performance of the algorithm. Surprisingly, on the synthetic data sets (not shown), the relative performance of the algorithms was quite different, and almost the same on the noisy data as on the noise-free data. (This suggests that the randomly generated data sets that are commonly used to evaluate ADCLUS algorithms do not accurately reflect the problems of interest to practitioners.) ewindclus performed best here, although it was only occasionally able to recover the original models from the noise-free data. Overall, it appears that current methods of additive clustering are quite sensitive to the type of problem they are run on and that there is little assurance that they can recover the underlying structure in the data, even for small problems. To address these problems, we turn now to a new approach. 3 A Purely Combinatorial Approach One common theme in indclus, sindclus, and ewindclus is their computation of each cluster and its weight in turn, at each step fitting only the residual similarity not accounted for by the other clusters. This forces memberships to be considered in a predetermined order and allows weights to become obsolete. Inspired in part by recent work of Lee (in press), we propose an orthogonal decomposition of the problem. Instead of computing the elements and weight of each cluster in succession, we first consider all the memberships and then derive all the weights using constrained regression. And where previous algorithms recompute all the memberships of one cluster simultaneously (and therefore independently), we will change memberships one by one in a dynamically determined order using simple heuristic search techniques, recomputing the weights after each step. (An incremental bounded least squares regression algorithm that took advantage of the previous solution would be ideal, but the algorithms tested below did not incorporate such an improvement.) From this perspective, one need only focus on changing the binary membership variables, and ADCLUS becomes a purely combinatorial optimization problem. We will evaluate three different algorithms based on this approach, all of which attempt to improve a random initial model. The first, indclus-hc, is a simple hill-climbing strategy which attempts to toggle individual memberships in an arbitrary order and the first change resulting in an improved model is accepted. The algorithm terminates when no membership can be changed to give an improvement. This strategy is reminiscent of a proposal by Clouse and Cottrell (1996), although here we are using the ADCLUS model of similarity. In the second algorithm, indclus-pbil, the PBIL algorithm of Baluja (1997) is used 2Table 3 shows one anomaly: no em-indclus run on animals resulted in a VAF ≥0. This also occurred on all synthetic problems with 32 or more objects (although very good solutions were found on the smaller problems). Tenenbaum (personal communication) suggests that the default annealing schedule in the em-indclus code may need to be modified for these problems. Table 4: The performance of the combinatorial algorithms on human data sets. indclus-hc ind-pbil ewind-klb indclus Name VAF IQR r VAF IQR VAF IQR r VAF IQR r animals-s 80 80–80 44 74 71–74 80 80–80 74 80 80–80 47 numbers 90 90–91 24 87 85–88 91 91–91 18 90 89–91 59 workers 88 88–89 16 86 84–87 89 89–89 13 88 88–89 53 consonants 86 85–87 11 80 76–82 92 92–92 9 91 91–91 61 animals 71 70–72 8 66 65–69 74 74–74 6 74 74–74 36 letters 70 69–71 3 66 64–68 76 74–78 2 82 81–82 57 to search for appropriate memberships. This is a simplification of the strategy suggested by Lee (in press), whose proposal also includes elements concerned with automatically controlling model complexity. We use the parameter settings he suggests but only allow the algorithm to generate 10,000 solutions. 3.1 KL Break-Out: A New Optimization Heuristic While the two approaches described above do not use any problem-specific information beyond solution quality, the third algorithm uses the gradient function from the ewindclus algorithm to guide the search. The move strategy is a novel combination of gradient ascent and the classic method of Kernighan and Lin (1970) which we call ‘KL break-out’. It proceeds by gradient ascent, changing the entry in F whose ewindclus gradient points most strongly to the opposite of its current value. When the ascent no longer results in an improvement, a local maximum has been reached. Motivated by results suggesting that good maxima tend to cluster (Boese, Kahng, and Muddu, 1994; Ruml et al., 1996), the algorithm tries to break out of the current basin of attraction and find a nearby maximum rather than start from scratch at another random model. It selects the least damaging variable to change, using the gradient as in the ascent, but now it locks each variable after changing it. The pool of unlocked variables shrinks, thus forcing the algorithm out of the local maximum and into another part of the space. To determine if it has escaped, a new gradient ascent is attempted after each locking step. If the ascent surpasses the previous maximum, the current break-out attempt is abandoned and the ascent is pursued. If the break-out procedure changes all variables without any ascent finding a better maximum, the algorithm terminates. The procedure is guaranteed to return a solution at least as good as that found by the original KL method (although it will take longer), and it has more flexibility to follow the gradient function. This algorithm, which we will call ewindclus-klb, surpassed the original KL method in time-equated tests. It is also conceptually simple and has no parameters that need to be tuned. 3.2 Evaluation of the Combinatorial Algorithms The time-equated performance of the combinatorial algorithms on the human data sets is shown in Table 4, with indclus, the best of the previous algorithms, shown for comparison. As one might expect, adding heuristic guidance to the search helps it enormously: ewindclus-klb surpasses the other combinatorial algorithms on every problem. It performs better than indclus on three of the human data sets (top panel), equals its performance on two, and performs worse on one data set, letters. (Results in which ewindclus-klb was not the best are marked with a box.) The variance of indclus on letters is very small, and the full distributions suggest that ewindclus-klb is the better choice on this data set if one can afford the time to take the best of 20 runs. (Experiments Table 5: ewindclus-klb and indclus on noisy synthetic data sets of increasing size. ewindclus-klb indclus n VAF IQR VAF IQR r 8 97 96–97 95 93–97 1 16 91 90–92 86 85–87 4 32 90 88–92 83 82–84 22 64 91 90–91 84 84–85 100 128 91 91–91 88 87–90 381 using 7 additional human data sets found that letters represented the weakest performance of ewindclus-klb.) Performance of a plain KL strategy (not shown) surpassed or equaled indclus on all but two problems (consonants and letters), indicating that the combinatorial approach, in tandem with heuristic guidance, is powerful even without the new ‘KL break-out’ strategy. While we have already seen that synthetic data does not predict the relative performance of algorithms on human data very well, it does provide a test of how well they scale to larger problems. On noise-free synthetic data, ewindclus-klb reliably recovered the original model on all data sets. It was also the best performer on the noisy synthetic data (a comparison with indclus is presented in Table 5. These results show that, in addition to performing best on the human data, the combinatorial approach retains its effectiveness on larger problems. In addition to being able to handle larger problems than previous methods, it is important to note that the higher VAF of the models induced by ewindclus-klb often translates into increased interpretability. In the model shown in Table 1, for instance, the best previously published model (Tenenbaum, 1996), whose VAF is only 1.6% worse, does not contain ˘s in the unvoiced cluster. 4 Conclusions We formalized the problem of constructing feature-based representations for cognitive modeling as the unsupervised learning of ADCLUS models from similarity data. In an empirical comparison sensitive to variance in solution quality and computation time, we found that several recently proposed methods for recovering such models perform worse than the original indclus algorithm of Arabie and Carroll (1980). We suggested a purely combinatorial approach to this problem that is simpler than previous proposals, yet more effective. By changing memberships one at a time, it makes fewer independence assumptions. We also proposed a novel variant of the Kernighan-Lin optimization strategy that is able to follow the gradient function more closely, surpassing the performance of the original. While this work has extended the reach of the additive clustering paradigm to large problems, it is directly applicable to feature construction of only those cognitive models whose representations encode similarity as shared features. (The cluster weights can be represented by duplicating strong features or by varying connection weights.) However, the simplicity of the combinatorial approach should make it straightforward to extend to models in which the absence of features can enhance similarity. Other future directions include using the output of one algorithm as the starting point for another, and incorporating measures of model complexity(Lee, in press). 5 Acknowledgments Thanks to Josh Tenenbaum, Michael Lee, and the Harvard AI Group for stimulating discussions; to Josh, Anil Chaturvedi, Henk Kiers, J. Douglas Carroll, and Phipps Arabie for providing source code for their algorithms; Josh, Michael, and Phipps for providing data sets; and Michael for sharing unpublished work. This work was supported in part by the NSF under grants CDA-94-01024 and IRI-9618848. References Arabie, Phipps and J. Douglas Carroll. 1980. MAPCLUS: A mathematical programming approach to fitting the adclus model. Psychometrika, 45(2):211–235, June. Baluja, Shumeet. 1997. Genetic algorithms and explicit search statistics. In Michael C. Mozer, Michael I. Jordan, and Thomas Petsche, editors, NIPS 9. Boese, Kenneth D., Andrew B. Kahng, and Sudhakar Muddu. 1994. A new adaptive multi-start technique for combinatorial global optimizations. Operations Research Letters, 16:101–113. Carroll, J. Douglas and Phipps Arabie. 1983. INDCLUS: An individual differences generalization of the ADCLUS model and the MAPCLUS algorithm. Psychometrika, 48(2):157–169, June. Chaturvedi, Anil and J. Douglas Carroll. 1994. An alternating combinatorial optimization approach to fitting the INDCLUS and generalized INDCLUS models. Journal of Classification, 11:155–170. Clouse, Daniel S. and Garrison W. Cottrell. 1996. Discrete multi-dimensional scaling. In Proceedings of the 18th Annual Conference of the Cognitive Science Society, pp. 290–294. Hojo, Hiroshi. 1983. A maximum likelihood method for additive clustering and its applications. Japanese Psychological Research, 25(4):191–201. Kernighan, B. and S. Lin. 1970. An efficient heuristic procedure for partitioning graphs. The Bell System Technical Journal, 49(2):291–307, February. Kiers, Henk A. L. 1997. A modification of the SINDCLUS algorithm for fitting the ADCLUS and INDCLUS models. Journal of Classification, 14(2):297–310. Lee, Michael D. in press. A simple method for generating additive clustering models with limited complexity. Machine Learning. Mechelen, I. Van and G. Storms. 1995. Analysis of similarity data and Tversky’s contrast model. Psychologica Belgica, 35(2–3):85–102. Noelle, David C., Garrison W. Cottrell, and Fred R. Wilms. 1997. Extreme attraction: On the discrete representation preference of attractor networks. In M. G. Shafto and P. Langley, eds, Proceedings of the 19th Annual Conference of the Cognitive Science Society, p. 1000. Ruml, Wheeler, J. Thomas Ngo, Joe Marks, and Stuart Shieber. 1996. Easily searched encodings for number partitioning. Journal of Optimization Theory and Applications, 89(2). Shepard, Roger N. and Phipps Arabie. 1979. Additive clustering: Representation of similarities as combinations of discrete overlapping properties. Psychological Review, 86(2):87–123, March. Stark, Philip B. and Robert L. Parker. 1995. Bounded-variable least-squares: An algorithm and applications. Computational Statistics, 10:129–141. Tenenbaum, Joshua B. 1996. Learning the structure of similarity. In D. S. Touretzky, M. C. Mozer, and M. E. Hasselmo, editors, NIPS 8.
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Products of Gaussians Christopher K. I. Williams Division of Informatics University of Edinburgh Edinburgh EH1 2QL, UK c.k. i. williams@ed.ac.uk http://anc.ed.ac.uk Felix V. Agakov System Engineering Research Group Chair of Manufacturing Technology Universitiit Erlangen-Niirnberg 91058 Erlangen, Germany F.Agakov@lft·uni-erlangen.de Stephen N. Felderhof Division of Informatics University of Edinburgh Edinburgh EH1 2QL, UK stephenf@dai.ed.ac.uk Abstract Recently Hinton (1999) has introduced the Products of Experts (PoE) model in which several individual probabilistic models for data are combined to provide an overall model of the data. Below we consider PoE models in which each expert is a Gaussian. Although the product of Gaussians is also a Gaussian, if each Gaussian has a simple structure the product can have a richer structure. We examine (1) Products of Gaussian pancakes which give rise to probabilistic Minor Components Analysis, (2) products of I-factor PPCA models and (3) a products of experts construction for an AR(l) process. Recently Hinton (1999) has introduced the Products of Experts (PoE) model in which several individual probabilistic models for data are combined to provide an overall model of the data. In this paper we consider PoE models in which each expert is a Gaussian. It is easy to see that in this case the product model will also be Gaussian. However, if each Gaussian has a simple structure, the product can have a richer structure. Using Gaussian experts is attractive as it permits a thorough analysis of the product architecture, which can be difficult with other models, e.g. models defined over discrete random variables. Below we examine three cases of the products of Gaussians construction: (1) Products of Gaussian pancakes (PoGP) which give rise to probabilistic Minor Components Analysis (MCA), providing a complementary result to probabilistic Principal Components Analysis (PPCA) obtained by Tipping and Bishop (1999); (2) Products of I-factor PPCA models; (3) A products of experts construction for an AR(l) process. Products of Gaussians If each expert is a Gaussian pi(xI8i) '" N(J1i' ( i), the resulting distribution of the product of m Gaussians may be expressed as By completing the square in the exponent it may be easily shown that p(xI8) N(/1;E, (2:), where (El = 2::1 (i l . To simplify the following derivations we will assume that pi(xI8i) '" N(O, (i) and thus that p(xI8) '" N(O, (2:). J12: i ° can be obtained by translation of the coordinate system. 1 Products of Gaussian Pancakes A Gaussian "pancake" (GP) is a d-dimensional Gaussian, contracted in one dimension and elongated in the other d - 1 dimensions. In this section we show that the maximum likelihood solution for a product of Gaussian pancakes (PoGP) yields a probabilistic formulation of Minor Components Analysis (MCA). 1.1 Covariance Structure of a GP Expert Consider a d-dimensional Gaussian whose probability contours are contracted in the direction w and equally elongated in mutually orthogonal directions VI , ... , vd-l.We call this a Gaussian pancake or GP. Its inverse covariance may be written as d - l ( - 1 = L ViV; /30 + wwT /3,;;, (1) i = l where VI, ... ,V d - l, W form a d x d matrix of normalized eigenvectors of the covariance C. /30 = 0"02 , /3,;; = 0";;2 define inverse variances in the directions of elongation and contraction respectively, so that 0"5 2 0"1. Expression (1) can be re-written in a more compact form as (2) where w = wJ/3,;; - /30 and Id C jRdxd is the identity matrix. Notice that according to the constraint considerations /30 < /3,;;, and all elements of ware real-valued. Note the similarity of (2) with expression for the covariance of the data of a 1factor probabilistic principal component analysis model ( = 0"21d + wwT (Tipping and Bishop, 1999), where 0"2 is the variance of the factor-independent spherical Gaussian noise. The only difference is that it is the inverse covariance matrix for the constrained Gaussian model rather than the covariance matrix which has the structure of a rank-1 update to a multiple of Id . 1.2 Covariance of the PoGP Model We now consider a product of m GP experts, each of which is contracted in a single dimension. We will refer to the model as a (I,m) PoGP, where 1 represents the number of directions of contraction of each expert. We also assume that all experts have identical means. From (1), the inverse covariance of the the resulting (I,m) PoGP model can be expressed as m C;;l = L Cil (3) i=l where columns of We Rdxm correspond to weight vectors of the m PoGP experts, and (3E = 2::1 (3~i) > o. 1.3 Maximum-Likelihood Solution for PoGP Comparing (3) with m-factor PPCA we can make a conjecture that in contrast with the PPCA model where ML weights correspond to principal components of the data covariance (Tipping and Bishop, 1999), weights W of the PoGP model define projection onto m minor eigenvectors of the sample covariance in the visible d-dimensional space, while the distortion term (3E Id explains larger variationsl . This is indeed the case. In Williams and Agakov (2001) it is shown that stationarity of the log-likelihood with respect to the weight matrix Wand the noise parameter (3E results in three classes of solutions for the experts' weight matrix, namely W 5 5W 0; CE ; CEW, W:j:. 0, 5:j:. CE, (4) where 5 is the covariance matrix of the data (with an assumed mean of zero). The first two conditions in (4) are the same as in Tipping and Bishop (1999), but for PPCA the third condition is replaced by C-lW = 5- lW (assuming that 5- 1 exists). In Appendix A and Williams and Agakov (2001) it is shown that the maximum likelihood solution for W ML is given by: (5) where R c Rmxm is an arbitrary rotation matrix, A is a m x m matrix containing the m smallest eigenvalues of 5 and U = [Ul , ... ,u m ] c Rdxm is a matrix of the corresponding eigenvectors of 5. Thus, the maximum likelihood solution for the weights of the (1, m) PoG P model corresponds to m scaled and rotated minor eigenvectors of the sample covariance 5 and leads to a probabilistic model of minor component analysis. As in the PPCA model, the number of experts m is assumed to be lower than the dimension of the data space d. The correctness of this derivation has been confirmed experimentally by using a scaled conjugate gradient search to optimize the log likelihood as a function of W and (3E. 1.4 Discussion of PoGP model An intuitive interpretation of the PoGP model is as follows: Each Gaussian pancake imposes an approximate linear constraint in x space. Such a linear constraint is that x should lie close to a particular hyperplane. The conjunction of these constraints is given by the product of the Gaussian pancakes. If m « d it will make sense to lBecause equation 3 has the form of a factor analysis decomposition, but for the inverse covariance matrix, we sometimes refer to PoGP as the rotcaf model. define the resulting Gaussian distribution in terms of the constraints. However, if there are many constraints (m > d/2) then it can be more efficient to describe the directions of large variability using a PPCA model, rather than the directions of small variability using a PoGP model. This issue is discussed by Xu et al. (1991) in what they call the "Dual Subspace Pattern Recognition Method" where both PCA and MCA models are used (although their work does not use explicit probabilistic models such as PPCA and PoGP). MCA can be used, for example, for signal extraction in digital signal processing (Oja, 1992), dimensionality reduction, and data visualization. Extraction of the minor component is also used in the Pisarenko Harmonic Decomposition method for detecting sinusoids in white noise (see, e.g. Proakis and Manolakis (1992), p. 911). Formulating minor component analysis as a probabilistic model simplifies comparison of the technique with other dimensionality reduction procedures, permits extending MCA to a mixture of MCA models (which will be modeled as a mixture of products of Gaussian pancakes), permits using PoGP in classification tasks (if each PoGP model defines a class-conditional density) , and leads to a number of other advantages over non-probabilistic MCA models (see the discussion of advantages of PPCA over PCA in Tipping and Bishop (1999)). 2 Products of PPCA In this section we analyze a product of m I-factor PPCA models, and compare it to am-factor PPCA model. 2.1 I-factor PPCA model Consider a I-factor PPCA model, having a latent variable Si and visible variables x. The joint distribution is given by P(Si, x) = P(si)P(xlsi). We set P(Si) '" N(O, 1) and P(XISi) '" N(WiSi' (]"2) . Integrating out Si we find that Pi(x) '" N(O, Ci ) where C = wiwT + (]"21d and (6) where (3 = (]"-2 and "(i = (3/(1 + (3llwiW). (3 and "(i are the inverse variances in the directions of contraction and elongation respectively. The joint distribution of Si and x is given by (7) (3 [s; T T ] exp - - 2x WiSi + X X . 2 "(i (8) Tipping and Bishop (1999) showed that the general m-factor PPCA model (mPPCA) has covariance C = (]"21d + WWT , where W is the d x m matrix of factor loadings. When fitting this model to data, the maximum likelihood solution is to choose W proportional to the principal components of the data covariance matrix. 2.2 Products of I-factor PPCA models We now consider the product of m I-factor PPCA models, which we denote a (1, m)-PoPPCA model. The joint distribution over 5 = (Sl' ... ,Srn)T and x is 13 m [s; T T ] P(x,s) ex: exp -"2 L ---:- - 2x W iSi + X X • i=l ,,(, (9) Let zT d~f (xT , ST). Thus we see that the distribution of z is Gaussian with inverse covariance matrix 13M, where -W) r - 1 , (10) and r = diag("(l , ... ,"(m)' Using the inversion equations for partitioned matrices (Press et al., 1992, p. 77) we can show that (11) where ~xx is the covariance of the x variables under this model. It is easy to confirm that this is also the result obtained from summing (6) over i = 1, ... ,m. 2.3 Maximum Likelihood solution for PoPPCA Am-factor PPCA model has covariance a21d + WWT and thus, by the Woodbury formula, it has inverse covariance j3ld - j3W(a2 lm + WT W) - lWT . The maximum likelihood solution for a m-PPCA model is similar to (5), i.e. W = U(A _a2Im)1/2 RT, but now A is a diagonal matrix of the m principal eigenvalues, and U is a matrix of the corresponding eigenvectors. If we choose RT = I then the columns of W are orthogonal and the inverse covariance of the maximum likelihood m-PPCA model has the form j3ld - j3WrwT. Comparing this to (11) (with W = W) we see that the difference is that the first term of the RHS of (11) is j3m1d, while for m-PPCA it is j3ld. In section 3.4 and Appendix C.3 of Agakov (2000) it is shown that (for m :::=: 2) we obtain the m-factor PPCA solution when m A<A' < --A ' m -I ' i = 1, ... ,m, (12) where A is the mean of the d - m discarded eigenvalues, and Ai is a retained eigenvalue; it is the smaller eigenvalues that are discarded. We see that the covariance must be nearly spherical for this condition to hold. For covariance matrices satisfying (12) , this solution was confirmed by numerical experiments as detailed in (Agakov, 2000, section 3.5). To see why this is true intuitively, observe that Ci 1 for each I-factor PPCA expert will be large (with value 13) in all directions except one. If the directions of contraction for each Ci 1 are orthogonal, we see that the sum of the inverse covariances will be at least (m - 1)13 in a contracted direction and m j3 in a direction in which no contraction occurs. The above shows that for certain types of sample covariance matrix the (1, m) PoPPCA solution is not equivalent to the m-factor PPCA solution. However, it is interesting to note that by relaxing the constraint on the isotropy of each expert's noise the product of m one-factor factor analysis models can be shown to be equivalent to an m-factor factor analyser (Marks and Movellan, 2001). (b) • • • • (c) • • • • (d) Figure 1: (a) Two experts. The upper one depicts 8 filled circles (visible units) and 4 latent variables (open circles), with connectivity as shown. The lower expert also has 8 visible and 4 latent variables, but shifted by one unit (with wraparound). (b) Covariance matrix for a single expert. (c) Inverse covariance matrix for a single expert. (d) Inverse covariance for product of experts. 3 A Product of Experts Representation for an AR(l) Process For the PoPPCA case above we have considered models where the latent variables have unrestricted connectivity to the visible variables. We now consider a product of experts model with two experts as shown in Figure l(a). The upper figure depicts 8 filled circles (visible units) and 4 latent variables (open circles), with connectivity as shown. The lower expert also has 8 visible and 4 latent variables, but shifted by one unit (with wraparound) with respect to the first expert. The 8 units are, of course, only for illustration- the construction is valid for any even number of visible units. Consider one hidden unit and its two visible children. Denote the hidden unit by s the visible units as Xl and Xr (l, r for left and right). Set s '" N(O, 1) and Xl = as + bWI Xr = ±as + bwr , (13) where WI and Wr are independent N(O, 1) random variables, and a, b are constants. (This is a simple example of a Gaussian tree-structured process, as studied by a number of groups including that led by Prof. Willsky at MIT; see e.g. Luettgen et al. (1993).) Then (xf) = (x;) = a2 + b2 and (XIXr ) = ±a2 • The corresponding 2 x 2 inverse covariance matrix has diagonal entries of (a2 + b2)j ~ and off-diagonal entries of =t=a2 j~ , where ~ = b2 (b2 + 2a2 ). Graphically, the covariance matrix of a single expert has the form shown in Figure l(b) (where we have used the + rather than - choice from (13) for all variables). Figure l(c) shows the corresponding inverse covariance for the single expert, and Figure 1 (d) shows the resulting inverse covariance for the product of the two experts, with diagonal elements 2(a2 + b2)j ~ and off-diagonal entries of =t=a2 j~. An AR(l) process of the circle with d nodes has the form Xi = aXi - 1 (mod d) + Zi, where Zi ~ N(O,v). Thusp(X) <X exp-21v L:i(Xi-aXi- 1 (mod d))2 and the inverse covariance matrix has a circulant tridiagonal structure with diagonal entries of (1 + ( 2 )/v and off-diagonal entries of -a/v. The product of experts model defined above can be made equivalent to the circular AR(I) process by setting (14) The ± is needed in (13) as when a is negative we require Xr = -as + bWr to match the inverse covariances. We have shown that there is an exact construction to represent a stationary circular AR(I) process as a product of two Gaussian experts. The approximation of other Gaussian processes by products of tree-structured Gaussian processes is further studied in (Williams and Felderhof, 2001). Such constructions are interesting because they may allow fast approximate inference in the case that d is large (and the target process may be 2 or higher dimensional) and exact inference is not tractable. Such methods have been developed by Willsky and coauthors, but not for products of Gaussians constructions. Acknowledgements This work is partially supported by EPSRC grant GR/L78161 Probabilistic Models for Sequences. Much of the work on PoGP was carried out as part of the MSc project of FVA at the Division of Informatics, University of Edinburgh. CW thanks Sam Roweis, Geoff Hinton and Zoubin Ghahramani for helpful conversations on the rotcaf model during visits to the Gatsby Computational Neuroscience Unit. FVA gratefully acknowledges the support of the Royal Dutch Shell Group of Companies for his MSc studies in Edinburgh through a Centenary Scholarship. SNF gratefully acknowledges additional support from BAE Systems. References Agakov, F. (2000). Investigations of Gaussian Products-of-Experts Models. Master's thesis, Division of Informatics, The University of Edinburgh. Available at http://'iI'iI'iI . dai.ed.ac.uk/homes/felixa/all.ps.gz. Hinton, G. E. (1999). Products of experts. In Proceedings of the Ninth International Conference on Artificial Neural Networks (ICANN gg), pages 1- 6. Luettgen, M., Karl, W., and Willsky, A. (1993). Multiscale Representations of Markov Random Fields. IEEE Trans. Signal Processing, 41(12):3377- 3395. Marks, T. and Movellan, J. (2001). Diffusion Networks, Products of Experts, and Factor Analysis. In Proceedings of the 3rd International Conference on Independent Component Analysis and Blind Source Separation. OJ a, E. (1992). Principal Components, Minor Components, and Linear Neural Networks. Neural Networks, 5:927 - 935. Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (1992). Numerical Recipes in C. Cambridge University Press, Second edition. Proakis, J. G. and Manolakis, D. G. (1992). Digital Signal Processing: Principles, Algorithms and Applications. Macmillan. Tipping, M. E. and Bishop, C. M. (1999). Probabilistic principal components analysis. J. Roy. Statistical Society B, 61(3) :611- 622. Williams, C. K. I. and Agakov, F. V. (2001). Products of Gaussians and Probabilistic Minor Components Analysis. Technical Report EDI-INF-RR-0043, Division of Informatics, University of Edinburgh. Available at http://'iI'iI'iI. informatics. ed. ac. ukl publications/report/0043.html. Williams, C. K. I. and Felderhof, S. N. (2001). Products and Sums of Tree-Structured Gaussian Processes. In Proceedings of the ICSC Symposium on Soft Computing 2001 (SOCO 2001). Xu, L. and Krzyzak, A. and Oja, E. (1991). Neural Nets for Dual Subspace Pattern Recogntion Method. International Journal of Neural Systems, 2(3):169- 184. A ML Solutions for PoGP Here we analyze the three classes of solutions for the model covariance matrix which result from equation (4) of section 1.3. The first case W = 0 corresponds to a minimum of the log-likelihood. In the second case, the model covariance e~ is equal to the sample covariance 5. From expression (3) for e i;l we find WWT = 5- 1 ;3~ ld. This has the known solution W = Um(A - 1 ;3~ lm)1 /2 RT , where Um is the matrix of the m eigenvectors of 5 with the smallest eigenvalues and A is the corresponding diagonal matrix of the eigenvalues. The sample covariance must be such that the largest d - m eigenvalues are all equal to ;3~; the other m eigenvalues are matched explicitly. Finally, for the case of approximate model covariance (5W = e~w, 5 =f. e~) we, by analogy with Tipping and Bishop (1999), consider the singular value decomposition of the weight matrix, and establish dependencies between left singular vectors of W = ULRT and eigenvectors of the sample covariance 5. U = [U1 , U2, ... , um] C lRdxm is a matrix of left singular vectors of W with columns constituting an orthonormal basis, L = diag(h,l2, ... ,lm) C lRmxm is a diagonal matrix of the singular values of Wand R C lRmxm defines an arbitrary rigid rotation of W. For this case equation (4) can be written as 5UL = e ~ UL , where e~ is obtained from (3) by applying the matrix inversion lemma [see e.g. Press et al. (1992)]. This leads to 5UL = e~UL (;3i;lld ;3i;l W(;3~ + WTW)-lWT)UL U(;3i;l lm ;3i;l LRT(;3~ lm + RL2RT)-l RL)L U(;3i;l lm ;3i;l(;3~ L -2 + Im)-l) L. (15) Notice that the term ;3;1 1m ;3;l(;3~ L -2 + Im)-l in the r.h.s. of equation (15) is just a scaling factor of U. Equation (15) defines the matrix form of the eigenvector equation, with both sides post-multiplied by the diagonal matrix L. If li =f. 0 then (15) implies that e~ U i = 5Ui = AiUi, Ai = ;3i;1(1 (;3~li2 + 1)- 1), (16) where Ui is an eigenvector of 5, and Ai is its corresponding eigenvalue. The scaling factor li of the ith retained expert can be expressed as li = (Ail ;3~)1/2 . Obviously, if li = 0 then Ui is arbitrary. If li = 0 we say that the direction corresponding to Ui is discarded, i.e. the variance in that direction is explained merely by noise. Otherwise we say that Ui is retained. All potential solutions of W may then be expressed as W = Um(D ;3~ lm)1 /2 RT , (17) where R C lRmxm is a rotation matrix, Um = [U1U2 ... um] C lRdxm is a matrix whose columns correspond to m eigenvectors of 5, and D = diag( d1 , d2 , ... , dm ) C lRm x m such that di = Ail if Ui is retained and di = ;3~ if Ui is discarded. It may further be shown (Williams and Agakov (2001)) that the optimal solution for the likelihood is reached when W corresponds to the minor eigenvectors of the sample covariance 5.
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