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Linear Hinge Loss and Average Margin Claudio Gentile DSI, Universita' di Milano, Via Comelico 39, 20135 Milano. Italy gentile@dsi.unimi.it Manfred K. Warmuth· Computer Science Department, University of California, 95064 Santa Cruz, USA manfred@cse.ucsc.edu Abstract We describe a unifying method for proving relative loss bounds for online linear threshold classification algorithms, such as the Perceptron and the Winnow algorithms. For classification problems the discrete loss is used, i.e., the total number of prediction mistakes. We introduce a continuous loss function, called the "linear hinge loss", that can be employed to derive the updates of the algorithms. We first prove bounds w.r.t. the linear hinge loss and then convert them to the discrete loss. We introduce a notion of "average margin" of a set of examples . We show how relative loss bounds based on the linear hinge loss can be converted to relative loss bounds i.t.o. the discrete loss using the average margin. 1 Introduction Consider the classical Perceptron algorithm. The hypothesis of this algorithm at trial t is a linear threshold function determined by a weight vector Wt E Rn. For an instance Xt ERn the linear activation at = Wt . Xt is passed through a threshold function (7 r which is -Ion arguments less than the threshold rand + 1 otherwise. Thus the prediction of the algorithm is binary and -1 , + 1 denote the two classes. The Perceptron algorithm is aimed at learning a classification problem where the examples have the form (X t , Yt ) E R n x {-I, +1}. After seeing T examples (Xt,Yt)1<t<T, the algorithm predicts with YT+1 = (7r(WT+1 . xT+ d on the next instance XT+1. Tfthe algorithm's prediction YT+l agrees with the label YT+ 1 on the instance xT +1, then its loss is zero. If the prediction and the label disagree, then the loss is one. We call this loss the discrete loss. The convergence of the Perceptron algonthm is established in the Perceptron convergence theorem. There is a second by now classical algorithm for learning with linear threshold functions: the Winnow algorithm of Nick Littlestone [Lit88]. This algorithm also maintains a weight vector and predicts with the same linear threshold function defined by the current weight vector Wt . However, the update of the weight vector W t = (Wt.l , ... , Wt ,n ) • Supported by NSF grant CCR-970020 1. 226 C. Gentile and M. K. Warmuth performed by the two algorithms is radically different: Perceptron: Wt+l := Wt 1] 6t Xt Winnow: In Wt+! ,i := In Wt,i - 1] 6t Xt ,i The Perceptron algorithm performs a simple additive update. The parameter 1] is a positive learning rate and 6t equals (fit - Yt) /2, which lies in {-1, 0, + 1 }. When 6t = 0 the prediction of the algorithm is correct and no update occurs. Both the Perceptron algorithm and Winnow update conservatively, i.e., they update only when the prediction of the algorithm is wrong. If fit = + 1 and Yt = -1 then the algorithm overshot and 6t = + 1. This causes the Perceptron to subtract 1] Xt from the current weight Wt. Similarly if fit = -1 and Yt = +1 then the algorithm undershot and 6t = -1. Now the Perceptron adds 1] Xt to the current weight Wt. We will later interpret 6t Xt as a gradient of a loss function. Winnow uses the same gradient but the update is done through the componentwise logarithm of the weight vector. One can also rewrite Winnow's update as Wt+l ,i := Wt ,i exp (-1] 6tXt,i), i = 1, ... , n, so that the gradient appears in the exponents of factors that multiply the old weights. The factors are now used to correct the weights in the right direction when the algorithm under or overshot. The algorithms are good for different purposes and, generally speaking, incomparable (see [KWA97] for a discussion). In [KW97] a framework was introduced for deriving simple on-line learning updates. This framework has been applied to a variety of different learning algorithms and differentiable loss functions [HKW95, KW98]. The updates are always derived by approximately solving the following minimization problem Wt+! := argminwU(w), where U(w) = d(w, Wt) + 1]loss(Yt, C1r (w · Xt )) . (1) Here loss denotes the chosen loss function. In our setting this would be the discrete loss. What is different now is that the prediction of the algorithm Yt = C1r (Wt . xd and the discrete loss are discontinuous in the weight vector Wt. We will return to this point later after discussing the other parts of the above minimization problem. The parameter TJ is the learning rate mentioned above and, most importantly, d(w , Wt) is a divergence measuring how far W is from Wt. The divergence function has two purposes. It motivates the update and it becomes the potential function in the amortized analysis used to prove loss bounds for the corresponding algorithm. The use of an amortized analysis in the context of learning essentially goes back to [Lit89] and the method for deriving updates based on the divergence was introduced in [KW97]. The divergence may be seen as a regularization term and may also serve as a barrier function in the optimization problem (1) for the purpose of keeping the weights in a particular region. The additive algorithms, such as gradient descent and the Perceptron algorithm, use d( w, wd = Ilw - Wt W /2 as the divergence. This can be used as a potential function for the proof of the Perceptron convergence theorem. Multiplicative update algorithms such as Winnow and various exponentiated gradient algorithms use entropy-based divergences as potential functions [HKW95, KW98]. The function U in (1) is minimized by differentiating w.r.t. w. This works very well when the loss function is convex and differentiable. For example for linear regression, when the loss function is the square loss (Wt . Xt - Yt)2/2 , then minimizing U( w ) with the divergence Ilw - Wt 112/2 gives the Widrow-Hoff update: Wt+! := W t -1](Wt+l . Xt - Yt )X t ~ Wt -1](Wt . Xt - Yt)Xt. Various exponentiated gradient algorithms [KW97] can be derived in the same way when entropic divergences are used instead. However, in our case we cannot differentiate the discrete loss since it is discontinuous. We asked ourselves which loss function motivates the Perceptron and Winnow algorithms in this framework. We will see that the loss function that achieves this is continuous and Linear Hinge Loss and Average Margin 227 its gradient w.r.t. Wt is Otxt. where Ot E {-I, 0, + I}. We call this loss the (linear) hinge loss (HL) and we believe this is the key tool for understanding linear threshold algorithms such as the Perceptron and Winnow. However, in the process of changing the discrete loss to the HL we also changed our learning problem from a classification to a regression problem. There are now two versions of each algorithm, a classification version and a regression version. The classification version predicts with a binary label using its linearly thresholded prediction. The loss function is the discrete loss. The regression version, on the other hand. predicts on the next instance Xt with its linear activation at = Wt ·Xt. In the classification problem the labels Yt of the examples are -1 and + 1, while in the regression problem the labels at are -00 and +00. We will see that both versions of each algorithm use the same rule to update the weight vector Wt. Another strong hint that the HL is related to Perceptron and Winnow comes from the fact that this loss may be seen as a limiting case of the entropic loss used in logistic regression. In logistic regression the threshold function Ur is replaced by the smooth tanh function. There is a technical way of associating a "matching loss function" with a given increasing transfer function [HKW95]. The matching loss for the tanh transfer function is the entropic loss. We will show that by making this transfer function steeper and by taking the right viewpoint of the matching loss, the entropic loss converges to the HL. In the limiting case the slope ofthe transferfunction is infinite, i.e., it becomes the threshold function U r • The question is whether this introduction of the HL buys us anything. We believe so. We can prove a unifying meta-theorem for the whole class of general additive algorithms [GLS97, KW98], when defined w.r.t. the HL. The bounds for the regression versions of the Perceptron and Winnow are simple special cases. These loss bounds can then be converted to loss bounds for the corresponding classification problems w.r.t. the discrete loss. This conversion is carried out through working with the "average margin" of a set of examples relative to a linear threshold classifier. The conversion of the HL described in this paper can then be considered a principled way of deriving average margin-based mistake bounds. The average margin reveals the inner structure of mistake bound results that have been proven thus far for conservative on-line algorithms. Previously used definitions, such as the deviation [FS98] and the attribute error [Lit91], can easily be related to the average margin or reinterpreted in terms of the HL and the average margin. 2 Preliminaries and the linear hinge loss We define two subsets of n n: the weight domain Wand the instance domain X. The weights W maintained by the algorithms always lie in the weight domain and the instances x of the examples always lie in the instance domain. We require W be convex. A general additive algorithm and a divergence are defined in terms of a link function f. Such a function is a vector valued function from the interior int W of the weight domain W onto nn. with the property that its Jacobian is strictly positive definite everywhere in int W. A link function f has a unique inverse f- 1 : nn -+ int W. We assume that f is the gradient of a (potential) function Pr from int W to n, i.e., f(w) = \7Pr(w) for W E int W. It is easy to extend the domain of Pr such that it includes the boundary of W. For any link function f, a (Bregman) divergence function dr : W x int W -+ [0, (0) is defined as [Bre67]: dr(u,w) = Pr(u) - Pr(w) - (u - w)· f(w). (2) Thus dr( u, w) is the difference between Pr{ u) and its first order Taylor expansion around w. Since f has a strictly positive definite Jacobian everywhere in int W, the potential Pr is strictly convex over W. Thus dr(u, w) ~ 0 with equality holding iff u = w. The Perceptron algorithm is motivated by the identity link f (w) = w, with weight domain W = nn. The corresponding divergence is dr(u, w) = Ilu - wW /2. For Winnow the 228 Ora 0 r Ur(a) = -1 Ur(a) = +1 Figure 1: HL( a, a) as a function of a for the two cases ur{a) = -1, +1. C. Gentile and M. K. Warmuth ii=u(a) Figure 2: The matching loss MLu-l (y, f). weight domain is W = [O,oo)n. The link function is the componentwise logarithm. The divergence related to this link function is the un-normalized relative entropy dr( U, w) = 2:~=1 Ui In ~ + Wi Ui. Note that now U E W, but w must lie in int W. The following key property immediately follows from the definition of the divergence dr. Lemma! [KW98JForanyu E Wandwl,w2 E intW: In this paper we focus on a single neuron using a hard threshold as the transfer function (see beginning of the introduction). We will view such a neuron in two ways. In the standard view the neuron is used for binary classification. It outputs f) = U r (a) trying to predict the desired label y using a threshold r. In the new view the neuron is a regressor. It outputs the linear activation a E 'R, and is trying to predict a E 'R,. For classification we use the discrete loss DL(y,f) = ~Ifj - yl E {0,1}. For regression we use the linear hinge loss (HL) parameterized by a threshold r: For any a, a E R: HLr{a, a) := ~(Ur(a) - CTr(a»(a - r) = DL(y, fj)la - rl· Note that the arguments in the two losses DL and HLr are switched. This is intentional and will be discussed later on. It can be easily shown that HLr{w . x, a) is convex in wand that the gradient of this loss w.r.t. w is 'VwHLr(w, x, a) = ~(ur{a) - ur(a» X. Note that 6 = (ur(a) ur (a»/2 can only take the three values 0, -1, and +1 mentioned in the introduction. Strictly speaking, this gradient is not defined when w . x equals the threshold r. But we will show in the subsequent sections that even in that case 6 x has the properties we need. Figure 1 provides a graphical representation of HLr. The threshold function CTr "transfers" the linear activation a = w . x to a prediction f) which is a hard classification in {-1, +1}. (For the remaining discussion of this section we can assume with no loss of generality that the threshold r is 0.) Smooth transfer functions such as the tanh are commonly used in neural networks, e.g., fj = tanh(a), and relative loss bounds have been proven when the comparison class consists of single neurons with any increasing differentiable transfer function CT [HKW95, KW98]. However, for this to work a loss function that "matches" the transfer function has to be used. This loss is defined] as follows [HKW95] (see Figure 2): MLu-l (y, fj) := f:~ll(~; u(z) - y dz = dU-1 (y, f). The matching loss for u(z) = z is the square loss (linear regression) and the matching loss for u(z) = tanh(z) is the entropic loss (logistic regression), which is defined as: lIn [HKW95] the notation Lu(Y, y) is used for the matching loss MLu-l (y, y). We use here the subscript a -1 instead of a to stress a connection between the matching loss and the divergence that is discussed at the end of this section. Linear Hinge Loss and Average Margin 229 MLO'-l (y, y) = ~(1 - y) In !=~ + t(1 + y) In !!~ . The entropic loss is finite when y E [-1, +1] and y = tanh(a) E (-1, +1). These are the ranges for y and Y needed for logistic regression. We now want to use this type of loss for classification with linear threshold functions, i.e., when y, y E {-I, + I} and the slope s of the tanh function is increased until in the limit it becomes the hard threshold (10 . Obviously, (1-1 ( -1) = - 00 and (1-1 (+ 1) = +00 for any slope s. Thus the matching loss is infinite for all slopes. Also, the known relative loss bounds based on the above notion of matching loss grow with the slope of the transfer function. Thus it seems to be impossible to use the matching loss when the transfer function is the hard threshold 170. However, we can still make sense of the matching loss by viewing the neuron as a regressor. The matching loss is now rewritten as another Bregman divergence: MLcr(a,a) = lil u(z) - a(a) dz = PO'(a) - PO'(a) - (a - a)a(a) = dcr((i,a), (3) where PO' is any function such that P;(a) = (1(a) . We now increase the slope of the transfer function tanh while keeping a and a fixed. In the limiting case (hard thr~old (70) the above loss becomes twice the linear hinge loss with threshold zero, i.e., MLO'o (a, a) = 2 HLo(a, a) = ((1o(a) - (1o(a))(a - 0). Finally, observe that the two views of the neuron are related to a duality property [AW98] of Bregman divergences: (4) 3 The algorithms In this paper we always associate two general additive algorithms with a given link function: a classification algorithm and a regression algorithm. Such algorithms, given in the next table, correspond to the two views of a linear threshold neuron discussed in the last section. For brevity, we will call the two algorithms "the classification algorithm" and "the regression algorithm", respectively. Gen. add. classification algorithm: Gen. add. regression algorithm: For t = 1,2, . . . For t = 1,2, .. . Instance: Xt E R n Instance: Xt E R n Prediction: Yt = ar(wt . Xt) Prediction: at = Wt . Xt Label: Yt E {-I, +1} Label:2 at = Ytoo Update: Update: Wt+1 =f-1 (f(wt) - ¥{f)t - Yt)xt} wt+l=f-1 (f(wt) - ¥(ar(at) -ur(ad)xt) Discrete loss: Linear hinge loss: DL(yt. Yt) = tlYt - Ytl HLr(at , at) = t(ar(at) -ar(at))(at - r) The classification algorithm receives a label Yt E {- I, + 1 }, while the regression algorithm receives the infinite label at with the sign of Yt. This assures that Yt = a r (ad. The classification algorithm predicts with Yt = ar(ad, and the regression algorithm with its linear activation at. The loss for the classification algorithm is the discrete loss DL(Yt, Yt), while for the regression algorithm we use HLr( at. at). The updates of the two algorithms are equivalent. The update of the regression algorithm is motivated by the minimization problem: Wt+1 := argminwU(w) where U(w) = dr(w, wd + 'T/ HLr(w . Xt. ad. By setting the gradient of U(w ) w.r.t. w to zero we get the following equilibrium equation that holds at the minimum of U(w): Wt -"-1 f- 1 (f(wt ) - ¥(ar(Wt+l . Xt) -o"r(at))xt} . We approximately solve this equation by replacing Wt+l ' Xt by at = Wt ' Xt, i.e., Wt+1 = f- 1 (f(wt) - ¥(ar(at}-ar(at))xt) . 2This is a short-hand meaning at = ;-00 if Yt = ;-1 and at = - 00 if Yt = -1 . 230 C. Gentile and M. K. Warmuth Both versions of the Perceptron and Winnow are obtained by using the link functions f(w) = wand f(w) = (In(wd, ... , In(wn )). respectively. 4 Relative loss bounds The following lemma relates the hinge loss of the regression algorithm to the hinge loss of an arbitrary linear predictor u. Lemma 2 For all U E W, W t E int W, Xt EX, at, r E Rand 1] > 0: HLr(at,at) HLr(u , xt,at) + HLr(u, xt,at) = * (dr(u,wt) - dr(u,wt+1) + dr(wt,wt+1)) = !(Yt - Yt) (at - U· xt} (5) Proof. We have dr(u, Wt) - dr(u, Wt+d + dr(wt , Wt+l) = (u - Wt) . (J(Wt+d j(wd) = (Wt - u) . ¥(O"r(at) - o"r(at)) Xt = ¥(O"r(at) - O"r(at)) (at - U . Xt) = 1] (HLr(at, at) - HLr(u . Xt, at) + HLr(u . Xt , ad) . The first equality follows Lemma 1 and the second follows from the update rule of the regression algorithm. The last equality uses HLr(at, at) as a divergence drTr (at , at) (see (4)) and again Lemma 1. 0 By summing the first equality of (5) over all trials t we could relate the total HLr of the regression algorithm to the total HLr of the regressor u. However, our goal is to obtain bounds on the number of mistakes on the classification algorithm. It is therefore natural to interpret u too as a linear threshold classifier. with the same threshold r used by the classification algorithm. We use the second equality of (5) and sum up over all T trials: L,;=I !(Yt - Yt) (a - u . Xt) = * (dr(u, wd - dr(u, wT+d + L,;=I dr(Wt, wt+d). Note that the sums in the above equality are unaffected by trials in which no mistake occurs. In such trials. Yt = Yt and Wt+1 = Wt . Thus the above is equivalent to the following. where M is the set of trials in which a mistake occurs: L,tEM !(Yt - Yt) (at - U· Xt) = ~ (dr(u, wd - dr (u , wT+d + L,tE.vt dr(wt, wt+d). Since t(Yt -Yt) = -Yt when t E J\It and dr(u, WT+1) :::: 0 we get the following theorem: Theorem 3 Let M ~ {I, .. . ,T} be the set a/trials in which the classification algorithm makes a mistake. Then/or every u E W we have L,tEM Yt (u . Xt - at) ~ ~ (dr(u, wt} + L,tEM dr(wt, wt+d) . 0 Throughout the rest of this section the classification algorithm is compared to the performance of a linear threshold classifier u with threshold r = O. We now apply Theorem 3 to the Perceptron algorithm with WI = 0, giving a bound i.t.o. the average margin of a linear threshold classifier u with threshold 0 on a trial sequence M: i'u ,M := ILl L,tEM YtU . Xt · Since Yt at ~ 0 for t E M . the I.h.s. of the inequality of Theorem 3 is at least I I A " ( ),,!i. 2 !i. "'2 M "(U,M' By the update rule. L."tEM dr wt, Wt+1 = L."tEM 211xtl12 ~ 21.Iv1IX2' where IIxI12 ~ X 2 for t E M . Since in Theorem 3 u is an arbitrary vector. we replace u by A u therein, and set A = .x~ 1) • When we solve the resulting inequality for 1.1v11 the I'U ,M dependence on 1] cancels out. This gives us the following bound on the number of mistakes: IMI ~ ( 1 1~ 1 12X)2 I'U. )vl Linear Hinge Loss and Average Margin 231 Note that in the usual mistake bound for the Perceptron algorithm the average 'Yu,/vt is replaced by mintEM Ytu, Xt.3 Also, observe that the predictions of the Perceptron algorithm with r = 0 and WI = 0 are not affected by 1]. Hence the previous bound holds for any 1] > O. Next, we apply Theorem 3 to a normalized version of Winnow. This version of Winnow keeps weights in the probability simplex and is obtained by a slight modification of Winnow's link function. We assume r = 0 and choose X = {x E nn : Ilxlloo ~ Xoo}. Unlike the Perceptron algorithm, a Winnow-like algorithm heavily depends on the learning rate, so a careful tuning is needed. One can show (details omitted due to space limitations) that if 1] is such that 1] 'YU,M + 1] X 00 - In ( e 2 '1 X2 °o +1) > 0 then this normalized version of Winnow achieves the bound IMI < dr(u, WI) -, X I (e 2'1Xoo +1) , 1]'YU,M +1] 00 n 2 where dr( u, wd is the relative entropy between the two probability vectors U and Wl. Conclusions: In the full paper we study the case when there is no consistent threshold U more carefully and give more involved bounds for the Winnow and normalized Winnow algorithms as well as for the p-norm Perceptron algorithm [GLS97]. References [AW98] K. Azoury and M. K. Warmuth", "Relative loss bounds and the exponential family of distributions", "1998", Unpublished manuscript. [Bre67] L.M. Bregman. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Computational Mathematics and Physics, 7 :200-217, 1967. [FS98] y. Freund and R. Schapire. Large margin classification using the perceptron algorithm. In 11th COLT, pp. 209-217, ACM, 1998. [GLS97] A. J. Grove, N. Littlestone, and D. Schuurmans. General convergence results for linear discriminant updates. In 10th COLT, pp. 171-183. ACM, 1997. [HKW95] D. P. Helmbold, 1. Kivinen. and M. K. Warmuth. Worst-case loss bounds for sigmoided linear neurons. In NIPS 1995, pp. 309-315. MIT Press, 1995. [KW97] J. Kivinen and M. K. Warmuth. Additive versus exponentiated gradient updates for linear prediction. Inform. and Comput., 132(1): 1-64. 1997. [KW98] 1. Kivinen and M. K. Warmuth. Relative loss bounds for multidimensional regression problems. In NIPS 10, pp. 287-293. MIT Press, 1998. [KWA97] J. Kivinen, M. K. Warmuth, and P. Auer. The perceptron algorithm vs. winnow: linear vs. logarithmic mistake bounds when few input variables are relevant. Artijiciallntelligence, 97:325-343,1997. [Lit88] N. Littlestone. Learning when irrelevant attributes abound: A new Iinearthreshold algorithm. Machine Learning, 2:285-318, 1988. [Lit89] N. Littlestone. Mistake Bounds and Logarithmic Linear-threshold Learning Algorithms. PhD thesis. Umversity of California Santa Cruz, 1989. [Lit91 J N. Littlestone. Redundant noisy attributes, attribute errors, and linear threshold learning using Winnow. In 4th COLT, pp. 147-156, Morgan Kaufmann, 1991. 3The average margin ~IU.M may be positive even though u is not consistent.
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Learning Instance-Independent Value Functions to Enhance Local Search Robert Moll Andrew G. Barto Theodore J. Perkins Department of Computer Science University of Massachusetts, Amherst, MA 01003 Richard S. Sutton AT&T Shannon Laboratory, 180 Park Avenue, Florham Park, NJ 07932 Abstract Reinforcement learning methods can be used to improve the performance of local search algorithms for combinatorial optimization by learning an evaluation function that predicts the outcome of search. The evaluation function is therefore able to guide search to low-cost solutions better than can the original cost function. We describe a reinforcement learning method for enhancing local search that combines aspects of previous work by Zhang and Dietterich (1995) and Boyan and Moore (1997, Boyan 1998). In an off-line learning phase, a value function is learned that is useful for guiding search for multiple problem sizes and instances. We illustrate our technique by developing several such functions for the Dial-A-Ride Problem. Our learning-enhanced local search algorithm exhibits an improvement of more then 30% over a standard local search algorithm. 1 INTRODUCTION Combinatorial optimization is of great importance in computer science, engineering, and operations research. We investigated the use of reinforcement learning (RL) to enhance traditionallocal search optimization (hillclimbing). Since local search is a sequential decision process. RL can be used to improve search performance by learning an evaluation function that predicts the outcome of search and is therefore able to guide search to low-cost solutions better than can the original cost function. Three approaches to using RL to improve combinatorial optimization have been described 1018 R. Moll, A. G. Barto, T. J Perkins and R. S. Sutton in the literature. One is to learn a value function over multiple search trajectories of a single problem instance. As the value function improves in its predictive accuracy, its guidance enhances additional search trajectories on the same instance. Boyan and Moore's STAGE algorithm (Boyan and Moore 1997, Boyan 1998) falls into this category, showing excellent performance on a range of optimization problems. Another approach is to learn a value function off-line and then use it over mUltiple new instances of the same problem. Zhang and Dietterich's (1995) application of RL to a NASA space shuttle mission scheduling problem takes this approach (although it does not strictly involve local search as we define it below). A key issue here is the need to normalize state representations and rewards so that trajectories from instances of different sizes and difficulties yield consistent training data. In each of the above approaches, a state of the RL problem is an entire solution (e.g., a complete tour in a Traveling Salesman Problem (TSP)) and the actions select next solutions from the current solutions' neighborhoods. A third approach, described by Bertsekas and Tsitsiklis (1996), uses a learned value function for guiding the direct construction of solutions rather than for moving between them. We focused on combining aspects of first two of these approaches with the goal of carefull y examining how well the TD(,\) algorithm can learn an instance-independent value function for a given problem to produce an enhanced local search algorithm applicable to all instances of that problem. Our approach combines an off-line learning phase with STAGE's alternation between using the learned value function and the original cost function to guide search. We present an extended case study of this algorithm's application to a somewhat complicated variant of TSP known as the Dial-A-Ride Problem, which exhibits some of the non-uniform structure present in real-world transportation and logistics problems. 2 ENHANCING LOCAL SEARCH The components of local search for combinatorial optimization are 1) a finite set ofJeasible solutions, S; 2) an objective, or cost,function, C : S -4 ~; and 3) a neighborhood Junction, A : S -4 P( S) (the power set of S). Local search starts with an initial feasible solution, So, ofa problem instance and then at each step k = 1,2, ... , it selects a solution Sk E A(Sk-d such that C(Sk) < c(sk-d. This process continues until further local improvement is impossible, and the current local optimum is returned. If the algorithm always moves to the first less expensive neighboring solution encountered in an enumeration of a neighborhood, it is called first improvement local search. Following Zhang and Dietterich (1995) and Boyan and Moore (1997), we note that local search can be viewed as a policy o£' a Markov decision process (MDP) with state set S and action sets A(s), S E S, where an action is identified with the neighboring solution selected. Local search selects actions which decrease the value of c, eventually absorbing at a state with a locally minimum cost. But C is not the optimal value function for the local search problem, whose objective is to reach the lowest-cost absorbing state (possibly including some tradeoff involving the number of search steps required to do so). RL used with a function approximator can learn an approximate optimal value function, V, thereby producing an enhanced search algorithm that is locally guided by V instead of by c. One way to do this is to give a small penalty, E, for each transition and a terminal reward upon absorption that is inversely related to the cost of the terminal state. Maximizing the expected undiscounted return accomplishes the desired tradeoff (determined by the value of E) between quality of final solution and search time (cf. Zhang and Dietterich, 1995). Since each instance of an optimization problem corresponds to a different MDP, a value Learning Instance-Independent Value Functions to Enhance Local Search 1019 function V learned in this way is instance-specific. Whereas Boyan's STAGE algorithm in effect uses such a V to enhance additional searches that start from different states of the same instance, we are interested in learning a V off-line, and then using it for arbitrary instances of the given problem. In this case, the relevant sequential decision problem is more complicated than a single-instance MDP since it is a summary of aspects of all problem instances. It would be extremely difficult to make the structure of this process explicit, but fortunately RL requires only the generation of sample trajectories, which is relatively easy in this case. In addition to their cost, secondary characteristics of feasible solutions can provide valuable information for search algorithms. By adjusting the parameters of a function approximation system whose inputs are feature vectors describing feasible solutions, an RL algorithm can produce a compact representation of V. Our approach operates in two distinct phases. In the learning phase, it learns a value function by applying the TD(A) algorithm to a number of randomly chosen instances of the problem. In the performance phase, it uses the resulting value function, now held fixed, to guide local search for additional problem instances. This approach is in principle applicable to any combinatorial optimization problem, but we describe its details in the context of the Dial-A-Ride problem. 3 THE DIAL-A-RIDE PROBLEM The Dial-a-Ride Problem (DARP) has the following formulation. A van is parked at a terminal. The driver receives calls from N customers who need rides. Each call identifies the location of a customer, as well as that customer's destination. After the calls have been received, the van must be routed so that it starts from the terminal, visits each pick-up and drop-off site in some order, and then returns to the terminal. The tour must pick up a passenger before eventually dropping that passenger off. The tour should be of minimal length. Failing this goal-and DARP is NP-complete, so it is unlikely that optimal DARP tours will be found easily-at least a good quality tour should be constructed. We assume that the van has unlimited capacity and that the distances between pick-up and drop-off locations are represented by a symmetric Euclidean distance matrix. We use the notation 012-13 - 3 - 2 to denote the following tour: "start at the terminal (0), then pick up 1, then 2, then drop off 1 (thus: - 1), pick up 3, drop off 3, drop off 2 and then return to the terminal (site 0)." Given a tour s, the 2-opt neighborhood of s, A2(S), is the set oflegal tours obtainable from s by subsequence reversal. For example, for the tour above, the new tour created by the following subsequence reversal 01 / 2 -13 / -3 - 2 --. 013 -12 -3-2 is an element of A2 (T). However, this reversal 012 / -13 -3/ -2 --. 012 - 33 - 1 - 2 leads to an infeasible tour, since it asserts that passenger 3 is dropped off first, then picked up. The neighborhood structure of DARP is highly non-uniform, varying between A2 neighborhood sizes of O(N) and O(N2). Let s be a feasible DARP tour. By 2-opt(s) we mean the tour obtained by first-improvement local search using the A2 neighborhood structure (presented in a fixed, standard enumeration), with tour length as the cost function. As with TSP, there is a 3-opt algorithm for 1020 R. Moll. A. G. Barto. T J Perkins and R. S. Sutton DARP, where a 3-opt neighborhood A3(S) is defined and searched in a fixed, systematic way, again in first-improvement style. This neighborhood is created by inserting three rather than two "breaks" in a tour. 3-opt is much slower than 2-opt, more than 100 times as slow for N = 50, but it is much more effective, even when 2-opt is given equal time to generate multiple random starting tours and then complete its improvement scheme. Psaraftis (1983) was the first to study 2-opt and 3-opt algorithms for DARP. He studied tours up to size N = 30, reporting that at that size, 3-opt tours are about 30% shorter on average than 2-opt tours. In theoretical studies of DARP, Stein (1978) showed that for sites placed in the unit square, the globally optimal tour for problem size N has a length that asymptotically approaches 1.02-/2N with probability 1 as N increases. This bound applies to our study-although we multiply position coordinates by 100 and then truncate to get integer distance matrices-and thus, for example, a value of 1020 gives us a baseline estimate of the globally optimal tour cost for N = 50. Healy and Moll (1995) considered using a secondary cost function to extend local search on DARP. In addition to primary cost (tour length) they considered as a secondary cost the ratio of tour cost to neighborhood size, which they called cost-hood. Their algorithm employed a STAGE-like alternation between these two cost functions: starting from a random tour s, it first found 20pt(s); then it performed a limited local search using the cost-hood function, which had the effect of driving the search to a new tour with a decent cost and a large neighborhood. These alternating processes were repeated until a time bound was exhausted, at which point the least cost tour seen so far was reported as the result of the search. This technique worked well, with effectiveness falling midway between that of 2-opt and 3-opt. 4 ENHANCED 2-0PT FOR DARP We restrict our description to a learning method for enhancing 2-opt for DARP, but the same method can be used for other problems. In the learning phase, after initializing the function approximator, we conduct a number training episodes until we are satisfied that the weights have stabilized. For each episode k, we select a problem size N at random (from a predetermined range) and generate a random DARP instance of that size, i.e., we generate a symmetric Euclidean distance matrix by generating random points in the plane inside the square bounded by the points (0,0), (0,100), (100,100) and (100,0). We set the "terminal site" to point (50,50) and the initial tour to a randomly generated feasible tour. We then conduct a modified first-improvement 2-opt local search using the negated current value function, - Vk, as the cost function. The modification is that termination is controlled by a parameter E > ° as follows: the search terminates at a tour s if there is no s' E A( s) such that Vk (s') > Vk (s) + E. In other words, a step is taken only if it produces an improvement of at least E according to the current value function. The episode returns a final tour sf. We run one unmodified 2-opt local search, this time using the DARP cost function c (tour length), from sf to compute 2-opt( sf). We then apply a batch version of undiscounted TD(A) to the saved search trajectory using the following immediate rewards: -E for each transition, and -c(2-opt( sf)) / Stein N as a terminal reward, where Stein N is the Stein estimate for instance size N. Normalization by SteinN helps make the terminal reward consistent across instance sizes. At the end of this learning phase, we have a final value function, V. V is used in the performance phase, which consists of applying the modified first-improvement 2-opt local search with cost function - Von new instances, followed by a 2-opt application to the resulting tour. The results described here were obtained using a simple linear approximator with a bias Learning Instance-Independent Value Functions to Enhance Local Search 1021 Table 1: Weight Vectors for Learned Value Functions. Value Function Weight Vector v < .951, .033, .0153 > V20 < .981, .019, .00017 > V30 < .984, .014, .0006 > V40 < .977, .022, .0009 > Vso < .980, .019, .0015 > V60 < .971 , .022, .0069 > weight and features developed from the following base features: 1) normcost N (s) = c(s)jSteinN ; 2) normhoodN = [A(s) [jaN' where aN is a normalization coefficient defined below; and 3) normprox N, which considers a list of the N j 4 least expensive edges of the distance matrix, as follows. Let e be one of the edges, with endpoints u and v. The normprOXN feature examines the current tour, and counts the number of sites on the tour that appear between u and v . normprOXN is the sum of these counts over the edges on the proximity list divided by a normalizing coefficient bN described below. Our function approximator is then give by Wo +normcostN j(normhoodN )2Wl + normproXN j(normhoodN )2W2 . The coefficients aN and bN are the result of running linear regression on randomly sampled instances of random sizes to determine coefficients that will yield the closest fit to a constant target value for normalized neighborhood size and proximity. The results were aN = .383N2 + .28.5N - 244.5 and bN = .43N2 + .736N - 68.9.jN + 181.75. The motivation for the quotient features comes from Healy and Moll (1995) who found that using a similar term improved 2-opt on DARP by allowing it to sacrifice cost improvements to gain large neighborhoods. 5 EXPERIMENTAL RESULTS Comparisons among algorithms were done at five representative sizes N = 20, 30, 40, 50, and 60. For the learning phase, we conducted approximately 3,000 learning episodes, each one using a randomly generated instance of size selected randomly between 20 and 60 inclusive. The result of the learning phase was a value function V . To assess the influence of this multi-instance learning, we also repeated the above learning phase 5 times, except that in each we held the instance size fixed to a different one of the 5 representative sizes, yielding in each case a distinct value function VN , where N is the training instance size. Table 1 shows the resulting weight vector < bias weight, costhoodN weight, proximitYN weight >. With the exception of the proximity,v weight, these are quite consistent across training instance size. We do not yet understand why training on multiple-sized instances led to this pattern of variation. Table 2 compares the tour quality found by six different local search algorithms. For the algorithms using learned value functions, the results are for the performance phase after learning using the algorithm listed. Table entries are the percent by which tour length exceeded SteinN for instance size N averaged over 100 instances of each representative size. Thus, 2-opt exceeded Stein20 = 645 on the 100 instance sample set by an average of 42%. The last row in the table gives the results of using the five different value functions VN , for the corresponding N . Results for TDC.8) are shown because they were better than 1022 R. Moll, A. G. Barto, T J. Perkins and R. S. Sutton Table 2: Comparison of Six Algorithms at Sizes N = 20, 30, 40, 50, 60. Entries are percentage above SteinN averaged over 100 random instances of size N. Algorithm N=20 N=30 N=40 N=50 N=60 2-opt 42 47 53 56 60 3-opt 8 8 11 10 10 TD(I) 28 31 34 39 40 TD(.8) E = 0 27 30 35 37 39 TD(.8) E = .Ol/N 29 35 37 41 44 TD(.8) E = 0, VN 29 30 32 36 40 Table 3: Average Relative Running Times. Times for 2-opt are in seconds; other entries give time divided by 2-opt time. Algorithm N=20 N=30 N=40 N=50 N=60 2-opt .237 .770 1.09 1.95 3.55 3-opt 32 45 100 162 238 TD(.8) E = 0 3.2 3.4 6.3 6.9 7.1 TD(.8) E = .01/ N 2.2 1.8 2.6 2.9 3.0 those for other values of .A. The learning-enhanced algorithms do well against 2-opt when running time is ignored, and indeed TD(.8), E = 0, is about 35% percent better (according to this measure) by size 60. Note that 3-opt clearly produces the best tours, and a non-zero E for TD(.8) decreases tour quality, as expected since it causes shorter search trajectories. Table 3 gives the relative running times of the various algorithms. The raw running times for 2-opt are given in seconds (Common Lisp on 266 Mhz Mac G-3) at each of five sizes in the first row. Subsequent rows give approximate running times divided by the corresponding 2-opt running time. Times are averages over 30 instances. The algorithms using learned value functions are slower mainly due to the necessity to evaluate the features. Note that TD(.8) becomes significantly faster with E non-zero. Finally. Table 4 gives the relative performance of seven algorithms. normalized for time, including the STAGE algorithm using linear regression with our features. We generated 20 random instances at each of the representative sizes, and we allowed each algorithm to run for the indicated amount of time on each instance. If time remained when a local optimum was reached, we restarted the algorithm at that point, except in the case of 2-opt, where we selected a new random starting tour. The restarting regime for the learningenhanced algorithms is the regime employed by STAGE. Each algorithm reports the best result found in the allotted time, and the chart reports the averages of these values across the 20 instances. Notice that the algorithms that take advantage of extensive off-line learning significantly outperform the other algorithms, including STAGE, which relies on singleinstance learning. 6 DISCUSSION We have presented an extension to local search that uses RL to enhance the local search cost function for a particular optimization problem. Our method combines aspects of work Learning Instance-Independent Value Functions to Enhance Local Search 1023 Table 4: Performance Comparisons, Equalized for Running Time. Size and Running Time N=20 N=30 N=40 N=50 N=60 Algorithm 10 sec 20 sec 40 sec 100 sec 150 sec 2-opt 16 29 28 30 38 STAGE 18 20 32 24 27 TD(.8) E = 0 12 13 16 22 20 TD(.8) E = .011N 13 11 14 24 28 by Zhang and Dietterich (1995) and Boyan and Moore (1997; Boyan 1998). We have applied our method to a relatively pure optimization problem-DARP-which possesses a relatively consistent structure across problem instances. This has allowed the method to learn a value function that can be applied across all problem instances at all sizes. Our method yields significant improvement over a traditional local search approach to DARP on the basis of a very simple linear approximator, built using a relatively impoverished set of features. It also improves upon Boyan and Moore's (1997) STAGE algorithm in our example problem, benefiting from extensive off-line learning whose cost was not included in our assessment. We think this is appropriate for some types of problems; since it is a one-time learning cost, it can be amortized over many future problem instances of practical importance. Acknowledgement We thank Justin Boyan for very helpful discussions of this subject. This research was supported by a grant from the Air Force Office of Scientific Research, Bolling AFB (AFOSR F49620-96-1-0254). References Boyan, J. A. (1998). Learning Evaluation Functions for Global Optimization. Ph.D. Thesis, Carnegie-Mellon University. Boyan, J. A., and Moore, A. W. (1997). Using Prediction to Improve Combinatorial Optimization Search. Proceedings of AI-STATS-97. D. P. Bertsekas, D. P., and Tsitsiklis, 1. N. (1996). Neuro-Dynamic Programming. Athena Scientific, Belmont, MA. Healy, P., and Moll, R. (1995). A New Extension to Local Search Applied to the Dial-ARide Problem. European Journal of Operations Research, 8: 83-104. Psaraftis, H. N. (1983). ~-interchange Procedures for Local Search in a PrecedenceConstrained Routing Problem. European Journal of Operations Research, 13:391-402. Zhang, W. and Dietterich, T. G. (1995). A Reinforcement Learning Approach to Job-Shop Scheduling. In Proceedings of the Fourteenth International Joint Conference on ArtifiCial Intelligence, pp. 1114-1120. Morgan Kaufmann, San Francisco. Stein, D. M. (1978). An Asymptotic Probabilistic Analysis of a Routing Problem. Math. Operations Res. J., 3: 89-101.
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Regular and Irregular Gallager-type Error-Correcting Codes Y. Kabashirna and T. Murayarna Dept. of Compt. IntI. & Syst. Sci. Tokyo Institute of Technology Yokohama 2268502, Japan D. Saad and R. Vicente Neural Computing Research Group Aston University Birmingham B4 7ET, UK Abstract The performance of regular and irregular Gallager-type errorcorrecting code is investigated via methods of statistical physics. The transmitted codeword comprises products of the original message bits selected by two randomly-constructed sparse matrices; the number of non-zero row/column elements in these matrices constitutes a family of codes. We show that Shannon's channel capacity may be saturated in equilibrium for many of the regular codes while slightly lower performance is obtained for others which may be of higher practical relevance. Decoding aspects are considered by employing the TAP approach which is identical to the commonly used belief-propagation-based decoding. We show that irregular codes may saturate Shannon's capacity but with improved dynamical properties. 1 Introduction The ever increasing information transmission in the modern world is based on reliably communicating messages through noisy transmission channels; these can be telephone lines, deep space, magnetic storing media etc. Error-correcting codes play a significant role in correcting errors incurred during transmission; this is carried out by encoding the message prior to transmission and decoding the corrupted received code-word for retrieving the original message. In his ground breaking papers, Shannon[l] analyzed the capacity of communication channels, setting an upper bound to the achievable noise-correction capability of codes, given their code (or symbol) rate, constituted by the ratio between the number of bits in the original message and the transmitted code-word. Shannon's bound is non-constructive and does not provide a recipe for devising optimal codes. The quest for more efficient codes, in the hope of saturating the bound set by Shannon, has been going on ever since, providing many useful but sub-optimal codes. One family of codes, presented originally by Gallager[2]' attracted significant interest recently as it has been shown to outperform most currently used techniques[3]. Gallager-type codes are characterized by several parameters, the choice of which defines a particular member of this family of codes. Current theoretical results[3] Regular and Irregular Gallager-type Error-Correcting Codes 273 offer only bounds on the error probability of various architectures, proving the existence of very good codes under some restrictions; decoding issues are examined via numerical simulations. In this paper we analyze the typical performance of Gallager-type codes for several parameter choices via methods of statistical mechanics. We then validate the analytical solution by comparing the results to those obtained by the TAP approach and via numerical methods. 2 The general framework In a general scenario, a message represented by an N dimensional Boolean vector e is encoded to the M dimensional vector JO which is transmitted through a noisy channel with some flipping probability p per bit (other noise types may also be studied). The received message J is then decoded to retrieve the original message. In this paper we analyze a slightly different version of Gallager-type codes termed the MN code[3] that is based on choosing two randomly-selected sparse matrices A and B of dimensionality M x N and M x M respectively; these are characterized by K and L non-zero unit elements per row and C and L per column respectively. The finite numbers K, C and L define a particular code; both matrices are known to both sender and receiver. Encoding is carried out by constructing the modulo 2 inverse of B and the matrix B-1 A (mod 2); the vector JO = B-1 A e (mod 2, e Boolean vector) constitutes the codeword. Decoding is carried out by taking the product of the matrix B and the received message J = JO +( (mod 2), corrupted by the Boolean noise vector (, resulting in Ae + B (. The equation Ae + B( = AS + B'T (mod 2) (1) is solved via the iterative methods of Belief Propagation (BP)[3] to obtain the most probable Boolean vectors Sand 'T; BP methods in the context of error-correcting codes have recently been shown to be identical to a TAP[4] based solution of a similar physical system[5]. The similarity between error-correcting codes of this type and Ising spin systems was first pointed out by Sourlas[6], who formulated the mapping of a simpler code, somewhat similar to the one presented here, onto an Ising spin system Hamiltonian. We recently extended the work of Sourlas, that focused on extensively connected systems, to the finite connectivity case[5] as well as to the case of MN codes [7]. To facilitate the current investigation we first map the problem to that of an Ising model with finite connectivity. We employ the binary representation (±1) of the dynamical variables Sand 'T and of the vectors J and JO rather than the Boolean (0,1) one; the vector JO is generated by taking products of the relevant binary message bits J2 = TIiE/.' ~i' where the indices J.L = (h, ... iK) correspond to the non-zero elements of B-1 A, producing a binary version of JO. As we use statistical mechanics techniques, we consider the message and codeword dimensionality (N and M respectively) to be infinite, keeping the ratio between them R = N 1M, which constitutes the code rate, finite. Using the thermodynamic limit is quite natural as Gallager-type codes are usually used for transmitting long (104 - 105) messages, where finite size corrections are likely to be negligible. To explore the system's capabilities we examine the Hamiltonian 274 Y. Kabashima, T. Murayama, D. Saad and R. Vicente The tensor product DlJ.uJ,.J.{Tl where JlJ.u = TIiEIJ. ~i TIjEu (j and u = (jl,'" iL), is the binary equivalent of Ae + B(, treating both signal (8 and index i) and noise (7" and index j) simultaneously. Elements of the sparse connectivity tensor D IJ.U take the value 1 if the corresponding indices of both signal and noise are chosen (Le., if all corresponding indices of the matrices A and Bare 1) and 0 otherwise; it has C unit elements per i-index and L per j-index representing the system's degree of connectivity. The f> function provides 1 if the selected sites' product TIiEIJ. Si TIjEu Tj is in disagreement with the corresponding element JIJ.U, recording an error, and 0 otherwise. Notice that this term is not frustrated, as there are M +N degrees of freedom and only M constraints from Eq.(l), and can therefore vanish at sufficiently low temperatures. The last two terms on the right represent our prior knowledge in the case of sparse or biased messages Fs and of the noise level Fr and require assigning certain values to these additive fields. The choice of f3 -+ 00 imposes the restriction of Eq.(l), limiting the solutions to those for which the first term of Eq.(2) vanishes, while the last two terms, scaled with f3, survive. Note that the noise dynamical variables 7" are irrelevant to measuring the retrieval success m = Jr (~~1 ~i sign (Si)!3 ) ~ . The latter monitors the normalized mean overlap between the Bayes-optimal retrieved message, shown to correspond to the alignment of (Si)!3 to the nearest binary value[6], and the original message; the subscript f3 denotes thermal averaging. Since the first part of Eq.(2) is invariant under the map Si -+ Si~i, Tj -+ Tj(j and JIJ.U -+ JIJ.U TIiEIJ. ~i TIjEu (j = 1, it is useful to decouple the correlation between the vectors 8, 7" and e, (. Rewriting Eq.(2) one obtains a similar expression apart from the last terms on the right which become Fs / f3 L:k Sk ~k and Fr / f3 ~k Tk (k. The random selection of elements in D introduces disorder to the system which is treated via methods of statistical physics. More specifically, we calculate the partition function Z(D, J) = Tr{8,7"} exp[-f31i] averaged over the disorder and the statistical properties of the message and noise, using the replica method[5, 8, 9]. Taking f3 -+ 00 gives rise to a set of order parameters q"",(3 •..• "Y = (~ tZi Sf Sf, .. ,S7) T"".(3, .. ,"Y = (~ ty; rj rf, .. ,r?) .=1 (3400 .=1 (3400 (2) where a, f3, .. represent replica indices, and the variables Zi and 1j come from enforcing the restriction of C and L connections per index respectively[5]: f> ( "D . . . - c) = i 21T dZ ZL:h .... i K f<i.i 2 •.. • h >-(C+l) L <t,t2 ,·· ,JL> 2 ' ( . .) 0 7r '2 ,·· ,'tK (3) and similarly for the restriction on the j indices. To proceed with the calculation one has to make an assumption about the order parameters symmetry. The assumption made here, and validated later on, is that of replica symmetry in the following representation of the order parameters and the related conjugate variables Qa,!3 .. -y aq / dx 7r(X) xl , Qa,!3 .. -y = aq-/ dx 1?(x) Xl (4) ra,!3 .. -y ar / dy p(y) yl , ra,!3 .. -y = a; / dy p(Y) yl , where l is the number of replica indices, a. are normalization coefficients, and 7r(x) , 1?(x), p(y) and p(Y) represent probability distributions. Unspecified integrals Regular and Irregular Gallager-type Error-Correcting Codes 275 are over the range [-1, + 1]. One then obtains an expression for the free energy per spin expressed in terms of these probability distributions liN (In Z)~,(,'D The free energy can then be calculated via the saddle point method. Solving the equations obtained by varying the free energy w.r.t the probability distributions 1T(X), 1?(x), p(y) and p(y), is difficult as they generally comprise both delta peaks and regular[9] solutions for the ferromagnetic and paramagnetic phases (there is no spin-glass solution here as the system is not frustrated). The solutions obtained in the case of unbiased messages (the most interesting case as most messages are compressed prior to transmission) are for the ferromagnetic phase: 1T(X) = 8(x - 1) , 1?(x) = 8(x - 1) , p(y) = 8(y - 1) , p(Y) = 8(Y - 1), (5) and for the paramagnetic phase: 1T(X) p(y) 8(x) , 1?(x) = 8(x) , p(Y) = 8(Y) 1 + tanh Fr r( _ h F ) 1 - tanh Fr r( h F ) 2 u y tan r + 2 u Y + tan r· (6) These solutions obey the saddle point equations. However, it is unclear whether the contribution of other delta peaks or of an additional continuous solution will be significant and whether the solutions (5) and (6) are stable or not. In addition, it is also necessary to validate the replica symmetric ansatz itself. To address these questions we obtained solutions to the system described by the Hamiltonian (2) via TAP methods of finitely connected systems[5]; we solved the saddle point equations derived from the free energy numerically, representing all probability distributions by up to 104 bin models and by carrying out the integrations via Monte-Carlo methods; finally, to show the consistency between theory and practice we carried out large scale simulations for several cases, which will be presented elsewhere. 3 Structure of the solutions The various methods indicate that the solutions may be divided to two different categories: K = L = 2 and either K ~ 3 or L ~ 3. We therefore treat them separately. For unbiased messages and either K ~ 3 or L ~ 3 we obtain the solutions (5) and (6) both by applying the TAP approach and by solving the saddle point equations numerically. The former was carried out at the value of Fr which corresponds to the true noise and input bias levels (for unbiased messages Fa = 0) and thus to Nishimori's condition[lO], where no replica symmetry breaking effects are expected. This is equivalent to having the correct prior within the Bayesian framework[6] and enables one to obtain analytic expressions for some observables as long as some gauge requirements are obeyed [10] . Numerical solutions show the emergence of stable dominant delta peaks, consistent with those of (5) and (6). The question of longitudinal mode stability (corresponding to the replica symmetric solution) was addressed by setting initial conditions for the numerical solutions close to the solutions (5) and (6), showing that they converge back to these solutions which are therefore stable. The most interesting quantity to examine is the maximal code rate, for a given corruption process, for which messages can be perfectly retrieved. This is defined in the case of K,L~3 by the value of R=KIC=NjM for which the free energy of the ferromagnetic solution becomes smaller than that of the paramagnetic solution, constituting a first order phase transition. A schematic description of the solutions obtained is shown in the inset of Fig.1a. The paramagnetic solution (m = 0) has a lower free energy than the ferromagnetic one (low Ihigh free energies are denoted 276 Y. Kabashima, T. Murayama, D. Saad and R. Vicente by the thick and thin lines respectively, there are no axis lines at m = 0,1) for noise levels P > Pc and vice versa for P ~ Pc; both solutions are stable. The critical code rate is derived by equating the ferromagnetic and paramagnetic free energies to obtain Rc = 1-H2(p) = 1+(plog2P+(1- p)log2(1- p)) . This coincides with Shannon's capacity. To validate these results we obtained TAP solutions for the unbiased message case (K = L = 3, C = 6) as shown in Fig.1a (as +) in comparison to Shannon's capacity (solid line). Analytical solutions for the saddle point equations cannot be obtained for biased patterns and we therefore resort to numerical methods ana the TAP approach. The maximal information rate (Le., code-rate xH2(fs = (1 + tanh Fs)/2) - the source redundancy) obtained by the TAP method (0) and numerical solutions of the saddle point equations (0), for each noise level, are shown in Fig.1a. Numerical results have been obtained using 103_104 bin models for each probability distribution and had been run for 105 steps per noise level point. The various results are highly consistent and practically saturate Shannon's bound for the same noise level. The MN code for K , L ~ 3 seems to offer optimal performance. However, the main drawback is rooted in the co-existence of the stable m = 1 and m = 0 solutions, shown in Fig.1a (inset), which implies that from some initial conditions the system will converge to the undesired paramagnetic solution. Moreover, studying the ferromagnetic solution numerically shows a highly limited basin of attraction, which becomes smaller as K and L increase, while the paramagnetic solution at m = 0 always enjoys a wide basin of attraction. Computer simulations (see also [3]) show that as initial conditions for the decoding process are typically of close-to-zero magnetization (almost no prior information about the original message is assumed) it is likely that the decoding process will converge to the paramagnetic solution. While all codes with K, L ~ 3 saturate Shannon's bound in their equilibrium properties and are characterized by a first order, paramagnetic to ferromagnetic, phase transition, codes with K = L = 2 show lower performance and different physical characteristics. The analytical solutions (5) and (6) are unstable at some flip rate levels and one resorts to solving the saddle point equations numerically and to TAP based solutions. The picture that emerges is sketched in the inset of Fig.1b: The paramagnetic solution dominates the high flip rate regime up to the point PI (denoted as 1 in the inset) in which a stable, ferromagnetic solution, of higher free energy, appears (thin lines at m = ±1). At a lower flip rate value P2 the paramagnetic solution becomes unstable (dashed line) and is replaced by two stable sub-optimal ferromagnetic (broken symmetry) solutions which appear as a couple of peaks in the various probability distributions; typically, these have a lower free energy than the ferromagnetic solution until P3, after which the ferromagnetic solution becomes dominant. Still, only once the sub-optimal ferromagnetic solutions disappear, at the spinodal point Ps, a unique ferromagnetic solution emerges as a single delta peak in the numerical results (plus a mirror solution). The point in which the sub-optimal ferromagnetic solutions disappear constitutes the maximal practical flip rate for the current code-rate and was defined numerically (0) and via TAP solutions (+) as shown in Fig.1b. Notice that initial conditions for TAP and the numerical solutions were chosen almost randomly, with a slight bias of 0(10-12), in the initial magnetization. The TAP dynamical equations are identical to those used for practical BP decoding[5], and therefore provide equivalent results to computer simulations with the same parameterization, supporting the analytical results. The excellent convergence results obtained point out the existence of a unique pair of global solutions to which the system converges (below Ps) from practically all initial conditions. This observation and the practical implications of using K = L = 2 code have not been obtained by Regular and Irregular Gallager-type Error-Correcting Codes 277 information theory methods (e.g.[3]}j these prove the existence of very good codes for C = L ~ 3, and examine decoding properties only via numerical simulations. 4 Irregular Constructions Irregular codes with non-uniform number of non-zero elements per column and uniform number of elements per row were recently introduced [11, 12] and were found to outperform regular codes. It is relatively straightforward to adapt our methods to study these particular constructions. The restriction of the number of connections per index can be replaced by a set of N restrictions of the form (1), enforcing Cj non-zero elements in the j-th column of the matrix A, and other M restrictions enforcing Ll non-zero elements in the l-th column of the matrix B. By construction these restrictions must obey the relations E.7=l Cj = M K and E~l Ll = M L. One can assume that a particular set of restrictions is generated independently by the probability distributions P(C) and P(L). With that we can compute average properties of irregularly constructed codes generated by arbitrary distributions. Proceeding along the same lines to those of the regular case one can find a very similar expression for the free energy which can be interpreted as a mixture of regular codes with column weights sampled with probabilities P(C) and P(L). As long as we choose probability distributions which vanish for C, L = 0 (avoiding trivial non-invertible matrices) and C, L = 1 (avoiding single checked bits), the solutions to the saddle point equations are the same as those obtained in the regular case (Eqs.5, 6) leading to exactly the same predictions for the maximum performance. The differences between regular and irregular codes show up in their dynamical behavior. In the irregular case with K > 2 and for biased messages the basin of attraction is larger for higher noise levels [13]. 5 Conclusion In this paper we examined the typical performance of Gallager-type codes. We discovered that for a certain choice of parameters, either K ~ 3 or L ~ 3, one potentially obtains optimal performance, saturating Shannon's bound. This comes at the expense of a decreasing basin of attraction making the decoding process increasingly impractical. Another code, K = L = 2, shows "close to optimal performance with a very large basin of attraction, making it highly attractive for practical purposes. The decoding performance of both code types was examined by employing the TAP approach, an iterative method identical to the commonly used BP. Both numerical and TAP solutions agree with the theoretical results. The equilibrium properties of regular and irregular constructions is shown to be the same. The improved performance of irregular codes reported in the literature can be explained as consequence of dynamical properties. This study examines the typical performance of these increasingly important error-correcting codes, from which optimal parameter choices can be derived, complementing the bounds and empirical results provided in the information theory literature. Important aspects that are yet to be investigated include other noise types, finite size effects and the decoding dynamics itself. Acknowledgement Support by the JSPS RFTF program (YK), The Royal Society and EPSRC grant GR/N00562 (DS) is acknowledged. 278 Y. Kabashima. T. Murayama. D. Saad and R. Vicente 1 1 Ferro 0.8 0.8 ~ 0.6 0.6 a: I a: 0.4 0.4 0.2 0.1 p 0.2 0.3 0.4 0.5 P 0.2 0 0 0.1 0.2 0.3 0.4 0.5 P Figure 1: Critical code rate as a function of the flip rate p, obtained from numerical solutions and the TAP approach (N = 104), and averaged over 10 different initial conditions with error bars much smaller than the symbols size. (a) Numerical solutions for K = L = 3, C = 6 and varying input bias fs (0) and TAP solutions for both unbiased (+) and biased (0) messages; initial conditions were chosen close to the analytical ones. The critical rate is multiplied by the source information content to obtain the maximal information transmission rate, which clearly does not go beyond R = 3/6 in the case of biased messages; for unbiased patterns H2(fs) = 1. Inset: The ferromagnetic and paramagnetic solutions as functions of p; thick and thin lines denote stable solutions of lower and higher free energies respectively. (b) For the unbiased case of K = L = 2; initial conditions for the TAP (+) and the numerical solutions (0) are of almost zero magnetization. Inset: The ferromagnetic (optimal/sub-optimal) and paramagnetic solutions as functions of p; thick and thin lines are as in (a), dashed lines correspond to unstable solutions. References [1] C.E. Shannon, Bell Sys. Tech.J., 27, 379 (1948); 27, 623 (1948). [2] R.G. Gallager, IRE Trans.Info. Theory, IT-8, 21 (1962). [3] D.J.C. MacKay, IEEE Trans.IT, 45, 399 (1999). [4] D. Thouless, P.W. Anderson and R.G. Palmer, Phil. Mag., 35, 593 (1977). [5] Y. Kabashima and D. Saad, Europhys.Lett., 44 668 (1998) and 45 97 (1999). [6] N. Sourlas, Nature, 339, 693 (1989) and Euro.Phys.Lett., 25, 159 (1994). [7] Y. Kabashima, T. Murayama and D. Saad, Phys.Rev.Lett., (1999) in press. [8] K.Y.M. Wong and D. Sherrington, J.Phys.A, 20, L793 (1987). [9] C. De Dominicis and P.Mottishaw, J.Phys.A, 20, L1267 (1987). [10] H. Nishimori, Prog. Theo.Phys., 66, 1169 (1981). [11] M. Luby et. ai, IEEE proceedings of ISIT98 (1998) and Analysis of Low Density Codes and Improved Designs Using Irregular Graphs, preprint. [12] D.J.C. MacKay et. al, IEEE Trans.Comm., 47, 1449 (1999). [13] R. Vicente et. ai, http://xxx.lanl.gov/abs/cond-mat/9908358 (1999).
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Building Predictive Models from Fractal Representations of Symbolic Sequences Peter Tioo Georg Dorffner Austrian Research Institute for Artificial Intelligence Schottengasse 3, A-101O Vienna, Austria {petert,georg}@ai.univie.ac.at Abstract We propose a novel approach for building finite memory predictive models similar in spirit to variable memory length Markov models (VLMMs). The models are constructed by first transforming the n-block structure of the training sequence into a spatial structure of points in a unit hypercube, such that the longer is the common suffix shared by any two n-blocks, the closer lie their point representations. Such a transformation embodies a Markov assumption - n-blocks with long common suffixes are likely to produce similar continuations. Finding a set of prediction contexts is formulated as a resource allocation problem solved by vector quantizing the spatial n-block representation. We compare our model with both the classical and variable memory length Markov models on three data sets with different memory and stochastic components. Our models have a superior performance, yet, their construction is fully automatic, which is shown to be problematic in the case of VLMMs. 1 Introduction Statistical modeling of complex sequences is a prominent theme in machine learning due to its wide variety of applications (see e.g. [5)). Classical Markov models (MMs) of finite order are simple, yet widely used models for sequences generated by stationary sources. However, MMs can become hard to estimate due to the familiar explosive increase in the number of free parameters when increasing the model order. Consequently, only low order MMs can be considered in practical applications. Some time ago, Ron, Singer and Tishby [4] introduced at this conference a Markovian model that could (at least partially) overcome the curse of dimensionality in classical MMs. The basic idea behind their model was simple: instead of fixed-order MMs consider variable memory length Markov models (VLMMs) with a "deep" memory just where it is really needed (see also e.g. [5][7]). The size of VLMMs is usually controlled by one or two construction parameters. Unfortunately, constructing a series of increasingly complex VLMMs (for example to enter a model selection phase on a validation set) by varying the construction parameters can be 646 P Tino and G. DorjJner a troublesome task [1). Construction often does not work "smoothly" with varying the parameters. There are large intervals of parameter values yielding unchanged VLMMs interleaved with tiny parameter regions corresponding to a large spectrum of VLMM sizes. In such cases it is difficult to fully automize the VLMM construction. To overcome this drawback, we suggest an alternative predictive model similar in spirit to VLMMs. Searching for the relevant prediction contexts is reformulated as a resource allocation problem in Euclidean space solved by vector quantization. A potentially prohibitively large set of alilength-L blocks is assigned to a much smaller set of prediction contexts on a suffix basis. To that end, we first transform the set of L-blocks appearing in the training sequence into a set of points in Euclidean space, such that points corresponding to blocks sharing a long common suffix are mapped close to each other. Vector quantization on such a set partitions the set of L-blocks into several classes dominated by common suffixes. Quantization centers play the role of predictive contexts. A great advantage of our model is that vector quantization can be performed on a completely self-organized basis. We compare our model with both classical MMs and VLMMs on three data sets representing a wide range of grammatical and statistical structure. First, we train the models on the Feigenbaum binary sequence with a very strict topological and metric organization of allowed subsequences. Highly specialized, deep prediction contexts are needed to model this sequence. Classical Markov models cannot succeed and the full power of admitting a limited number of variable length contexts can be exploited. The second data set consists of quantized daily volatility changes of the Dow Jones Industrial Average (DnA). Predictive models are used to predict the direction of volatility move for the next day. Financial time series are known to be highly stochastic with a relatively shallow memory structure. In this case, it is difficult to beat low-order classical MMs. One can perform better than MMs only by developing a few deeper specialized contexts, but that, on the other hand, can lead to overfitting. Finally, we test our model on the experiments of Ron, Singer and Tishby with language data from the Bible [5]. They trained classical MMs and a VLMM on the books of the Bible except for the book of Genesis. Then the models were evaluated on the bases of negative log-likelihood on an unseen text from Genesis. We compare likelihood results of our model with those of MMs and VLMMs. 2 Predictive models We consider sequences S = 8182 .. . over a finite alphabet A = {I, 2, ... , A} generated by stationary sources. The set of all sequences over A with exactly n symbols is denoted by An. An information source over A = {I, 2, ... , A} is defined by a family of consistent probability measures Pn on An, n = 0,1,2, ... , :LIEA Pn+1 (ws) = Pn(w), for all wEAn (AO = {A} and Po(A) = 1, A denotes the empty string). In applications it is useful to consider probability functions Pn that are easy to handle. This can be achieved, for example, by assuming a finite source memory of length at most L, and formulating the conditional measures P(slw) = PL+1(WS)/PL(w), WEAL, using a function c : AL ~ C, from L-blocks over A to a (presumably small) finite set C of prediction contexts: P(slw) = P(sjc(w)). (1) In Markov models (MMs) of order n :s; L, for all L-blocks w E A L, c( w) is the length-n Predictive Models from Fractal Representations of Sequences 647 suffix ofw, i.e. c(uv) = v, v E An, U E A L- n. In variable memory length Markov models (VLMMs), the suffices c( w) of L-blocks w E AL can have different lengths, depending on the particular L-block w. For strategies of selecting and representing the prediction contexts through prediction suffix trees and/or probabilistic suffix automata see, for example, [4](5]. VLMM construction is controlled by one, or several parameters regulating selection of candidate contexts and growing/pruning decisions. Prediction context function c : AL -+ C in Markov models of order n ~ L, can be interpreted as a natural homomorphism c : AL -+ AL 1£ corresponding to the equivalence relation E ~ AL X AL on L-blocks over A: two L-blocks u, v are in the same class, i.e. ( U, v) E E, if they share the same suffix of length n. The factor set ALI £ = C = An consists of all n-blocks over A. Classical MMs define the equivalence E on the suffix bases, but regardless of the suffix structure present in the training data. Our idea is to keep the Markov-motivated suffix strategy for constructing E, but at the same time take into an account the data suffix structure. Vector quantization on a set of B points in a Euclidean space positions N < < B codebook vectors (CV s), each CV representing a subset of points that are closer to it than to any other CV, so that the overall error of substituting CVs for points they represent is minimal. In other words, CVs tend to represent points lying close to each other (in a Euclidean metric). In order to use vector quantization for determining relevant predictive contexts we need to do two things: 1. Define a suitable metric in the sequence space that would correspond to Markov assumptions: (a) two sequences are "close" if they share a common suffix (b) the longer is the common suffix the closer are the sequences 2. Define a uniformly continuous map from the sequence metric space to the Euclidean space, i.e. sequences that are close in the sequence space (i.e. share a long common suffix) are mapped close to each other in the Euclidean space. In [6] we rigorously study a class of such spatial representations of symbolic structures. Specifically, a family of distances between two L-blocks U = UIU2 ... UL-IUL and v = VI V2·· . V L-l V L over A = {I, 2, ... , A}, expressed as L 1 dk(u, v) = L kL - i+1c5 (Ui, Vi), k $ 2' (2) i=l with c5(i,j) = 1 if i = j, and c5(i,j) = ° otherwise, correspond to Markov assumption. The parameter k influences the rate of "forgetting the past". We construct a map from the sequence metric space to the Euclidean space as follows: Associate with each symbol i E A a map (3) operating on a unit D-dimensional hypercube [0, l]D. Dimension of the hypercube should be large enough so that each symbol i is associated with a unique vertex, i.e. D = flog2 A 1 and tj #- tj whenever i #- j. The map u : AL -+ [0, l]D, from L-blocks VIV2 ... VL over A to the unit hypercube, U(VI V2 ... VL) = VdVL-l( .,.(V2(VI(X*))) ... )) = (VL 0 VL-l 0 . . . 0 V2 0 vt}(x*), (4) 648 P. Tina and G. DarjJner where x· {~}D is the center of the hypercube, is "unifonnly continuous". Indeed, whenever two sequences u, v share a common suffix of length Q, the Euclidean distance between their point representations O'(u) and O'(v) is less than V2kQ. Strictly speaking, for a mathematically correct treatment of unifonn continuity, we would need to consider infinite sequences. Finite blocks of symbols would then correspond to cylinder sets (see [6]). For sake of simplicity we only deal with finite sequences. As with classical Markov models, we define the prediction context function c : A L -t C via an equivalence £ on L-blocks over A: two L-blocks u, v are in the same class if their images under the map 0' are represented by the same codebook vector. In this case, the set of prediction contexts C can be identified with the set of codebook vectors {bI , b2 , ... , b N }, hi E ~D, i = 1,2, ... , N. We refer to predictive models with such a context function as prediction/ractal machines (PFMs). The prediction probabilities (1) are determined by N(i, s) P(slbd = L: N(' )' sEA, aEA Z, a (5) where N(i, a) is the number of (L+l)-blocks ua, a E AL, a E A, in the training sequence, such that the point 0'( u) is allocated to the codebook vector bi . 3 Experiments In all experiments we constructed PFMs using a contraction coefficient k = ~ (see eq. (3» and K-means as a vector quantization tool. The first data set is the Feigenbaum sequence over the binary alphabet A = {1,2}. This sequence is well-studied in symbolic dynamics and has a number of interesting properties. First, the topological structure of the sequence can only be described using a context sensitive tool - a restricted indexed context-free grammar. Second, for each block length n = 1, 2, .. . , the distribution of n-blocks is either unifonn, or has just two probability levels. Third, the n-block distributions are organized in a self-similar fashion (see [2]). The sequence can be specified by the subsequence composition rule ' (6) We chose to work with the Feigenbaum sequence, because increasingly accurate modeling of the sequence with finite memory models requires a selective mechanism for deep prediction contexts. We created a large portion of the Feigenbaum sequence and trained a series of classical MMs, variable memory length MMs (VLMMs), and prediction fractal machines (PFMs) on the first 260,000 symbols. The following 200,000 symbols fonned a test set. Maximum memory length L for VLMMs and PFMs was set to 30. As mentioned in the introduction, constructing a series of increasingly complex VLMMs . by varying the construction parameters appeared to be a troublesome task. We spent a fair amount of time finding "critical" parameter values at which the model size changed. In contrast, a fully automatic construction of PFMs involved sliding a window of length L = 30 through the training set; for each window position, mapping the L-block u appearing in the window to the point 0'( u) (eq. (4», vector-quantizing the resulting set of points (up to 30 codebook vectors). After the quantization step we computed predictive probabilities according to eq. (5). Predictive Models from Fractal Representations of Sequences 649 Table I: Normalized negative log-likelihoods (NNL) on the Feigenbaum test set. model # contexts NNL captured block distribution PFM 2-4 0.6666 1-3 5-7 0.3333 1-6 8-22 0.1666 1-12 230.0833 1-24 VLMM 2-4 0.6666 1-3 5 0.3333 1-6 11 0.1666 1-12 23 0.0833 1-24 MM 2,4,8,16,32 0.6666 1-3 Negative log-likelihoods per symbol (the base oflogarithm is always taken to be the number of symbols in the alphabet) of the test set computed using the fitted models exhibited a steplike increasing tendency shown in Table 1. We also investigated the ability of the models to reproduce the n-block distribution found in the training and test sets. This was done by letting the models generate sequences of length equal to the length of the training sequence and for each block length n = 1,2, ... , 30, computing the L1 distance between the n-block distribution of the training and model-generated sequences. The n-block distributions on the test and training sets were virtually the same for n = 1,2, ... 30. In Table I we show block lengths for which the L1 distance does not exceed a small threshold~ . We set ~ = 0.005, since in this experiment, either the L1 distance was less 0.005, or exceeded 0.005 by a large amount. An explanation of the step-like behavior in the log-likelihood and n-block modeling behavior of VLMMs and PFMs is out of the scope of this paper. We briefly mention, however, that by combining the knowledge about the topological and metric structur~s of the Feigenbaum sequence (e.g. [2]) with a careful analysis of the models, one can show why and when an inclusion of a prediction context leads to an abrupt improvement in the modeling performance. In fact, we can show that VLMMs and PFMs constitute increasingly better approximations to the infinite self-similar Feigenbaum machine known in symbolic dynamics [2]. The classical MM totally fails in this experiment, since the context length 5 is far too small to enable the MM to mimic the complicated subsequence structure in the Feigenbaum sequence. PFMs and VLMMs quickly learn to explore a limited number of deep prediction contexts and perform comparatively well. In the second experiment, a time series {xtJ of the daily values ofthe Dow Jones Industrial Average (DJIA) from Feb. 1 1918 until April 1 1997 was transformed into a time series of returns rt = log Xt+1 - log Xt, and divided into 12 partially overlapping epochs, each containing about 2300 values (spanning approximately 9 years). We consider the squared return r; a volatility estimate for day t. Volatility change forecasts (volatility is going to increase or decrease) based on historical returns can be interpreted as a buying or selling signal for a straddle (see e.g. [3]). If the volatility decreases we go short (straddle is sold), if it increases we take a long position (straddle is bought). In this respect, the quality of a volatility model can be measured by the percentage of correctly predicted directions of daily volatility differences. 650 P Tino and G. DorfJner Table 2: Prediction perfonnance on the DJIA volatility series. Percent correct on test set model 1 2 3 4 5 6 PPM 71.08 70.39 69.70 70.05 72.12 72.46 VLMM 68.67 68.18 68.79 69.25 69.41 68.29 MM 68.56 69.11 69.78 68.28 69.50 73.13 7 8 9 10 11 12 PPM 74.01 71.77 73.84 73.84 71,77 74.19 VLMM 69.83 67.00 67.96 70.76 69.80 70.25 MM 74.16 71.96 69.95 69.16 71.74 71.07 The series {r~+1 r~} of differences between the successive squared returns is transfonned into a sequence {Dt} over 4 symbols by quantizing the series {r~+1 - rn as follows: { I (extreme down), if rr+1 - rr < 01 < 0 D 2 (nonnal down), if 01 ~ r;+1 r~ < a t 2 2 3 (nonnal up), if a ~ rt~1 r t < O2 4 (extreme up), if 02 ~ rt+1 - r;, (7) where the parameters 01 and ()2 correspond to Q percent and (100 - Q) percent sample quantiles, respectively. So, the upper (lower) Q% of all daily volatility increases (decreases) in the sample are considered extremal, and the lower (upper) (50 - Q)% of daily volatility increases (decreases) are viewed as nonnal. Each epoch is partitioned into training, validation and test parts containing 110, 600 and 600 symbols, respectively. Maximum memory length L for VLMMs and PFMs was set to 10 (two weeks). We trained classical MMs, VLMMs and PFMs with various numbers of prediction contexts (up to 256) and extremal event quantiles Q E {5, 10, 15, ... , 45}. For each model class, the model size and the quantile Q to be used on the test set were'selected according to the validation set perfonnance. Perfonnance of the models was quantified as the percentage of correct guesses of the volatility change direction for the next day. If the next symbol is 1 or 2 (3 or 4) and the sum of conditional next symbol probabilities for 1 and 2 (3 and 4) given by a model is greater than 0.5, the model guess is considered correct. Results are shown in Table 2. Paired t-test reveals that PFMs significantly (p < 0.005) outperfonn both VLMMs and classical MMs. Of course, fixed-order MMs are just special cases of VLMMs, so theoretically, VLMMs cannot perfonn worse than MMs. We present separate results for MMs and VLMMs to illustrate practical problems in fitting VLMMs. Besides familiar problems with setting the construction parameter values, one-parameter-schemes (like that presented in [4] and used here) operate only on small subsets of potential VLMMs. On data sets with a rather shallow memory structure, this can have a negative effect. The third experiment extends the work of Ron, Singer and Tishby [5]. They tested classical MMs and VLMMs on the Bible. The alphabet is English letters and the blank character (27 symbols). The training set consisted of the Bible except for the book of Genesis. The test set was a portion of 236 characters from the book of Genesis. They set the maximal memory depth to L = 30 and constructed a VLMM with about 3000 contexts. Summarizing the results in [5], classical MMs of order 0, 1, 2 and 3 achieved negative log-likelihoods per Predictive Models from Fractal Representations of Sequences 651 character (NNL) of 0.853, 0.681, 0.560 and 0.555, respectively. The authors point out a huge difference between the number of states in MMs of order 2 and 3: 273 - 272 = 18954. VLMM performed much better and achieved an NNL of 0.456. In our experiments, we set the maximal memory length to L = 30 (the same maximal memory length was used for VLMM construction in [5]). PFMs were constructed by vector quantizing a 5-dimensional (alphabet has 27 symbols) spatial representation of 3D-blocks appearing in the training set. On the test set, PFMs with 100, 500, 1O(}(} and 3000 predictive contexts achieved an NNL of 0.622, 0.518, 0.510 and 0.435. 4 Conclusion We presented a novel approach for building finite memory predictive models similar in spirit to variable memory length Markov models (VLMMs). Constructing a series of VLMMs is often a troublesome and highly time-consuming task requiring a lot of interactive steps. Our predictive models, prediction fractal machines (PFMs), can be constructed in a completely automatic and intuitive way - the number of codebook vectors in the vector quantization PFM construction step corresponds to the number of predictive contexts. We tested our model on three data sets with different memory and stochastic components. VLMMs excel over the classical MMs on the Feigenbaum sequence requiring deep prediction contexts. On this sequence, PFMs achieved the same performance as their rivals - VLMMs. On financial time series, PFMs significantly outperform the purely symbolic Markov models - MMs and VLMMs. On natural language Bible data, our PFM outperforms a VLMM of comparable size. Acknowledgments This work was supported by the Austrian Science Fund (FWF) within the research project "Adaptive Information Systems and Modeling in Economics and Management Science" (SFB 010) and the Slovak Academy of Sciences grant SAY 2/6018/99. The Austrian Research Institute for Artificial Intelligence is supported by the Austrian Federal Ministry of Science and Transport. References [1] P. BOhlmann. Model selection for variable length Markov chains and tuning the context algorithm. Annals of the Institute of Statistical Mathematics, (in press), 1999. [2] 1. Freund, W. Ebeling, and K. Rateitschak. Self-similar sequences and uniVersal scaling of dynamical entropies. PhysicaL Review E, 54(5), pp. 5561-5566, 1996. [3] 1. Noh, R.F. Engle, and A. Kane. Forecasting volatility and option prices of the s&p 500 index. JournaL of Derivatives, pp. 17-30, 1994. [4] D. Ron, Y. Singer, and N. Tishby. The power of amnesia. In Advances in Neural Information Processing Systems 6, pp. 176-183. Morgan Kaufmann, 1994. [5] D. Ron, Y. Singer, and N. Tishby. The power of amnesia. Machine Learning, 25,1996. [6] P. Tino. Spatial representation of symbolic sequences through iterative function system. IEEE Transactions on Systems. Man. and Cybernetics Part A: Systems and Humans, 29(4), pp. 386-392, 1999. [7] M.1. Weinberger, 1.1. Rissanen, and M. Feder. A universal finite memory source. IEEE Transactions on Information Theory, 41 (3), pp. 643-652,1995.
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Constructing Heterogeneous Committees Using Input Feature Grouping: Application to Economic Forecasting Yuansong Liao and John Moody Department of Computer Science, Oregon Graduate Institute, P.O.Box 91000, Portland, OR 97291-1000 Abstract The committee approach has been proposed for reducing model uncertainty and improving generalization performance. The advantage of committees depends on (1) the performance of individual members and (2) the correlational structure of errors between members. This paper presents an input grouping technique for designing a heterogeneous committee. With this technique, all input variables are first grouped based on their mutual information. Statistically similar variables are assigned to the same group. Each member's input set is then formed by input variables extracted from different groups. Our designed committees have less error correlation between its members, since each member observes different input variable combinations. The individual member's feature sets contain less redundant information, because highly correlated variables will not be combined together. The member feature sets contain almost complete information, since each set contains a feature from each information group. An empirical study for a noisy and nonstationary economic forecasting problem shows that committees constructed by our proposed technique outperform committees formed using several existing techniques. 1 Introduction The committee approach has been widely used to reduce model uncertainty and improve generalization performance. Developing methods for generating candidate committee members is a very important direction of committee research. Good candidate members of a committee should have (1) good (not necessarily excellent) individual performance and (2) small residual error correlations with other members. Many techniques have been proposed to reduce residual correlations between members. These include resampling the training and validation data [3], adding randomness to data [7], and decorrelation training [8]. These approaches are only effective for certain models and problems. Genetic algorithms have also been used to generate good and diverse members [6]. Input feature selection is one of the most important stages of the model learning process. It has a crucial impact both on the learning complexity and the general922 Y. Liao and J. Moody ization performance. It is essential that a feature vector gives sufficient information for estimation. However, too many redundant input features not only burden the whole learning process, but also degrade the achievable generalization performance. Input feature selection for individual estimators has received a lot of attention because of its importance. However, there has not been much research on feature selection for estimators in the context of committees. Previous research found that giving committee members different input features is very useful for improving committee performance [4], but is difficult to implement [9]. The feature selection problem for committee members is conceptually different than for single estimators. When using committees for estimation, as we stated previously, committee members not only need to have reasonable performance themselves, but should also make decisions independently. When all committee members are trained to model the same underlying function, it is difficult for committee members to optimize both criteria at the same time. In order to generate members that provide a good balance between the two criteria, we propose a feature selection approach, called input feature grouping, for committee members. The idea is to give each member estimator of a committee a rich but distinct feature sets, in the hope that each member will generalize independently with reduced error correlations. The proposed method first groups input features using a hierarchical clustering algorithm based on their mutual information, such that features in different groups are less related to each other and features within a group are statistically similar to each other. Then the feature set for each committee member is formed by selecting a feature from each group. Our empirical results demonstrate that forming a heterogeneous committee using input feature grouping is a promising approach. 2 Committee Performance Analysis There are many ways to construct a committee. In this paper, we are mainly interested in heterogeneous committees whose members have different input feature sets. Committee members are given different subsets of the available feature set. They are trained independently, and the committee output is either a weighted or unweighted combination of individual members' outputs. In the following, we analyze the relationship between committee errors and average member errors from the regression point of view and discuss how the residual correlations between members affect the committee error. We define the training data V = {(X.B, y.B);;:3 = 1,2, . . . N} and the test data T = {(XI', YI'); JL = 1,2, ... oo}, where both are assumed to be generated by the model: Y = t(X) + f. , f. '" N(o, (72) . The data V and T are independent, and inputs are drawn from an unknown distribution. Assume that a committee has K members. Denote the available input features as X = [Xl, X2, . .. ,Xm ], the feature sets for the ith and jth members as Xi = [Xiu Xi2' . .. , xm;] and Xj = [Xjl) Xj2' ... ,xmj ] respectively, where Xi EX, Xj E X and Xi =I X j, and the mapping function of the ith and lh member models trained on data from V as fi(Xd and fi(Xj). Define the model error ef = tl' - h(Xn , for all JL = 1,2,3, ... ,00 and i = 1,2, ... , K. Constructing Heterogeneous Committees for Economic Forecasting 923 The MSE of a committee is K K = ~2 L £11 [(en2] + ~2 L £11 [efej] , i=l i#j (1) and the average MSE made by the committee members acting individually is K Eave = ~ L £11 [(en21 , i=l (2) where £[.] denotes the expectation over all test data T. Using Jensen's inequality, we get Ec ~ Eave, which indicates that the performance of a committee is always equal to or better than the average performance of its members. We define the average model error correlation as C = K(i -1) l:~j £11 [efejl , and then have 1 K-1 1 K-1 Ec = KEave + ~C = (K + ~q)Eave , (3) where q = Be . We consider the following four cases of q: ave • Case 1: - K~l ~ q < O. In this case, the model errors between members are anti-correlated, which might be achieved through decorrelation training. • Case 2: q = O. In this case, the model errors between members are uncorrelated, and we have: Ec = k Eave. That is to say, a committee can do much better than the average performance of its members. • Case 3: 0 < q < 1. If Eave is bounded above, when the committee size K -t 00, we have Ec = qEave . This gives the asymptotic limit of a committee's performance. As the size of a committee goes to infinity, the committee error is equal to the average model error correlation C. The difference between Ec and Eave is determined by the ratio q. • Case 4: q = 1. In this case, Ec is equal to Eave. This happens only when ei = ej, for all i,j = 1, ... ,K. It is obvious that there is no advantage to combining a set of models that act identically. It is clear from the analyses above that a committee shows its advantage when the ratio q is less than one. The smaller the ratio q is, the better the committee performs compared to the average performance of its members. For the committee to achieve substantial improvement over a single model, committee members not only should have small errors individually, but also should have small residual correlations between each other. 3 Input Feature Grouping One way to construct a feature subset for a committee member is by randomly picking a certain number of features from the original feature set. The advantage of this method is that it is simple. However, we have no control on each member's performance or on the residual correlation between members by randomly selecting subsets. 924 Y. Liao and J. Moody Instead of randomly picking a subset of features for a member, we propose an input feature grouping method for forming committee member feature sets. The input grouping method first groups features based on a relevance measure in a way such that features between different groups are less related to one another and features within a group are more related to one another. After grouping, there are two ways to form member feature sets. One method is to construct the feature set for each member by selecting a feature from · each group. Forming a member's feature set in this way, each member will have enough information to make decision, and its feature set has less redundancy. This is the method we use in this paper. Another way is to use each group as the feature set for a committee member. In this method each member will only have partial information. This is likely to hurt individual member's performance. However, because the input features for different members are less dependent, these members tend to make decisions more independently. There is always a trade-off between increasing members' independence and hurting individual members' performance. If there is no redundancy among input feature representations, removing several features may hurt individual members' performance badly, and the overall committee performance will be hurt even though members make decisions independently. This method is currently under investigation. The mutual information [(Xi; X j) between two input variables Xi and X j is used as the relevance measure to group inputs. The mutual information [(Xi; Xj) , which is defined in equation 4, measures the dependence between the two random variables. (4) If features Xi and X j are highly dependent, [(Xi; X j) will be large. Because the mutual information measures arbitrary dependencies between random variables, it has been effectively used for feature selections in complex prediction tasks [1], where methods bases on linear relations like the correlation are likely to make mistakes. The fact that the mutual information is independent of the coordinates chosen permits a robust estimation. 4 Empirical Studies We apply the input grouping method to predict the one-month rate of change of the Index of Industrial Production (IP), one of the key measures of economic activity. It is computed and published monthly. Figure 4 plots monthly IP data from 1967 to 1993. Nine macroeconomic time series, whose names are given in Table 1, are used for forecasting IP. Macroeconomic forecasting is a difficult task because data are usually limited, and these series are intrinsically very noise and nonstationary. These series are preprocessed before they are applied to the forecasting models. The representation used for input series is the first difference on one month time scales of the logged series. For example, the notation IP.L.Dl represents IP.L.Dl == In(IP(t)) -In(IP(t-l)). The target series is IP.L.FDl, which is defined as IP.L.FDI == In(IP(t+l)) - In(IP(t)). The data set has been one of our benchmarks for various studies [5, 10]. Constructing Heterogeneous Committees for Economic Forecasting 925 Index of Industrial Production: 1967 • 1993 Figure 1: U.S. Index of Industrial Production (IP) for the period 1967 to 1993. Shaded regions denote official recessions, while unshaded regions denote official expansions. The boundaries for recessions and expansions are determined by the National Bureau of Economic Research based on several macroeconomic series. As is evident for IP, business cycles are irregular in magnitude, duration, and structure, making prediction of IP an interesting challenge. Series Description IP Index of Industrial Production SP Standard & Poor's 500 DL Index of Leading Indicators M2 Money Supply CP Consumer Price Index CB Moody's Aaa Bond Yield HS HOUSing Starts TB3 3-month Treasury Bill Yield Tr Yield Curve Slope: (10-Year Bond Composite)-(3-Month Treasury Bill) Table 1: Input data series. Data are taken from the Citibase database. During the grouping procedure, measures of mutual information between all pairs of input variables are computed first. A simple histogram method is used to calculate these estimates. Then a hierarchical clustering algorithm [2] is applied to these values to group inputs. Hierarchical clustering proceeds by a series of successive fusions of the nine input variables into groups. At any particular stage, the process fuses variables or groups of variables which are closest, base on their mutual information estimates. The distance between two groups is defined as the average of the distances between all pairs of individuals in the two groups. The result is presented by a tree which illustrates the fusions made at each successive level (see Figure 2). From the clustering tree, it is clear that we can break the input variables into four groups: (IP.L.Dl, DL.L.Dl) measure recent economic changes, (SP.L.Dl) reflects recent stock market momentum, (CB.D1, TB3.D1, Tr.D1) give interest rate information, and (M2.L.D1, CP.L.D1, HS.L.D1) provide inflation information. The grouping algorithm meaningfully clusters the nine input series. 926 Y. Liao and J. Moody ~ :; ~ ::l ::I ~ ~ • :;; ~ ~ es ~ S ~ Figure 2: Variable grouping based on mutual information. Y label is the distance. Eighteen differept subsets of features can be generated from the four groups by selecting a feature from each group. Each subset is given to a committee member. For example, the subsets (IP.L.Dl, SP.L.Dl, CB.Dl, M2.L.Dl) and (DL.L.Dl, SP.L.Dl, TB3.Dl, M2.L.Dl) are used as feature sets for different committee members. A committee has totally eighteen members. Data from Jan. 1950 to Dec. 1979 is used for training and validation, and from Jan. 1980 to Dec. 1989 is used for testing. Each member is a linear model that is trained using neural net techniques. We compare the input grouping method with three other committee member generating methods: baseline, random selection, and bootstrapping. The baseline method is to train a committee member using all the input variables. Members are only different in their initial weights. The bootstrapping method also trains a member using all the input features, but each member has different bootstrap replicates of the original training data as its training and validation sets. The random selection method constructs a feature set for a member by randomly picking a subset from the available features. For comparison with the grouping method, each committee generated by these three methods has 18 members. Twenty runs are performed for each of the four methods in order to get reliable performance measures. Figure 3 shows the boxplots of normalized MSE for the four methods. The grouping method gives the best result, and the performance improvement is significant compared to other methods. The grouping method outperforms the random selection method by meaningfully grouping of input features. It is interesting to note that the heterogeneous committee methods, grouping and random selection, perform better than homogeneous methods for this data set. One of the reasons for this is that giving different members different input sets increases their model independence. Another reason could be that the problem becomes easier to model because of smaller feature sets. 5 Conclusions The performance of a committee depends on both the performance of individual members and the correlational structure of errors between members. An empirical study for a noisy and nonstationary economic forecasting problem has demonstrated that committees constructed by input variable grouping outperform committees formed by randomly selecting member input variables. They also outperform committees without any input variable manipulation. Constructing Heterogeneous Committees Jor Economic Forecasting 0.84 0.82 UJ ~ 0.8 ) J 0.78 0.75 0.74 Commltt .. Performanc. Comp.rteon (20 rurw) I I ! 8; 1 8 9 I -L 1 2 3 1 :Gr~lng. 2:Random •• lectlon. 3.BaM"n ..... Bootatrtpplng I -I927 Figure 3: Comparison between four different committee member generating methods. The proposed grouping method gives the best result, and the performance improvement is significant compared to the other three methods. References [1] R. Battiti. Using mutual information for selecting features in supervised neural net learning. IEEE TI-ans. on Neural Networks, 5(4), July 1994. [2] B.Everitt. Cluster Analysis. Heinemann Educational Books, 1974. [3] L. Breiman. Bagging predictors. Machine Learning, 24(2):123-40, 1996. [4] K.J. Cherkauer. Human expert-level performance on a scientific image analysis task by a system using combined artificai neural networks. In P. Chan, editor, Working Notes of the AAAI Workshop on Integrating Multiple Learned Models, pages 15-2l. 1996. [5] J. Moody, U. Levin, and S. Rehfuss. Predicting the U.S. index of industrial production. In proceedings of the 1993 Parallel Applications in Statistics and Economics Conference, Zeist, The Netherlands. Special issue of Neural Network World, 3(6):791794, 1993. [6] D. Opitz and J. Shavlik. Generating accurate and diverse members of a neuralnetwork ensemble. In D. Touretzky, M. Mozer, and M. Hasselmo, editors, Advances in Neural Information Processing Systems 8. MIT Press, Cambridge, MA, 1996. [7] Y. Raviv and N. Intrator. Bootstrapping with noise: An effective regularization technique. Connection Science, 8(3-4):355-72, 1996. [8] B. E. Rosen. Ensemble learning using decorrelated neural networks. Connection Science, 8(3-4):373-83, 1996. [9] K. Tumer and J. Ghosh. Error correlation and error reduction in ensemble classifiers. Connection Science, 8(3-4):385-404, December 1996. [10] L. Wu and J . Moody. A smoothing regularizer for feedforward and recurrent neural networks. Neural Computation, 8.3:463- 491, 1996.
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An Analysis of Turbo Decoding with Gaussian Densities Paat Rusmevichientong and Benjamin Van Roy Stanford University Stanford, CA 94305 {paatrus, bvr} @stanford.edu Abstract We provide an analysis of the turbo decoding algorithm (TDA) in a setting involving Gaussian densities. In this context, we are able to show that the algorithm converges and that - somewhat surprisingly - though the density generated by the TDA may differ significantly from the desired posterior density, the means of these two densities coincide. 1 Introd uction In many applications, the state of a system must be inferred from noisy observations. Examples include digital communications, speech recognition, and control with incomplete information. Unfortunately, problems of inference are often intractable, and one must resort to approximation methods. One approximate inference method that has recently generated spectacular success in certain coding applications is the turbo decoding algorithm [1, 2], which bears a close resemblance to message-passing algorithms developed in the coding community a few decades ago [4]. It has been shown that the TDA is also related to well-understood exact inference algorithms [5, 6], but its performance on the intractable problems to which it is applied has not been explained through this connection. Several other papers have further developed an understanding of the turbo decoding algorithm. The exact inference algorithms to which turbo decoding has been related are variants of belief propagation [7J. However, this algorithm is designed for inference problems for which graphical models describing conditional independencies form trees, whereas graphical models associated with turbo decoding possess many loops. To understand the behavior of belief propagation in the presence of loops, Weiss has analyzed the algorithm for cases where only a single loop is present [11]. Other analyses that have shed significant light on the performance of the TDA in its original coding context include [8, 9, 10]. In this paper, we develop a new line of analysis for a restrictive setting in which underlying distributions are Gaussian. In this context, inference problems are tractable and the use of approximation algorithms such as the TDA are unnecessary. However, studying the TDA in this context enables a streamlined analysis that generates new insights into its behavior. In particular, we will show that the algorithm converges and that the mean of the resulting distribution coincides with that of the 576 P. Rusmevichientong and B. V. Roy desired posterior distribution. While preparing this paper, we became aware of two related initiatives, both involving analysis of belief propagation when priors are Gaussian and graphs possess cycles. Weiss and Freeman [12] were studying the case of graphs possessing only cliques of size two. Here, they were able to show that, if belief propagation converges, the mean of the resulting approximation coincides with that of the true posterior distribution. At the same time, Frey [3] studied a case involving graphical structures that generalize those employed in turbo decoding. He also conducted an empirical study. The paper is organized as follows. In Section 2, we provide our working definition of the TDA. In Section 3, we analyze the case of Gaussian densities. Finally, a discussion of experimental results and open issues is presented in Section 4. 2 A Definition of Turbo Decoding Consider a random variable x taking on values in ~n distributed according to a density PO. Let YI and Y2 be two random variables that are conditionally independent given x. For example, YI and Y2 might represent outcomes of two independent transmissions of the signal x over a noisy communication channel. If YI and Y2 are observed, then one might want to infer a posterior density f for x conditioned on YI and Y2. This can be obtained by first computing densities pi and P2, where the first is conditioned on YI and the second is conditioned on Y2. Then, f = a (P~:2), where a is a "normalizing operator" defined by 9 ag = J g(x)dX' and multiplication/division are carried out pointwise. Unfortunately, the problem of computing f is generally intractable. The computational burden associated with storing and manipulating high-dimensional densities appears to be the primary obstacle. This motivates the idea of limiting attention to densities that factor. In this context, it is convenient to define an operator 71' that generates a density that factors while possessing the same marginals as another density. In particular, this operator is defined by ("9)(') '" !! l ..... I •• ~ •• J 9(x)dX A dXi for all densities 9 and all a E ~n, where dx /\ dXi = dXI'" dXi-Idxi+I ... dXn. One may then aim at computing 7l'f as a proxy for f. Unfortunately, even this problem is generally intractable. The TDA can be viewed as an iterative algorithm for approximating 71' f. Let operators FI and F2 be defined by FIg = a ( ( 7l'~:) ~ ) , and An Analysis o/Turbo Decoding with Gaussian Densities 577 for any density g. The TDA is applicable in cases where computation of these two operations is tractable. The algorithm generates sequences qik) and q~k) according to (HI) _ F (k) d (HI) _ D (k) ql 1 q2 an q2 - r2qI . initialized with densities qiO) and q~O) that factor. The hope is that Ci.(qik)q~k) /Po) converges to an approximation of 7r f. 3 The Gaussian Case We will consider a setting in which joint density of x, Yl, and Y2, is Gaussian. In this context, application of the TDA is not warranted - there are tractable algorithms for computing conditional densities when priors are Gaussian. Our objective, however, is to provide a setting in which the TDA can be analyzed and new insights can be generated. Before proceeding, let us define some notation that will facilitate our exposition. We will write 9 "-' N(/-Lg, ~g) to denote a Gaussian density 9 whose mean vector and covariance matrix are /-Lg and ~g, respectively. For any matrix A, b"(A) will denote a diagonal matrix whose entries are given by the diagonal elements of A. For any diagonal matrices X and Y, we write X ~ Y if Xii ~ Yii for all i. For any pair of nonsingular covariance matrices ~u and ~v such that ~;; 1 + ~; 1 - I is nonsingular, let a matrix AEu .E" be defined by A == (~-l + ~-l _ I)-I. Eu.E" u v To reduce notation, we will sometimes denote this matrix by Auv. When the random variables x, Yt, and Y2 are jointly Gaussian, the densities pi, P2' f, and Po are also Gaussian. We let pi "-' N(/-Ll, ~l)' P2 "-' N(/-L2, ~2)' f "-' N(/-L, ~), and assume that both ~l and ~2 are symmetric positive definite matrices. We will also assume that Po "-' N(O, I) where I is the identity matrix. It is easy to show that A E1 •E2 is well-defined. The following lemma provides formulas for the means and covariances that arise from multiplying and rescaling Gaussian densities. The result follows from simple algebra, and we state it without proof. Lemma 1 Let u "-' N(/-Lu, ~u) and v "-' N(/-Lv, ~v), where ~u and ~v are positive definite. If ~;;l + ~;l - I is positive definite then Ci. (;~) "-' N (Auv (~~l /-Lu + ~;l/-Lv) ,Auv) . One immediate consequence of this lemma is an expression for the mean of f: /-L = AE1.E2 (~ll/-Ll + ~2l/-L2). Let S denote the set of covariance matrices that are diagonal and positive definite. Let 9 denote the set of Gaussian densities with covariance matrices in S. We then have the following result, which we state without proof. Lemma 2 The set 9 is closed under Fl and F2 • If the TDA is initialized with qiO), q~O) E g, this lemma allows us to represent all iterates using appropriate mean vectors and covariance matrices. 578 P. Rusmevichientong and B. V. Roy 3.1 Convergence Analysis Under suitable technical conditions, it can be shown that the sequence of mean vectors and covariance matrices generates by the TDA converges. Due to space limitations, we will only present results pertinent to the convergence of covariance matrices. FUrthermore, we will only present certain central components of the analyses. For more complete results and detailed analyses, we refer the reader to our upcoming full-length paper. Recall that the TDA generates sequences qik) and q~k) according to (HI) F (k) d (HI) D (k) qI = Iq2 an q2 = L'2qI . As discussed earlier, if the algorithm is initialized with elements of 9, by Lemma 2, q(k) '" N (m(k) E(k)) and q(k) '" N (m(k) ~(k)) 1 1 , 1 2 2 '~2 , for appropriate sequences of mean vectors and covariance matrices. It turns out that there are mappings 7i : S 1--+ S and 72 : S 1--+ S such that Eik+1) = 7i (E~k)) and E~k+1) = 72 (Elk)) , for all k. Let T == 7i 072. To establish convergence of Elk) and E~k), it suffices to show that Tn(E~O)) converges. The following theorem establishes this and further points out that the limit does not depend on the initial iterates. Theorem 1 There exists a matrix V* E S such that lim m(V) = V*, n->oo for all V E S. 3.1.1 Preliminary Lemmas Our proof of Theorem 1 relies on a few lemmas that we will present in this section. We begin with a lemma that captures important abstract properties of the function T. Due to space constraints, we omit the proof, even though it is nontrivial. Lemma 3 (a) There exists a matrix DES such that for all DES, D ::; T(D) ::; f. (b) For all X, YES, if X::; Y then T(X) ::; T(Y). ( c) The function T is continuous on S. (d) For all f3 E (0,1) and DES, (f3 + o:)T (D) ::; T (f3D) for some 0: > o. The following lemma establishes convergence when the sequence of covariance matrices is initialized with the identity matrix. Lemma 4 The sequence Tn (f) converges in S to a fixed point of T. Proof; By Lemma 3(a), T(1) ::; f, and it follows from monotonicity of T (Lemma 3(b)) that Tn+1(I) ::; Tn(I) for all n. Since Tn(I) is bounded below by a matrix DES, the sequence converges in S. The fact that the limit is a fixed point of T follows from the continuity of T (Lemma 3( c) ). • Let V* = limn->oo Tn(I). This matrix plays the following special role. Lemma 5 The matrix V* is the unique fixed point in S of T. An Analysis of Turbo Decoding with Gaussian Densities 579 Proof: Because Tn (1) converges to V* and T is monotonic, no matrix V E S with V i= V* and V* ::; V ::; I can be a fixed point. Furthermore, by Lemma 3(a), no matrix V E S with V ~ I and V i= I can be a fixed point. For any V E S with V::; V*, let f3v = sup {f3 E (0, 111f3V* ::; V} . For any V E S with V i= V* and V ::; V*, we have f3v < 1. For such a V, by Lemma 3(d), there is an a > 0 such that T(f3vV*) ~ (f3v + a)V*, and therefore T(V) i= V. The result follows. • 3.1.2 Proof of Theorem 1 Proof: For V E S with V* ::; V ::; I convergence to V* follows from Lemma 4 and monotonicity (Lemma 3(b)). For V E S with V ~ I, convergence follows from the fact that V* ::; T(V) ::; I, which is a consequence of the two previously invoked lemmas together with Lemma 3(a). Let us now address the case of V E S with V ::; V*. Let f3v be defined as in the proof of Lemma 5. Then, f3v V* ::; T (f3v V*). By monotonicity, Tn (f3v V*) ::; Tn+I(f3v V*) ::; V* for all n. It follows that Tn(f3v V*) converges, and since T is continuous, the limit must be the unique fixed point V*. We have established convergence for elements V of S satisfying V ::; V* or V ~ V*. For other elements of S, convergence follows from the monotonicity of T. • 3.2 Analysis of the Fixed Point As discussed in the previous section, under suitable conditions, FI 0 F2 and F2 0 FI each possess a unique fixed point, and the TDA converges on these fixed points. Let qi ,...., N (f-Lq~ , Eq~) and q2 ,...., N (f-Lq2 ' Eq* ) denote the fixed points of FI 0 F2 and F2 0 FI, respectively. Based on Theorem 1, Eq~ and Eqi are in S. The following lemma provides an equation relating means associated with the fixed points. It is not hard to show that Aq*q*, AEI E *' and AE * E2' which are used in 1 2 ' q2 ql ' the statement, are well-defined. Lemma 6 Aq~qi (E;~lf-Lq; + E~lf-Lqi) = AE1 ,Eq2 (E1lf-LI + E~If-Lq2) = AEq~,E2 (E;;lf-Lq~ + E2"If-L2) Proof: It follows from the definitions of FI and F2 that, if qi = Fl q2 and q2 = F2qi, * * * * * * a ql q2 = a7rPI q2 = a7r qlP2 . Po Po Po The result then follows from Lemma 1 and the fact that 7r does not alter the mean of a distribution. • We now prove a central result of this paper: the mean of the density generated by the TDA coincides with the mean f-L of the desired posterior density f. Theorem 2 a (qi q2/ po) ,...., N (f-L, Aq; qi ) Proof: By Lemma 1, f-L = AE 1,E2 (E1lf-LI + E2"If-L2) , while the mean of a(qiq2lpo) is Aq~q2 (E;~l f-Lq; + E;i f-Lqi)' We will show that these two expressions are equal. 580 P Rusmevichientong and B. V. Roy , . . ~\ Figure 1: Evolution of errors. Multiplying the equations from Lemma 6 by appropriate matrices, we obtain Aq*q* A~1 E Aq*q* (2:-.I j.Lq* + 2:-.1 j.Lq*) = Aq*q* (2:11 j.Ll + 2:-} j.Lq*) , 1 2 1 , q:i 1 2 ql 1 q2 2 1 2 q2 2 and It follows that ( Aq~q:i (A~;,Eq:i +A~:~ ,E2) - I ) Aq~q:i (2:;il j.Lq~ + 2:;:i1 j.Lq:i) = Aqiq:i (2:11 j.Ll + 2:2'1 j.L2) , and therefore (A~IE +A~1 E -Aq-.Iq*) Aq*q* (2:q-.Ij.Lq* +2:q-.Ij.Lq*) = 2:11j.Ll+2:2'1j.L2' 1, q:i q~ , 2 1 2 1 2 1 1 2 2 • 4 Discussion and Experimental Results The limits of convergence qi and q2 of the TDA provide an approximation a( qi q2 / po) to 7r f. We have established that the mean of this approximation coincides with that of the desired density. One might further expect that the covariance matrix of a(qiq2/PO) approximates that of 7r f, and even more so, that qi and q2 bear some relation to pi and P2' Unfortunately, as will be illustrated by experimental results in this section, such expectations appear to be inaccurate. We performed experiments involving 20 and 50 dimensional Gaussian densities (Le., x was either 20 or 50 dimensional in each instance). Problem instances were sampled randomly from a fixed distribution. Due to space limitations, we will not describe the tedious details of the sampling mechanism. Figure 1 illustrates the evolution of certain "errors" during representative runs of the TDA on 20-dimensional problems. The first graph plots relative errors in means of densities a(q~n)q~n) /po) generated by iterates of the TDA. As indicated by our analysis, these errors converge to zero. The second chart plots a measure of relative error for the covariance of a(q~n)q~n) /po) versus that of 7rf for representative runs. Though these covariances converge, the ultimate errors are far from zero. The two An Analysis of Turbo Decoding with Gaussian Densities 581 Figure 2: Errors after 50 iterations. final graphs plot errors between the means of qin) and q~n) and those of pi and pi, respectively. Again, though these means converge, the ultimate errors can be large. Figure 2 provides plots of the same sorts of errors measured on 1000 different instances of 50-dimensional problems after the 50th iteration of the TDA. The horizontal axes are labeled with indices of the problem instances. Note that the errors in the first graph are all close to zero (the units on the vertical axis must be multiplied by 10- 5 and errors are measured in relative terms). On the other hand, errors in the other graphs vary dramatically. It is intriguing that - at least in the context of Gaussian densities - the TDA can effectively compute conditional means without accurately approximating conditional densities. It is also interesting to note that, in the context of communications, the objective is to choose a code word x that is comes close to the transmitted code x. One natural way to do this involves assigning to x the code word that maximizes the conditional density J, i.e., the one that has the highest chance of being correct. In the Gaussian case that we have studied, this corresponds to the mean of J - a quantity that is computed correctly by the TDA! It will be interesting to explore generalizations of the line of analysis presented in this paper to other classes of densities. References [lJ S. Benedetto and G. Montorsi, "Unveiling turbo codes: Some results on parallel concatenated coding schemes," in IEEE Trans. Inform. Theory, vol. 42, pp. 409-428, Mar. 1996. {2] G. Berrou, A. Glavieux, a.nd P. Thitimajshima, "Near Shannon limit error-correcting coding: 'TUrbo codes," in Proc. 1998 Int. Conf. Commun., Geneva, Switzerland, May 1993, pp. 1064-1070. [3J B. Frey, "Turbo Factor Analysis." To appear in Advances in Neural Information Processing Systems 11J. [4J R. G. Gallager, Low-Density Parity-Check Codes. Cambridge, MA: MIT Press, 1963. [5J F. R. Kschischang and B . J. Frey, "Iterative Decoding of Compound Codes by Probability Propagation in Graphical Models," in IEEE Journal on Selected Areas in Commun., vol. 16, 2, pp. 219-230, Feb. 1998. [6J R. J. McEliece, D. J . C. MacKay, and J-F. Cheng, "Turbo Decoding as an Instance of Pearl's "Belief Propagation" Algorithm," in IEEE Journal on Selected Areas in Commun., vol. 16, 2, pp. 140-152, Feb. 1998. [7] J. Pearl, Probabuistic Reasoning in Intelligent Systems: Networks of Plausible Inference. San Mateo, CA: Morgan Kaufmann, 1988. [8] T . Richardson, "The Geometry of Turbo-Decoding Dynamics," Dec. 1998. To appear in IEEE Trans. Inform. Theory. [9J T. Richardson and R. Urbanke, "The Capacity of Low-Density Parity Check Codes under Message-Passing Decoding", submitted to the IEEE Trans. on Information Theory. [10J T. Richardson, A. Shokrollahi, and R . Urbanke, "Design of Provably Good Low-Density Parity Check Codes," submitted to the IEEE Trans. on Information Theory. [l1J Y . Weiss, "Belief Propagation and Revision in Networks with Loops," November 1997. Available by ftp to publications.ai.mit.edu. [12] Y . Weiss and W. T . Freeman, "Correctness of belief propagation in Gaussian graphical m odels of arbitrary topology." To appear in Advances >n Neural Information Processtng Systems 11J.
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Probabilistic methods for Support Vector Machines Peter Sollich Department of Mathematics, King's College London Strand, London WC2R 2LS, U.K. Email: peter.sollich@kcl.ac.uk Abstract I describe a framework for interpreting Support Vector Machines (SVMs) as maximum a posteriori (MAP) solutions to inference problems with Gaussian Process priors. This can provide intuitive guidelines for choosing a 'good' SVM kernel. It can also assign (by evidence maximization) optimal values to parameters such as the noise level C which cannot be determined unambiguously from properties of the MAP solution alone (such as cross-validation error) . I illustrate this using a simple approximate expression for the SVM evidence. Once C has been determined, error bars on SVM predictions can also be obtained. 1 Support Vector Machines: A probabilistic framework Support Vector Machines (SVMs) have recently been the subject of intense research activity within the neural networks community; for tutorial introductions and overviews of recent developments see [1, 2, 3]. One of the open questions that remains is how to set the 'tunable' parameters of an SVM algorithm: While methods for choosing the width of the kernel function and the noise parameter C (which controls how closely the training data are fitted) have been proposed [4, 5] (see also, very recently, [6]), the effect of the overall shape of the kernel function remains imperfectly understood [1]. Error bars (class probabilities) for SVM predictions important for safety-critical applications, for example are also difficult to obtain. In this paper I suggest that a probabilistic interpretation of SVMs could be used to tackle these problems. It shows that the SVM kernel defines a prior over functions on the input space, avoiding the need to think in terms of high-dimensional feature spaces. It also allows one to define quantities such as the evidence (likelihood) for a set of hyperparameters (C, kernel amplitude Ko etc). I give a simple approximation to the evidence which can then be maximized to set such hyperparameters. The evidence is sensitive to the values of C and Ko individually, in contrast to properties (such as cross-validation error) of the deterministic solution, which only depends on the product CKo. It can thfrefore be used to assign an unambiguous value to C, from which error bars can be derived. 350 P. Sollich I focus on two-class classification problems. Suppose we are given a set D of n training examples (Xi, Yi) with binary outputs Yi = ±1 corresponding to the two classes. The basic SVM idea is to map the inputs X onto vectors c/>(x) in some high-dimensional feature space; ideally, in this feature space, the problem should be linearly separable. Suppose first that this is true. Among all decision hyperplanes w·c/>(x) + b = 0 which separate the training examples (Le. which obey Yi(W'c/>(Xi) + b) > 0 for all Xi E Dx , Dx being the set of training inputs), the SVM solution is chosen as the one with the largest margin, Le. the largest minimal distance from any of the training examples. Equivalently, one specifies the margin to be one and minimizes the squared length of the weight vector IIwI12 [1], subject to the constraint that Yi(W'c/>(Xi) + b) 2:: 1 for all i. If the problem is not linearly separable, 'slack variables' ~i 2:: 0 are introduced which measure how much the margin constraints are violated; one writes Yi(W'c/>(Xi) + b) 2:: 1 ~i' To control the amount of slack allowed, a penalty term C Ei ~i is then added to the objective function ~ IIwI12 , with a penalty coefficient C. Training examples with Yi(w ·c/>(xd + b) 2:: 1 (and hence ~i = 0) incur no penalty; all others contribute C[l - Yi(W 'c/>(Xi) + b)] each. This gives the SVM optimization problem: Find wand b to minimize ~llwl12 + C Ei l(Yi[W'c/>(Xi) + b]) where l(z) is the (shifted) 'hinge loss', l(z) = (1- z)8(1- z). (1) To interpret SVMs probabilistically, one can regard (1) as defining a (negative) log-posterior probability for the parameters wand b of the SVM, given a training set D. The first term gives the prior Q(w,b) "" exp(-~llwW ~b2B-2). This is a Gaussian prior on W; the components of W are uncorrelated with each other and have unit variance. I have chosen a Gaussian prior on b with variance B2; the flat prior implied by (1) can be recovered! by letting B -+ 00. Because only the 'latent variable' values O(x) = w·c/>(x) + b rather than wand b individually appear in the second, data dependent term of (1), it makes sense to express the prior directly as a distribution over these. The O(x) have a joint Gaussian distribution because the components ofw do, with covariances given by (O(x)O(x')) = (( c/>(x) ·w) (w·c/>(x'))) + B2 = c/>(x)·c/>(x') + B2. The SVM prior is therefore simply a Gaussian process (GP) over the functions 0, with covariance function K(x,x') = c/>(x) ·c/>(x') + B2 (and zero mean). This correspondence between SVMs and GPs has been noted by a number of authors, e.g. [6, 7, 8, 9, 10J. The second term in (1) becomes a (negative) log-likelihood if we define the probability of obtaining output Y for a given X (and 0) as Q(y =±llx, 0) = ~(C) exp[-Cl(yO(x))] (2) We set ~(C) = 1/[1 + exp(-2C)] to ensure that the probabilities for Y ±1 never add up to a value larger than one. The likelihood for the complete data set is then Q(DIO) = It Q(Yilxi, O)Q(Xi), with some input distribution Q(x) which remains essentially arbitrary at this point. However, this likelihood function is not normalized, because lI(O(x)) = Q(llx, 0) + Q( -llx, 0) = ~(C){ exp[ -Cl(O(x))] + exp[-Cl( -O(x))]} < 1 lIn the probabilistic setting, it actually makes more sense to keep B finite (and small); for B -+ 00, only training sets with all Yi equal have nonzero probability. Probabilistic Methods for Support Vector Machines 351 except when IO(x)1 = 1. To remedy this, I write the actual probability model as P(D,9) = Q(DI9)Q(9)/N(D). (3) Its posterior probability P(9ID) '" Q(DI9)Q(9) is independent Qfthe normalization factor N(D); by construction, the MAP value of 9 is therefore the SVM solution. The simplest choice of N(D) which normalizes P(D, 9) is D-independent: N = Nn = Jd9Q(9)Nn(9), N(9) = JdxQ(x)lI(O(x)). (4) Conceptually, this corresponds to the following procedure of sampling from P(D, 9): First, sample 9 from the GP prior Q(9) . Then, for each data point, sample x from Q(x). Assign outputs Y = ±1 with probability Q(ylx,9), respectively; with the remaining probability l-lI(O(x)) (the 'don't know' class probability in [11]), restart the whole process by sampling a new 9. Because lI(O(x)) is smallest2 inside the 'gap' IO(x)1 < 1, functions 9 with many values in this gap are less likely to 'survive' until a dataset of the required size n is built up. This is reflected in an n-dependent factor in the (effective) prior, which follows from (3,4) as P(9) '" Q(9)Nn(9). Correspondingly, in the likelihood P(ylx,9) = Q(ylx, 9)/1I(O(x)), P(xI9) '" Q(x) lI(O(x)) (5) (which now is normalized over y = ±1), the input density is influenced by the function 9 itself; it is reduced in the 'uncertainty gaps' IO(x)1 < 1. To summarize, eqs. (2-5) define a probabilistic data generation model whose MAP solution 9* = argmax P(9ID) for a given data set D is identical to a standard SVM. The effective prior P(9) is a GP prior modified by a data set size-dependent factor; the likelihood (5) defines not just a conditional output distribution, but also an input distribution (relative to some arbitrary Q(x)). All relevant properties of the feature space are encoded in the underlying GP prior Q(9), with covariance matrix equal to the kernel K(x, Xl). The log-posterior of the model In P(9ID) = -t J dx dxl O(X)K-l(X, Xl) O(XI) - C 'Ei l(YiO(xi)) + const (6) is just a transformation of (1) from wand b to 9. By differentiating w.r.t. the O(x) for non-training inputs, one sees that its maximum is of the standard form O*(x) = Ei (}:iYiK(X, Xi); for YiO*(Xi) > 1, < 1, and = lone has (}:i = 0, (}:i = C and (}:i E [0, C] respectively. I will call the training inputs Xi in the last group marginal; they form a subset of all support vectors (the Xi with (}:i > 0). The sparseness of the SVM solution (often the number of support vectors is « n) comes from the fact that the hinge loss l(z) is constant for z > 1. This contrasts with other uses of GP models for classification (see e.g. [12]), where instead of the likelihood (2) a sigmoidal (often logistic) 'transfer function' with nonzerO gradient everywhere is used. Moreover, in the noise free limit, the sigmoidal transfer function becomes a step function, and the MAP values 9* will tend to the trivial solution O*(x) = O. This illuminates from an alternative point of view why the margin (the 'shift' in the hinge loss) is important for SVMs. Within the probabilistic framework, the main effect of the kernel in SVM classification is to change the properties of the underlying GP prior Q(9) in P(9) '" 2This is true for C > In 2. For smaller C, v( O( x» is actually higher in the gap, and the model makes less intuitive sense. 352 P. Sollich (e) (h) Figure 1: Samples from SVM priors; the input space is the unit square [0,1]2. 3d plots are samples 8(x) from the underlying Gaussian process prior Q(8). 2d greyscale plots represent the output distributions obtained when 8(x) is used in the likelihood model (5) with C = 2; the greyscale indicates the probability of y = 1 (black: 0, white: 1). (a,b) Exponential (Ornstein-Uhlenbeck) kernel/covariance function Koexp(-Ix - x/l/l), giving rough 8(x) and decision boundaries. Length scale l = 0.1, Ko = 10. (c) Same with Ko = 1, i.e. with a reduced amplitude of O(x); note how, in a sample from the prior corresponding to this new kernel, the grey 'uncertainty gaps' (given roughly by 18(x)1 < 1) between regions of definite outputs (black/white) have widened. (d,e) As first row, but with squared exponential (RBF) kernel Ko exp[-(x X I )2/(2l2)], yielding smooth 8(x) and decision boundaries. (f) Changing l to 0.05 (while holding Ko fixed at 10) and taking a new sample shows how this parameter sets the typical length scale for decision regions. (g,h) Polynomial kernel (1 + x·xl)P, with p = 5; (i) p = 10. The absence of a clear length scale and the widely differing magnitudes of 8(x) in the bottom left (x = [0,0]) and top right (x = [1,1]) corners of the square make this kernel less plausible from a probabilistic point of view. Probabilistic Methods for Support Vector Machines 353 Q(O)Nn(o). Fig. 1 illustrates this with samples from Q(O) for three different types of kernels. The effect of the kernel on smoothness of decision boundaries, and typical sizes of decision regions and 'uncertainty gaps' between them, can clearly be seen. When prior knowledge about these properties of the target is available, the probabilistic framework can therefore provide intuition for a suitable choice of kernel. Note that the samples in Fig. 1 are from Q(O), rather than from the effective prior P(O). One finds, however, that the n-dependent factor Nn(o) does not change the properties of the prior qualitatively3. 2 Evidence and error bars Beyond providing intuition about SVM kernels, the probabilistic framework discussed above also makes it possible to apply Bayesian methods to SVMs. For example, one can define the evidence, i.e. the likelihood of the data D, given the model as specified by the hyperparameters C and (some parameters defining) K(x, x'). It follows from (3) as P(D) = Q(D)/Nn, Q(D) = J dO Q(DIO)Q(O). (7) The factor Q(D) is the 'naive' evidence derived from the unnormalized likelihood model; the correction factor Nn ensures that P(D) is normalized over all data sets. This is crucial in order to guarantee that optimization of the (log) evidence gives optimal hyperparameter values at least on average (M Opper, private communication). Clearly, P(D) will in general depend on C and K(x,x') separately. The actual SVM solution, on the other hand, i.e. the MAP values 0*, can be seen from (6) to depend on the product C K (x, x') only. Properties of the deterministically trained SVM alone (such as test or cross-validation error) cannot therefore be used to determine C and the resulting class probabilities (5) unambiguously. I now outline how a simple approximation to the naive evidence can be derived. Q(D) is given by an integral over all B(x), with the log integrand being (6) up to an additive constant. After integrating out the Gaussian distributed B( x) with x ¢ Dx , an intractable integral over the B(Xi) remains. However, progress can be made by expanding the log integrand around its maximum B*(Xi)' For all non-marginal training inputs this is equivalent to Laplace's approximation: the first terms in the expansion are quadratic in the deviations from the maximum and give simple Gaussian integrals. For the remaining B(Xi), the leading terms in the log integrand vary linearly near the maximum. Couplings between these B(Xi) only appear at the next (quadratic) order; discarding these terms as subleading, the integral factorizes over the B(xd and can be evaluated. The end result of this calculation is: InQ(D) ~ -! LiYi<liB*(Xi) - CLil(YiB*(xd) - nln(l + e-2C ) - ! Indet(LmKm) (8) The first three terms represent the maximum of the log integrand, In Q(DIO*); the last one comes from the integration over the fluctuations of the B(x). Note that it only contains information about the marginal training inputs: Km is the corresponding submatrix of K(x, x'), and Lm is a diagonal matrix with entries 3Quantitative changes arise because function values with IO(x)1 < 1 are 'discouraged' for large nj this tends to increase the size of the decision regions and narrow the uncertainty gaps. I have verified this by comparing samples from Q(O) and P(O). 354 9(x) 2 1 o -1 -2 0.2 0.4 x 0.6 0.8 1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0.8 0.6 0.4 0.2 o o P. Sollich 1 2 C 3 4 I I P(y=llx) I I I I I I I I I \ I , \. J \. }; 0.2 0.4 x 0.6 0.8 1 Figure 2: Toy example of evidence maximization. Left: Target 'latent' function 8(x) (solid line). A SVM with RBF kernel K(x, Xl) = Ko exp[-(x - XI)2 /(2[2)], [ = 0.05, CKo = 2.5 was trained (dashed line) on n = 50 training examples (circles). Keeping CKo constant, the evidence P(D) (top right) was then evaluated as a function of C using (7,8). Note how the normalization factor Nn shifts the maximum of P(D) towards larger values of C than in the naive evidence Q(D). Bottom right: Class probability P(y = 11x) for the target (solid), and prediction at the evidence maximum C ~ 1.8 (dashed). The target was generated from (3) with C=2. 27r[ai(C -ai)/C]2. Given the sparseness ofthe SVM solution, these matrices should be reasonably small, making their determinants amenable to numerical computation or estimation [12]. Eq. (8) diverges when ai -+ a or -+ C for one of the marginal training inputs; the approximation of retaining only linear terms in the log integrand then breaks down. I therefore adopt the simple heuristic of replacing det(LmKm) by det(1 + LmKm), which prevents these spurious singularities (I is the identity matrix). This choice also keeps the evidence continuous when training inputs move in or out of the set of marginal inputs as hyperparameters are varied. Fig. 2 shows a simple application of the evidence estimate (8). For a given data set, the evidence P(D) was evaluated4 as a function of C. The kernel amplitude Ko was varied simultaneously such that C Ko and hence the SVM solution itself remained unchanged. Because the data set was generated artificially from the probability model (3), the 'true' value of C = 2 was known; in spite of the rather crude approximation for Q(D), the maximum of the full evidence P(D) identifies C ~ 1.8 quite close to the truth. The approximate class probability prediction P(y = 11x, D) for this value of C is also plotted in Fig. 2; it overestimates the noise in the target somewhat. Note that P(ylx, D) was obtained simply by inserting the MAP values 8*(x) into (5). In a proper Bayesian treatment, an average over the posterior distribution P(OID) should of course be taken; I leave this for future work. 4The normalization factor Nn was estimated, for the assumed uniform input density Q(x) of the example, by sampling from the GP prior Q(9) . If Q(x) is unknown, the empirical training input distribution can be used as a proxy, and one samples instead from a multivariate Gaussian for the 9(xd with covariance matrix K(Xi , Xj). This gave very similar values of In Nn in the example, even when only a subset of 30 training inputs was used. Probabilistic Methods for Support Vector Machines 355 In summary, I have described a probabilistic framework for SVM classification. It gives an intuitive understanding of the effect of the kernel, which determines a Gaussian process prior. More importantly, it also allows a properly normalized evidence to be defined; from this, optimal values of hyperparameters such as the noise parameter C, and corresponding error bars, can be derived. Future work will have to include more comprehensive experimental tests of the simple Laplacetype estimate of the (naive) evidence Q(D) that I have given, and comparison with other approaches. These include variational methods; very recent experiments with a Gaussian approximation for the posterior P(9ID), for example, seem promising [6]. Further improvement should be possible by dropping the restriction to a 'factor-analysed' covariance form [6]. (One easily shows that the optimal Gaussian covariance matrix is (D + K- 1 )-1, parameterized only by a diagonal matrix D.) It will also be interesting to compare the Laplace and Gaussian variational results for the evidence with those from the 'cavity field' approach of [10]. Acknowledgements It is a pleasure to thank Tommi Jaakkola, Manfred Opper, Matthias Seeger, Chris Williams and Ole Winther for interesting comments and discussions, and the Royal Society for financial support through a Dorothy Hodgkin Research Fellowship. References [1] C J C Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2:121-167, 1998. [2] A J Smola and B Scholkopf. A tutorial on support vector regression. 1998. Neuro COLT Technical Report TR-1998-030; available from http://svm.first.gmd.de/. [3] B Scholkopf, C Burges, and A J Smola. Advances in Kernel Methods: Su.pport Vector Machines. MIT Press, Cambridge, MA, 1998. [4) B Scholkopf, P Bartlett, A Smola, and R Williamson. Shrinking the tube: a new support vector regression algorithm. In NIPS 11. [5] N Cristianini, C Campbell, and J Shawe-Taylor. Dynamically adapting kernels in support vector machines. In NIPS 11. [6] M Seeger. Bayesian model selection for Support Vector machines, Gaussian processes and other kernel classifiers. Submitted to NIPS 12. [7] G Wahba. Support vector machines, reproducing kernel Hilbert spaces and the randomized GACV. Technical Report 984, University of Wisconsin, 1997. [8] T S Jaakkola and D Haussler. Probabilistic kernel regression models. In Proceedings of The 7th International Workshop on Artificial Intelligence and Statistics. To appear. [9] A J Smola, B Scholkopf, and K R Muller. The connection between regularization operators and support vector kernels. Neu.ral Networks, 11:637-649, 1998. [10] M Opper and 0 Winther. Gaussian process classification and SVM: Mean field results and leave-one-out estimator. In Advances in Large Margin Classifiers. MIT Press. To appear. [11] P Sollich. Probabilistic interpretation and Bayesian methods for Support Vector Machines. Submitted to ICANN 99. [12] C K I Williams. Prediction with Gaussian processes: From linear regression to linear prediction and beyond. In M I Jordan, editor, Learning and Inference in Graphical Models, pages 599-621. Kluwer Academic, 1998.
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Neural System Model of Human Sound Localization Craig T. Jin Department of Physiology and Department of Electrical Engineering, Univ. of Sydney, NSW 2006, Australia Simon Carlile Department of Physiology and Institute of Biomedical Research, Univ. of Sydney, NSW 2006, Australia Abstract This paper examines the role of biological constraints in the human auditory localization process. A psychophysical and neural system modeling approach was undertaken in which performance comparisons between competing models and a human subject explore the relevant biologically plausible "realism constraints". The directional acoustical cues, upon which sound localization is based, were derived from the human subject's head-related transfer functions (HRTFs). Sound stimuli were generated by convolving bandpass noise with the HRTFs and were presented to both the subject and the model. The input stimuli to the model was processed using the Auditory Image Model of cochlear processing. The cochlear data was then analyzed by a time-delay neural network which integrated temporal and spectral information to determine the spatial location of the sound source. The combined cochlear model and neural network provided a system model of the sound localization process. Human-like localization performance was qualitatively achieved for broadband and bandpass stimuli when the model architecture incorporated frequency division (or tonotopicity), and was trained using variable bandwidth and center-frequency sounds. 1 Introduction The ability to accurately estimate the location of a sound source has obvious evolutionary advantages in terms of avoiding predators and finding prey. Indeed, humans are very accurate in their ability to localize broadband sounds. There has been a considerable amount of psychoacoustical research into the auditory processes involved in human sound localization (recent review [1]). Furthermore, numerous models of the human and animal sound localization process have been proposed (recent reviews [2,3]). However, there still remains a large gap between the psychophysical and the model explanations. Principal congruence between the two approaches exists for localization performance under restricted conditions, such as for narrowband sounds where spectral integration is not required, or for restricted regions of space. Unfortunately, there is no existing computational model that accounts well for human sound localization performance for a wide-range of sounds (e.g., varying in bandwidth and center-frequency). Furthermore, the biological constraints pertinent to sound localization have generally not been explored by these models. These include the spectral resolution of the auditory system in terms of the number and bandwidth of 762 C. T. Jin and S. Carlile frequency channels and the role of tonotopic processing. In addition, the perfonnance requirements of such a system are substantial and involve, for example, the accomodation of spectrally complex sounds, the robustness to irregularity in the sound source spectrum, and the channel based structure of spatial coding as evidenced by auditory spatial after-effects [4]. The crux of the matter is the notion that "biologically-likely realism", if built into a model, provides for a better understanding of the underlying processes. This work attempts to bridge part of this gap between the modeling and psychophysics. It describes the development and use (for the first time, to the authors ' knowledge) of a timedelay neural network model that integrates both spectral and temporal cues for auditory sound localization and compares the perfonnance of such a model with the corresponding human psychophysical evidence. 2 Sound Localization The sound localization perfonnance of a nonnal hearing human subject was tested using stimuli consisting of three different band-passed sounds: (1) a low-passed sound (300 2000 Hz) (2) a high-passed sound (2000 - 14000 Hz) and (3) a broadband sound (300 14000 Hz). These frequency bands respectively cover conditions in which either temporal cues, spectral cues, or both dominate the localization process (see [1]). The subject perfonned five localization trials for each sound condition, each with 76 test locations evenly distributed about the subject's head. The detailed methods used in free-field sound localization can be found in [5]. A short summary is presented below. 2.1 Sound Localization Task Human sound localization experiments were carried out in a darkened anechoic chamber. Free-field sound stimuli were presented from a loudspeaker carried on a semicircular robotic ann. These stimuli consisted of "fresh" white Gaussian noise appropriately bandpassed for each trial. The robotic ann allowed for placement of the speaker at almost any location on the surface of an imaginary sphere, one meter in radius, centered on the subject's head. The subject indicated the location of the sound source by pointing his nose in the perceived direction of the sound. The subject's head orientation was monitored using an electromagnetic sensor system (Polhemus, Inc.). 2.2 Measurement and Validation of Outer Ear Acoustical Filtering The cues for sound localization depend not only upon the spectral and temporal properties of the sound stimulus, but also on the acoustical properties of the individual's outer ears. It is generally accepted that the relevant acoustical cues (i.e., the interaural time difference, ITO; interaurallevel difference, ILD; and spectral cues) to a sound's location in the free-field are described by the head-related transfer function (HRTF) which is typically represented by a finite-length impulse response (FIR) filter [1]. Sounds filtered with the HRTF should be localizable when played over ear-phones which bypass the acoustical filtering of the outer ear. The illusion of free-field sounds using head-phones is known as virtual auditory space (VAS). Thus in order to incorporate outer ear filtering into the modelling process, measurements of the subject's HRTFs were carried out in the anechoic chamber. The measurements were made for both ears simultaneously using a ''blocked ear" technique [1]. 393 measurements were made at locations evenly distributed on the sphere. In order to establish that the HRTFs appropriately indicated the direction of a sound source the subject repeated the localization task as above with the stimulus presented in VAS. Neural System Model of Human Sound Localization 763 2.3 Human Sound Localization Performance The sound localization performance of the human subject in three different stimulus conditions (broadband, high-pass, low-pass) was examined in both the free-field and in virtual auditory space. Comparisons between the two (using correlational statistics, data not shown, but see [3]) across all sound conditions demonstrated their equivalence. Thus the measured HRTFs were highly effective. Localization data across all three sound conditions (single trial VAS data shown in Fig. la) shows that the subject performed well in both the broadband and high-pass sound conditions and rather poorly in the low-pass condition, which is consistent with other studies [6]. The data is illustrated using spherical localization plots which well demonstrates the global distribution of localization responses. Given the large qualitative differences in the data sets presented below, this visual method of analysis was sufficient for evaluating the competing models. For each condition, the target and response locations are shown for both the left (L) and right (R) hemispheres of space. It is clear that in the low-pass condition, the subject demonstrated gross mislocalizations with the responses clustering toward the lower and frontal hemispheres. The gross mislocalizations correspond mainly to the traditional cone of confusion errors [6]. 3 Localization Model The sound localization model consisted of two basic system components: (1) a modified version of the physiological Auditory Image Model [7] which simulates the spectrotemporal characteristics of peripheral auditory processing, and (2) the computational architecture of a time-delay neural network. The sounds presented to the model were filtered using the sUbject's HRTFs in exactly the same manner as was used in producing VAS. Therefore, the modeling results can be compared with human localization performance on an individual basis. The modeling process can be broken down into four stages. In the first stage a sound stimulus was generated with specific band-pass characteristics. The sound stimulus was then filtered with the subject's right and left ear HRTFs to render an auditory stimulus originating from a particular location in space. The auditory stimulus was then processed by the Auditory Image Model (AIM) to generate a neural activity profile that simulates the output of the inner hair cells in the organ of Corti and indicates the spiking probability of auditory nerve fibers. Finally, in the fourth and last stage, a time-delay neural network (TDNN) computed the spatial direction of the sound input based on the distribution of neural activity calculated by AIM. A detailed presentation of the modeling process can be found in [3], although a brief summary is presented here. The distribution of cochlear filters across frequency in AIM was chosen such that the minimum center frequency was 300 Hz and the maximum center frequency was 14 kHz with 31 filters essentially equally spaced on a logarithmic scale. In order to fully describe a computational layer of the TDNN, four characteristic numbers must be specified: (l) the number of neurons; (2) the kernel length, a number which determines the size of the current layer's time-window in terms of the number of time-steps of the previous layer; (3) the kernel width, a number which specifies how many neurons in the previous layer with which there are actual connections; and (4) the undersampling factor, a number describing the multiplicative factor by which the current layer's time-step interval is increased from the previous layer's. Using this nomenclature, the architecture of the different layers of one TDNN is summarized in Table 1, with the smallest time-step being 0.15 ms. The exact connection arrangement of the network is described in the next section. 764 C. T. Jin and S. Carlile Table I: The Architecture of the TDNN. Layer Neurons Kernel Length Kernel Width Undersampling Input Hidden I Hidden 2 Output 62 50 15 6 2 28 10 4,5,6 2 393 4 28 1 The spatial location of a sound source was encoded by the network as a distributed response with the peak occurring at the output neuron representing the target location of the input sound. The output response would then decay away in the fonn of a two-dimensional Gaussian as one moves to neurons further away from the target location. This derives from the well-established paradigm that the nervous system uses overlapping receptive fields to encode properties of the physical world. 3.1 Networks with Frequency Division and Tonotopicity The major auditory brainstem nuclei demonstrate substantial frequency division within their structure. The tonotopic organization of the primary auditory nerve fibers that innervate the cochlea carries forward to the brainstem's auditory nuclei. This arrangement is described as a tonotopic organization. Despite this fact and to our knowledge, no previous network model for sound localization incorporates such frequency division within its architecture. Typically (e.g., [8]) all of the neurons in the first computational layer are fully connected to all of the input cochlear frequency channels. In this work, different architectures were examined with varying amounts of frequency division imposed upon the network structure. The network with the architecture described above had its network connections constrained by frequency in a tonotopic like arrangement. The 31 input cochlear frequency channels for each ear were split into ten overlapping groups consisting generally of six contiguous frequency channels. There were five neurons in the first hidden layer for each group of input channels. The kernel widths of these neurons were set, not to the total number of frequency channels in the input layer, but only to the six contiguous frequency channels defining the group. Infonnation across the different groups of frequency channels was progressively integrated in the higher layers of the network. 3.2 Network Training Sounds with different center-frequency and bandwidth were used for training the networks. In one particular training paradigm, the center-frequency and bandwidth of the noise were chosen randomly. The center-frequency was chosen using a unifonn probability distribution on a logarithmic scale that was similar to the physiological distribution of output frequency channels from AIM. In this manner, each frequency region was trained equally based on the density of neurons in that frequency region. During training, the error backpropagation algorithm was used with a summed squared error measure. It is a natural feature of the learning rule that a given neuron's weights are only updated when there is activity in its respective cochlear channels. So, for example, a training sound containing only low frequencies will not train the high-frequency neurons and vice versa. All modeling results correspond with a single tonotopically organized TDNN trained using random sounds (unless explicitly stated otherwise). Neural System Model of Human Sound Localization 765 4 Localization Performance of a Tonotopic Network Experimentation with the different network architectures clearly demonstrated that a network with frequency division vastly improved the localization performance of the TDNNs (Figure I). In this case, frequency division was essential to producing a reasonable neural system model that would localize similarly to the human subject across all of the different band-pass conditions. For any single band-pass condition, it was found that the TDNN did not require frequency division within its architecture to produce quality solutions when trained only on these band-passed sounds. As mentioned above it was observed that a tonotopic network, one that divides the input frequency channels into different groups and then progressively interconnects the neurons in the higher layers across frequency, was more robust in its localization performance across sounds with variable center-frequency and bandwidth than a simple fully connected network. There are two likely explanations for this observation. One line of reasoning argues that it was easier for the tonotopic network to prevent a narrow band of frequency channels from dominating the localization computation across the entire set of sound stimuli. Or expressed slightly differently, it may have been easier for it to incorporate the relevant information across the different frequency channels. A second line of reasoning argues that the tonotopic network structure (along with the training with variable sounds) encouraged the network to develop meaningful connections for all frequencies. (a) SUBJECT VAS L R (b) TONOTOPIC NETWORK L R (c) NETWORK without FREQUENCY DIVISION • //'~~~;" " .~::,:<;- '., >~ , , , .>t .. ' .' .~ : ' '" ~;~::;;;,L R Figure I: Comparison of the subject's VAS localization performance and the model's localization performance both with and without frequency division. The viewpoint is from an outside observer, with the target location shown by a cross and the response location shown by a black dot. 766 C. T. Jin and S. Carlile 5 Matched Filtering and Sound Localization A number of previous sound localization models have used a relatively straight-forward matched filter or template matching analysis [9]. In such cases, the lTD and spectrum of a given input sound is commonly cross-correlated with the lTD and spectrum of an entire database of sounds for which the location is known. The location with the highest correlation is then chosen as the optimal source location. Matched filtering analysis is compared with the localization performance of both the human subject and the neural system model using a bandpass sound with restricted highfrequencies (Figure 2). The matched filtering localizes the sounds much better than the subject or the TDNN model. The matched filtering model used the same number of cochlear channels as the TDNNs and therefore contained the same inherent spectral resolution. This spectral resolution (31 cochlear channels) is certainly less than the spectral resolution of the human cochlea. This shows that although there was sufficient information to localize the sounds from the point of view of matched filtering, neither the human nor TDNN demonstrated such ability in their performance. In order for the TDNN to localize similarly to the matched filtering model, the network weights corresponding to a given location need to assume the form of the filter template for that location. As all of the training sounds were flat-spectrum, the TDNN received no ambiguity as far as the source spectrum was concerned. Thus it is likely that the difference in the distribution of localization responses in Figure 2b, as compared with that in Figure 2c, has been encouraged by using training sounds with random center-frequency and bandwidth, providing a partial explanation as to why the human localization performance is not optimal from a matched filtering standpoint. Figure 2: Comparison of the localization performances of the subject, the TDNN model, and a matched filtering model. Details as in Fig. I. 6 Varying Sound Levels and the ILD Cue The training ofthe TDNNs was performed in such a fashion, that for any particular location in space, the sound level (67 dB SPL) did not vary by more than 1 dB SPL during repeated presentations of the sound. The localization performance of the neural system model was then examined, using a broadband sound source, across a range of sound levels varying from 60 dB SPL to 80 dB SPL. The spherical correlation coefficient between the target and response locations ([10], values above 0.8 indicate "high" correlation) remained above 0.8 between 60 and 75 dB SPL demonstrating that there was a graceful degradation in localization performance over a range in sound level of 15 dB. The network was also tested on broadband sounds, 10 dB louder in one ear than the other. The results of these tests are shown in Figure 3 and clearly illustrate that the localization responses were pulled toward the side with the louder sound. While the magnitude of this effect is certainly not human-like, such behaviour suggests that interaurallevel difference Neural System Model of Human Sound Localization 767 cues were a prominent and constant feature of the data that conferred a measure of robustness to sound level variations. Figure 3: Model's localization performance with a 10 dB increase in sound level: (a,b) monaurally, (c) binaurally. 7 Conclusions A neural system model was developed in which physiological constraints were imposed upon the modeling process: (I) a TDNN model was used to incorporate the important role of spectral-temporal processing in the auditory nervous system, (2) a tonotopic structure was added to the network, (3) the training sounds contained randomly varying centerfrequencies and bandwidths. This biologically plausible model provided increased understanding of the role that these constraints play in determining localization performance. Acknowledgments The authors thank Markus Schenkel and Andre van Schaik for valuable comments. This research was supported by the NHMRC, ARC, and a Dora Lush Scholarship to CJ. References [I] S. Carlile, Virtual auditory space: Generation and applications. New York: Chapman and Hall, 1996. [2] R. H. Gilkey and T. R. Anderson, Binaural and Spatial Hearing in real and virtual environments. Mahwah, New Jersey: Lawrence Erlbaum Associates, Publishers, 1997. [3] C. Jin, M. Schenkel, and S. Carlile, "Neural system identification model of human sound localisation," (Submitted to J. Acoust. Soc. Am.), 1999. [4] S. Hyams and S. Carlile, "After-effects in auditory localization: evidence for channel based processing," Submitted to the J. Acoust. Soc. Am., 2000. [5] S. Carlile, P. Leong, and S. Hyams, "The nature and distribution of errors in the localization of sounds by humans," Hearing Research, vol. 114, pp. 179-196, 1997. [6] S. Carlile, S. Delaney, and A. Corderoy, "The localization of spectrally restricted sounds by human listeners," Hearing Research, vol. 128, pp. 175-189, 1999. [7] C. Giguere and P. C. Woodland, "A computational model of the auditory periphery for speech and hearing research. i. ascending path," J. Acoust. Soc. Am., vol. 95, pp. 331-342, 1994. [8] C. Neti, E. Young, and M. Schneider, "Neural network models of sound localization based on directional filtering by the pinna," J. Acoust. Soc. Am., vol. 92, no. 6, pp. 3140-3156, 1992. [9] J. Middlebrooks, "Narrow-band sound localization related to external ear acoustics," J. Acoust. Soc. Am., vol. 92, no. 5, pp. 2607-2624, 1992. [10] N. Fisher, I, T. Lewis, and B. J. J. Embleton, Statistical analysis of spherical data. Cambridge: Cambridge University Press, 1987.
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A Neuromorphic VLSI System for Modeling the Neural Control of Axial Locomotion Girish N. Patel girish@ece.gatech.edu Edgar A. Brown ebrown@ece.gatech.edu Stephen P. De Weerth steved@ece.gatech.edu School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, Ga. 30332-0250 Abstract We have developed and tested an analog/digital VLSI system that models the coordination of biological segmental oscillators underlying axial locomotion in animals such as leeches and lampreys. In its current form the system consists of a chain of twelve pattern generating circuits that are capable of arbitrary contralateral inhibitory synaptic coupling. Each pattern generating circuit is implemented with two independent silicon Morris-Lecar neurons with a total of 32 programmable (floating-gate based) inhibitory synapses, and an asynchronous address-event interconnection element that provides synaptic connectivity and implements axonal delay. We describe and analyze the data from a set of experiments exploring the system behavior in terms of synaptic coupling. 1 Introduction In recent years, neuroscientists and modelers have made great strides towards illuminating structure and computational properties in biological motor systems. For example, much progress has been made toward understanding the neural networks that elicit rhythmic motor behaviors, including leech heartbeat, crustacean stomatogastric mill and lamprey swimming (a good review on these is in [1] and [2]). It is thought that these same mechanisms form the basis for more complex motor behaviors. The neural substrate for these control mechanisms are called central pattern generators (CPG). In the case of locomotion these circuits are distributed along the body (in the spinal cord of vertebrates or in the ganglia of invertebrates) and are richly interactive with sensory input and descending connections from the brain, giving rise to a highly distributed system as shown in Figure 1. In cases in which axial locomotion is involved, such as leech and lamprey swimming, synaptic interconnection patterns among autonomous segmental oscillators along the animal's axis produce coordinated motor patterns. These intersegmental coordination architectures have been well studied through both physiological experimentation and mathematical modeling. In addition, undulatory gaits in snakes have also been studied from a robotics perspective [3]. However, a thorough understanding of the computational principles in these systems is still lacking. A Neuromorphic System for Modeling Axial Locomotion 725 Figure 1: Neuroanatomy of segmented animals. In order to better understand the computational paradigms that mediate intersegmental coordination and the resulting neural control of axial locomotion (and other motor patterns), we are using neuromorphic very large-scale integrated (VLSI) circuits to develop models of these biological systems. The goals in our research are (i) to study how the properties of individual neurons in a network affect the overall system behavior; (ii) to facilitate the validation of the principles underlying intersegmental coordination; and (iii) to develop a real-time, low power, motion control system. We want to exploit these principles and architectures both to improve our understanding of the biology and to design artificial systems that perform autonomously in various environments. Embedded Controller Event Output " Parameter Input Address-Event Communication Network 12 segments / Figure 2: Block-level diagram of the implemented system. The intersegmental communications network facilitates communication among the intrasegmental units with pipelined stages. In this paper, we present a VLSI model of intersegmental coordination as shown in Figure 2. Each segment in our system is implemented with a custom Ie containing a CPG consisting of two silicon model neurons, each one with 16 inhibitory synapses whose values are stored on chip and are continuously variable; an asynchronous address event communications IC that implements the queuing and delaying of events providing synaptic connectivity and thus simulating axonal properties; and a microcontroller (with internal AID converter and timer) that facilitates the modification of individual parameters through a serial bus. The entire system consists of twelve such segments linked to a computer on which a graphical user interface (GUI) is implemented. By using the GUI, we are able to control all of the synaptic connections in the system and to measure the result726 G. N Patel, E. A. Brown and S. P. DeWeerth ing neural outputs. We present the system model, and we investigate the role of synaptic coupling in the establishment of phase lags along this chain of neural oscillators. 2 Pattern generating circuits The smallest neural system capable of generating the basic alternating activity that characterizes the swimming ePGs is the half-center oscillator, essentially two bursting neurons with reciprocally inhibitory connections [1] as shown in Figure 3a. In biological systems, the associated neurons have both slow and fast time constants to facilitate the fast spiking (action potentials) and the slower bursting oscillations that control the elicited movements as shown in Figure 3b. To simplify the parameter space of our system, we use reduced two-state silicon neurons [4]. The output of each silicon neuron is an oscillation that represents the envelope of the bursting activity (i.e. the spiking activity and corresponding fast time constants are eliminated) as shown in Figure 3c. Each neuron also has 16 analog synapses that receive off-chip input. The synaptic parameters are stored in an array of floating-gate transistors [5] that provide nonvolatile analog memory. CPG 1 JW~WID~,J,U~l.v B 2 111 111111111111 1111111111111111 11111 111111 1111 1111 III 1 C 2 A Figure 3: Half-center oscillator and the generation of events in spiking and nonspiking silicon neurons. Events are generated by detecting rapid rises in the membrane potential of spiking neurons or by detecting rapid rises and falls in nonspiking neurons. 3 Intersegmental communication Our segmented system consists of an array of ePG circuits interconnected via an communication network that implements an asynchronous, address-event protocol [6][7]. Each ePG is connected to one node of this address-event intersegmental communication system as illustrated in Figure 2. This application-specific architecture uses a pipelined broadcast scheme that is based upon its biological counterpart. The principal advantage of using this custom scheme is that requisite addresses and delays are generated implicitly based upon the system architecture. In particular the system implements distance-dependent delays and relative addressing. The delays, which are thought to be integral to the network computation, replicate the axonal delays that result as action potentials propagate down an animal's body [2]. The relative addressing greatly simplifies the implementation of synaptic spread [8], the hypothesized translational invariance in the intersegmental connectivity in biological axial locomotion systems. Thus, we can set the synaptic parameters identically at every segment, greatly reducing system complexity. In this architecture (which is described in more depth in [4]), each event is passed from segment to neighboring segment bidirectionally down the length of the one-dimensional A Neuromorphic System for Modeling Axial Locomotion 727 communications network. By delaying each event at every segment, the pipeline architecture facilitates the creation of distance-dependent delays. The other primary advantage of this architecture is that it can easily generate a relative addressing scheme. Figure 4 illustrates the event-passing architecture with respect to the relative addressing and distancedependent delays. Each event, generated at a particular node (the center node, in this example), is transmitted bidirectionally down the length of the network. It is delayed by time /). T at each segment, not including the initiating segment. t=to+2/).T t=to+/).T t=to+/).T t=to+2/).T '8"'O"'8~8~c;( A = -2 A = -1 A = 0 A = 1 V Figure 4: Relative addressing and distance-dependent delays. The events are generated by the neurons in each segment. Because these are not spiking neurons, we could not use the typical scheme of generating one event per action potential. Instead, we generate one event at the beginning and end of each burst (as illustrated in Figure 3) and designate the individual events as rising or falling. In each segment the events are stored in a queue (Figure 5), which implements delay based upon uniform conduction velocities. As an event arrives at each new segment, it is time stamped, its relative address is incremented (or decremented), and then it is stored in the queue for the /). T interval. As the event exits the queue, its data is decoded by the intrasegmental units, and synaptic inputs are applied to the appropriate intrasegmental neurons. events from rostral segment (clo ser to head) .. ... 9: .. ev ents from caudal segment (closer to tail) QUEUE .. (event storage and processing) events from intrasegmental unit l. 9: .. ... .. ... to ro stral and cau dal segments intrasegmental and unit Figure 5: Block-level diagram of a communications node illustrating how events enter and exit each stage of the pipeline. 4 Experiments and Discussion We have implemented the complete system shown in Figure 2, and have performed a number of experiments on the system. In Figure 6, we show the behaviors the system exhibits when it is configured with asymmetrical nearest-neighbor connections. The system displays traveling waves whose directions depend on the direction of the dominant coupling. Note that the intersegmental phase lags vary for different swim frequencies. One important set of experiments focussed on the role of long-distance connections on the system behaviors. In these experiments, we configured the system with strong descending (towards the tail) connections such that robust rearward traveling waves (forward swimming) are observed. The long-distance connections are weak enough to avoid any bifurcations in behavior (different type of behavior). Thus, the traveling wave solution resulting from the nearest-neighbor connections persists as we progressively add long-distance connections. In Figure 7 we show the dependency of the swim frequency and the total phase lag (summation of the normalized intersegmental phase lags, where 1 == 3600 ) on the extent of the connections. The results show a clear difference in behav728 stronger ascending coupling 1 0.5 o -0.2 B ......... . .. , -0.1 o 0.1 time (sec) 0.2 o 0.05 time (sec) ..... 1O&-" 8 ~ 6 (l) 4 CI) 2 G. N. Patel, E. A. Brown and S. P. DeWeerth stronger descending coupling c :: . 1 o.~ ~t=~~~~~ -0.2 -0.1 1 0.5 o 0.1 time (sec) 0.2 o .L.fI==::1.L:;:::1l:::=:,:..:-.J -0.05 o 0.05 time (sec) Figure 6: Traveling waves in the system with asymmetrical, nearest-neighbor connections. Plots are cross-correlations between rising edge events generated by a neuron in segment six and events generated by homolog neurons in each segment. Stronger ascending connections (A & B) produce forward traveling waves (backward swimming) and stronger descending connections (C & D) produce rearward traveling waves (forward swimming). An externally applied current (lext) controls the swim frequency. At small values of lext (6.7 nA) the periods of the swim cycles are approximately 0.180 ms and 0.150 ms for A & C, respectively; for large values of lext (32.8 nA), the periods of the swim cycles are approximately 36 ms and 33 ms for B & D, respectively. iors between the lowest tonic drive (lext = 21.9 nA) and the two higher tonic drives. (By tonic drive, we mean a constant dc current is applied to all neurons.) In the former, the sensitivity of long-distance connections on frequency and intersegmental phase lags is considerably greater than in the latter. The demarcation in behavior may be attributed to different behaviors at different tonic drives. For lower tonic drive, the long-distance connections tend to synchronize the system (decrease the intersegmental phase lags). At the higher tonic drives, long-distance connections do not affect the system considerably. For lext = 32.8 nA, connections that span up to four segments aid in producing uniformity in the intersegmental phase lags. Although this does not hold for lext = 48.1 nA, long-distance connections playa more significant role in preserving the total phase difference. At lext = 32.8 nA and lext = 48.1 nA, the system with short-distance connections produces a total phase difference of 1.19 and 1.33, respectively. In contrast, for lext = 32.8 nA and lext = 48.1 nA, the system with long-distance connections that span up to seven segments produces a total phase difference of 1.20 and 1.25, respectively. In the above experiments, we have demonstrated that, in a specific parameter regime, weak long-distance connections can affect the intersegmental phase lags. However, these weight profiles should not be construed as a possible explanation on what the weight profiles in a biological system might be. The parameter regime in which we observed this behavior is small; at moderate strengths of coupling, the traveling wave solutions disappear and move towards synchronous behavior. Recent experiments done on spinalized lampreys reveal that long-distance connections are moderately strong [10]. Thus, our CUTrent model is unable to replicate this aspect of intersegmental coordination. There are several explanations that may account for this discrepancy. A Neuromorphic System for Modeling Axial Locomotion A 40 r---~-~-~----' 30 i)' c ~ 20 cO> .... - 10 o~--~----~--~--~ o 2 4 extent 6 8 B 1.5 .---~-~-~--, m 1 ~ .c c. ~ :§ 0.5 o'----~----~--~----' o 2 4 extent 6 8 729 Figure 7: Effects of weak long-distance connections on swimming frequency (A), on the total phase difference (summation of the normalized intersegmental phase lags) (B), and on the standard deviation of the intersegmental phase lags (C). 5 < = denote Iext = 48.1 nA, 32.8 nA, and 21 nA, respectively. In the segmental CPG network of the animal, there are many classes of neurons that send projections to many other classes of neurons. The phase a connection imposes is determined by which neuron class connects with which other neuron class. In our system, the segmental CPG network has only a single class of neurons upon which the long-distance connections can impose their phase. Depending on where in parameter space we operate our system, the long-distance connections have too little or too great an effect on the behavior of the system. At high tonic drives, the sensitivity of the weak long-distance connections on the intersegmental phase lags is small, whereas for small tonic drives, the long-distance connections have a great effect on the intersegmental phase lags. It has been shown that if the waveform of the oscillators is sinusoidal (i.e., the time scales of the two state variables are not too different), traveling wave solutions exist and have a large basin of attraction [11]. However, as the disparity between the two time scales is made larger (i.e., the neurons are stiff and the waveform of the oscillations appears square-wave like), the system will move towards synchrony. In our implementation, to facilitate accurate communication of events, we bias the neurons with relatively large differences in the time scales. Thus, this restriction reduces the parameter regime in which we can observe stable traveling waves. Another factor that determines the range of parameters in which stable traveling waves are observed is the slope of our synaptic coupling function. When the slope of the coupling function is steep, the total synaptic current over a cycle can increase significantly, causing weak connections to appear strong. This has an overall effect of synchronizing the network [11]. For coupling functions whose slopes are shallow, the total synaptic current over a cycle is reduced; therefore, the connections appear weak and larger intersegmental phase lags are possible. Thus, the sharp synaptic coupling function in our implementation, which is necessary for communication, is another factor that diminishes the parameter regime in which we can observe stable traveling waves. The above factors limit the parameter range in which we observe traveling waves. However, all of these issues can be addressed by improving our CPG network. The first issue can be addressed by increasing the number of neuron classes or adding more segments. The second and third issues can be addressed by adding spiking neurons in our CPG network so that the form of the oscillations can be coded in the spike train and the synaptic coupling functions can be implemented on the receiving side of the CPG chip. The fourth 730 G. N. Patel, E. A. Brown and S. P. DeWeerth issue can be addressed by designing self-adapting neurons that tune their internal parameters so that their waveforms and intrinsic frequencies are matched. Although weak coupling may not be biologically plausible, producing traveling waves based on phase oscillators would be an interesting research direction. 5 Conclusions and Future Work In this paper, we described a functional, neuromorphic VLSI system that implements an array of neural oscillators interconnected by an address-event communication network. This system represents our most ambitious neuromorphic VLSI effort to date, combining 24 custom ICs, a special-purpose asynchronous communication architecture designed analogously to its biological counterpart, large-scale synaptic interconnectivity with parameters stored using floating-gate devices, and a computer interface for setting the parameters and for measuring the neural activity. The working system represents the culmination of a four-year effort, and now provides a testbed for exploring a variety of biological hypotheses and theoretical predictions. Our future directions in the development of this system are threefold. First, we will continue to explore, in depth, the operation of the present system, comparing it to theoretical predictions and biological hypotheses. Second, we are implementing a segmented mechanical system that will provide a moving output and will facilitate the implementation of sensory feedback. Third, we are developing new CPG model centered around sensory feedback and motor learning. The modular design of the system, which puts all of the neural and synaptic specificity on the CPG IC, allows us to design a completely new CPG and to replace it in the system without changing the communication architecture. References [1] E. Marder & R.L. Calabrese. Principles of rhythmic motor pattern generation. Physiological Reviews 76 (3): 687-717,1996. [2] A.H. Cohen, G.B. Ermentrout, T Kiemel, N. Kopell, K.A. Sigvardt, & TL. Williams. Modeling of intersegmental coordination in the lamprey central pattern generator for locomotion. TINS 15:434-438, 1992. [3] S. Hirose. Biologically Inspired Robots: Snake-like Locomotors and Manipulators. Oxford University Press, 1993. [4] S. DeWeerth, G. Patel, D. Schimmel, M. Simoni, & R.L. Calabrese. A VLSI Architecture for Modeling Intersegmental Coordination. In Proceedings of the Seventeenth Conference on Advanced Research in VLSI, R.B. Brown and A.T Ishii (eds), Los Alamitos, CA: IEEE Computer Society, 182-200, 1997. [5] P. Hasler, B.A. Minch, and C. Diorio. Adaptive circuits using pFet floating-gate devices. In Scott Wills and Stephen DeWeerth editors, 20th Conference of Advanced Research in VLSI, pages 215-230, Los Alamitos, California, CA: IEEE Computer Society, 1999. [6] M.A. Mahowald. VLSI Analogs of Neuronal Visual Processing: A Synthesis of Form and Function. Ph.D. Thesis, California Institute of Technology, Pasadena, CA, 1992. [7] K.A. Boahen. Communicating Neuronal Ensembles between Neuromorphic Chips. Analog Integrated Circuits and Signal Processing, 1997. [8] T Wiliams. Phase Coupling and Synaptic Spread in Chains of Coupled Neuronal Oscillators. Science, vol. 258, 1992. [9] G. Patel. A Neuromorphic Architecture for Modeling Intersegmental Coordination. Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA, 1999. [10] A. H. Cohen. Personal communication. [11] D. Somers & N. Kopell. Waves and synchrony in networks of oscillators of relaxation and nonrelaxation type. Phyica D, 89:169-183,1995. [12] N. Kopell & G.B. Ermentrout. Coupled oscillators and the design of central pattern generators. Mathematical Biosciences, 90:87-109, 1988.
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Graded grammaticality in Prediction Fractal Machines Shan Parfitt, Peter Tiilo and Georg Dorffner Austrian Research Institute for Artificial Intelligence, Schottengasse 3, A-IOIO Vienna, Austria. { shan,petert,georg} @ai. univie. ac. at Abstract We introduce a novel method of constructing language models, which avoids some of the problems associated with recurrent neural networks. The method of creating a Prediction Fractal Machine (PFM) [1] is briefly described and some experiments are presented which demonstrate the suitability of PFMs for language modeling. PFMs distinguish reliably between minimal pairs, and their behavior is consistent with the hypothesis [4] that wellformedness is 'graded' not absolute. A discussion of their potential to offer fresh insights into language acquisition and processing follows. 1 Introduction Cognitive linguistics has seen the development in recent years of two important, related trends. Firstly, a widespread renewal of interest in the statistical, 'graded' nature of language (e.g. [2]-[4]) is showing that the traditional all-or-nothing notion of well-formedness may not present an accurate picture of how the congruity of utterances is represented internally. Secondly, the analysis of state space trajectories in artificial neural networks (ANNs) has provided new insights into the types of processes which may account for the ability of learning devices to acquire and represent language, without appealing to traditional linguistic concepts [5]-[7]. Despite the remarkable advances which have come out of connectionist research (e.g. [8]), and the now common use of recurrent networks, and Simple Recurrent Networks (SRNs) [9] especially, in the study of language (e.g. [10]), recurrent neural networks suffer from particular problems which make them imperfectly suited to language tasks. The vast majority of work in this field employs small networks and datasets (usually artificial), and although many interesting linguistic issues may be thus tackled, real progress in evaluating the potentials of state trajectories and graded 'grammaticality' to uncover the underlying processes responsible for overt linguistic phenomena must inevitably be limited whilst the experimental tasks remain so small. Nevertheless, there are certain obstacles to the scaling-up of networks trained by back-propagation (BP). Such networks tend towards ever Graded Grammaticality in Prediction Fractal Machines 53 longer training times as the sizes of the input set and of the network increase, and although Real-Time Recurrent Learning (RTRL) and Back-propagation Through Time are potentially better at modeling temporal dependencies, training times are longer still [11]. Scaling-up is also difficult due to the potential for catastrophic interference and lack of adaptivity and stability [12]-[14]. Other problems include the rapid loss of information about past events as the distance from the present increases [15] and the dependence of learned state trajectories not only on the training data, but also upon such vagaries as initial weight vectors, making their analysis difficult [16]. Other types of learning device also suffer problems. Standard Markov models require the allocation of memory for every n-gram, such that large values of n are impractical; variable-length Markov models are more memory-efficient, but become unmanageable when trained on large data sets [17]. Two important, related concerns in cognitive linguistics are thus (a) to find a method which allows language models to be scaled up, which is similar in spirit to recurrent neural networks, but which does not encounter the same problems of scale, and (b) to use such a method to evince new insights into graded grammaticality from the state trajectories which arise given genuinely large, naturally-occurring data sets. Accordingly, we present a new method of generating state trajectories which avoids most of these problems. Previously studied in a financial prediction task, the method creates a fractal map of the training data, from which state machines are built. The resulting models are known as Prediction Fractal Machines (PFMs) [18] and have some useful properties. The state trajectories in the fractal representation are fast and computationally efficient to generate, and are accurate and well-understood; it may be inferred that, even for very large vocabularies and training sets, catastrophic interference and lack of adaptivity and stability will not be a problem, given the way in which representations are built (demonstrating this is a topic for future work); training times are significantly less than for recurrent networks (in the experiments described below, the smallest models took a few minutes to build, while the largest ones took only around three hours; in comparison, all of the ANNs took longer - up to a day - to train); and there is little or no loss of information over the course of an input sequence (allowing for the finite precision of the computer). The scalability of the PFM was taken advantage of by training on a large corpus of naturally-occurring text. This enabled an assessment of what potential new insights might arise from the use of this method in truly large-scale language tasks. 2 Prediction Fractal Machines (PFMs) A brief description of the method of creating a PFM will now be given. Interested readers should consult [1], since space constraints preclude a detailed examination here. The key idea behind our predictive model is a transformation F of symbol sequences from an alphabet (here, tagset) {I, 2, ... , N} into points in a hypercube H = [0, I]D. The dimensionality D of the hypercube H should be large enough for each symbol 1, 2, ... , N to be identified with a unique vertex of H. The particular assignment of symbols to vertices is arbitrary. The transformation F has the crucial property that symbol sequences sharing the same suffix (context) are mapped close to each other. Specifically, the longer the common suffix shared by two sequences, the smaller the (Euclidean) distance between their point representations. The transformation F used in this study corresponds to an Iterative Function System [19] 54 S. Parfitt, P TIno and G. Dorffner consisting of N affine maps i : H -+ H, i = 1,2, ... , N, i(x) = ~(x + ti), tj E {a, l}D, ti =F tj for i =F j. (1) Given a sequence 5182 ... 5L of L symbols from the alphabet 1,2, ... , N, we construct its point representation as where x* is the center {l}D of the hypercube H. (Note that as is common in the Iterative Function Systems literature, i refers either to a symbol or to a map, depending upon the context.) PFMs are constructed on point representations of subsequences appearing in the training sequence. First, we slide the window of length L > 1 over the training sequence. At each position we transform the sequence of length L appearing in the window into a point. The set of points obtained by sliding through the whole training sequence is then partitioned into several classes by k-means vector quantization (in the Euclidean space), each class represented by a particular codebook vector. The number of code book vectors required is chosen experimentally. Since quantization classes group points lying close together, sequences having point representations in the same class (potentially) share long suffixes. The quantization classes may then be treated as prediction contexts, and the corresponding predictive symbol probabilities computed by sliding the window over the training sequence again and counting, for each quantization class, how often a sequence mapped to that class was followed by a particular symbol. In test mode, upon seeing a new sequence of L symbols, the transformation F is again performed, the closest quantization center found, and the corresponding predictive probabilities used to predict the next symbol. 3 An experimental comparison of PFMs and recurrent networks The performance of the PFM was compared against that of a RTRL-trained recurrent network on a next-tag prediction task. Sixteen grammatical tags and a 'sentence start' character were used. The models were trained on a concatenated sequence (22781 tags) of the top three-quarters of each of the 14 sub-corpora of the University of Pennsylvania 'Brown' corpus1 . The remainder was used to create test data, as follows. Because in a large training corpus of naturally-occurring data, contexts in most cases have more than one possible correct continuation, simply counting correctly predicted symbols is insufficient to assess performance, since this fails to count correct responses which are not targets. The extent to which the models distinguished between grammatical and ungrammatical utterances was therefore additionally measured by generating minimal pairs and comparing their negative log likelihoods (NLLs) per symbol with respect to the model. Likelihood is computed by sliding through the test sequence and for each window position, determining the probability of the symbol that appears immediately beyond it. As processing progresses, these probabilities are multiplied. The negative of the natural logarithm is then taken and divided by the number of symbols. Significant differences in NLLs Ihttp://www.ldc.upenn.edu/ Graded Grammaticality in Prediction Fractal Machines 55 are much harder to achieve between members of minimal pairs than between grammatical and random sequences, and are therefore a good measure of model validity. Minimal pairs generated by theoretically-motivated manipulations tend to be no longer ungrammatical given a small tagset, because the removal of grammatical sub-classes necessarily also removes a large amount of information. Manipulations were therefore performed by switching the positions of two symbols in each sentence in the test sets. Symbols switched could be any distance apart within the sentence, as long as the resulting sentence was ungrammatical under all surface instantiations. By changing as little as possible to make the sentence ungrammatical, the goal was retained that the task of distinguishing between grammatical and ungrammatical sequences be as difficult as possible. The test data then consisted of 28 paired grammatical/ungrammatical test sets (around 570 tags each), plus an ungrammatical, 'meaningless' test set containing all 17 codes listed several times over, used to measure baseline performance. Ten 1st-order randomly-initialised networks were trained for 100 epochs using RTRL. The networks consisted of 1 input and 1 output layer, each with 17 units corresponding to the 17 tags, 2 hidden layers, each with 10 units, and 1 context layer of 10 units connected to the first hidden layer. The second hidden layer was used to increase the flexibility of the maps between the hidden representations in the recurrent portion and the tag activations at the output layer. A logistic sigmoid activation function was used, the learning rate and momentum were set to 0.05, and the training sequence was presented at the rate of one tag per clock tick. The PFMs were derived by clustering the fractal representation of the training data ten times for various numbers of codebook vectors between 5 and 200. More experiments were performed using PFMs than neural networks because in the former case, experience in choosing appropriate numbers of codebook vectors was initially lacking for this type of data. The results which follow are given as averages, either over all neural networks, or else over all PFMs derived from a given number of codebook vectors. The networks correctly predicted 36.789% and 32.667% of next tags in the grammatical and ungrammatical test sets, respectively. The PFMs matched this performance at around 30 codebook vectors (37.134% and 32.814% respectively), and exceeded it for higher numbers of vectors (39.515% and 34.388% respectively at 200 vectors). The networks generated mean NLLs per symbol of 1.966 and 2.182 for the grammatical and ungrammatical test sets, respectively (a difference of 0.216) and 4.157 for the 'meaningless' test set (the difference between NLLs for grammatical and 'meaningless' data = 2.191). The PFMs matched this difference in NLLs at 40 codebook vectors (NLL grammatical = 1.999, NLL ungrammatical = 2.217; difference = 0.218). The NLL for the 'meaningless' data at 40 codebook vectors was 6.075 (difference between NLLs for grammatical and 'meaningless' data = 4.076). The difference between NLLs for grammatical and ungrammatical, and for grammatical and 'meaningless' data sets, became even larger with increased numbers of codebook vectors. The difference in performance between grammatical and ungrammatical test sets was thus highly significant in all cases (p < .0005): all the models distinguished what was grammatical from what was not. This conclusion is supported by the fact that the mean, NLLs for the 'meaningless' test set were always noticeably higher than those for the minimal pair sets. 56 S. Parfitt, P. Tina and G. Daiffner 4 Discussion The PFMs exceeded the performance of the networks for larger numbers of codebook vectors, but it is possible that networks with more hidden nodes would also do better. In terms of ease of use, however, as well as in their scaling-up potential, PFMs are certainly superior. Their other great advantage is that the representations created are dependable (see section 1), making hypothesis creation and testing not just more rapid, but also more straightforward: the speed with which PFMs may be trained made it possible to make statistically significant observations for a large number of clustering runs. In the introduction, 'graded' wellformedness was spoken of as being productive of new hypotheses about the nature of language. Our use of minimal pairs, designed to make a clear-cut distinction between grammatical and ungrammatical utterances, appears to leave this issue to one side. But in reality, our results were rather pertinent to it, as the use of the likelihood measure might indeed imply. The Brown corpus consists of subcorpora representative of 14 different discourse types, from fiction to government documents. Whereas traditional notions of grammaticality would lead us to treat all of the 'ungrammatical' sentences in the minimal pair test sets as equally ungrammatical, the NLLs in our experiments tell a different story. The grammatical versions consistently had a lower associated NLL (higher probability) than the ungrammatical versions, but the difference between these was much smaller than that between the 'meaningless' data and either the grammatical or the ungrammatical data. This supports the concept of 'graded grammaticality', and NLLs for 'meaningless' data such as ours might be seen as a sort of benchmark by which to measure all lesser degrees ofungrammaticality. (Note incidentally that the PFMs appear to associate with the 'meaningless' data a significantly higher NLL than did the networks, even though the difference between the NLLs of the grammatical and ungrammatical data was the same. This is suggestive of PFMs having greater powers of discrimination between grades of wellformedness than the recurrent networks used, but further research will be needed to ascertain the validity of this.) Moreover, the NLL varied not just between grammatical and ungrammatical test sets, but also from sentence to sentence, from word to word and from discourse style to discourse style. While it increased, often dramatically, when the manipulated portion of an ungrammatical sentence was encountered, some words in grammatical sentences exhibited a similar effect: thus, if a subsequence in a well-formed utterance occurs only rarely - or never - in a training set, it will have a high associated NLL in the same way as an ungrammatical one does. This is likely to happen even for very large corpora, since some grammatical structures are very rare. This is consistent with recent findings that, during human sentence processing, well-formedness is linked to conformity with expectation [20] as measured by CLOZE scores. Interesting also was the remarkable variation in NLL between discourse styles. Although the mean NLL across all discourse styles (test sets) is lower for the grammatical than for the ungrammatical versions, it cannot be guaranteed that the grammatical version of one test set will have a lower NLL than the ungrammatical version of another. Indeed, the grammatical and ungrammatical NLLs interleave, as may be observed in figure 1, which shows the NLLs for the three discourse styles which lie at the bottom, middle and top of the range. Even more interestingly, if the NLLs for the grammatical versions of all discourse styles are ordered according to where they lie within this range, it becomes clear that NLL is a predictor of discourse style. Styles which linguists class as 'formal', e.g. those of Graded Grammaticality in Prediction Fractal Machines :::I z : ~ NU,s associaIed wi1h grammatical and ungrammalical versions of 3 discourse types 3r----------r~~------~--------~------~--~ 2.8 2.8 Leamed text: grammatical -leamed text ungrammalical ....... Romantic: fiction: grammatical .•.... Romantfc fiction: ungrammalical -+Science fiction: grammatical •.•.• Science fiction: ungrammatical -D .• .. ....... _a·a__ ----'" ........ --"'G. e _.-a ... ·..£t· .. ·q ·.. __ .. _ .. ---... 1:).. B:Ie...... ._ .. ----...2.4 \. .--+-'---........ _ .... -.-.-.--t::::~~~::-::-:-·-=~·· ....... ' 2.2 2 \'--_.. ___ --•• _.-•• ~~~~~~:::.:~~~:~~~~-::~~: •. :;:::-:::::.::::.:.:;:::.:;:::.:;;::::;;;:::;:::.::~::"::,. ~ 1.8 o 50 100 150 200 No. of codebook vectors 57 Figure 1: NLLs of minimal pair test sets containing different discourse styles suggest grades of wellformedness based upon prototypicality. the Learned and Government Document test sets, have the lowest NLLs, with the three Press test sets clustering just above, and the Fiction test sets, exemplifying creative language use, clustering at the high end. Similarly, that the Learned and Government test sets have the lowest NLLs conforms with the intuition that their usage lies closest to what is grammatically 'prototypical' - even though in the training set, 6 out of the 14 test sets are fiction and thus might be expected to contribute more to the prototype. That they do not, suggests that usage varies significantly across fiction test sets. 5 Conclusion Work on the use of PFMs in language modeling is at an early stage, but as results to date show, they have a lot to offer. A much larger project is planned, which will examine further Allen and Seidenberg's hypothesis that 'graded grammaticality' (or wellformedness) applies not only to syntax, but also to other language subdomains such as semantics, an integral part of this being the use of larger corpora and tagsets, and the identification of vertices with semantic/syntactic features rather than atomic symbols. Identifying the possibilities of combining PFMs with ANNs, for example as a means of bypassing the normal method of creating state-space trajectories, is the subject of current study. Acknowledgments This work was supported by the Austrian Science Fund (FWF) within the research project "Adaptive Information Systems and Modeling in Economics and Management Science" (SFB 010). The Austrian Research Institute for Artificial Intelligence is supported by the Austrian Federal Ministry of Science and Transport. 58 S. Parfitt, P Tino and G. DorjJner References [1] P. Tino & G. Dorffner (1998). Constructing finite-context sources from fractal representations of symbolic sequences. Technical Report TR-98-18, Austrian Research Institute for AI, Vienna. [2] J. R. Taylor (1995). Linguistic categorisation: Prototypes in linguistic theory. Clarendon, Oxford. [3] J. R. Saffran, R. N. Aslin & E. L. Newport (1996). Statistical cues in language acquisition: Word segmentation by infants. In Proc. of the Cognitive Science Society Conference, 376-380, La Jolla, CA. [4] J. Allen & M. S. Seidenberg (in press). The emergence of grammaticality in connectionist networks. In B. Macwhinney (ed.), Emergentist approaches to language: Proc. of the 28th Carnegie Symposium on cognition. Erlbaum. [5] S. Parfitt (1997). Aspects of anaphora resolution in artificial neural networks: Implications for nativism. PhD thesis, Imperial College, London. [6] D. Servan-Schreiber et al (1989). Graded state machines: The representation of temporal contingencies in Simple Recurrent Networks. In Advances in Neural Information Processing Systems, 643-652. [7] W. Tabor & M. Tanenhaus (to appear). Dynamical models of sentence processing. Cognitive Science. [8] J. L. Elman et al (1996). Rethinking innateness: A connectionist perspective on development. Bradford. [9] J . L. Elman (1990). Finding structure in time. In: Cognitive Science, 14: 179-211. [10] S. Lawrence, C. Lee Giles & S. Fong (in press). Natural language grammatical inference with recurrent neural networks. IEEE Trans. on knowledge and data engineering. [11] J. Hertz, A. Krogh & R. G. Palmer (1991). Introduction to the theory of neural computation. Addison Wesley. [12] M. McCloskey & N. J. Cohen (1989). Catastrophic interference in connectionist networks: The sequential learning problem. In G. Bower (ed.), The psychology of learning and motivation, vol 24. Academic, NY. [13] J. K. Kruschke (1991). ALCOVE: A connectionist model of human category learning. In R. P. Lippman et al (eds.), Advances in Neural Information Processing 9, 649-655. Kaufmann, San Mateo, CA. [14] S. Grossberg (ed.) (1988). Neural networks and natural intelligence. Bradford, MIT, Cambs, MA. [15] Y. Bengio, P. Simard & P. Frasconi (1994). Learning long-term dependencies with gradient descent is difficult. IEEE Trans. on neural networks, 5(2). [16] M. P. Casey (1996). The dynamics of discrete-time computation, with application to recurrent neural networks and finite-state machine extraction. Neural Computation, 8(6):1135-1178. [17] D. Ron, Y. Singer & N. Tishby (1996). The power of amnesia. Machine Learning, 25. [18] P. Tino, B. G. Horne, C. Lee Giles & P. C. Collingwood (1998). Finite state machines and recurrent neural networks - automata and dynamical systems approaches. In J. E. Dayhoff & O. Omidvar (eds.), Neural Networks and Pattern Recognition, 171- 220. Academic. [19] M. F. Barnsley (1988). Fractals everywhere. Academic, NY. [20] S. Coulson, J. W. King & M. Kutas (1998). Expect the unexpected: Responses to morphosyntactic violations. Language and Cognitive Processes, 13(1).
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Learning sparse codes with a mixture-of-Gaussians prior Bruno A. Olshausen Department of Psychology and Center for Neuroscience, UC Davis 1544 Newton Ct. Davis, CA 95616 baolshausen@ucdavis.edu K. Jarrod Millman Center for Neuroscience, UC Davis 1544 Newton Ct. Davis, CA 95616 kjmillman@ucdavis. edu Abstract We describe a method for learning an overcomplete set of basis functions for the purpose of modeling sparse structure in images. The sparsity of the basis function coefficients is modeled with a mixture-of-Gaussians distribution. One Gaussian captures nonactive coefficients with a small-variance distribution centered at zero, while one or more other Gaussians capture active coefficients with a large-variance distribution. We show that when the prior is in such a form, there exist efficient methods for learning the basis functions as well as the parameters of the prior. The performance of the algorithm is demonstrated on a number of test cases and also on natural images. The basis functions learned on natural images are similar to those obtained with other methods, but the sparse form of the coefficient distribution is much better described. Also, since the parameters of the prior are adapted to the data, no assumption about sparse structure in the images need be made a priori, rather it is learned from the data. 1 Introduction The general problem we address here is that of learning a set of basis functions for representing natural images efficiently. Previous work using a variety of optimization schemes has established that the basis functions which best code natural images in terms of sparse, independent components resemble a Gabor-like wavelet basis in which the basis functions are spatially localized, oriented and bandpass in spatial-frequency [1, 2, 3, 4]. In order to tile the joint space of position, orientation, and spatial-frequency in a manner that yields useful image representations, it has also been advocated that the basis set be overcomplete [5], where the number of basis functions exceeds the dimensionality of the images being coded. A major challenge in learning overcomplete bases, though, comes from the fact that the posterior distribution over the coefficients must be sampled during learning. When the posterior is sharply peaked, as it is when a sparse prior is imposed, then conventional sampling methods become especially cumbersome. 842 B. A. Olshausen and K. J. Millman One approach to dealing with the problems associated with overcomplete codes and sparse priors is suggested by the form of the resulting posterior distribution over the coefficients averaged over many images. Shown below is the posterior distribution of one of the coefficients in a 4 x's overcomplete representation. The sparse prior that was imposed in learning was a Cauchy distribution and is overlaid (dashed line). It would seem that the coefficients do not fit this imposed prior very well, and instead want to occupy one of two states: an inactive state in which the coefficient is set nearly to zero, and an active state in which the coefficient takes on some significant non-zero value along a continuum. This suggests that the appropriate choice of prior is one that is capable of capturing these two discrete states. II I I • I I I I \ . -2 0 2 coo!IIdon1v .... Figure 1: Posterior distribution of coefficients with Cauchy prior overlaid. Our approach to modeling this form of sparse structure uses a mixture-of-Gaussians prior over the coefficients. A set of binary or ternary state variables determine whether the coefficient is in the active or inactive state, and then the coefficient distribution is Gaussian distributed with a variance and mean that depends on the state variable. An important advantage of this approach, with regard to the sampling problems mentioned above, is that the use of Gaussian distributions allows an analytical solution for integrating over the posterior distribution for a given setting of the state variables. The only sampling that needs to be done then is over the binary or ternary state variables. We show here that this problem is a tractable one. This approach differs from that taken previously by Attias [6] in that we do not use variational methods to approximate the posterior, but rather we rely on sampling to adequately characterize the posterior distribution over the coefficients. 2 Mixture-of-Gaussians model An image, I(x, y), is modeled as a linear superposition of basis functions, ¢i(X, y), with coefficients ai, plus Gaussian noise II( x, y): (1) In what follows this will be expressed in vector-matrix notation as 1= q. a + II. The prior probability distribution over the coefficients is factorial, with the distribution over each coefficient ai modeled as a mixture-of-Gaussians distribution with either two or three Gaussians (fig. 2). A set of binary or ternary state variables Si then determine which Gaussian is used to describe the coefficients. The total prior over both sets of variables, a and s, is of the form (2) Learning Sparse Codes with a Mixture-of-Gaussians Prior Two Gaussians (binary state variables) P(lIj) ai Three Gaussians (ternary state variables) P(lIj) _,;=-1 .<;=1 Figure 2: Mixture-of-Gaussians prior. 843 ai where P(Si) determines the probability of being in the active or inactive states, and P(ailsi) is a Gaussian distribution whose mean and variance is determined by the current state Si. The total image probability is then given by where P(IIO) = L P(s/O) J P(I/a, O)P(als, O)da s P(Ila, 0) P(als,O) P(sIO) 1 -~II-4>aI2 --e 2 ZAN _l_e-t(a-Il(s))t Aa(s) (a-Il(s)) ZAa(s) 1 _1s t A s --e 2 • ZA. (3) (4) (5) (6) and the parameters 0 include AN, 4), Aa(s), f.L(s), and As. Aa(s) is a diagonal inverse covariance matrix with elements Aa(S)ii = Aa; (Si). (The notations Aa(s) and f.L(s) are used here to explicitly reflect the dependence of the means and variances of the ai on sd As is also diagonal (for now) with elements ASii = As;. The model is illustrated graphically in figure 3. Si (binary or ternary) Figure 3: Image model. 844 B. A. Olshausen and K. J. Millman 3 Learning The objective function for learning the parameters of the model is the average log-likelihood: £ = (log P(IIO)) (7) Maximizing this objective will minimize the lower bound on coding length. Learning is accomplished via gradient ascent on the objective, £. The learning rules for the parameters As, Aa (s), J-t( s) and ~ are given by: ex = {)£ {)>"Si 1 2 [(Si)P(SiI9) - (si)P(sII,9)] {)£ {)>"ai (u) ! [(8(Si - u))P(sII,9)_ 2 >"ai (u) (8(Si - u) (Kii(U) - 2ai(U)J-ti(U) + J-t~(u)))P(sII,9)] {)£ {)J-ti( u) >"ai (u) (8(Si - u) (ai(u) - J-ti(U))) {)£ {)~ >"N [I (a(s)) P(sII,9) - ~ (K(s)) P(SII,9)] (8) (9) (10) (11) where u takes on values 0,1 (binary) or -1,0,1 (ternary) and K(s) H-1 (s) + a(s) a(s)T. (a and H are defined in eqs. 15 and 16 in the next section.) Note that in these expressions we have dropped the outer brackets averaging over images simply to reduce clutter. Thus, for each image we must sample from the posterior P(sll, 0) in order to collect the appropriate statistics needed for learning. These statistics must be accumulated over many different images, and then the parameters are updated according to the rules above. Note that this approach differs from that of Attias [6] in that we do not attempt to sum over all states, s, or to use the variational approximation to approximate the posterior. Instead, we are effectively summing only over those states that are most probable according to the posterior. We conjecture that this scheme will work in practice because the posterior has significant probability only for a small fraction of states s, and so it can be well-characterized by a relatively small number of samples. Next we present an efficient method for Gibbs sampling from the posterior. 4 Sampling and inference In order to sample from the posterior P(sll,O), we first cast it in Boltzmann form: P(sll,O) ex e-E(s) (12) where E(s) = -logP(s,IIO) = -logP(sIO) J P(lla,O)P(als,O)da Learning Sparse Codes with a Mixture-of-Gaussians Prior 845 ~ST Ass + 10gZAa(S) + Eals(a,s) + ~IOgdetH(s) + const. (13) and a = argminEals(a,s) a H(s) = \7\7 aEals(a, s) = )\Nif!T if! + Aa(s) (14) (15) (16) Gibbs-sampling on P(sII, (}) can be performed by flipping state variables Si according to P(Si t- sa) = P(Si t- sa) = l+e~E(·i~·a) 1 (binary) (ternary) (17) (18) Where sa = Si in the binary case, and sa and sf3 are the two alternative states in the ternary case. AE(Si t- sa) denotes the change in E(s) due to changing Si to sa and is given by: (19) where ASi = sa - Si, AAai = Aai (sa) - Aai (Si), J = H- 1, and Vi = Aai (Si) J.Li(Si). Note that all computations for considering a change of state are local and involve only terms with index i. Thus, deciding whether or not to change state can be computed quickly. However, if a change of state is accepted, then we must update J. Using the Sherman-Morrison formula, this can be kept to an O(N2) computation: J t- J [ AAak ] J k Jk 1 + AAak Jkk (20) As long as accepted state changes are rare (which we have found to be the case for sparse distributions), then Gibbs sampling may be performed quickly and efficiently. In addition, Hand J are generally very sparse matrices, so as the system is scaled up the number of elements of a that are affected by a flip of Si will be relatively few. In order to code images under this model, a single state of the coefficients must be chosen for a given image. We use for this purpose the MAP estimator: argmaxP(aII,s, (}) a arg max P(sII, (}) s (21) (22) Maximizing the posterior distribution over s is accomplished by assigning a temperature, P(sII, (}) ex e-E(s)/T (23) and gradually lowering it until there are no more state changes. 846 B. A. Olshausen and K. 1. Millman 5 Results 5.1 Test cases We first trained the algorithm on a number of test cases containing known forms of both sparse and non-sparse (bi-modal) structure, using both critically sampled (complete) and 2x's overcomplete basis sets. The training sets consisted of 6x6 pixel image patches that were created by a sparse superposition of basis functions (36 or 72) with P(ISil = 1) = 0.2, Aa; (0) = 1000, and Aa; (1) = 10. The results of these test cases confirm that the algorithm is capable of correctly extracting both sparse and non-sparse structure from data, and they are not shown here for lack of space. 5.2 Natural images We trained the algorithm on 8x8 image patches extracted from pre-whitened natural images. In all cases, the basis functions were initialized to random functions (white noise) and the prior was initialized to be Gaussian (both Gaussians of roughly equal variance). Shown in figure 4a, b are the results for a set of 128 basis functions (2 x 's overcomplete) in the two-Gausian case. In the three-Gaussian case, the prior was initialized to be platykurtic (all three Gaussians of equal variance but offset at three different positions). Thus, in this case the sparse form of the prior emerged completely from the data. The resulting priors for two of the coefficients are shown in figure 4c, with the posterior distribution averaged over many images overlaid. For some of the coefficients the posterior distribution matches the mixture-of-Gaussians prior well, but for others the tails appear more Laplacian in form. Also, it appears that the extra complexity offered by having three Gaussians is not utilized: Both Gaussians move to the center position and have about the same mean. When a non-sparse, bimodal prior is imposed, the basis function solution does not become localized, oriented, and bandpass as it does with sparse priors. 5.3 Coding efficiency We evaluated the coding efficiency by quantizing the coefficients to different levels and calculating the total coefficient entropy as a function of the distortion introduced by quantization. This was done for basis sets containing 48, 64, 96, and 128 basis functions. At high SNR's the overcomplete basis sets yield better coding efficiency, despite the fact that there are more coefficients to code. However, the point at which this occurs appears to be well beyond the point where errors are no longer perceptually noticeable (around 14 dB). 6 Conclusions We have shown here that both the prior and basis functions of our image model can be adapted to natural images. Without sparseness being imposed, the model both seeks distributions that are sparse and learns the appropriate basis functions for this distribution. Our conjecture that a small number of samples allows the posterior to be sufficiently characterized appears to hold. In all cases here, averages were collected over 40 Gibbs sweeps, with 10 sweeps for initialization. The algorithm proved capable of extracting the structure in challenging datasets in high dimensional spaces. The overcomplete image codes have the lowest coding cost at high SNR levels, but at levels that appear higher than is practically useful. On the other hand, the Learning Sparse Codes with a Mixture-of-Gaussians Prior 847 a. c. ::~~::~~ . - '~ ". ~:. " " 10'" . " 10.... .\ '0" -. 10~ J./ "'\. -2 -1 0 I 2 -Z _I 0 1 2 Figure 4: An overcomplete set of 128 basis functions (a) and priors (b, vertical axis is log-probability) learned from natural images. c, Two of the priors learned from a three-Gaussian mixture using 64 basis functions, with the posterior distribution averaged over many coefficients overlaid. d, Rate distortion curve comparing the coding efficiency of different learned basis sets. sum of marginal entropies likely underestimates the true entropy of the coefficients considerably, as there are certainly statistical dependencies among the coefficients. So it may still be the case that the overcomplete bases will show a win at lower SNR's when these dependencies are included in the model (through the coupling term As). Acknowledgments This work was supported by NIH grant R29-MH057921. References [1] Olshausen BA, Field DJ (1997) "Sparse coding with an overcomplete basis set: A strategy employed by VI?" Vision Research, 37: 3311-3325. [2] Bell AJ, Sejnowski TJ (1997) "The independent components of natural images are edge filters," Vision Research, 37: 3327-3338. [3] van Hateren JH, van der Schaaff A (1997) "Independent component filters of natural images compared with simple cells in primary visual cortex," Proc. Royal Soc. Lond. B, 265: 359-366. [4] Lewicki MS, Olshausen BA (1999) "A probabilistic framework for the adaptation and comparison of image codes," JOSA A, 16(7): 1587-160l. [5] Simoncelli EP, Freeman WT, Adelson EH, Heeger DJ (1992) "Shiftable multiscale transforms," IEEE Transactions on Information Theory, 38(2): 587-607. [6] Attias H (1999) "Independent factor analysis," Neural Computation, 11: 803-852.
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Semiparametric Approach to Multichannel Blind Deconvolution of Nonminimum Phase Systems L.-Q. Zhang, S. Amari and A. Cichocki Brain-style Information Systems Research Group, BSI The Institute of Physical and Chemical Research Wako shi, Saitama 351-0198, JAPAN zha@open.brain.riken.go.jp {amari,cia }@brain.riken.go.jp Abstract In this paper we discuss the semi parametric statistical model for blind deconvolution. First we introduce a Lie Group to the manifold of noncausal FIR filters. Then blind deconvolution problem is formulated in the framework of a semiparametric model, and a family of estimating functions is derived for blind deconvolution. A natural gradient learning algorithm is developed for training noncausal filters. Stability of the natural gradient algorithm is also analyzed in this framework. 1 Introduction Recently blind separation/deconvolution has been recognized as an increasing important research area due to its rapidly growing applications in various fields, such as telecommunication systems, image enhancement and biomedical signal processing. Refer to review papers [7] and [13] for details. A semi parametric statistical model treats a family of probability distributions specified by a finite-dimensional parameter of interest and an infinite-dimensional nuisance parameter [12]. Amari and Kumon [10] have proposed an approach to semiparametric statistical models in terms of estimating functions and elucidated their geometric structures and efficiencies by information geometry [1]. Blind source separation can be formulated in the framework of semi parametric statistical models. Amari and Cardoso [5] applied information geometry of estimating functions to blind source separation and derived an admissible class of estimating functions which includes efficient estimators. They showed that the manifold of mixtures is m-curvature free, so that we can design algorithms of blind separation without taking much care of misspecification of source probability functions. The theory of estimating functions has also been applied to the case of instantaneous mixtures, where independent source signals have unknown temporal correlations [3]. It is also applied to derive efficiency and superefficiency of demixing learning algorithms [4]. Most of these theories treat only blind source separation of instantaneous mixtures. It is only recently that the natural gradient approach has been proposed for multichannel blind 364 L.-Q. Zhang, S. Amari and A. Cichocki deconvolution [8], [18]. The present paper extends the geometrical theory of estimating functions to the semiparametric model of multichannel blind deconvolution. For the limited space, the detailed derivations and proofs are left to a full paper. 2 Blind Deconvolution Problem In this paper, as a convolutive mixing model, we consider a multichannel linear timeinvariant (LTI) systems, with no poles on the unit circle, of the form 00 x(k) = L Hps(k - p), (1) p=-oo where s(k) is an n-dimensional vector of source signals which are spatially mutually independent and temporarily identically independently distributed, and x(k) is an n-dimensional sensor vector at time k, k = 1,2, . . '. We denote the unknown mixing filter by H(z) = 2::-00 Hpz-p. The goal of multichannel blind deconvolution is to retrieve source signals s(k) only using sensor signals x(k), k = 1,2"", and certain knowledge of the source signal distributions and statistics. We carry out blind deconvolution by using another multichannel LTI system of the form y(k) = W(z)x(k), (2) where W(z) = 2:~= -N Wpz-P, N is the length of FIR filter W(z), y(k) [Yl (k), ... ,Yn(k)V is an n-dimensional vector of the outputs, which is used to estimate the source signals. When we apply W(z) to the sensor signal x(k), the global transfer function from s(k) to y(k) is defined by G(z) = W(z)H(z). The goal of the blind deconvolution task is to find W(z) such that G(z) = PAD(z), where P E R nxn is a permutation matrix, D(z) = diag{z-d 1 , ••• ,z- dn }, and A E R n x n is a nonsingular diagonal scaling matrix. 3 Lie Group on M (N, N) In this section, we introduce a Lie group to the manifold of noncausal FIR filters. The Lie group operations playa crucial role in the following discussion. The set of all the noncausal FIR filters W (z) of length N, having the constraint that W is nonsingular, is denoted by M(N,N) = {W(Z) I W(z) = .tN W.z- · , det(W) # o}, (3) where W is an N x N block matrix, ... W_N+ll ... W - N+2 . . . . Wo (4) M(N, N) is a manifold of dimension n 2(2N + 1). In general, multiplication of two filters in M(N, N) will enlarge the filter length and the result does belong to M(N, N) anymore. This makes it difficult to introduce the Riemannian structure to the manifold of noncausal FIR filters. In order to explore possible geometrical structures of M(N, N) which will lead to effective learning algorithms for W (z), we define algebraic operations of filters in the Lie group framework. First, we introduce a novel filter decomposition of noncausal filters in M (N, N) into a product of two one-sided FIR filters [19], which is illustrated in Fig. 1. Blind Deconvolution of Nonminimum Phase Systems 365 Unknown :s(k) H(z) x(k) R(Z") u(k) L(z) y(k) n : n ~ n i i Mixing model Demixing model Figure 1: Illustration of decomposition of noncausal filters in M (N, N) Lemma 1 [19] If the matrix W is nonsingular, any noncausalfilter W(z) in M(N,N) has the decomposition W(z) = R(z)L(z-l), where R(z) = L::=o Rpz-P, L(Z-l) = L::=o LpzP are one-sided FIR filters. In the manifold M(N, N), Lie operations, multiplication * and inverse t, are defined as follows: For B(z), C(z) E M(N, N), B(z) * C(z) = [B(z)C(z)]N' Bt(z) = Lt(Z-l)Rt(z), (5) where [B(Z)]N is the truncating operator that any terms with orders higher than N in the polynomial B (z) are truncated, and the inverse of one-side FIR filters is recurrently defined by ~ = ROl, at = - L::=l Rt_qRqROl , p = 1,'" ,N. Refer to [18] for the detailed derivation. With these operations, both B(z) * C(z) and Bt (z) still remain in the manifold M (N, N). It is easy to verify that the manifold M (N, N) with the above operations forms a Lie Group. The identity element is E(z) = I. 4 Semiparametric Approach to Blind Deconvolution We first introduce the basic theory of semiparametric models, and formulate blind deconvolution problem in the framework of the semiparametric models. 4.1 Semiparametric model Consider a general statistical model {p( Xj 6, en, where x is a random variable whose probability density function is specified by two parameters, 6 and e, 6 being the parameter of interest, and e being the nuisance parameter. When the nuisance parameter is of infinite dimensions or of functional degrees of freedom, the statistical model is called a semiparametric model [12]. The gradient vectors of the log likelihood u(Zj 6, e) = 81ogp(z;6.e) v(z, 6 ~) 81ogp(z;6,e) are called the score functions of the parameter 80 ' ,,'it 8( , of interest or shortly 6-score and the nuisance score or shortly e -score, respectively. In the semiparametric model, it is difficult to estimate both the parameters of interest and nuisance parameters at the same time, since the nuisance parameter e is of infinite degrees of freedom. The semiparametric approach suggests to use an estimating function to estimate the parameters of interest, regardless of the nuisance parameters. The estimating function is a vector function z(x, 6), independent of nuisance parameters e, satisfying the following conditions 1) Eo,dz(x,6)] = 0, 8z(x,6) 2) det(lC) i= 0, where IC = Eo,d 88 ]. (6) (7) 366 L.-Q. Zhang, S. Amari and A. Cichocki 3) Eo,dz(x, 8)zT (x, 8)] < 00, (8) for all 8 and e. Generally speaking, it is difficult to find an estimating function. Amari and Kawanabe [9] studied the information geometry of estimating functions and provided a novel approach to find all the estimating functions. In this paper, we follow the approach to find a family of estimating functions for bind deconvolution. 4.2 Semiparametric Formulation for Blind Deconvolution Now we tum to formulate the blind deconvolution problem in the framework of semi parametric models. From the statistical point of view, the blind deconvolution problem is to estimate H(z) or H- 1(z) from the observed data VL = {x(k), k = 1, 2", .}. The estimate includes two unknowns: One is the mixing filter H(z) which is the parameter of interest, and the other is the probability density function p(s) of sources, which is the nuisance parameter in the present case. FOIf blind deconvolution problem, we usually assume that source signals are zero-mean, E[sil' = 0, for i = 1"", n. In addition, we generally impose constraints on the recovered signals to remove the indeterminacy, (9) A typical example of the constraint is ki ( Si) = sf -1. Since the source signals are spatially mutually independent and temporally iid, the pdfr(s) can be factorized into a product form r(s) = TI~l r(si). The purpose of this paper is to find a family of estimating functions for blind deconvolution. Remarkable progress has been made recently in the theory of the semiparametric approach [9],[12]. It has been shown that the efficient score itself is an estimating function for blind separation. 5 Estimating Functions In this section, we give an explicit form of the score function matrix of interest and the nuisance tangent space, by using a local nonholonomic reparameterization. We then derive a family of estimating functions from it. 5.1 Score function matrix and its representation Since the mixing model is a matrix filter, we write an estimating function in the same matrix filter format N F(x;H(z)) = L Fp(x;H(z))z-P, (10) p= -N where F p(x; H(z)) are n x n-matrices. In orderto derive the explicit form ofthe H-score, we reparameterize the filter in a small neighborhood of H (z) by using a new variable matrix filter as H(z) * (I - X(z)), where 1 is the identity element of the manifold M(N, N). The variation X(z) represents a local coordinate system at the neighborhoodNH of H(z) on the manifold M(N, N). The variation dH(z) of H(z) is represented as dH(z) = -H(z) * dX(z). Letting W(z) = Ht(z), we have dX(z) = dW(z) * wt(z) , (11 ) which is a nonholonomic differential variable [6] since (11) is not integrable. With this representation of the parameters, we can obtain learning algorithms having the equivariant property [14] since the deviation dX(z) is independent of a specific H(z). The relative or the natural gradient of a cost function on the manifold can be automatically derived from this representation [21, [14], [18]. Blind Deconvolution of Nonminimum Phase Systems 367 {p(x;e,;)} {p(x;9,e)} Figure 2: Illustration of orthogonal decomposition of score functions The derivative of any cost function l(H(z)) with respect to a noncausal filter X(z) E:==-N Xpz-P is defined by N 8l(H(z» _ L 8l(H(z» z-p aX(z) p==-N axp Now we can easily calculate the score function matrix of non causal filter X(z), N alogp(XiH(z),r) _ '"' () T(k _ ) -p aX(z) L.J lP Y Y P z , p=-N where lP(y) = ('Pi(Yi),"', 'Pn(Yn»T, 'Pi(Yi) = dlO~;:(II/). and y = Ht(z)x. S.2 Efficient scores (12) (13) The efficient scores, denoted by UE(s; H(z), r), can be obtained by projecting the score function to the space orthogonal to the nuisance tangent space TJ'{z},r' which is illustrated in figure 2. In this section, we give an explicit form of the efficient scores for blind deconvolution. Lemma 2 [5} The tangent nuisance space TJ'{z),r is a linear space spanned by the nuisance score junctions, denoted by TJ'{z),r = {E:=I CiOi(Si)} , where Ci are coefficients, and ai(si) are arbitrary junctions, satisfying the/ollowing conditions E[ai(si)2] < 00, E[sai(si)] = 0, E[k(si)ai(si») = O. (14) We rewrite the score function (13) into the form U(s; H(z), r) = E!-N Upz-P, where Up = (cp(si(k»sj(k - P»nxn. Lemma 3 The off-diagonal elements UO,ij(S; H(z), r), i =/: j, and the delay elements Up,ij(S; H(z), r), P =/: 0, 0/ the score junctions are orthogonal to the nuisance tangent space TJ(z),r' Lemma 4 The projection 0/ UO,ii to the space orthogonal to the nuisance tangent space TJ(z),r is o/the/orm W(Si) = Co + CISi + C2k(Si), where Cj are any constants. 368 L.-Q. Zhang, S. Amari and A. Cichocki In summary we have the following theorem Theorem 1 The efficient score, UE(s; H(z), r) = L::=-N U: z-P, is given by U: <p(s)sT(k - p), for p:f. 0; (15) U~ for off diagonal elements, for diagonal elements. (16) For the instantaneous mixture case, it has been proven [9] that the semiparametric model for blind separation is information m-curvature free. This is also true in the multichannel blind deconvolution case. As a result, the efficient score function is an estimating function for blind deconvolution. Using this result, we easily derive a family of estimating functions for blind deconvolution N F(x(k); W(z)) = L c.p(y(k))y(k - pf z-P - I, (17) p=-N where y(k) = W(z)x(k), and <p is a given function vector. The estimating function is the efficient score function, when Co = Cl = 0, C2 = 1 and ki(Si) = c.pi(sdsi - 1. 6 Natural Gradient Learning and its Stability Ordinary stochastic gradient methods for parameterized systems suffer from slow convergence due to the statistical correlations of the processes signals. While quasi-Newton and related methods can be used to improve convergence, they also suffer from the mass computation and numerical instability, as well as local convergence. The natural gradient approach was developed to overcome the drawback of the ordinary gradient algorithm in the Riemannian spaces [2, 8, 15]. It has been proven that the natural gradient algorithm is an efficient algorithm in blind separation and blind deconvolution [2]. The efficient score function ( the estimating function) gives an efficient search direction for updating filter X(z) . Therefore, the updating rule for X(z) is described by Xk+l(Z) = Xk(z) -1]F(x(k), Wk(Z)), (18) where 1] is a learning rate. Since the new parameterization X(z) is defined by a nonholonomic transformation dX (z) = dW (z) * wt (z ), the deviation of W (z) is given by ~ W(z) = ~X(z) * W(z). (19) Hence, the natural gradient learning algorithm for W (z) is described as Wk+l(Z) = Wk(Z) -1]F(x(k), Wk(z)) * Wk(z) , (20) where F(x, W (z)) is an estimating function in the form (17). The stability ofthe algorithm (20) is equivalent to the one of algorithm (18). Consider the averaged version of algorithm (18) ~X(z) = -1]E[F(x(k), Wk(Z))] . (21) Analyzing the variational equation of the above equation and using the mutual independence and i.i.d. properties of source signals, we derive the stability conditions of learning algorithm (21) at vicinity of the true solution mi + 1 > 0, K.i > 0, K.iK.ja;aJ > 1, (22) for i ,j = 1," ' , n, where mi = E(c.p'(Yi(k))y;(k)], K.i = E[c.p~(Yi)], a; = E[IYiI 2]. Therefore, we have the following theorem: Theorem 2 If the conditions (22) are satisfied, then the natural gradient learning algorithm (20) is locally stable. Blind Deconvolution of Nonminimum Phase Systems 369 References [1] S. Amari. Differential-geometrical methods in statistics, Lecture Notes in Statistics, volume 28. Springer, Berlin, 1985. [2] S. Amari. Natural gradient works efficiently in learning. Neural Computation, 10:251-276, 1998. [3 J S. Amari. ICA of temporally correlated signals - Learning algorithm. In Proceeding of 1st Inter. Workshop on Independent Component Analysis and Signal Separation, pages 37-42, Aussois, France, January, 11-15 1999. [4] S. Amari. Superefficiency in blind source separation. IEEE Trans. on Signal Processing, 47(4):936-944, April 1999. [5] S. Amari and J.-F. Cardoso. Blind source separation- semiparametric statistical approach. IEEE Trans. Signal Processing, 45:2692-2700, Nov. 1997. [6] S. Amari, T. Chen, and A. Cichocki. Nonholonomic orthogonal constraints in blind source separation. Neural Comput., to be published. [7] S. Amari and A. Cichocki. Adaptive blind signal processing- neural network approaches. Proceedings of the IEEE, 86(10):2026-2048,1998. [81 S. Amari, S. Douglas, A. Cichocki, and H. Yang. Multichannel blind deconvolution and equalization using the natural gradient. In Proc. IEEE Workshop on Signal Processing Adv. in Wireless Communications, pages 101-104, Paris, France, April 1997. [9] S. Amari and M. Kawanabe. Estimating functions in semiparametric statistical models. In I. V. Basawa, v.P. Godambe, and R.L. Taylor, editors, Estimating Functions, volume 32 of Monograph Series, pages 65-81. IMS, 1998. [10] S. Amari and M. Kumon. Estimation in the presence of infinitely many nuisance parameters in semiparametric statistical models. Ann. Statistics, 16: 1044-1068, 1988. [11] A.l. Bell and T.l. Sejnowski. An information maximization approach to blind separation and blind deconvolution. Neural Computation, 7: 1129-1159, 1995. [12] P. Bickel, C. Klaassen, Y. Ritov, and J. Wellner. Efficient and Adaptive Estimation for Semiparametric Models. The Johns Hopkins Univ. Press, Baltimore and London, 1993. [13] J.-F Cardoso. Blind signal separation: Statistical principles. Proceedings of the IEEE, 86(10):2009-2025,1998. [14] J.-F. Cardoso and B. Laheld. Equivariant adaptive source separation. IEEE Trans. Signal Processing, SP-43: 30 17-3029, Dec 1996. [15] A. Cichocki and R. Unbehauen. Robust neural networks with on-line learning for blind identification and blind separation of sources. IEEE Trans Circuits and Systems I: Fundamentals Theory and Applications, 43(11):894-906, 1996. [16] L. Tong, R.W. Liu, v.c. Soon, and Y.F. Huang. Indeterminacy and identifiability of blind identification. IEEE Trans. Circuits, Syst., 38(5):499-509, May 1991. [17] H. Yang and S. Amari. Adaptive on-line learning algorithms for blind separation: Maximum entropy and minimal mutual infonnation. Neural Comput., 9: 1457-1482, 1997. [18] L. Zhang, A. Cichocki, and S. Amari. Geometrical structures of FIR manifold and their application to multichannel blind deconvolution. In Proceeding of NNSP'99, pages 303-312, Madison, Wisconsin, August 23-25 1999. [19] L. Zhang, A. Cichocki, and S. Amari. Multichannel blind deconvolution of nonminimum phase systems using information backpropagation. In Proceedings of the Fifth International Conference on Neural Information Processing(ICONIP'99), page 210-216, Perth, Australia, Nov. 16-20 1999.
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Application of Blind Separation of Sources to Optical Recording of Brain Activity Holger Schoner, Martin Stetter, Ingo Schie61 Department of Computer Science Technical University of Berlin Germany {hjsch,moatl,ingos}@cs.tu-berlin.de John E. W. Mayhew University of Sheffield, UK j. e.mayhew@sheffield.ac.uk Jennifer S. Lund, Niall McLoughlin Institute of Ophthalmology University College London, UK {j.lund,n.mcloughlin}@ucl.ac.uk Klaus Obermayer Department of Computer Science, Technical University of Berlin, Germany oby@cs.tu-berlin.de Abstract In the analysis of data recorded by optical imaging from intrinsic signals (measurement of changes of light reflectance from cortical tissue) the removal of noise and artifacts such as blood vessel patterns is a serious problem. Often bandpass filtering is used, but the underlying assumption that a spatial frequency exists, which separates the mapping component from other components (especially the global signal), is questionable. Here we propose alternative ways of processing optical imaging data, using blind source separation techniques based on the spatial decorre1ation of the data. We first perform benchmarks on artificial data in order to select the way of processing, which is most robust with respect to sensor noise. We then apply it to recordings of optical imaging experiments from macaque primary visual cortex. We show that our BSS technique is able to extract ocular dominance and orientation preference maps from single condition stacks, for data, where standard post-processing procedures fail. Artifacts, especially blood vessel patterns, can often be completely removed from the maps. In summary, our method for blind source separation using extended spatial decorrelation is a superior technique for the analysis of optical recording data. 1 Introduction One approach in the attempt of comprehending how the human brain works is the analysis of neural activation patterns in the brain for different stimuli presented to a sensory system. An example is the extraction of ocular dominance or orientation preference maps from recordings of activity of neurons in the primary visual cortex of mammals. A common technique for extracting such maps is optical imaging (01) of intrinsic signals. Currently this is the imaging technique with the highest spatial resolution (~ 100 J1m) for mapping of the cortex. This method is explained e.g. in [1], for similar methods using voltage sensitive dyes see [2, 3]. 01 uses changes in light reflection to estimate spatial patterns of stimulus 950 H. Sch6ner. M Stetter. I. Schiej3l, J E. Mayhew, J Lund, N. Mcloughlin and K. Obermayer answers. The overall change recorded by a CCD or video camera is the total signal. The part of the total signal due to local neural activity is called the mapping component and it derives from changes in deoxyhemoglobin absorption and light scattering properties of the tissue. Another component of the total signal is a "global" component, which is also correlated with stimulus presentation, but has a much coarser spatial re~olution . It derives its part from changes in the blood volume with the time. Other components are blood vessel artifacts, the vasomotor signal (slow oscillations of neural activity), and ongoing activity (spontaneous, stimulus-uncorrelated activity). Problematic for the extraction of activity maps are especially blood vessel artifacts and sensor noise, such as photon shot noise. A procedure often used for extracting the activity maps from the recordings is bandpass filtering, after preprocessing by temporal, spatial, and trial averaging. Lowpass filtering is unproblematic, as the spatial resolution of the mapping signal is limited by the scattering properties of the brain tissue, hence everything above a limiting frequency must be noise. The motivation for highpass filtering, on the other hand, is questionable as there is no specific spatial frequency separating local neural activity patterns and the global signal [4]). A different approach, Blind Source Separation (BSS), models the components of the recorded image frames as independent sources, and the observations (recorded image frames) as noisy linear mixtures of the unknown sources. After performing the BSS the mapping component should ideally be concentrated in one estimated source, the global signal in another, and blood vessel artifacts, etc. in further ones. Previous work ([5]) has shown that BSS algorithms, which are based on higher order statistics ([6, 7, 8]), fail for optical imaging data, because of the high signal to noise ratio. In this work we suggest and investigate versions of the M&S algorithm [9, 10], which are robust against sensor noise, and we analyze their performance on artificial as well as real optical recording data. In section 2 we describe an improved algorithm, which we later compare to other methods in section 3. There an artificial data set is used for the analysis of noise robustness, and benchmark results are presented. Then, in section 4, it is shown that the newly developed algorithm is very well able to separate the different components of the optical imaging data, for ocular dominance as well as orientation preference data from monkey striate cortex. Finally, section 5 provides conclusions and perspectives for future work. 2 Second order blind source separation Let m be the number of mixtures and r the sample index, i.e. a vector specifying a pixel in the recorded images. The observation vectors y(r) = (Y1(r) , ... ,Ym¥')f are assumed to be linear mixtures of m unknown sources s(r) = (Sl (r) , . . . ,Sm (r)) with A being the m x m mixing matrix and n describing the sensor noise: y(r) = As(r) + n (1) The goal of BSS is to obtain optimal source estimates s(r) under the assumption that the original sources are independent. In the noiseless case W = A -1 would be the optimal demixing matrix. In the noisy case, however, W also has to compensate for the added noise: s(r) == Wy(r) = W . A . s(r) + W . n. BSS algorithms are generally only able to recover the original sources up to a permutation and scaling. Extended Spatial Decorrelation (ESD) uses the second order statistics of the observations to find the source estimates. If sources are statistical independent all source crosscorrelations Ci(,~) (D.r) = (si(r)Sj(r+ D.r))r = ~ LSi(r)Sj(r+ D.r) , i =F j (2) r Application of BSS to Optical Recording of Brain Activity 951 must vanish for all shifts ~r, while the autocorrelations (i = j) of the sources remain (the variances). Note that this implies that the sources must be spatially smooth. Motivated by [to] we propose to optimize the cost function, which is the sum of the squared cross-correlations of the estimated sources over a set of shifts {~r}, E(W) = L L ((WC(~r)WT)i,jr (3) 6r i~j = L L\Si(r)Sj(r + ~r))~ , 6r itj with respect to the demixing matrix W. The matrix Ci,j(~r) = (Yi (r)Yj(r + ~r))r denotes the mixture cross-correlations for a shift ~r. This cost function is minimized using the Polak Ribiere Conjugate Gradient technique, where the line search is substituted by a dynamic step width adaptation ([11]). To keep the demixing matrix W from converging to the zero matrix, we introduce a constraint which keeps the diagonal elements of T = W-l (in the noiseless case and for non-sphered data T is an estimate of the mixing matrix, with possible permutations) at a value of 1.0. Convergence properties are improved by sphering the data (transforming their correlation matrix for shift zero to an identity matrix) prior to decorrelating the mixtures. Note that use of multiple shifts ~r allows to use more information about the auto- and cross-correlation structure of the mixtures for the separation process. Two shifts provide just enough constraints for a unique solution ([to]). Multiple shifts, and the redundancy they introduce, additionally allow to cancel out part of the noise by approximate simultaneous diagonalization of the corresponding cross correlation matrices. In the presence of sensor noise, added after mixing, the standard sphering technique is problematic. When calculating the zero-shift cross-correlation matrix the variance of the noise contaminates the result, and sphering using a shifted cross-correlation matrix, is recommended ([12]). For spatially white sensor noise and sources with reasonable auto correlations this technique is more appropriate. In the following we denote the standard algorithm by dpaO, and the variant using noise robust sphering by dpa1. 3 Benchmarks for artificial data The artificial data set used here, whose sources are approximately uncorrelated for all shifts, is shown in the left part of figure 1. The mixtures were produced by generating a random mixing matrix (in this case with condition number 3.73), applying it to the sources, and finally adding white noise of different variances. In order to measure the performance on the artificial data set we measure a reconstruction error (RE) between the estimated and the correct sources via (see [l3]): RE(W) = od(L §(r)sT(r)) , r 1 ~ 1 ( IC· ·1 ) od(C) = N ~ N _ 1 L maXk'I~i , kl - 1 I J (4) The correlation between the real and the estimated sources (the argument to "od"), should be close to a permutation matrix, if the separation is successful. If the maxima of two rows are in the same column, the separation is labeled unsuccessful. Otherwise, the normalized absolute sum of non-permutation (cross-correlation) elements is computed and returned as the reconstruction error. We now compare the method based on optimization of (3) by gradient descent with the following variants of second order blind source separation: (1) standard spatial decorrelation 952 H. Schaner; M Stetter; I. Schiej3l, J. E. Mayhew, J. Lund, N. McLoughlin and K. Obermayer -~-.-opt hI -.. .. . __ . jacO , ... 0.5 mean 0.5 jacl g <; ''I, ': cor JiO.4 " dpaO ~0.4 . ~ . , dpal -. 1 § c: '. 0 gOJ 'gO.3 % l:l ' •. b ~ ~ H, 80.2 -" ~02 " " 0:: 0:: ~ 0.1 '. 0.1 , ~ '!i ___ :. , -----o 0 5 10 15 20 25 00 5 10 15 20 25 Signal to Noise Ratio (dB) Signal to Noise Ratio (dB) Figure 1: The set of three approximately uncorrelated source images of the artificial data set (left). The two plots (middle, right) show the reconstruction error versus signal to noise ratio for different separation algorithms. In the right plot jac1 and dpa1 are very close together. using the optimal single shift yielding the smallest reconstruction error (opt). (2) Spatial decorrelation using the shift selected by norm (C(.6.r) - diag (C(.6.r))) .6.rcor = argmax{.D.r} norm (diag (C(.6.r)))·' (5) where "diag" sets all off-diagonal elements of its argument matrix to zero, and "norm" computes the largest singular value of its argument matrix (cor). .6.rcor is the shift for which the cross correlations are largest, i.e. whose signal to noise ratio (SNR) should be best. (3) Standard spatial decorrelation using the average reconstruction error for all successful shifts in a 61 x 61 square around the zero shift (mean). (4) A multi-shift algorithm ([12]), using several elementary rotations (Jacobi method) to build an orthogonal demixing matrix, which optimizes the cost function (3). The variants using standard sphering and noise robust sphering are denoted by dacO) and dac1). cor, opt, and mean use two shifts for their computation; but as one of those is always the zero-shift, there is only one shift to choose and they are called single-shift algorithms here. Figure 1 gives two plots which show the reconstruction error (4) versus the SNR (measured in dB) for single shift (middle) and multi-shift (right) algorithms. The error bars indicate twice the standard error of the mean (2x SEM), for 10 runs with the same mixing matrix, but newly generated noise of the given noise level. In each of these runs, the best result of three was selected for the gradient descent method. This is because, contrary to the other algorithms, the gradient descent algorithm depends on the initial choice of the demixing matrix. All multi-shift algorithms (all except opt and mean), used 8 shifts (±r, ±r), (±r, 0), and (0, ±r) for each r E {I, 3, 5, 10,20, 30}, so 48 all together. Several points are noticeable in the plots. (i) The cor algorithm is generally closer to the optimum than to the average successful shift. (ii) A comparison between the two plots shows that the multi-shift algorithms (right plot) are able to perform much better than even the optimal single-shift method. For low to medium noise levels this is even the case when using the standard sphering method combined with the gradient descent algorithm. (iii) The advantage of the noise robust sphering method, compared to the standard sphering, is obvious: the reconstruction error stays very low for all evaluated noise levels, for both the jac1 and dpa1 algoritlnns. (iv) The gradient descent technique is more robust than the Jacobi method For the standard sphering its performance is much better than that of the Jacobi method. Figure 1 shows results which were produced using a single mixing matrix. However, our simulations show that the algorithms compare qualitatively similar when using mixing maApplication of BSS to Optical Recording of Brain Activity 953 t = 1 sec. t = 2 sec. t = 3 sec. t = 4 sec. t = 5 sec. t = 6 sec. t = 7 sec. Figure 2: Optical imaging stacks. The top stack is a single condition stack from ocular dominance experiments, the lower one a difference stack from orientation preference experiments (images for 90° gratings subtracted from those for 0° gratings). The stimulus was present during recording images 2-7 in each row. Two large blood vessels in the top and left regions of the raw images were masked out prior to the analysis. trices with condition numbers between 2 and 10. The noise robust versions of the multishift algorithms generally yield the best separation results of all evaluated algorithms. 4 Application to optical imaging We now apply extended spatial decorrelation to the analysis of optical imaging data. The data consists of recordings from the primary visual cortex of macaque monkeys. Each trial lasted 8 seconds, which were recorded with frame rates of 15 frames per second. A visual stimulus (a drifting bar grating of varying orientation) was presented between seconds 2 and 8. Trials were separated by a recovery period of 15 seconds without stimulation. The cortex was illuminated at a wavelength of 633 nm. One pixel corresponds to about 15 J.Lm on the cortex; the image stacks used for further processing, consisting of 256 x 256 pixels, covered an area of cortex of approximately 3.7 mm2 . Blocks of 15 consecutive frames were averaged, and averaging over 8 trials using the same visual stimulus further improved the SNR. First frame analysis (subtraction of the first, blank, frame from the others) was then applied to the resulting stack of 8 frames, followed by lowpass filtering with 14 cycles/mm. Figure 2 shows the resulting image stacks for an ocular dominance and an orientation preference experiment. One observes strong blood vessel artifacts (particularly in the top row of images), which are superimposed to the patchy mapping component that pops up over time. Figure 3 shows results obtained by the application of extended spatial decorrelation (using dpaO). Only those estimated sources containing patterns different from white noise are shown. Backprojection of the estimated sources onto the original image stack yields the amplitude time series of the estimated sources, which is very useful in selecting the mapping component: it can be present in the recordings only after the stimulus onset (starting at t = 2 sec.). The middle part shows four estimated sources for the ocular dominance single condition stack. The mapping component (first image) is separated from the global component (second image) and blood vessel artifacts (second to fourth) quite well. The time course of the mapping component is plausible as well: calculation of a plausibility index (sum of squared differences between the normalized time series and a step function, which is 0 before and 1 after the stimulus onset) gives 0.5 for the mapping component and 2.31 for the next best one. Results for the gradient descent algorithm are similar for this data set, regardless of the sphering technique used. The Jacobi method also gives similar results, but a small blood vessel artifact is remaining in the resulting map. The cor algorithm usually gives much worse separation results. In the right part of figure 3 two es954 H Schaner, M Stetter, I SchieJ3l, 1. E. Mayhew, 1. Lund, N McLoughlin and K. Obermayer Figure 3: Left: Summation technique for ocular dominance (aD) experiment (upper) and orientation preference (OP) experiment (lower). Middle, Right: dpaO algorithm applied to the same aD single condition (middle) and OP (right) stacks. The images show the 4 (aD) and 2 (OP) estimated components, which are visually different from white noise. In the bottom row the respective time courses of the estimated sources are given. timated sources (those different from white noise) for the orientation preference difference stack can be seen. Here the proposed algorithm (dpaO) again works very well (plausibility index is 0.56 for mapping component, compared to 3.04 for the best other component). It generally has to be applied a few times (usually around 3 times) to select the best separation result Uudging by visual quality of the separation and the time courses of the estimated sources), because of its dependence on parameter initialization; in return it yields the best results of all algorithms used, especially when compared to the traditional summation technique. The similar results when using standard and noise robust sphering, and the small differences between the gradient descent and the Jacobi algorithms indicate, that not sensor noise is the limiting factor for the quality of the extracted maps. Instead it seems that, assuming a linear mixing model, no better results can be obtained from the used image stacks. It will remain for further research to analyze, how appropriate the linear mixing model is, and whether the underlying biophysical components are sufficiently uncorrelated. In the meantime the maps obtained by the ESD algorithm are superior to those obtained using conventional techniques like summation of the image stack. 5 Conclusion The results presented in the previous sections show the advantages of the proposed algorithm: In the comparison with other spatial decorrelation algorithms the benefit in using multiple shifts compared to only two shifts is demonstrated. The robustness against sensor noise is improved, and in addition, the selection of multiple shifts is less critical than selecting a single shift, as the resulting multi-shift system of equations contains more redundancy. In comparison with the Jacobi method, which is restricted to find only orthogonal demixing matrices, the greater tolerance of demixing by a gradient descent technique concerning noise and incorrect sphering are demonstrated. The application of second order blind separation of sources to optical imaging data shows that these techniques represent an important alternative to the conventional approach, bandpass filtering followed by summation of the image stack, for extraction of neural activity maps. Vessel artifacts can be separated from the mapping component better than using classical approaches. The spatial decorrelation algorithms are very well adapted to the optical imaging task, because of their use of spatial smoothness properties of the mapping and other biophysical components. An important field for future research concerning BSS algorithms is the incorporation of prior knowledge about sources and the mixing process, e.g. that the mixing has to be causal: the mapping signal cannot occur before the stimulus is presented. Assumptions Application of BSS to Optical Recording of Brain Activity 955 about the time course of signals could also be helpful, as well as knowledge about their spatial statistics. Smearing and scattering limit the resolution of recordings of biological components, and, depending on the wavelength of the light used for illumination, the mapping component constitutes only a certain percentage of the changes in total light reflections. Acknowledgments This work has been supported by the Wellcome Trust (050080IZJ97). References [I] T. Bonhoeffer and A. Grinvald. Optical imaging based on intrinsic signals: The methodology. In A. Toga and J. C. Maziotta, editors, Brain mapping: The methods, pages 55-97, San Diego, CA, 1996. Academic Press, Inc. [2] G. G. Blasdel and G. Salama. Voltage-sensitive dyes reveal a modular organization in monkey striate cortex. Nature, 321 :579-585, 1986. [3] G. G. Blasdel. Differential imaging of ocular dominance and orientation selectivity in monkey striate cortex. 1. Neurosci., 12:3115-3138, 1992. [4] M. Stetter, T. Otto, T. Mueller, F. Sengpiel, M. Huebener, T. Bonhoeffer, and K. Obermayer. Temporal and spatial analysis of intrinsic signals from cat visual cortex. Soc. Neurosci. Abstr., 23:455,1997. [5] I. SchieGl, M. Stetter, J. E. W. Mayhew, S. Askew, N. McLoughlin, J. B. Levitt, J. S. Lund, and K. Obermayer. Blind separation of spatial signal patterns from optical imaging records. In J .-F. Cardoso, C. Jutten, and P. Loubaton, editors, Proceedings of the lCA99 workshop, volume I, pages 179-184, 1999. [6] A. J. Bell and T. J. Sejnowski. An information-maximization approach to blind separation and blind deconvolution. Neural Computation, 7: 1129-1159, 1995. [7] S. Amari. Neural learning in structured parameter spaces - natural riemannian gradient. In M. C. Mozer, M. I. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems, volume 9, 1996. [8] A. Hyvtlrinen and E. Oja. A fast fixed point algorithm for independent component analysis. Neural Comput., 9:1483-1492,1997. [9] J. C. Platt and F. Faggin. Networks for the separation of sources that are superimposed and delayed. In 1. E. Moody, S. 1. Hanson, and R. P. Lippmann, editors, Advances in Neurallnformation Processing Systems, volume 4, pages 730--737, 1991. [10] L. Molgedey and H. G. Schuster. Separation of a mixture of independent signals using time delayed correlations. Phys. Rev. Lett., 72:3634-3637, 1994. [II] S. M. Riiger. Stable dynamic parameter adaptation. In D. S. Touretzky, M. C. Mozer, and M. E. Hasselmo, editors, Advances in Neural Information Processing Systems., volume 8, pages 225231 . MIT Press Cambridge, MA, 1996. [12] K.-R. Miiller, Philips P, and A. Ziehe. Jadetd: Combining higher-order statistics and temporal information for Blind Source Separation (with noise). In J.-F. Cardoso, C. Jutten, and P. Loubaton, editors, Proceedings of the 1. lCA99 Workshop, Aussois, volume I, pages 87-92, 1999. [13] B.-U. Koehler and R. Orglmeister. Independent component analysis using autoregressive models. In 1.-F. Cardoso, C. Jutten, and P. Loubaton, editors, Proceedings of the lCA99 workshop, volume I, pages 359-363, 1999.
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Spectral Cues in Human Sound Localization Craig T. Jin Department of Physiology and Department of Electrical Engineering, Univ. of Sydney, NSW 2006, Australia Simon Carlile Department of Physiology and Institute of Biomedical Research Univ. of Sydney, NSW 2006, Australia Anna Corderoy Department of Physiology Univ. of Sydney, NSW 2006, Australia Andre van Schaik Department of Electrical Engineering, Univ. of Sydney, NSW 2006, Australia Abstract The differential contribution of the monaural and interaural spectral cues to human sound localization was examined using a combined psychophysical and analytical approach. The cues to a sound's location were correlated on an individual basis with the human localization responses to a variety of spectrally manipulated sounds. The spectral cues derive from the acoustical filtering of an individual's auditory periphery which is characterized by the measured head-related transfer functions (HRTFs). Auditory localization performance was determined in virtual auditory space (VAS). Psychoacoustical experiments were conducted in which the amplitude spectra of the sound stimulus was varied independentlyat each ear while preserving the normal timing cues, an impossibility in the free-field environment. Virtual auditory noise stimuli were generated over earphones for a specified target direction such that there was a "false" flat spectrum at the left eardrum. Using the subject's HRTFs, the sound spectrum at the right eardrum was then adjusted so that either the true right monaural spectral cue or the true interaural spectral cue was preserved. All subjects showed systematic mislocalizations in both the true right and true interaural spectral conditions which was absent in their control localization performance. The analysis of the different cues along with the subjects' localization responses suggests there are significant differences in the use of the monaural and interaural spectral cues and that the auditory system's reliance on the spectral cues varies with the sound condition. 1 Introduction Humans are remarkably accurate in their ability to localize transient, broadband noise, an ability with obvious evolutionary advantages. The study of human auditory localization has a considerable and rich history (recent review [I]) which demonstrates that there are three general classes of acoustical cues involved in the localization process: (1) interaural time differences, ITDs; (2) interaurallevel differences, ILDs; and (3) the spectral cues resulting Spectral Cues in Human Sound Localization 769 from the auditory periphery. It is generally accepted that for humans, the lTD and ILD cues only specify the location of the sound source to within a "cone of confusion" [I], i.e., a locus of points approximating the surface of a cone symmetric with respect to the interaural axis. It remains, therefore, for the localization system to extract a more precise sound source location from the spectral cues. The utilization of the outer ear spectral cues during sound localization has been analyzed both as a statistical estimation problem, (e.g., [2]) and as optimization problem, often using neural networks, (e.g., [3]). Such computational models show that sufficient localization information is provided by the spectral cues to resolve the cone of confusion ambiguity which corroborates the psychoacoustical evidence. Furthermore, it is commonly argued that the interaural spectral cue, because of its natural robustness to level and spectral variations, has advantages over the monaural spectral cues alone. Despite these observations, there is still considerable contention as to the relative role or contribution of the monaural versus the interaural spectral cues. In this study, each subject's spectral cues were characterized by measuring their head related transfer functions (HRTFs) for 393 evenly distributed positions in space. Measurements were carried out in an anechoic chamber and were made for both ears simultaneously using a "blocked ear" technique [I]. Sounds filtered with the HRTFs and played over earphones, which bypass the acoustical filtering of the outer ear, result in the illusion of free-field sounds which is known as virtual auditory space (VAS). The HRTFs were used to generate virtual sound sources in which the spectral cues were manipulated systematically. The recorded HRTFs along with the Glasberg and Moore cochlear model [4] were also used to generate neural excitation patterns (frequency representations of the sound stimulus within the auditory nerve) which were used to estimate the different cues available to the subject during the localization process. Using this analysis, the interaural spectral cue was characterized and the different localization cues have been correlated with each subjects' VAS localization responses. 2 VAS Sound Localization The sound localization performance of four normal hearing subjects was examined in VAS using broadband white noise (300 - 14 000 Hz). The stimuli were filtered under three differing spectral conditions. (1) control: stimuli were filtered with spectrally correct left and right ear HRTFs for a given target location, (2) veridical interaural: stimuli at the left ear were made spectrally flat with an appropriate dB sound level for the given target location, while the stimuli at the right ear were spectrally shaped to preserve the correct interaural spectrum, (3) veridical right monaural: stimuli at the left ear were spectrally flat as in the second condition, while the stimuli at the right ear were filtered with the correct HRTF for the given target location, resulting in an inappropriate interaural spectral difference. For each condition, a minimum-phase filter spectral approximation was made and the interaural time difference was modeled as an all-pass delay [5]. Sounds were presented at approximately 70 dB SPL and with duration 150 ms (with 10 ms raised-cosine onset and offset ramps). Each subject performed five trials at each of 76 test positions for each stimulus condition. Detailed sound localization methods can be found in [1]. A short summary is presented below. 2.1 Sound Localization Task The human localization experiments were carried out in a darkened anechoic chamber. Virtual auditory sound stimuli were presented using earphones (ER-2, Etymotic Research, with a flat frequency response, within 3 dB, between 200-16 000 Hz). The perceived location of the virtual sound source was indicated by the subject pointing hislher nose in 770 C. T. lin, A. Corderoy, S. Carlile and A. v. Schaik the direction of the perceived source. The subject's head orientation and position were monitored using an electromagnetic sensor system (Polhemus, Inc.). 2.2 Human Sound Localization Performance The sound localization performance of two subjects in the three different stimulus conditions are shown in Figure 1. The pooled data across 76 locations and five trials is presented for both the left (L) and right (R) hemispheres of space from the viewpoint of an outside observer. The target location is shown by a cross and the centroid of the subjects responses for each location is shown by a black dot with the standard deviation indicated by an ellipse. Front-back confusions are plotted, although, they were removed for calculating the standard deviations. The subjects localized the control broadband sounds accurately (Figure 1a). In contrast, the subjects demonstrated systematic mislocalizations for both the veridical interaural and veridical monaural spectral conditions (Figures I b,c). There is clear pulling of the localization responses to particular regions of space with evident intersubject variations. (8) Subject 1: Broadband Control Subject 2: Broadband Control Ellipse: Standard Deviation L:i;;:~ses ~JC~~ ~~i.iif. ,':.,;':::::::.><" (b) Subject 1: Veridicallnteraural Spectrum Subject 2: Veridical Interaural Spectrum L R L. ,.c._:. e'·· R . ~...:. . .." .-. ';'-"'" , . ',. ' ~I~;> ;::· . (c) Subject 1: Veridical Right Monaural Spectrum Subject 2: Veridical Right Monaural Spectrum Le '.. - ' . . " , ~l~ . :.:. _ . .;.' · .... -.L ... .,. ~~.j;~;" • ":':"$ . '~ R /...., " ~ . -, . ' .~ • . .•. ~...... .t ' ~ i ~: ' • ...... ~ • .;;.:J;...;,1"" L~~R V~ Figure 1: Localization performance for two subjects in the three sound conditions: (a) control broadband; (b) veridical interaural; (c) veridical monaural. See text for details. 3 Extraction of Acoustical Cues With accurate measurements of each individual's outer ear filtering, the different acoustical cues can be compared with human localization performance on an individual basis. In order to extract the different acoustical cues in a biologically plausible manner, a model of peripheral auditory processing was used. A virtual source sound stimulus was prepared as described in Secion 2 for a particular target location. The stimulus was then filtered using a cochlear model based on the work of Glasberg and Moore [4]. This cochlear model consisted of a set of modified rounded-exponential auditory filters. The width and shape of the auditory filters change as a function of frequency (and sound level) in a manner Spectral Cues in Human Sound Localization 771 consistent with the known physiological and psychophysical data. These filters were logarithmically spaced on the frequency axis with a total of 200 filters between 300 Hz and 14 kHz. The cochlea's compressive non-linearity was modelled mathematically using a logarithmic function. Thus the logarithm of the output energy ofa given filter indicated the amount of neural activity in that particular cochlear channel. The relative activity across the different cochlear channels was representative of the neural excitation pattern (EP) along the auditory nerve and it is from this excitation pattern that the different spectral cues were estimated. For a given location, the left and right EPs themselves represent the monaural spectral cues. The difference in the total energy (calculated as the area under the curve) between the left and right EPs was taken as a measure of the interaural level difference and the interaural spectral shape cue was calculated as the difference between the left and right EPs. The fourth cue, interaural time difference, is a measure of the time lag between the signal in one ear as compared to the other and depends principally upon the geometrical relationship between the sound source and the human subject. This time delay was calculated using the acoustical impulse response for both ears as measured during the HRTF recordings. 4 Correlation of Cues and Location For each stimulus condition and location, the acoustical cues were calculated as described above for all 393 HRTF locations. Locations at which a given cue correlates well with the stimulus cue for a particular target location were taken as analytical predictions of the subject's response locations according to that cue. As the spectral content of the signal is varied, the cue(s) available may strongly match the cue(s) normally arising from locations other than the target location. Therefore the aim of this analysis is to establish for which locations and stimulus conditions a given response most correlated with a particular cue. The following analyses (using a Matlab toolbox developed by the authors) hinge upon the calculation of "cue correlation values". To a large extent, these calculations follow the examples described by [6] and are briefly described here. For each stimulus condition and target location, the subject performed five localizations trials. For each of the subject's five response locations, each possible cue was estimated (Section 3) assuming a flat-spectrum broadband Gaussian white noise as the stimulus. A mathematical quantity was then calculated which would give a measure of the similarity of the response location cues with the corresponding stimulus cues. The method of calculation depended on the cue and several alternative methods were tried. Generally, for a given cue, these different methods demonstrated the same basic pattern and the term "cue correlation value" has been given to the mathematical quantity that was used to measure cue similarity. The methods are as follows. For the ITO cue, the negative of the absolute value of the difference between possible response location ITDs and the stimulus ITO was used as the ITO cue correlation values (the more positive a value, the higher its correlation). The ILD cue correlation value was calculated in a similar fashion. The cue correlation values for the left and right monaural spectral cues (in this case, the shape of the neural excitation pattern) was calculated by taking the difference between the stimulus EP and the possible response location EPs and then summing across frequency the variation of this difference about its mean value. For the interaural spectral cue, the vector difference between the left and right EPs was calculated for both the stimulus and the possible response locations. The dot product between the stimulus and the possible response location vectors gave the ISO cue correlation values. The cue correlation values were normalized in order to facilitate meaningful comparisons across the different acoustical cues. Following Middlebrooks [6], a "z-score normalized" cue value, for each response location corresponding to a given target location, was obtained by subtracting the mean correlation value (across all possible locations) and dividing by the 772 C. T. Jin, A. Corderoy, S. Carlile and A. v. Schaik standard deviation. For these new cue values, termed the cue z-score values, a score of 1.0 or greater indicates p'oocl correlation. 5 Relationship between the ISD and the Cone-of-Confusion The distribution of a given cue's z-score values around the sphere of space surrounding the subject reveals the spatial directions for that cue that correlate best with the given stimulus and target location being examined. An examination of the interaural spectral cue indicated that, unlike the other cues, the range of its cue z-score variation was relatively restricted on the ipsilateral hemisphere of space relative to the sound stimulus (values on the ipsilateral side were approximately 1.0, those on the contralateral side, -1.0). This was the first indication of the more moderate variation of the ISO cue across space as compared with the monaural spectral cues. Closer examination of the ISO cue revealed more detailed variational properties. In order to facilitate meaningful comparisons with the other cues, the ISO cue z-score values were adjusted such that all negative values (i.e., those values at locations generally contralateral to the stimulus) were set to 0.0 and the cue z-score values recalculated. The spatial distribution of the rescaled ISO cue z-score values, as compared with the cue z-score values for the other cues, is shown in Figure 2. The cone of confusion described by the ITO and ILO is clearly evident (Fig. 2a,b) and it can be seen that the ISO cue is closely aligned with these cues (Fig. 2c). Furthermore, the ISO cue demonstrates significant asymmetry along the front-back dimensions. These novel observations demonstrate that while previous work [3, 2] indicates that the ISO cue provides sufficient information to determine a sound's 10eation exactly along the cone of confusion, the variation of the cue z-score values along the cone is substantially less than that for the monaural spectral cues (Fig. 2d), suggest;:-:3 perhaps that this acts to make the monaural spectral cue a more salient cue. (8) Interaural Time Difference (c) Interaural Spectrum 40 40 1.5 o o c: .40 -40 ,g 0 90 180 0 90 180 ~ (b) Interaurai Level Difference (d) Ipsilateral Monaural Spectrum iIi 40 40 o o -40 -40 1.1 o 90 180 0 90 180 Azimuth Figure 2: Spatial plot of the cue z-score values for a single target location (460 azimuth, 20 0 elevation) and broadband sound condition. Gray-scale color values indicate the cue's correlation in different spatial directions with the stimulus cue at the target location. (Zscore values for the ISO cue have been rescaled, see text.) 6 Analysis of Subjects Responses using Cue Z-score Values A given cue's z-score values for the subject's responses across all 76 test locations and five trials were averaged. The mean and standard deviation are presented in a bar graph (Fig. 3). The subjects' response locations correlate highly with the ITO and ILD cue and Spectral Cues in Human Sound Localization 773 the standard deviation of the correlation was low (Fig. 3a,b). In other words, subjects' responses stayed on the cone of confusion of the target location. A similar analysis of the more restricted, rescaled version of the interaural spectral cue shows that despite the spectral manipulations and systematic mislocalizations, subject's were responding to locations which were highly correlated with the interaural spectral cue (Fig. 3c). The bar graphs for the monaural spectral cues ipsilateral and contralateral to the target location show the average correlation of the subjects' responses with these cues varied considerably with the stimulus condition (Fig. 3d-g) and to a lesser extent across subjects. rn Q) :::l (ij 2 > ~ 1 8 If 0 N Q) :::l U 2 o 2 Control Veridical VeridiCal Broadband Interaural Right Mona .... 1 1 ILD =~= o Control VeridiCal VeridiCal Broadband Interaural Right Monaural 2 ISD Control Veridical Veridical Broadband Interaural Right Monaural o (d) Left Contralateral Spectrum Control Veridical Veridical Broadband Interaural Right Monaural (f) Left Ipsilateral Spectrum Control VeridiCal VeridiCal Broadband Interaurai Right Monaural (e) Right Contralateral Spectrum 2 o Control Veridical Veridical Broadband InterauraJ Right Monaural (g) Right Ipsilateral Spectrum 2 o Control V9!idical VeridiCal Broadband I nteraural Right Monaural Figure 3: Correlation of the four subjects' (indicated by different gray bars) localization responses with the different acoustical cues for each stimulus condition. The bar heights indicates the mean cue z-score value, while the error bars indicate standard deviation. 7 Spatial Plots of Correlation Regions As the localization responses tended to lie along the cone of confusion, the relative importance of the spectral cues along the cone of confusion was examined. The correlation values for the spectral cues associated with the subjects' responses were recalculated as a z-score value using only the distribution of values restricted to the cone of confusion. This demonstrates whether the spectral cues associated with the subjects' response locations were better correlated with the stimulus cues, than for any random location on the cone of confusion. Spatial plots of the recalculated response cue z-score values for the spectral cues of one subject (similar trends across subjects), obtained for each stimulus location and across the three different sound conditions, is shown in Figure 4. Spatial regions of both high and low correlation are evident that vary with the stimulus spectrum. The z-score values for the ISD cue shows greater bilateral correlation across space in the veridical interaural condition (Fig. 4d) than for the veridical monaural condition (Fig. 4g), while the right monaural spectral cue demonstrates higher correlation in the right hemisphere of space for the veridical monaural condition (Fig. 4i) as opposed to the veridical interaural condition (Fig. 4t). This result (although not surprising) demonstrates that the auditory system is extracting cues to source location in a manner dependent on the input sound spectrum and in a manner consistent with the spectral infonnation available in the sound spectrum. Figures 4e,h clearly demonstrate that the flat sound spectrum in the left ear was strongly correlated with and influenced the subject's localization judgements for specific regions of space. 774 C. T. lin, A. Corderoy, S. Carlile and A. v. Schaik c: .Q ~ iIi Broadband (a) Veridical Interaural (d) Veridical Monaural (9) 40 a -40 ·180 40 a (e) a (1) 180 40 a -40 180 ·180 ". , . '~'t;' 1 40 a f ., ~ 4 ·40 180 ·180 ·180 a Azimuth (h) a (i) o E -=:2 Ol-Ial Cl. U) 180 180 § .5 .~ Ol::J c: 0 co u 1 >Ol 0 ~ Ol 8 c: I/) 0 , u 0.5 N 0 Ol::J Ol () .~ 15 g Figure 4: Spatial plot of the spectral cue z-score values for one subject's localization responses across the three different sound conditions. 8 Conclusions The correlation of human sound localization responses with the available acoustical cues across three spectrally.. different sound conditions has provided insights into the human auditory system and its integration of cues to produce a coherent percept of spatial location. These data suggest an interrelationship between the interaural spectral cue and the cone of confusion. The ISO cue is front-back asymmetrical along the cone and its cue correlation values vary more moderately as a function of space than those of the monaural spectral cues. These data shed light on the relative role and importance of the interaural and monaural spectral cues. Acknowledgments This research was supported by the ARC, NHMRC, and Dora Lush Scholarship to CJ. References [1] S. Carlile, Virtual auditory space: Generation and applications. New York: Chapman and Hall, 1996. [2] R. O. Duda, "Elevation dependence of the interaural transfer function," in Binaural and spatial hearing in real and virtual environments (R. H. Gilkey and T. R. Anderson, eds.), ch. 3, pp. 4975, Mahwah, New Jersey: Lawrence Erlbaum Associates, 1997. [3] J. A. Janko, T. R. Anderson, and R. H. Gilkey, "Using neural networks to evaluate the viability of monaural and interaural cues for sound localization," in Binaural and Spatial Hearing in real and virtual environments (R. H. Gilkey and T. R. Anderson, eds.), ch. 26, pp. 557-570, Mahwah, New Jersey: Lawrence Erlbaum Associates, 1997. [4] B. Glasberg and B. Moore, "Derivation of auditory filter shapes from notched-noise data," Hearing Research, vol. 47, no. 1-2, pp. 103-138, 1990. [5] F. Wightman and D. Kistler, "The dominant role oflow-frequency interaural time differences in sound localization," 1. Acoust. Soc. Am., vol. 91, no. 3, pp. 1648-1661, 1992. [6] J. Middlebrooks, "Narrow-band sound localization related to external ear acoustics," 1. Acoust. Soc. Am., vol. 92, no. 5, pp. 2607-2624, 1992.
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Neural Computation with Winner-Take-All as the only Nonlinear Operation Wolfgang Maass Institute for Theoretical Computer Science Technische UniversWit Graz A-8010 Graz, Austria email: maass@igi.tu-graz.ac.at http://www.cis.tu-graz.ac.atiigi/maass Abstract Everybody "knows" that neural networks need more than a single layer of nonlinear units to compute interesting functions. We show that this is false if one employs winner-take-all as nonlinear unit: • Any boolean function can be computed by a single k-winner-takeall unit applied to weighted sums of the input variables. • Any continuous function can be approximated arbitrarily well by a single soft winner-take-all unit applied to weighted sums of the input variables. • Only positive weights are needed in these (linear) weighted sums. This may be of interest from the point of view of neurophysiology, since only 15% of the synapses in the cortex are inhibitory. In addition it is widely believed that there are special microcircuits in the cortex that compute winner-take-all. • Our results support the view that winner-take-all is a very useful basic computational unit in Neural VLS!: o it is wellknown that winner-take-all of n input variables can be computed very efficiently with 2n transistors (and a total wire length and area that is linear in n) in analog VLSI [Lazzaro et at., 1989] o we show that winner-take-all is not just useful for special purpose computations, but may serve as the only nonlinear unit for neural circuits with universal computational power o we show that any multi-layer perceptron needs quadratically in n many gates to compute winner-take-all for n input variables, hence winner-take-all provides a substantially more powerful computational unit than a perceptron (at about the same cost of implementation in analog VLSI). Complete proofs and further details to these results can be found in [Maass, 2000]. 294 W. Maass 1 Introduction Computational models that involve competitive stages have so far been neglected in computational complexity theory, although they are widely used in computational brain models, artificial neural networks, and analog VLSI. The circuit of [Lazzaro et aI., 1989] computes an approximate version of winner-take-all on n inputs with just 2n transistors and wires oflength O(n), with lateral inhibition implemented by adding currents on a single wire of length O( n). Numerous other efficient implementations of winner-take-all in analog VLSI have subsequently been produced. Among them are circuits based on silicon spiking neurons ([Meador and Hylander, 1994], [Indiveri, 1999]) and circuits that emulate attention in artificial sensory processing ([Horiuchi et aI., 1997], [Indiveri, 1999]). Preceding analytical results on winner-take-all circuits can be found in [Grossberg, 1973] and [Brown, 1991]. We will analyze in section 4 the computational power of the most basic competitive computational operation: winner-take-all (= l-WTAn). In section 2 we will discuss the somewhat more complex operation k-winner-take-all (k-WTAn ), which has also been implemented in analog VLSI [Urahama and Nagao, 1995]. Section 3 is devoted to soft winner-take-all, which has been implemented by [Indiveri, 1999] in analog VLSJ via temporal coding of the output. Our results shows that winner-take-all is a surprisingly powerful computational module in comparison with threshold gates (= McCulloch-Pitts neurons) and sigmoidal gates. Our theoretical analysis also provides answers to two basic questions that have been raised by neurophysiologists in view of the well-known asymmetry between excitatory and inhibitory connections in cortical circuits: how much computational power of neural networks is lost if only positive weights are employed in weighted linear sums, and how much learning capability is lost if only the positive weights are subject to plasticity. 2 Restructuring Neural Circuits with Digital Output We investigate in this section the computational power of a k-winner-take-all gate computing the function k - WT An : ~n -+ {a, l}n k- WTAn ... with bi = 1 +-+ Xi is among the k largest ofthe inputs Xl, ... ,Xn . [precisely: bi = 1 +-+ Xj > Xi holds for at most k - 1 indices j] E~ E {a, I} Neural Computation with Winner-Take-All 295 Theorem 1. Any two-layer feedf01ward circuit C (with m analog or binary input variables and one binary output variable) consisting of threshold gates (=perceptrons) can be simulated by a circuit W consisting of a single k-winner-take-all gate k-WTAn I applied to weighted sums of the input variables with positive weights. This holds for all digital inputs. and for analog inputs except for some set S ~ IR.m ~f inputs that has measure O. In particular, any booleanfunction f : {D, l}m -+ {O, I} can be computed by a single k-winner-take-all gate applied to positive weighted sums of the input bits. Remarks I. If C has polynomial size and integer weights, whose size is bounded by a polynomial in m, then the number oflinear gates S in W can be bounded by a polynomial in m, and all weights in the simulating circuit W are natural numbers whose size is bounded by a polynomial in m. 2. The exception set of measure D in this result is a union of finitely many hyperplanes in lRm. One can easily show that this exception set S of measure D in Theorem 1 is necessary. 3. Any circuit that has the structure ofW can be converted back into a 2-layerthreshold circuit, with a number of gates that is quadratic in the number of weighted sums (=1inear gates) in W . This relies on the construction in section 4. Proof of Theorem 1: Since the outputs of the gates on the hidden layer of C are from {O, I}, we can assume without loss of generality that the weights a1 , . .. ,an of the output gate G of C are from { - 1, 1} (see for example [Siu et al., 1995] for details; one first observes that it suffices to use integer weights for threshold gates with binary inputs, one can then nonnalize these weights to values in { -1,1} by duplicating gates on the hidden n layer of C). Thus for any circuit input & E IR.m we have C(&) = 1 ¢:} L: ajG j (&) 2: e, j=1 where G1, ... ,Gn are the threshold gates on the hidden layer of C, a1 , .. . , an are from {-I, I}, and e is the threshold of the output gate G. In order to eliminate the negative weights in G we replace each gate G j for which a j = -1 by another threshold gate (; j so that (;j(&) = 1 - Gj (&) for all & E IR.m except on some hyperpJane.2 We set Gj := Gj for all j E {I, . . . ,n} with a j = 1. Then we have for all & E lRm , except for & from some exception set S consisting of up to n hyperplanes, n n 2: a j Gj(&) = 2: (;j(&) -I{j E {I, ... ,n}: aj = -1}1· j=1 j=1 n , , Hence C(&) = 1 ¢:} L: Gj (&) 2: k for all Z E IR.m - S, for some suitable kE N. j=1 Let w{ , ... , win E lR be the weights and ej E IR. be the threshold of gate (; j ,j = 1, .. . , n. I of which we only use its last output bit 2We exploit here that --, I:7:1 W iZi ;::: 0 <=? I:7:1 (-W i )Zi > -0 for arbitrary Wi , Zi, 0 E R . 296 b ZI Zm b '" andback i:w{>O i:wt <0 and i:w{<O W. Maass c G1 , ••. ,Gn are arbitrary threshold gates, G is a threshold gate with weights from {-I, I} w SI, ... ,Sn+1 are linear gates (with positive weights only, which are sums of absolute values of weights from the gates G 1 , . .• ,G n) for j = 1, ... ,n l#j i:wf>o n Sn+1 := L L Iw11zi j=1 i:w1>o we have for every j E {I, ... ,n} and every £ E ~m : Sn+l ~ Sj ¢:} L Iw11zi - L Iw11zi > ej ¢:} Gj (£) = 1 . i:w{>O i:w{<O This implies that the (n + l}st output bn+1 of the k-winner-take-all gate k-WTAn+1 for Neural Computation with Winner-Take-All k := n - k + 1 applied to Sl, ... , Sn+l satisfies bn+1 = 1 ¢:> Ib E {I, ... ,n+ I}: Sj > Sn+dl ~ n - k ¢:> Ib E {I, ... ,n+ I}: Sn+1 ~ Sj}1 ~ k+ 1 ¢:> Ib E {I, ... ,n}: Sn+1 ~ Sj}1 ~ k n A A ¢:> L: Gj(~) ~ k j=l ¢:> C(~) = 1 . Note that all the coefficients in the sums Sl, ... , Sn+1 are positive. 3 Restructuring Neural Circuits with Analog Output 297 • In order to approximate arbitrary continuous functions with values in [0, 1] by circuits that have a similar structure as those in the preceding section, we consider here a variation of a winner-take-all gate that outputs analog numbers between 0 and I, whose values depend on the rank of the corresponding input in the linear order of all the n input numbers. One may argue that such gate is no longer a "winner-take-all" gate, but in agreement with common terminology we refer to it as a soft winner-take-all gate. Such gate computes a function from m.n into [0, l]n Xn ElR soft winner-take-all ... E [0,1] whose ith output Ti E [0,1] is roughly proportional to the rank of Xi among the numbers Xl, ••. , X n . More precisely: for some parameter TEN we set l{jE{I, ... ,n}: xi~xj}I-~ Ti = T ' rounded to 0 or 1 if this value is outside [0,1]. Hence this gate focuses on those inputs Xi whose rank among the n input numbers Xl, • •. ,Xn belongs to the set {~, ~ + 1, ... , min{n, T + ~}}. These ranks are linearly scaled into [0, 1].3 Theorem 2. Circuits consisting oj a single soft winner-take-all gate (oJ which we only use its first output T1) applied to positive weighted sums oj the input variables are universal approximatorsJor arbitrary continuousJunctionsJrom lRm into [0, 1]. • 3It is shown in [Maass, 2000] that actually any continuous monotone scaling into [0,1] can be used instead. 298 W Maass A circuit of the type considered in Theorem 2 (with a soft winner-take-all gate applied to n positive weighted sums 51, ... ,5n ) has a very simple geometrical interpretation: Over each point &: of the input "plane" Rm we consider the relative heights of the n hyperplanes HI, ... ,Hn defined by the n positive weighted sums 51, .. . ,5n. The circuit output depends only on how many ofthe otherhyperplanesH2 , ... , Hn are above HI at this point£. 4 A Lower Bound Result for Winner-Take-All One can easily see that any k-WTA gate with n inputs can be computed by a 2-layer threshold circuit consisting of (~) + n threshold gates: I ": , bl bi , , , ? X · > X· l _ J , ? L:~n-k Xn I I b· J bn Hence the following result provides an optima/lower bound. G) threshold gates n threshold gates Theorem 3. Any JeedJmward threshold circuit (=multi-Iayer perceptron) that computes l-WTAJor n inputs needs to have at least (~) + n gates. • 5 Conclusions The lower bound result of Theorem 3 shows that the computational power of winner-takeall is quite large, even if compared with the arguably most powerful gate commonly studied in circuit complexity theory: the threshold gate (also referred to a McCulloch-Pitts neuron or perceptron). Neural Computation with Winner-Take-All 299 It is well known ([Minsky and Papert, 1969]) that a single threshold gate is not able to compute certain important functions, whereas circuits of moderate (i.e., polynomial) size consisting of two layers of threshold gates with polynomial size integer weights have remarkable computational power (see [Siu et aI., 1995]). We have shown in Theorem 1 that any such 2-layer(i.e., I hidden layer) circuit can be simulated by a single k-winner-take-all gate, applied to polynomially many weighted sums with positive integer weights of polynomial size. We have also analyzed the computational power of soft winner-take-all gates in the context of analog computation. It is shown in Theorem 2 that a single soft winner-take-all gate may serve as the only nonlinearity in a class of circuits that have universal computational power in the sense that they can approximate any continuous functions. Furthermore our novel universal approximators require only positive linear operations besides soft winner-take-all, thereby showing that in principle no computational power is lost if in a biological neural system inhibition is used exclusively for unspecific lateral inhibition, and no adaptive flexibility is lost if synaptic plasticity (i.e., "learning") is restricted to excitatory synapses. Our somewhat surprising results regarding the computational power and universality of winner-take-all point to further opportunities for low-power analog VLSI chips, since winner-take-all can be implemented very efficiently in this technology. References [Brown, 1991] Brown, T. X. (1991). Neural Network Design for Switching Network Control .. Ph.-D.-Thesis, CAL TECH. [Grossberg, 1973] Grossberg, S. (1973). Contour enhancement, short term memory, and constancies in reverberating neural networks. Studies in Applied Mathematics, vol. 52, 217-257. [Horiuchi et aI., 1997] Horiuchi, T. K., Morris, T. G., Koch, C., DeWeerth, S. P. (1997). Analog VLSI circuits for attention-based visual tracking. Advances in Neural Information Processing Systems, vol. 9, 706-712. [Indiveri, 1999] Indiveri, G. (1999). Modeling selective attention using a neuromorphic analog VLSI device, submitted for publication. [Lazzaro et aI., 1989] Lazzaro, 1., Ryckebusch, S., Mahowald, M. A., Mead, C. A. (1989). Winner-take-all networks of O( n) complexity. Advances in Neural Information Processing Systems, vol. I, Morgan Kaufmann (San Mateo), 703-711. [Maass,2000] Maass, W. (2000). On the computational power of winner-take-all, Neural Computation, in press. [Meador and Hylander, 1994] Meador, J. L., and Hylander, P. D. (1994). Pulse coded winner-take-all networks. In: Silicon Implementation of Pulse Coded Neural Networks, Zaghloul, M. E., Meador, 1., and Newcomb, R. W., eds., Kluwer Academic Publishers (Boston),79-99. [Minsky and Papert, 1969] Minsky, M. C., Papert, S. A. (1969). Perceptrons, MIT Press (Cambridge). [Siu et aI., 1995] Siu, K.-Y., Roychowdhury, v., Kailath, T. (1995). Discrete Neural Computation: A Theoretical Foundation. Prentice Hall (Englewood Cliffs, NJ, USA). [Urahama and Nagao, 1995] Urahama, K., and Nagao, T. (1995). k-winner-take-all circuit with O(N) complexity. IEEE Trans. on Neural Networks, vol.6, 776--778.
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Robust Full Bayesian Methods for Neural Networks Christophe Andrieu* Cambridge University Engineering Department Cambridge CB2 1PZ England ca226@eng.cam.ac.uk J oao FG de Freitas UC Berkeley Computer Science 387 Soda Hall, Berkeley CA 94720-1776 USA jfgf@cs.berkeley.edu Abstract Arnaud Doucet Cambridge University Engineering Department Cambridge CB2 1PZ England ad2@eng.cam.ac.uk In this paper, we propose a full Bayesian model for neural networks. This model treats the model dimension (number of neurons), model parameters, regularisation parameters and noise parameters as random variables that need to be estimated. We then propose a reversible jump Markov chain Monte Carlo (MCMC) method to perform the necessary computations. We find that the results are not only better than the previously reported ones, but also appear to be robust with respect to the prior specification. Moreover, we present a geometric convergence theorem for the algorithm. 1 Introduction In the early nineties, Buntine and Weigend (1991) and Mackay (1992) showed that a principled Bayesian learning approach to neural networks can lead to many improvements [1,2]. In particular, Mackay showed that by approximating the distributions of the weights with Gaussians and adopting smoothing priors, it is possible to obtain estimates of the weights and output variances and to automatically set the regularisation coefficients. Neal (1996) cast the net much further by introducing advanced Bayesian simulation methods, specifically the hybrid Monte Carlo method, into the analysis of neural networks [3]. Bayesian sequential Monte Carlo methods have also been shown to provide good training results, especially in time-varying scenarios [4]. More recently, Rios Insua and Muller (1998) and Holmes and Mallick (1998) have addressed the issue of selecting the number of hidden neurons with growing and pruning algorithms from a Bayesian perspective [5,6]. In particular, they apply the reversible jump Markov Chain Monte Carlo (MCMC) algorithm of Green [7] to feed-forward sigmoidal networks and radial basis function (RBF) networks to obtain joint estimates of the number of neurons and weights. We also apply the reversible jump MCMC simulation algorithm to RBF networks so as to compute the joint posterior distribution of the radial basis parameters and the number of basis functions. However, we advance this area of research in two important directions. Firstly, we propose a full hierarchical prior for RBF networks. That * Authorship based on alphabetical order. 380 C. Andrieu, J. F G. d. Freitas and A. Doucet is, we adopt a full Bayesian model, which accounts for model order uncertainty and regularisation, and show that the results appear to be robust with respect to the prior specification. Secondly, we present a geometric convergence theorem for the algorithm. The complexity of the problem does not allow for a comprehensive discussion in this short paper. We have, therefore, focused on describing our objectives, the Bayesian model, convergence theorem and results. Readers are encouraged to consult our technical report for further results and implementation details [Sp. 2 Problem statement Many physical processes may be described by the following nonlinear, multivariate input-output mapping: (1) where Xt E ~d corresponds to a group of input variables, Yt E ~c to the target variables, TIt E ~c to an unknown noise process and t = {I, 2,' .. } is an index variable over the data. In this context, the learning problem involves computing an approximation to the function f and estimating the characteristics of the noise process given a set of N input-output observations: 0 = {Xl, X2, ... ,XN, YI, Y2, ... ,Y N } Typical examples include regression, where YI:N,1:C2 is continuous; classification, where Y corresponds to a group of classes and nonlinear dynamical system identification, where the inputs and targets correspond to several delayed versions of the signals under consideration. We adopt the approximation scheme of Holmes and Mallick (199S), consisting of a mixture of k RBFs and a linear regression term. Yet, the work can be easily extended to other regression models. More precisely, our model Mis: Mo : Yt = b + f3'Xt + TIt k = 0 Mk: Yt = ~~=l aj¢(IIx t - 11)1) + b + f3'Xt + TIt k ~ 1 (2) where" ./1 denotes a distance metric (usually Euclidean or Mahalanobis), I-Lj E ~d denotes the j-th RBF centre for a model with k RBFs, aj E ~c the j-th RBF amplitude and b E ~c and f3 E ~d X ~c the linear regression parameters. The noise sequence TIt E ~c is assumed to be zero-mean white Gaussian. It is important to mention that although we have not explicitly indicated the dependency of b, f3 and TIt on k, these parameters are indeed affected by the value of k. For convenience, we express our approximation model in vector-matrix form: b1 . . , be Yl,l . . . Y1,e Y2,1 . .. Y2,e 1 X1,1 .. . X1 ,d </J(X1 , 1'1)' .. </J(X1, JLk) f31,1 ... f31 ,e 1 X2,1 ... X2,d </J(X2, 1'1) .. . </J(X2, JLk) = f3d,l ... f3d,e +n1:N a1,1 .. , a1,e YN,l .. . YN,c 1 XN,l" ' XN,d </J(XN, 1'1) ' " </J(XN, JLk) ak,l .,. ak,e 1The software is available at http://vvv . cs . berkeley. edur jfgf. 2Y1 :N,1:e is an N by c matrix, where N is the number of data and c the number of outputs. We adopt the notation Y1:N,j ~ (Y1,j,Y2,j, .. . ,YN,j)' to denote all the observations corresponding to the j-th output (j-th column of y). To simplify the notation, Yt is equivalent to Yt,l :c' That is, if one index does not 'appear, it is implied that we are referring to all of its possible values. Similarly, Y is equivalent to Y1:N,1:c, We will favour the shorter notation and only adopt the longer notation to avoid ambiguities and emphasise certain dependencies. Robust Full Bayesian Methods for Neural Networks 381 where the noise process is assumed to be normally distributed Dt '" N(o, un for i = 1, ... ,c. In shorter notation, we have: Y = D(J.Ll:k,l:d,Xl:N,l:d)Ol:l+d+k,l:C + Dt (3) We assume here that the number k of RBFs and their parameters (J ~ {Ol:m,l:c, J.Ll:k,l:d' uI:J, with m = 1 + d + k, are unknown. Given the data set {x,y}, our objective is to estimate k and (J E 8 k . 3 Bayesian model and aims We follow a Bayesian approach where the unknowns k and (J are regarded as being drawn from appropriate prior distributions. These priors reflect our degree of belief on the relevant values of these quantities [9]. Furthermore, we adopt, a hierarchical prior structure that enables us to treat the priors' parameters (hyper-parameters) as random variables drawn from suitable distributions (hyper-priors). That is, instead of fixing the hyper-parameters arbitrarily, we acknowledge that there is an inherent uncertainty in what we think their values should be. By devising probabilistic models that deal with this uncertainty, we are able to implement estimation techniques that are robust to the specification of the hyper-priors. The overall parameter space 8 x 'II can be written as a finite union of subspaces 8 x 'II = (U~:ir {k} x 8 k) X 'II where 8 0 ~ (JRd+l)C X (JR+) C and 8 k ~ (JRd+l+k)c X (JR+)C X !lk for k E {I, ... ,kmax }. That is, ° E (JRd+l+k)C, 0' E (JR+)C and J.L E !lk. The hyper-parameter space 'II ~ (1R+)C+l, with elements 'l/J ~ {A, 82}, will be discussed at the end of this section. The space of the radial basis centres !lk is defined as a compact set including the input data: !lk ~ {J.L; J.Ll:k,i E [min(xl:N,i) -tSi, max(xl:N,i) +tSdk for i = 1, ... ,d with J.Lj,i ofJ.L1 i for j of- l}. Si = II max(xl:N,i) - min(xl:N,i)1I denotes the Euclidean distance for the i-th dimension of the input and t is a user specified parameter that we only need to consider if we wish to place basis functions outside the region where the input data lie. That is, we allow !lk to include the space of the input data and extend it by a factor which is proportional to the spread of the input data. The hyper-volume of this space is: ~k ~ (rrf=l (1 + 2t)Si)k. The maximum number of basis functions is defined as kmax ~ (N - (d + 1)) We also define !l ~ U~:(nk} x !lk with !lo ~ 0. Under the assumption of independent outputs given (k, (J), the likelihood p(ylk, (J, 'l/J, x) for the approximation model described in the previous section is: D. (2,,-.,.1) -N /2 exp ( - 2!; (hN,i - D(I'"" x)a',m,,)' (Y',N" - D(I'H, x)a',m,;) ) We assume the following structure for the prior distribution: p(k, (J, 'l/J) = p( ol:mlk, 0'2, ( 2)p(J.Ll:k Ik)p(kIA)p(u2)p(A)p( ( 2) where the scale parameters 0';, are assumed to be independent of the hyperparameters (i. e. p( u 2 1A, ( 2 ) = p( 0'2)), independent of each other (p( 0'2) = rr~=l p(u;)) and distributed according to conjugate inverse-Gamma prior distributions: 0'; '" I9 (~, ~). When 'Uo = 0 and /0 = 0, we obtain Jeffreys' uninformative prior [9]. For a given 0'2, the prior distribution p(k, Glom, J.Ll:kIU2, A, 82 ) is: [lI C I 2 2 -1/2 1 I )] [][o(k,J.Ll:k)] [ Aklk! ] 27ru i 8 i 1m I exp (- -2-2 0l'm iOl:m i o.k k. . 20'.8. .,' :s " max AJI)'! t=l t t 6 J=0 382 C. Andrieu, J. F. G. d. Freitas and A. Doucet where 1m denotes the identity matrix of size m x m and IIn(k, ILI:k) is the indicator function of the set n (1 if (k,IL1:k) En, 0 otherwise). The prior model order distribution p(kIA) is a truncated Poisson distribution. Conditional upon k, the RBF centres are uniformly distributed. Finally, conditional upon (k, IL1:k), the coefficients 01:m,i are assumed to be zero-mean Gaussian with variance c5~ ur The hyper-parameters c52 E (IR.+)C and A E IR.+ can be respectively interpreted as the expected signal to noise ratios and the expected number of radial basis. We assume that they are independent of each other, i.e. p(A, c52) = p(A)p(c52 ). Moreover, p(c5 2 ) = TI~=1 p(c5~). As c52 is a scale parameter, we ascribe a vague conjugate prior density to it: c5~ "" YQ (a,p ,(302) for i = 1, ... ,c, with a02 = 2 and (302 > O. The variance of this hyper-prior with a02 = 2 is infinite. We apply the same method to A by setting an uninformative conjugate prior [9]: A"" Qa(1/2 +Cl,c2) (ci« 1 i = 1,2). 3.1 Estimation and inference aims The Bayesian inference of k, 0 and 1/J is based on the joint posterior distribution p(k, 0, 1/Jlx, y) obtained from Bayes' theorem. Our aim is to estimate this joint distribution from which, by standard probability marginalisation and transformation techniques, one can "theoretically" obtain all posterior features of interest. We propose here to use the reversible jump MCMC method to perform the necessary computations, see [8] for details. MCMC techniques were introduced in the mid 1950's in statistical physics and started appearing in the fields of applied statistics, signal processing and neural networks in the 1980's and 1990's [3,5,6,10,11]. The key idea is to build an ergodic Markov chain (k(i) , O(i), 1/J(i»)iEN whose equilibrium distribution is the desired posterior distribution. Under weak additional assumptions, the P » 1 samples generated by the Markov chain are asymptotically distributed according to the posterior distribution and thus allow easy evaluation of all posterior features of interest. For example: .-.... 1 P . .-... . ~~ 0(i)1I . (k(i») p(k = Jlx, y) = p I){j}(k(t») and IE(Olk = J, x, y) = t-~ {J}. i=1 ~i=1 lI{j}(k(t») (4) In addition, we can obtain predictions, such as: P .-... 1 ~ (i) (i) IE(YN+llxl:N+l,Yl:N) = p L...,.D(ILl:k,XN+I)OI:m i=1 (5) 3.2 Integration of the nuisance parameters According to Bayes theorem, we can obtain the posterior distribution as follows: p(k, 0, 1/Jlx, y) ex p(Ylk, 0, 1/J, x)p(k, 0, 'ljJ) In our case, we can integrate with respect to 01:m (Gaussian distribution) and with respect to u; (inverse Gamma distribution) to obtain the following expression for the posterior: (k A c521 ) [rrC (c52)-m/2IM' 11/2 (TO + Y~:N,iP i,kYl:N,i ) (_ N~VQ)] P ,IL1:k" x,yex t t,k 2 X i=1 [lIn(k,lLk)][ Ak/k!. ][rrC (c5~)-(062 +l)exp(- (362 )][(A)(Cl-l/2)exP (-C2A)] <;Sk "kmax AJ / ., t c52 LJj=O J. i=l t (6) Robust Full Bayesian Methods for Neural Networks 383 It is worth noticing that the posterior distribution is highly non-linear in the RBF centres I-'k and that an expression of p(klx,y) cannot be obtained in closed-form. 4 Geometric convergence theorem It is easy to prove that the reversible jump MCMC algorithm applied to our model converges, that is, that the Markov chain (k( i) , I-'l( ~~, A (i) , 82( i») is ergodic. We . iEN present here a stronger result, namely that (k(i)'I-'(I~LA(i),82(i») converges to . iEN the required posterior distribution at a geometric rate: Theorem 1 Let (k(i), I-'i~)k' A(i), 82(i») be the Markov chain whose transition . iEN kernel has been described in Section 3. This Markov chain converges to the probability distribution p (k, I-'l :k, A, 82 1 x, y). Furthermore this convergence occurs at a geometric rate, that is, for almost every initial point (k(O), I-'i~k, A (0),82(0») E 11 x 'II there exists a function of the initial states Co > 0 and a constant and p E [0,1) such that IIp(i) (k,I-'l:k,A,82) -p(k,I-'l:k,A, 82 I x ,y)IITV ~ CopLi/kmaxJ (7) where p(i) (k,I-'l:k,A,82) is the distribution of (k(i)'l-'i~~,A(i),82(i») and II·IITV is the total variation norm [11]. Proof. See [8] • Corollary 1 If for each iteration i one samples the nuisance parameters (Ol:m, u%) then the distribution of the series (k(i), oi~~, I-'i~~, u~(i), A(i), 8Z(i»)iEN converges geometrically towards p(k,Ol:m,I-'l:k,ULA,82Ix,y) at the same rate p. 5 Demonstration: robot arm data This data is often used as a benchmark to compare learning algorithms3 . It involves implementing a model to map the joint angle of a robot arm (Xl, X2) to the position of the end of the arm (Yl, yz). The data were generated from the following model: Yl = 2.0cos(xt} + 1.3COS(Xl +X2) + El Y2 = 2.0sin(xt} + 1.3sin(xl +X2) +E2 where Ei '" N(O, 0"2); 0" = 0.05. We use the first 200 observations of the data set to train our models and the last 200 observations to test them. In the simulations, we chose to use cubic basis functions. Figure 1 shows the 3D plots of the training data and the contours of the training and test data. The contour plots also include the typical approximations that were obtained using the algorithm. We chose uninformative priors for all the parameters and hyper-parameters (Table 1). To demonstrate the robustness of our algorithm, we chose different values for (382 (the only critical hyper-parameter as it quantifies the mean of the spread 8 of Ok)' The obtained mean square errors and probabilities for 81, 82, u~ k' u~ k and k, shown in Figure 2, clearly indicate that our algorithm is robust with 'respe~t to prior specification. Our mean square errors are of the same magnitude as the ones reported by other researchers [2,3,5,6]; slightly better (Not by more than 10%). Moreover, our algorithm leads to more parsimonious models than the ones previously reported. 3The robot arm data set can be found in David Mackay's home page: http://vol.ra.phy.cam.ac.uk/mackay/ 384 C. Andrieu. J. F G. d. Freitas and A. Doucet 5 .' . >.0 .. .. .. .. ... ... ........ ..... -~ .' . ' .' 2 x2 o -2 o x1 4r---~--~--~----, oL---~------~--~ -2 -1 o 2 4r---~--~--~----, o~--~------~--~ -2 -1 o 2 2 5 ~ 0 ·· · ··· · -5 ... ... 4 2 x2 o -2 4r---~--~--~----, 0 -2 -1 0 2 4 :~ 0 -2 -1 0 2 Figure 1: The top plots show the training data surfaces corresponding to each coordinate of the robot arm's position. The middle and bottom plots show the training and validation data [- -] and the respective REF network mappings [-]. Table 1: Simulation parameters and mean square test errors. a 02 /30 2 Vo ')'0 CI C2 MS ERROR 2 0.1 0 0 0.0001 0.0001 0.00505 2 10 0 0 0.0001 0.0001 0.00503 2 100 0 0 0.0001 0.0001 0.00502 6 Conclusions We presented a general methodology for estimating, jointly, the noise variance, parameters and number of parameters of an RBF model. In adopting a Bayesian model and the reversible jump MCMC algorithm to perform the necessary integrations, we demonstrated that the method is very accurate. Contrary to previous reported results, our experiments indicate that our model is robust with respect to the specification of the prior. In addition, we obtained more parsimonious RBF networks and better approximation errors than the ones previously reported in the literature. There are many avenues for further research. These include estimating the type of basis functions, performing input variable selection, considering other noise models and extending the framework to sequential scenarios. A possible solution to the first problem can be formulated using the reversible jump MCMC framework. Variable selection schemes can also be implemented via the reversible jump MCMC algorithm. We are presently working on a sequential version of the algorithm that allows us to perform model selection in non-stationary environments. References [1] Buntine, W.L. & Weigend, A.S. (1991) Bayesian back-propagation. Complex Systems 5:603-643. Robust Full Bayesian Methods for Neural Networks 385 'fl and 'fl 1 2 ~and~ k 0.06 0.8 0 0.2 0.6 0.04 II 0.4 ~0. 1 0.02 :!:l. 0.2 I 0 0 I 100 200 0 12 14 16 0.06 O.B 0.04 0.6 0.4 '" no 0.1 0.02 1 0.2 0 0 100 200 0 12 14 16 0.06 0.8 8 0.2 0.6 ,... 0.4 "'00 0.1 0.02 :!:l. 0.2 I 0 0 100 200 0 2 4 6 12 14 16 x 10-" Figure 2: Histograms of smoothness constraints (~1 and 82), noise variances (O'i k and O'~ k) and model order (k) for the robot arm data using 3 different values f~r {382. The plots confirm that the algorithm is robust to the setting of {382. [2] Mackay, D.J.C. (1992) A practical Bayesian framework for backpropagation networks. Neural Computation 4:448-472. [3] Neal, R.M. (1996) Bayesian Learning for Neural Networks. New York: Lecture Notes in Statistics No. 118, Springer-Verlag. [4] de Freitas, J.F.G., Niranjan, M., Gee, A.H. & Doucet, A. (1999) Sequential Monte Carlo methods to train neural network models. To appear in Neural Computation. [5] Rios Insua, D. & Miiller, P. (1998) Feedforward neural networks for nonparametric regression. Technical report 98-02. Institute of Statistics and Decision Sciences, Duke University, http://vtw.1 . stat. duke. edu. [6] Holmes, C.C. & Mallick, B.K. (1998) Bayesian radial basis functions of variable dimension. Neural Computation 10:1217-1233. [7] Green, P.J. (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82:711-732. [8] Andrieu, C., de Freitas, J.F.G. & Doucet, A. (1999) Robust full Bayesian learning for neural networks. Technical report CUED/F-INFENG/TR 343. Cambridge University, http://svr-www.eng.cam.ac.uk/. [9] Bernardo, J.M. & Smith, A.F.M. (1994) Bayesian Theory. Chichester: Wiley Series in Applied Probability and Statistics. [10] Besag, J., Green, P.J., Hidgon, D. & Mengersen, K. (1995) Bayesian computation and stochastic systems. Statistical Science 10:3-66. [11] Tierney, L. (1994) Markov chains for exploring posterior distributions. The Annals of Statistics. 22(4):1701-1762.
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A Neurodynamical Approach to Visual Attention Gustavo Deco Siemens AG Corporate Technology Neural Computation, ZT IK 4 Otto-Hahn-Ring 6 81739 Munich, Germany Gustavo.Deco@mchp.siemens.de JosefZihl Institute of Psychology Neuropsychology Ludwig-Maximilians-University Munich Leopoldstr. 13 80802 Munich, Germany Abstract The psychophysical evidence for "selective attention" originates mainly from visual search experiments. In this work, we formulate a hierarchical system of interconnected modules consisting in populations of neurons for modeling the underlying mechanisms involved in selective visual attention. We demonstrate that our neural system for visual search works across the visual field in parallel but due to the different intrinsic dynamics can show the two experimentally observed modes of visual attention, namely: the serial and the parallel search mode. In other words, neither explicit model of a focus of attention nor saliencies maps are used. The focus of attention appears as an emergent property of the dynamic behavior of the system. The neural population dynamics are handled in the framework of the mean-field approximation. Consequently, the whole process can be expressed as a system of coupled differential equations. 1 Introduction Traditional theories of human vision considers two functionally distinct stages of visual processing [1]. The first stage, termed the preattentive stage, implies an unlimitedcapacity system capable of processing the information contained in the entire visual field in parallel. The second stage is termed the attentive or focal stage, and is characterized by the serial processing of visual information corresponding to local spatial regions. This stage of processing is typically associated with a limited-capacity system which allocates its resources to a single particular location in visual space. The designed psychophysical experiments for testing this hypothesis consist of visual search tasks. In a visual search test the subject have to look at the display containing a frame filled with randomly positioned items in order to seek for an a priori defined target item. All other items in a frame which are not the target are called distractors. The number of items in a frame is called the frame size. The relevant variable to be measured is the reaction time as a function of the frame size. In this context, the Feature Integration Theory, assumes that the two stage processes operate sequentially [1]. The first early pre attentive stage runs in parallel over the complete visual field extracting single primitive features without A Neurodynamical Approach to Visual Attention J1 integrating them. The second attentive stage has been likened to a spotlight. This metaphor alludes that attention is focally allocated to a local region of the visual field where stimuli are processed in more detail and passed to higher level of processing, while, in the other regions not illuminated by the attentional spotlight, no further processing occurs. Computational models formulated in the framework of feature integration theory require the existence of a saliency or priority map for registering the potentially interesting areas of the retinal input, and a gating mechanism for reducing the amount of incoming visual information, so that limited computational resources in the system are not overloaded. The priority map serves to represent topographically the relevance of different parts of the visual field, in order to have a mechanism for guiding the attentional focus on salient regions of the retinal input. The focused area will be gated, such that only the information within will be passed further to yet higher levels, concerned with object recognition and action. The disparity between these two stages of attentional visual processing originated a vivid experimental disputation. Duncan and Humphreys [2] have postulated a hypothesis that integrates both attentional modes (parallel and serial) as an instantiates of a common principle. This principle sustains in both schemes that a selection is made. In the serial focal scheme, the selection acts on in the space dimension, while in the parallel spread scheme the selection concentrates in feature dimensions, e.g. color. On the other hand, Duncan's attentional theory [3] proposed that after a first parallel search a competition is initiated, which ends up by accepting only one object namely the target. Recently, several electrophysiological experiments have been performed which seems to support this hypothesis [4]. Chelazzi et al. [4] measured IT (inferotemporal) neurons in monkeys observing a display containing a target object (that the monkey has seen previously) and a distractor. They report a short period during which the neuron's response is enhanced. After this period the activity level of the neuron remains high if the target is the neuron's effective stimulus, and decay otherwise. The challenging question is therefore: is really the linear increasing reaction time observed in some visual search tests due to a serial mechanism? or is there only parallel processing followed by a dynamical time consuming latency? In other words, are really priority maps and spotlight paradigm required? or can a neurodynamical approach explain the observed psychophysical experiments? Furthermore, it should be clarified if the feature dimension search is achieved independently in each feature dimension or is done after integrating the involved feature dimensions. We study in this paper these questions from a computational perspective. We formulate a neurodynamical model consisting in interconnected populations of biological neurons specially designed for visual search tasks. We demonstrate that it is plausible to build a neural system for visual search, which works across the visual field in parallel but due to the different intrinsic dynamics can show the two experimentally observed modes of visual attention, namely: the serial focal and the parallel spread over the space mode. In other words, neither explicit serial focal search nor saliency maps should be assumed. The focus of attention is not ·included in the system but just results after convergence of the dynamical behavior of the neural networks. The dynamics of the system can be interpreted as an intrinsic dynamical routing for binding features if top-down information is available. Our neurodynamical computational model requires independent competition mechanism along each feature dimension for explaining the experimental data, implying the necessity of the independent character of the search in separated and not integrated feature dimensions. The neural population dynamics are handled in the framework of the meanfield approximation yielding a system of coupled differential equations. 2 Neurodynamical model We extend with the present model the approach of Usher and Niebur [5], which is based on the experimental data of Chelazzi et al. [4], for explaining the results of visual search experiments. The hierarchical architecture of our system is shown in Figure 1. The input retina is given as a matrix of visual items. The location of each item at the retina is 12 G. Deco and J. Zihl specified by two indices ij meaning the position at the row i and the column j . The dimension of this matrix is SxS, i.e. the frame size is also SxS . The information is processed at each spatial location in parallel. Different feature maps extract for the item at each position the local values of the features. In the present work we hypothesize that selective attention is guided by an independent mechanism which corresponds to the independent search of each feature. Let us assume that each visual item can be defined by K features. Each feature k can adopt L(k) values, for example the feature color can have the values red or green (in this case L( color) =2). For each feature map k exist L(k) layers of neurons for characterizing the presence of each feature value. 1-I I I I Stern SpikIng Neurons Figure 1: Hierarchical architecture of spiking neural modules for visual selective attention. Solid arrows denote excitatory connections and dotted arrows denote inhibitory connections A cell assembly consisting in a population of full connected excitatory integrate-and-fire spiking neurons (pyramidal cells) is allocated in each layer and for each item location for encoding the presence of a specific feature value (e.g. color red) at the corresponding position. This corresponds to a sparse distributed representation. The feature maps are topographically ordered, i.e. the receptive fields of the neurons belonging to the cell assembly ij at one of these maps are sensible to the location ij at the retinal input. We further assume that the cell assemblies in layers corresponding to a feature dimension are mutually inhibitory. Inhibition is modeled, according to the constraint imposed by Dale's principle, by a different pool of inhibitory neurons. Each feature dimension has therefore an independent pool of inhibitory neurons. This accounts for the neurophysiological fact that the response of V 4 neurons sensible to a specific feature value is enhanced and the activity of the other neurons sensible to other feature values are suppressed. A high level map consisting also in a topographically ordered excitatory cell assemblies is introduced for integration of the different feature dimension at each item location, i.e. for binding the features of each item. These cell assemblies are also mutually inhibited through a common A Neurodynamical Approach to Visual Attention 13 pool of inhibitory neurons. This layer corresponds to the modeling of IT neurons, which show location specific enhancement of activity by suppression of the responses of the cell assemblies associated to other locations. This fact would yield a dynamical formation of a focus of attention without explicitly assuming any spotlight. Top-down information consisting in the feature values at each feature dimension of the target item is feed in the system by including an extra excitatory input to the corresponding feature layers. The whole system analyzes the information at all locations in parallel. Larger reaction times correspond to slower dynamical convergence at all levels, i.e. feature map and integration map levels. Instead of solving the explicit set of integrate-and-fire neural equations, the Hebbian cell assemblies adopted representation impels to adopt a dynamic theory whose dependent variables are the activation levels of the cell popUlations. Assuming an ergodic behavior [5] it is possible to derive the dynamic equations for the cell assembly activities level by utilizing the mean-field approximation [5]. The essential idea consists in characterizing each cell assembly by means of each activity x, and an input current that is characteristic for all cells in the popUlation, denoted by I, which satisfies: x = F(l) (I) which is the response function that transforms current into discharge rates for an integrateand-fire spiking neuron with deterministic input, time membrane constant 't and absolute refractory time T r • The system of differential equations describing the dynamics of the feature maps are: P F A -bF(l k(t))+Io+I ijkl+I kl+V a S S L(k) 'tpa/Pk(t) = -IPk(t) + eLL L F(lijk/(t)) i-lj-lk-l -dF(lPk(t)) where Iijk/(t) is the input current for the population with receptive field at location ij of the feature map k that analysis the value feature I, I P k( t) is the current in the inhibitory pool bounded to the feature map layers of the feature dimension k. The frame size is S. The additive Gaussian noise v considered has standard deviation 0.002. The synaptic time constants were 't = 5 msec for the excitatory populations and 'tp = 20 for the inhibitory pools. The synaptic weights chosen were: a = 0.95, b = 0.8, c = 2. and d = 0.1 . 10 = 0.025 is a diffuse spontaneous background input, IF ijk/ is the sensory input to the cells in feature map k sensible to the value I and with receptive fields at the location ij at the retina. This input characterizes the presence of the respective feature value at the corresponding position. A value of 0.05 corresponds to the presence of the respective feature value and a value of 0 to the absence of it. The top-down target information IA kl was equal 0.005 for the layers which code the target properties and 0 otherwise. The higher level integrating assemblies are described by following differential equation system: 14 a H _ PH 'tHa/ ij(t) = - [ij(t) + a F(lij(t» - b F(l (t» K L(k) lo+w L LF(lijk/(t»+V k - 11- 1 s s a PH PH _ ~ H 'tP'ra"/ (t) = -I (t) + c £.J L F(l ij(t» i - Ij- 1 G. Deco and J. Zihl where [Hij(t) is the input current for the population with receptive field at location ij of the high level integrating map, IPH (t) is the associated current in the inhibitory pool. The synaptic time constants were 'tH "" 5 msec for the excitatory populations and 'tpH = 2C for the inhibitory pools. The synaptic weights chosen were: ..-.. ...-.... ..-......-... ..-.. a = 0.95, b .. 0.8, w .. 1, c .. 1. and d - 0.1 . These systems of differential equations were integrated numerically until a convergence criterion were reached. This criterion were that the neurons in the high level map are polarized, i.e. H F(/' . (t» lmaxlnrax 1_·~-,im"",a"-,xJ;....·~....::;j~ma,,,-x ____ > e (S2 - 1) where the index imaxjmax denotes the cell assembly in the high level map with maximal activity and the threshold e was chosen equal to 0.1. The second in the l.h.s measure the mean distractor activity. At each feature dimension the fixed point of the dynamic is given by the activity of cell assemblies at the layers with a common value with the target and corresponding to items having this value. For example, if the target is red, at the color map, the activity at the green layer will be suppressed and the cell assemblies corresponding to red items will be enhanced. At the high-level map, the populations corresponding to location which are maximally in all feature dimensions activated will be enhanced by suppressing the others. In other words, the location that shows all feature dimension equivalent at what top-down is stimulated and required, will be enhanced when the target is at this location. 3 Simulations of visual search tasks In this section we present results simulating the visual search experiments involving feature and conjunction search [1]. Let us define the different kinds of search tasks by given a pair of numbers m and n , where m is the number of feature dimensions by which the dis tractors differ from the target and n is the number of feature dimensions by which the distractor groups simultaneously differ from the target. In other words, the feature search corresponds to aI, I-search; a standard conjunction search corresponds to a 2,1search; a triple conjunction search can be a 3,1 or a 3,2-search if the target differs from all distractor groups in one or in two features respectively. We assume that the items are defined by three feature dimensions (K = 3, e.g. color, size and position), each one A Neurodynamical Approach to Visual Attention 15 having two values (L(k) = 2 for k = 1, 2,3). At each size we repeat the experiment 100 times, each time with different randomly generated distractors and target. T 100 00' 0.00 ..•... (a) 3,I-search 085 080 2,I-search 0.15 010 065 060 055 050 045 3,2-search 0.40 0.35 0.30 ... -_ . .,;. '. . . __ . I, I-search - ':""..; ~,.....-.::.---=--.:~-~: - ---- --- ---- -- ---- - -- - -- - --10 .00 2000 3000 4000 50 .00 Frame size Figure 2: Search times for feature and conjunction searches obtained utilizing the presented model. We plot as result the mean value T of the 100 simulated reaction times (in msec) as a function of the frame size. In Figure 2, the results obtained for I, 1 ~ 2, 1 ~ 3, I and 3,2searches are shown. The slopes of the reaction time vs. frame size curves for all simulations are absolutely consistent with the existing experimental results[I). The experimental work reports that in feature search (1,1) the target is detected in parallel across the visual field. Furthermore, the slopes corresponding to · standard conjunction search and triple conjunction search are a linear function of the frame size, where by the slope of the triple conjunction search is steeper or very flat than in the case of standard search (2,1) if the target differs from the distractors in one (3,1) or two features (3,2) respectively. In order to analyze more carefully the dynamical evolution of the system, we plot in Figure 3 the temporal evolution of the rate activity corresponding to the target and to the distractors at the high-level integrating map and also separately for each feature dimension level for a parallel (I, I-search) and a serial (3,I-search) visual tasks. The frame size used is 25. It is interesting to note that in the case of I, I-search the convergence time in all levels are very small and therefore this kind of search appears as a parallel search. In the case of 3, I-search the latency of the dynamic takes more time and therefore this kind of search appears as a serial one, in spite that the underlying mechanisms are parallel. In this case (see Figure 3-c) the large competition present in each feature dimension delays the convergence of the dynamics at each feature dimension and therefore also at the highlevel map. Note in Figure 3-c the slow suppression of the distractor activity that reflects the underlying competition. 16 (High-Level-Map) Rates dOOOO I="T---,--.-----,--......---=. ~ Thrget-Activlty (l,l-search) G. Deco and J. Zihl Thrget-Activlty (3,I-search) (a) Distractors-Activity (l,l-search) \ \ Distractors-Activity (3,I-search) , ' ~~~~=*=:~~£~:;~:;;;,:~::,j.~C:~~-~:~ :JC)aoo Time Rates (b) l,l-search Rates (c) 3~1-search ;.t\ ....... f ...... ·::I(. ........ .: ·¥·:·..:;-;~-:..::·'r-J..~ ... .;./ '.~ ... :~""~+ .. (:. ... ~· !. ",,-- Distractors-Activit~ ..... ,,'\,..... ---..-------. .. _ .............. _-_. -.---Time Time Figure 3: Activity levels during visual search experiments. (a) High-level-map rates for target F(lHimaxjmax(t» and mean distractors-activity. (b) Feature-level map rates for target and one distractor activity for 1,1-search. There is one curve for each feature dimension (i.e. 3 for target and 3 for distractor. (c) the same as (b) but for 3,1search. References [1] Treisman, A. (1988) Features and objects: The fourteenth Barlett memorial lecture. The Quarterly Journal of Experimental Psychology, 4OA, 201-237. [2] Duncan, J. and Humphreys, G. (1989) Visual search and stimulus similarity. Psychological Review, 96, 433-458. [3] Duncan, J. (1980) The locus of interference in the perception of simultaneous stimuli. Psychological Review, 87, 272-300. [4] Chelazzi, L., Miller, E., Duncan, J. and Desimone, R. (1993) A neural basis for visual search in inferior temporal cortex. Nature (London), 363, 345-347. [5] Usher, M. and Niebur, E. (1996) Modeling the temporal dynamics of IT neurons in visual search: A mechanism for top-down selective attention. Journal of Cognitive Neuroscience, 8, 311-327.
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Dynamics of Supervised Learning with Restricted Training Sets and Noisy Teachers A.C.C. Coolen Dept of Mathematics King's College London The Strand, London WC2R 2LS, UK tcoolen@mth.kc1.ac.uk C.W.H.Mace Dept of Mathematics King's College London The Strand, London WC2R 2LS, UK cmace@mth.kc1.ac.uk Abstract We generalize a recent formalism to describe the dynamics of supervised learning in layered neural networks, in the regime where data recycling is inevitable, to the case of noisy teachers. Our theory generates reliable predictions for the evolution in time of training- and generalization errors, and extends the class of mathematically solvable learning processes in large neural networks to those situations where overfitting can occur. 1 Introduction Tools from statistical mechanics have been used successfully over the last decade to study the dynamics of learning in layered neural networks (for reviews see e.g. [1] or [2]). The simplest theories result upon assuming the data set to be much larger than the number of weight updates made, which rules out recycling and ensures that any distribution of relevance will be Gaussian. Unfortunately, both in terms of applications and in terms of mathematical interest, this regime is not the most relevant one. Most complications and peculiarities in the dynamics of learning arise precisely due to data recycling, which creates for the system the possibility to improve performance by memorizing answers rather than by learning an underlying rule. The dynamics of learning with restricted training sets was first studied analytically in [3] (linear learning rules) and [4] (systems with binary weights). The latter studies were ahead of their time, and did not get the attention they deserved just because at that stage even the simpler learning dynamics without data recycling had not yet been studied. More recently attention has moved back to the dynamics of learning in the recycling regime. Some studies aimed at developing a general theory [5, 6, 7], some at finding exact solutions for special cases [8]. All general theories published so far have in common that they as yet considered realizable scenario's: the rule to be learned was implementable by the student, and overfitting could not yet occur. The next hurdle is that where restricted training sets are combined with unrealizable rules. Again some have turned to non-typical but solvable cases, involving Hebbian rules and noisy [9] or 'reverse wedge' teachers [10]. More recently the cavity method has been used to build a general theory [11] (as yet for batch learning only). In this paper we generalize the general theory launched in [6,5,7], which applies to arbitrary learning rules, to the case of noisy teachers. We will mirror closely the presentation in [6] (dealing with the simpler case of noise-free teachers), and we refer to [5, 7] for background reading on the ideas behind the formalism. 238 A. C. C. Coolen and C. W. H. Mace 2 Definitions As in [6, 5] we restrict ourselves for simplicity to perceptrons. A student perceptron operates a linear separation, parametrised by a weight vector J E iRN : S:{-I,I}N -t{-I,I} S(e) = sgn[J·e] It aims to emulate a teacher o~erating a similar rule, which, however, is characterized by a variable weight vector BE iR ,drawn at random from a distribution P(B) such as output noise: P(B) = >'6[B+B*] + (1->')6[B-B*] (1) Gaussian weight noise: P(B) = [~~/NrN e- tN(B-B')2/E2 (2) The parameters>. and ~ control the amount of teacher noise, with the noise-free teacher B = B* recovered in the limits>. -t 0 and ~ -t O. The student modifies J iteratively, using examples of input vectors e which are drawn at random from a fixed (randomly composed) training set containing p = aN vectors e E {-I, I}N with a> 0, and the corresponding values of the teacher outputs. We choose the teacher noise to be consistent, i.e. the answer given by the teacher to a question e will remain the same when that particular question re-appears during the learning process. Thus T(e·) = sgn[BJL . e], with p teacher weight vectors BJL, drawn randomly and independently from P(B), and we generalize the training set accordingly to jj = Hel , B l ), . .. , (e, BP)}. Consistency of teacher noise is natural in terms of applications, and a prerequisite for overfitting phenomena. Averages over the training set will be denoted as ( ... ) b; averages over all possible input vectors e E {-I, I}N as ( ... ) e. We analyze two classes of learning rules, of the form J (£ + 1) = J (£) + f).J (£): on-line: f).J(£) = 11 {e(£) 9 [J(£)·e(£), B(£)·e(£)] - ,J(£) } batch : f).J(£) = 11 { (e 9 [J(£)·e, B·eDl> - ,J(m) } (3) In on-line learning one draws at each step £ a question/answer pair (e (£), B (£)) at random from the training set. In batch learning one iterates a deterministic map which is an average over all data in the training set. Our performance measures are the training- and generalization errors, defined as follows (with the step function O[x > 0] = 1, O[x < 0] = 0): Et(J) = (O[-(J ·e)(B ·em b Eg(J) = (O[-(J ·e)(B* ·e)])e (4) We introduce macroscopic observables, taylored to the present problem, generalizing [5, 6]: Q[J]=J2, R[J]=J·B*, P[x,y,z;J]=(6[x-J·e]6[y-B*·e]6[z-B·eDl> (5) As in [5, 6] we eliminate technical subtleties by assuming the number of arguments (x, y, z) for which P[x, y, z; J] is evaluated to go to infinity after the limit N -t 00 has been taken. 3 Derivation of Macroscopic Laws Upon generalizing the calculations in [6, 5], one finds for on-line learning: ! Q = 2'f} !dXdydZ P[x, y, z] xg[x, z] - 2'f},Q + 'f}2 !dXdYdZ P[x, y, z] g2[x, z] (6) ! R = 'f} !dXdydZ P[x, y, z] y9[x, z]- 'f},R (7) :t P[x, y, z] = ~ ! dx' P[x', y, z] {6[x-x' -'f}G[x', z]] -6[x-x']} -'f}! / dx'dy'dz' / dx'dy'dz'9[x', z]A[x, y, z; x',y', z'] + 'f}, :x {xP[x , y, z]} 1 ! EP +'i'f}2 dx'dy'dz' P[x', y', z']92[x', z'] 8x2 P[x, y, z] (8) Supervised Learning with Restricted Training Sets The complexity of the problem is concentrated in a Green's function: A[x, y, Zj x', y', z'] = lim N-+oo 239 (( ([1-6ee, ]6[x-J·e]6[y-B*·e]6[z-B·e] (e·e')6[x' -J·e']6[y' - B*·e']6[y' - B·e'])i»i> )QW;t It involves a conditional average of the form (K[J])QW;t = J dJ Pt(JIQ,R,P)K[J], with Pt(J) 6[Q-Q[J]]6[R- R[J]] nXYZ 6[P[x, y, z] -P[x, y, Zj J]] Pt(JIQ,R,P) = J dJ Pt(J) 6[Q - Q[J]]6[R- R[J]] nXYZ 6[P[x, y, z] - P[x, y, z; J]] in which Pt (J) is the weight probability density at time t. The solution of (6,7,8) can be used to generate the N -+ 00 performance measures (4) at any time: Et = / dxdydz P[x, y, z]O[-xz] Eg = 11"-1 arccos[RIVQ] (9) Expansion of these equations in powers of"" and retaining only the terms linear in "" gives the corresponding equations describing batch learning. So far this analysis is exact. 4 Closure of Macroscopic Laws As in [6, 5] we close our macroscopic laws (6,7,8) by making the two key assumptions underlying dynamical replica theory: (i) For N -+ 00 our macroscopic observables obey closed dynamic equations. (ii) These equations are self-averaging with respect to the specific realization of D. (i) implies that probability variations within {Q, R, P} subshells are either absent or irrelevant to the macroscopic laws. We may thus make the simplest choice for Pt (J I Q, R, P): Pt(JIQ,R,P) -+ 6[Q-Q[J]] 6[R-R[J]] II 6[P[x,y,z]-P[x,y,ZjJ]] (10) xyz The procedure (10) leads to exact laws if our observables {Q, R, P} indeed obey closed equations for N -+ 00. It is a maximum entropy approximation if not. (ii) allows us to average the macroscopic laws over all training sets; it is observed in simulations, and proven using the formalism of [4]. Our assumptions (10) result in the closure of (6,7,8), since now the Green's function can be written in terms of {Q, R, Pl. The final ingredient of dynamical replica theory is doing the average of fractions with the replica identity / J dJ W[JID]GIJID]) = lim /dJ I ••• dJn (G[J 1ID] IT W[JO<ID])sets \ J dJ W[JID] sets n-+O a=1 Our problem has been reduced to calculating (non-trivial) integrals and averages. One finds that P[x, y, z] = P[x, zly]P[y] with Ply] = (211")-!exp[-!y21 With the short-hands Dy = P[y]dy and (f(x, y, z)) = J Dydxdz P[x, zly]f(x, y, z) we can write the resulting macroscopic laws, for the case of output noise (1), in the following compact way: d d dt Q = 2",(V - ,Q) + rJ2 Z dtR = ",(W - ,R) (11) [) 1/ { } 1 [)2 [)tP[x,zly] = a dx'P[x',zly] 6[x-x'-",G[x',z]]-6[x-x'] +2",2Z [)x2P[x,zIY] -",:x {P[x,zly] [U(x-RY)+Wy-,x+[V-RW-(Q-R2)U]~[x,y,z])} (12) with U = (~[x, y, z]9[x, z]), v = (x9[x, z]), W = (y9[x, z]), Z = (92[x, z]) The solution of (12) is at any time of the following form: P[x,zly] = (1-,x)6[y-z]P+[xly] + ,x6[y+z]P-[xly] (13) 240 A. C. C. Coolen and C. W. H. Mace Finding the function <I> [x, y, z] (in replica symmetric ansatz) requires solving a saddle-point problem for a scalar observable q and two functions M±[xly]. Upon introducing B = ....:..V...,...q.,-Q ___ R,-2 Q(I-q) (f[x, y])± = J dx M±[xly]eBxs J[x, y] * J dx M±[xly]eBxs (with J dx M±[xly] = 1 for all y) the saddle-point equations acquire the fonn for all X, y : p±[Xly] = ! Ds (O[X -xl); (14) ((x-Ry)2) + (qQ-R2)[I-!:.] = qQ+Q-2R2 !DYDS S[(I-A)(X); + A(X);] (15) a ..jqQ_R2 The equations (14) which detennine M±[xly] have the same structure as the corresponding (single) equation in [5, 6], so the proofs in [5, 6] again apply, and the solutions M±[xly], given a q in the physical range q E [R2/Q, 1], are unique. The function <I> [x, y, z] is then given by <I> [X, y, z] =! Ds s {(I-A)O[Z-y](o[X -x)); + AO[Z+Y](o[X -xl);} ..jqQ_R2 P[X, zly] (16) Working out predictions from these equations is generally CPU-intensive, mainly due to the functional saddle-point equation (14) to be solved at each time step. However, as in [7] one can construct useful approximations of the theory, with increasing complexity: (i) Large a approximation (giving the simplest theory, without saddle-point equations) (ii) Conditionally Gaussian approximation for M[xly] (with y-dependent moments) (iii) Annealed approximation of the functional saddle-point equation 5 Benchmark Tests: The Limits a --+ 00 and ,\ --+ 0 We first show that in the limit a --+ 00 our theory reduces to the simple (Q, R) formalism of infinite training sets, as worked out for noisy teachers in [12]. Upon making the ansatz p±[xly] = P[xly] = [27r(Q-R2)]-t e-t [x- Rv]2/(Q-R2) (17) one finds M±[xly] = P[xly], <I>[x,y,Z] = (x-Ry)/(Q-R2) Insertion of our ansatz into (12), followed by rearranging of terms and usage of the above expression for <I> [x, y, z], shows that (12) is satisfied. The remaining equations (11) involve only averages over the Gaussian distribution (17), and indeed reduce to those of [12]: ~! Q = (I-A) { 2(x9[x, y)) + 1}{92[x, y)) } + A {2(x9[x,-y)) + 1}(92[x,-y)) } - 2,Q 1 d --d R = (I-A)(y9[x,y)) + A(y9[x,-yl) -,R 1} t Next we turn to the limit A --+ 0 (restricted training sets & noise-free teachers) and show that here our theory reproduces the fonnalism of [6,5]. Now we make the following ansatz: P+[xly] = P[xly], P[x, zly] = o[z-y]P[xIY] (18) Insertion shows that for A = 0 solutions of this fonn indeed solve our equations, giving <p[x, y, z]--+ <I> [x, y] and M+[xly] = M[xly), and leaving us exactly with the fonnalism of [6, 5] describing the case of noise-free teachers and restricted training sets (apart from some new tenns due to the presence of weight decay, which was absent in [6, 5]). Supervised Learning with Restricted Training Sets 241 0. , r------~--__, 0..4 11>=0.' 0..4 ~-------_____I 11>=0.' 0;=1 ~-------~ 0;=1 0;=2 0..3 0..3 a=4 :::----------1 0;=2 ':::::========:::j a=4 a=4 a=4 = =-=--=-=--=-=--=-=-=--=-= - -=-=-_oed 0;=2 0..2 ------0;=2 0.2 -_ __ ___ _____ _ a=1 - ------0;=1 0. , , 0., , ----------------11>=0, 0..0. no. I 0. ,0. " 0. ,0. " Figure 1: On-line Hebbian learning: conditionally Gaussian approximation versus exact solution in [9] (.,., = 1, ,X = 0.2). Left: "I = 0.1, right: "I = 0.5. Solid lines: approximated theory, dashed lines: exact result. Upper curves: Eg as functions of time (here the two theories agree), lower curves: E t as functions of time. 6 Benchmark Tests: Hebbian Learning The special case of Hebbian learning, i.e. Q[x, z] = sgn(z), can be solved exactly at any time, for arbitrary {a, ,x, "I} [9], providing yet another excellent benchmark for our theory. For batch execution of Hebbian learning the macroscopic laws are obtained upon expanding (11,12) and retaining only those terms which are linear in.,.,. All integrations can now be done and all equations solved explicitly, resulting in U =0, Z = 1, W = (I-2,X)J2/7r, and Q = Qo e-2rryt + 2Ro(I-2'x) e-17"Yt[I_e-rrrt] f{ + [~(I-2,X)2+.!.] [I-e- 17"YtF "I V:; 7r a "12 R = Ro e- 17"Yt+(I-2'x)J2/7r[I-e-17"Yt]/"I q = [aR2+(I_e- 17"Yt)2 i'l]/aQ p±[xIY] = [27r(Q-R2)] -t e-tlz-RH sgn(y)[1-e-"..,t]/a"Y]2/(Q-R2) (19) From these results, in tum, follow the performance measures Eg = 7r- 1 arccos[ R/ JQ) and E = ! - !(1-,X)!D erf[IYIR+[I-e-77"Yt]/a"l] + !,X!D erf[IYIR-[I-e-17"Yt]/a"l] t 2 2 Y J2(Q-R2) 2 y J2(Q-R2) Comparison with the exact solution, calculated along the lines of [9] or, equivalently, obtained upon putting t « .,.,-2 in [9], shows that the above expressions are all exact. For on-line execution we cannot (yet) solve the functional saddle-point equation in general. However, some analytical predictions can still be extracted from (11,12,13): Q = Qo e-217"Yt + 2Ro(I-2,X) e-77"Yt[I_e-17"Yt] f{ + [~(I-2,X)2+.!.] [I_e- 17"Yt]2 "I V:; 7r a "12 R = Ro e-17"Yt + (I-2,X)J2/7r[I-e- 17"Yt]/"I J dx xP±[xIY] = Ry ± sgn(y)[I-e-17"Yt]/a"l + !L[I_e-217"Yt] 2"1 with U =0, W = (I-2,X)J2/7r, V = W R+[I-e-17"Yt]/a"l, and Z = 1. Comparison with the results in [9] shows that the above expressions, and thus also that of E g , are all fully exact, at any time. Observables involving P[x, y, z] (including the training error) are not as easily solved from our equations. Instead we used the conditionally Gaussian approximation (found to be adequate for the noiseless Hebbian case [5, 6, 7]). The result is shown in figure 1. The agreement is reasonable, but significantly less than that in [6]; apparently teacher noise adds to the deformation of the field distribution away from a Gaussian shape. 242 A. C. C. Coolen and C. W H. Mac 0.4 000000 0.6 ~ 0.4 E 0.2 ~ I i 0.0 0.0 0 2 4 6 10 -3 -2 -I 0 X 0.6 f 0.4 E 0.4 [ 0.2 0.2 0.0 L-o!i6iIII."""""',-"--~_~~ __ --' -3 -2 -I 0 2 3 X Figure 2: Large a approximation versus numerical simulations (with N = 10,000), for ,= 0 and A = 0.2. Top row: Perceptron rule, with.,., = ~. Bottom row: Adatron rule, with.,., = ~. Left: training errors E t and generalisation errors Eg as functions of time, for aE {~, 1, 2}. Lines: approximated theory, markers: simulations (circles: Et , squares: Eg). Right: joint distributions for student field and teacher noise p±[x] = J dy P[x, y, z = ±y] (upper: P+[x], lower: P-[x]). Histograms: simulations, lines: approximated theory. 7 Non-Linear Learning Rules: Theory versus Simulations In the case of non-linear learning rules no exact solution is known against which to test our formalism, leaving numerical simulations as the yardstick. We have evaluated numerically the large a approximation of our theory for Perceptron learning, 9[x, z] = sgn(z)O[-xz], and for Adatron learning, 9[x, z] = sgn(z)lzIO[-xz]. This approximation leads to the following fully explicit equation for the field distributions: d 1/ ' 1 ~ -p±[xly] = dx' p±[x'ly]{o[x-x'-.,.,.1'[x', ±y]] -o[x-x]} + _.,.,2 Z!:I 2 p±[xly] dt a 2 uX _ ~ {P[ I ] [W _ U[X±(y)-RY]+(V-RW)[X-X±(y)]]} . .,., 8 x y y ,X + Q _ R2 With X U = J Dydx {(I-A)P+[xly][x-P(y)]9[x,Y]+AP-[xly][x-x-(y)]9[x,-y]) V = ! Dydx x {(I-A)P+[xly]9[x, Y]+AP-[xly]9[x,-y]) W = 1 Dydx y {(1-A)P+[xly]9[x, Y]+AP-[xly]9[x,-y]) Z = 1 Dydx {(I-A)P+[xly]92[x, Y]+AP-[xly]92[x,-yJ) Supervised Learning with Restricted Training Sets 243 and with the short-hands X±(y) = J dx xP±[xly). The result of our comparison is shown in figure 2. Note: Et increases monotonically with a, and Eg decreases monotonically with a, at any t. As in the noise-free formalism [7], the large a approximation appears to capture the dominant terms both for a -7 00 and for a -7 O. The predicting power of our theory is mainly limited by numerical constraints. For instance, the Adatron learning rule generates singularities at x = 0 in the distributions P±[xly) (especially for small "I) which, although predicted by our theory, are almost impossible to capture in numerical solutions. 8 Discussion We have shown how a recent theory to describe the dynamics of supervised learning with restricted training sets (designed to apply in the data recycling regime, and for arbitrary online and batch learning rules) [5, 6, 7] in large layered neural networks can be generalized successfully in order to deal also with noisy teachers. In our generalized approach the joint distribution P[x, y, z) for the fields of student, 'clean' teacher, and noisy teacher is taken to be a dynamical order parameter, in addition to the conventional observables Q and R. From the order parameter set {Q, R, P} we derive the generalization error Eg and the training error Et . Following the prescriptions of dynamical replica theory one finds a diffusion equation for P[x, y, z], which we have evaluated by making the replica-symmetric ansatz. We have carried out several orthogonal benchmark tests of our theory: (i) for a -7 00 (no data recycling) our theory is exact, (ii) for A -7 0 (no teacher noise) our theory reduces to that of [5, 6, 7], and (iii) for batch Hebbian learning our theory is exact. For on-line Hebbian learning our theory is exact with regard to the predictions for Q, R, Eg and the y-dependent conditional averages J dx xP±[xly), at any time, and a crude approximation of our equations already gives reasonable agreement with the exact results [9] for Et . For non-linear learning rules (Perceptron and Adatron) we have compared numerical solution of a simple large a aproximation of our equations to numerical simulations, and found satisfactory agreement. This paper is a preliminary presentation of results obtained in the second stage of a research programme aimed at extending our theoretical tools in the arena of learning dynamics, building on [5, 6, 7]. Ongoing work is aimed at systematic application of our theory and its approximations to various types of non-linear learning rules, and at generalization of the theory to multi-layer networks. References [1] Mace C.W.H. and Coolen AC.C (1998), Statistics and Computing 8, 55 [2] Saad D. (ed.) (1998), On-Line Learning in Neural Networks (Cambridge: CUP) [3] Hertz J.A., Krogh A and Thorgersson G.I. (1989), J. Phys. A 22, 2133 [4] HomerH. (1992a), Z. Phys. B 86, 291 and Homer H. (1992b), Z. Phys. B 87,371 [5] Coolen A.C.C. and Saad D. (1998), in On-Line Learning in Neural Networks, Saad D. (ed.), (Cambridge: CUP) [6] Coolen AC.C. and Saad D. (1999), in Advances in Neural Information Processing Systems 11, Kearns D., Solla S.A., Cohn D.A (eds.), (MIT press) [7] Coolen A.C.C. and Saad D. (1999), preprints KCL-MTH-99-32 & KCL-MTH-99-33 [8] Rae H.C., Sollich P. and Coolen AC.C. (1999), in Advances in Neural Information Processing Systems 11, Kearns D., Solla S.A., Cohn D.A. (eds.), (MIT press) [9] Rae H.C., Sollich P. and Coolen AC.C. (1999),J. Phys. A 32, 3321 [10] Inoue J.I. (1999) private communication [11] Wong K.YM., Li S. and Tong YW. (1999),preprint cond-mat19909004 [12] Biehl M., Riegler P. and Stechert M. (1995), Phys. Rev. E 52, 4624
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Audio-Vision: Using Audio-Visual Synchrony to Locate Sounds John Hershey .. jhershey~cogsci.ucsd.edu Department of Cognitive Science University of California, San Diego La Jolla, CA 92093-0515 Javier Movellan movellan~cogsci.ucsd.edu Department of Cognitive Science University of California, San Diego La Jolla, CA 92093-0515 Abstract Psychophysical and physiological evidence shows that sound localization of acoustic signals is strongly influenced by their synchrony with visual signals. This effect, known as ventriloquism, is at work when sound coming from the side of a TV set feels as if it were coming from the mouth of the actors. The ventriloquism effect suggests that there is important information about sound location encoded in the synchrony between the audio and video signals. In spite of this evidence, audiovisual synchrony is rarely used as a source of information in computer vision tasks. In this paper we explore the use of audio visual synchrony to locate sound sources. We developed a system that searches for regions of the visual landscape that correlate highly with the acoustic signals and tags them as likely to contain an acoustic source. We discuss our experience implementing the system, present results on a speaker localization task and discuss potential applications of the approach. Introd uction We present a method for locating sound sources by sampling regions of an image that correlate in time with the auditory signal. Our approach is inspired by psychophysical and physiological evidence suggesting that audio-visual contingencies play an important role in the localization of sound sources: sounds seem to emanate from visual stimuli that are synchronized with the sound. This effect becomes particularly noticeable when the perceived source of the sound is known to be false, as in the case of a ventriloquist's dummy, or a television screen. This phenomenon is known in the psychophysical community as the ventriloquism effect, defined as a mislocation of sounds toward their apparent visual source. The effect is robust in a wide variety of conditions, and has been found to be strongly dependent on the degree of "synchrony" between the auditory and visual signals (Driver, 1996; Bertelson, Vroomen, Wiegeraad & de Gelder, 1994). "1'0 whom correspondence should be addressed. 814 J. Hershey and J. R. Movellan The ventriloquism effect is in fact less speech-specific than first thought. For example the effect is not disrupted by an upside-down lip signal (Bertelson, Vroomen, Wiegeraad & de Gelder, 1994) and is just as strong when the lip signals are replaced by light flashes that are synchronized with amplitude peaks in the audio signal (Radeau & Bertelson, 1977). The crucial aspect here is correlation between visual and auditory intensity over time. When the light flashes are not synchronized the effect disappears. The ventriloquism effect is strong enough to produce an enduring localization bias, known as the ventriloquism aftereffect. Over time, experience with spatially offset auditory-visual stimuli causes a persistent shift in subsequent auditory localization. Exposure to audio-visual stimuli offset from each other by only 8 degrees of azimuth for 20-30 minutes is sufficient to shift auditory localization by the same amount. A corresponding shift in neural processing has been detected in macaque monkeys as early as primary auditory cortex(Recanzone, 1998). In barn owls a misalignment of visual and auditory stimuli during development causes the realignment of the auditory and visual maps in the optic tectum(Zheng & Knudsen, 1999; Stryker, 1999; Feldman & Knudsen, 1997). The strength of the psychophysical and physiological evidence suggests that audiovisual contingency may be used as an important source of information that is currently underutilized in computer vision tasks. Visual and auditory sensor systems carry information about the same events in the world, and this information must be combined correctly in order for a useful interaction of the two modalities. Audiovisual contingency can be exploited to help determine which signals in different modalities share a common origin. The benefits are two-fold: the two signals can help localize each other, and once paired can help interpret each other. To this effect we developed a system to localize speakers using input from a camera and a single microphone. The approach is based on searching for regions of the image which are "synchronized" with the acoustic signal. Measuring Synchrony The concept of audio-visual synchrony is not well formalized in the psychophysical literature, so for a working definition we interpret synchrony as the degree of mutual information between audio and spatially localized video signals. Ultimately it is a causal relationship that we are often interested in, but causes 'can only be inferred from effects such as synchrony. Let a(t) E IRn be a vector describing the acoustic signal at time t. The components of a(t) could be cepstral coefficients, pitch measurements, or the outputs of a filter bank. Let v(x, y, t) E IRm be a vector describing the visual signal at time t, pixel (x,y). The components ofv(x,y,t) could represent Gabor energy coefficients, RGB color values, etc. Consider now a set of s audio and visual vectors S = (a(tl), v(x, y, tl»l=k-s-l,. .. ,k sampled at times tk-s-l,'" ,tk and at spatial coordinates (x, y). Given this set of vectors our goal is to provide a number that describes the temporal contingency between audio and video at time tk' The approach we take is to consider each vector in S as an independent sample from a joint multivariate Gaussian process (A(tk), V(x, y, tk» and define audio-visual synchrony at time tk as the estimate of the mutual information between the audio and visual components of the process. Let A(tk) ,..., Nn(ltA(tk), ~A(tk»' and V(x,y, tk) ,..., Nm(ltv(x, y, t), ~v(x,y, tk)), where It represents means and ~ covariance matrices. Let A(tk) and V(x, y, tk) be jointly Gaussian, i.e., (A(tk), V(x, y, tk» ,..., Nn+m(ltA,V (x, y, tk), ~A,V(X, y, tk)' Audio Vision: Using Audio-Vzsual Synchrony to Locate Sounds 815 The mutual information between A(x, y, tk) and V(tk) can be shown to be as follows [(A(tk); V(x, y, tk)) = H(A(tk)) + H(V(x, y, tk)) - H(A(tk), V(x, y, tk)) 1 1 "2log(27re)nIEA(tk)1 + "2log(27re)mIEv(x, y, tk)1 (1) 1 -"2log(27re)n+mIEA,v(x,y, tk)1 11 IEA(tk)IIEv(x,y,tk)1 - og "'-----:-::::---'--'-'--;----'-----'-::-:--':"':' 2 IEA,V(X,y,tk)I' In the special case that n = m = 1, then (2) (3) (4) where p(x, y, tk) is the Pearson correlation coefficient between A(tk) and V(x, y, tk)' For each triple (x, y, tk) we estimate the mutual information between A( tk) and V(x, y, tk) by considering each element of S as an independent sample from the random vector (A(tk), V(x, y, tk))' This amounts to computing estimates of the joint covariance matrix EA,V (x, y, tk). For example the estimate of the covariance between the ith audio component and the jLh video component would be as follows 1 8-1 SAi,v; (x, y, tk) = s _ 1 I)ai(tk-l) - ai(tk))(Vj(X,y, tk-l) - Vj (x, y, tk)), (5) 1=0 where (6) (7) (8) These simple covariance estimates can be computed recursively in constant time with respect to the number of timepoints. The independent treatment of pixels would lend well to a parallel implementation. To measure performance, a secondary system produces a single estimate of the auditory location, for use with a database of labeled solitary audiovisual sources. Unfortunately there are many ways of producing such estimates so it becomes difficult to separate performance of the measure from the underlying system. The model used here is a centroid computation on the mutual information estimates, with some enhancements to aid tracking and reduce background noise. Implementation Issues A real time system was prototyped using a QuickCam on the Linux operating system and then ported to NT as a DirectShow filter. l'his platform provides input from real-time audio and video capture hardware as well as from static movie files. The video output could also be rendered live or compressed and saved in a movie file. The implementation was challenging in that it turns out to be rather difficult 816 -'.'----f:.20--"~--:':"--""=:----:':''':--~'20 F,_ (a) M is talking. J. Hershey and J. R. Movellan -2 ~.~~'~.~20~-~"=:---"~~~-~"-=~-~"~"· F" ... (b) J is talking. Figure 1: Normalized audio and visual intensity across sequences of frames in which a sequence of four numbers is spoken. The top trace is the contour of the acoustic energy from one of two speakers, M or J, and the bottom trace is the contour of intensity values for a single pixel, (147,100), near the mouth of J. to process precisely time-synchronized audio and video on a serial machine in real time. Multiple threads are required to read from the peripheral audio and visual devices. By the time the audio and visual streams reach the AV filter module, they are quite separate and asynchronous. The separately threaded auditory and visual packet streams must be synchronized, buffered, and finally matched and aligned by time-stamps before they can finally be processed. It is interesting that successful biologial audiovisual systems employ a parallel architecture and thus avoid this problem. Results To obtain a performance baseline we first tried the simplest possible approach: A single audio and visual feature per location: n = m = 1, v(x, y, t) E IR is the intensity of pixel (x, y) at time t, and a(t) E IR is the average acoustic energy over the interval [t - 6.t, tJ, where 6.t = 1/30 msec , the sampling period for the NTSC video signal. Figure 1 illustrates the time course of these signals for a non-synchronous and a synchronous pair of acoustic energy and pixel intensity. Notice in particular that in the synchonous pair, 1 (b), where the sound and pixel values come from the same speaker, the relationship between the signals changes over time. There are regions of positive and negative covariance strung together in succession. Clearly the relationship over the entire sequence is far from linear. However over shorter time periods a linear relationship looks like a better approximation. Our window size of 16 samples (Le., s = 16 in 5 coincides approximately with this time-scale. Perhaps by averaging over many small windows we can capture on a larger scale what would be lost to the same method applied with a larger window. Of course there is a trade-off in the time-scale between sensitivity to spurious transients, and the response time of the system. We applied this mutual information measure to all the pixels in a movie, in the spirit of the perceptual maps of the brain. The result is a changing topographic map of audiovisual mutual information. Figure 2 illustrates two snapshots in which Audio Vision: Using Audio-Visual Synchrony to Locate Sounds 817 (a) Frame 206: M (at left) is talking. (b) Frame 104: J (at right) is talking. Figure 2: Estimated mutual information between pixel intensity and audio intensity (bright areas indicate greater mutual information) overlaid on stills from the video where one person is in mid-utterance. different parts of the face are synchronous (possibly with different sign) with the sound they take part in producing. It is interesting that the synchrony is shared by some parts, such as the eyes, that do not directly contribute to the sound, but contribute to the communication nonetheless. ___ To estimate the position of the speaker we computed a centroid were each point was weighted by the estimated mutual information between the correpsonding pixel and the audio signal. At each time step the mutual information was estimated using 16 past frames (Le., s = 16) In order to reduce the intrusion of spurious correlations from competing targets, once a target has been found, we employ a Gaussian influence function. (Goodall, 1983) The influence function reduces the weight given to mutual information from locations far from the current centroid when computing the next centroid. To allow for the speedy disengagement from a dwindling source of mutual information we set a threshold on the mutual information. Measurements under the threshold are treated as zero. This threshold also reduces the effects of unwanted background noise, such as camera and microphone jitter. A Lx L x 8(1og(1 - f} (x, y, t)))'I/;(X, Sx(t - 1)) Sx(t) = Y A Lx L y 8(log(1- p2(X,y,t)))'I/;(x,Sx(t -1)) (9) where Sx(t) represents the estimate of the x coordinate for the position of the speaker at time t. 8(.) is the thresholding function, and 'I/;(x, Sx(t - 1)) is the influence function, which depends upon the 'position x of the pixel being sampled and the prior estimate Sx(t-1). p2(X, y, t) is the estimate of the correlation between the intensity in pixel (x,y) and the acoustic enery, when using the 16 past video frames. -~ log(l- p2(x, y, t)) is the corresponding estimate of mutual information (the factor, -~ cancels out in the quotient after adjusting the threshold function accordingly. ) We tried the approach on a movie of two people (M and J) taking turns while saying random digits. Figure 3 shows the estimates of the actual positions of the speaker 818 J. Hershey and J. R. Movellan as a function of time. The estimates clearly provide information that could be used to localize the speaker, especially in combination with other approaches (e.g., flesh detection) . 180 180 ~ ~ 140 ~ ~ 120 i ~ 100 ~ _! 80 40 20~----~----~----~----~----~------~--~ o 100 200 300 400 500 600 700 Frame Number Figure 3: Estimated and actual position of speaker at each frame for six hundred frames. The sources, M and J, took turns uttering a series of four digits, for three turns each. The actual positions and alternation times were measured by hand from the video recording Conclusions We have presented exploratory work on a system for localizing sound sources on a video signal by tagging regions of the image that are correlated in time with the auditory signal. The approach was motivated by the wealth of evidence in the psychophysical and physiological literature showing that sound localization is strongly influenced by synchrony with the visual signal. We presented a measure of local synchrony based on modeling the audio-visual signal as a non-stationary Gaussian process. We developed a general software tool that accepts as inputs all major video and audio file formats as well as direct input from a video camera. We tested the tool on a speaker localization task with very encouraging results. The approach could have practical applications for localizing sound sources in situations where where acoustic stereo cues are inexistent or unreliable. For example the approach could be used to help localize the actor talking in a video scene and put closed-captioned text near the audio source. The approach could also be used to guide a camera in teleconferencing applications. While the results reported here are very encouraging, more work needs to be done before practical applications are developed. For example we need to investigate more sophisticated methods for processing the audio and video signals. At this point we use average energy to represent the video and thus changes in the fundamental frequency that do not affect the average energy would not be captured by our model. Similarly local video decompositions, like spatio-temporal Gabor filtering, or approaches designed to enhance the lip regions may be helpful. The Audio Vision: Using Audio-Visual Synchrony to Locate Sounds 819 changing symmetry observed between audio and video signals might be addressed rectifying or squaring the normalized signals and derivatives. Finally, relaxing the Gaussian constraints in our measure of audio-visual contingency may help improve performance. While the work shown here is exploratory at this point, the approach is very promising: It emphasizes the idea of machine perception as a multimodal process it is backed by psychophysical evidence, and when combined with other approaches it may help improve robustness in tasks such as localization and separation of sound sources. References Bertelson, P., Vroomen, J., Wiegeraad, G., & de Gelder, B. (1994). Exploring the relation between McGurk interference and ventriloquism. In Proceedings of the 1994 International Conference on Spoken Language Processing, volume 2, pages 559-562. Driver, J. (1996). Enhancement of selective listening by illusory mislocation of speech sounds due to lip-reading. Nature, 381, 66-68. Feldman, D. E. & Knudsen, E. I. (1997). An anatomical basis for visual calibration of the auditiory space map in the barn owl's midbrain. The Journal of Neuroscience, 17(17), 6820-6837. Goodall, C. (1983). M-Estimators of Location: an outline of the theory. Wiley series in probability and mathematical statistics. Applied probability and statistics. Radeau, M. & Bertelson, P. (1977). Adaptation to auditory-visual discordance and ventriloquism in semi-realistic situations. Perception and Psychophysics, 22, 137-146. Recanzone, G. H. (1998). Rapidly induced auditory plasticity: The ventriloquism aftereffect. Proceedings of the National Academy of Sciences, USA, 95, 869- 875. Stryker, M. P. (1999). Sensory Maps on the Move. Science, 925-926. Zheng, W. & Knudsen, E. I. (1999). Functional Selection of Adaptive Auditory Space Map by GABAA-Mediated Inhibition, 962-965.
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Predictive Sequence Learning in Recurrent Neocortical Circuits* R.P.N.Rao Computational Neurobiology Lab and Sloan Center for Theoretical Neurobiology The Salk Institute, La Jolla, CA 92037 rao@salk.edu T. J. Sejnowski Computational Neurobiology Lab and Howard Hughes Medical Institute The Salk Institute, La Jolla, CA 92037 terry@salk.edu Abstract Neocortical circuits are dominated by massive excitatory feedback: more than eighty percent of the synapses made by excitatory cortical neurons are onto other excitatory cortical neurons. Why is there such massive recurrent excitation in the neocortex and what is its role in cortical computation? Recent neurophysiological experiments have shown that the plasticity of recurrent neocortical synapses is governed by a temporally asymmetric Hebbian learning rule. We describe how such a rule may allow the cortex to modify recurrent synapses for prediction of input sequences. The goal is to predict the next cortical input from the recent past based on previous experience of similar input sequences. We show that a temporal difference learning rule for prediction used in conjunction with dendritic back-propagating action potentials reproduces the temporally asymmetric Hebbian plasticity observed physiologically. Biophysical simulations demonstrate that a network of cortical neurons can learn to predict moving stimuli and develop direction selective responses as a consequence of learning. The space-time response properties of model neurons are shown to be similar to those of direction selective cells in alert monkey VI. 1 INTRODUCTION The neocortex is characterized by an extensive system of recurrent excitatory connections between neurons in a given area. The precise computational function of this massive recurrent excitation remains unknown. Previous modeling studies have suggested a role for excitatory feedback in amplifying feedforward inputs [1]. Recently, however, it has been shown that recurrent excitatory connections between cortical neurons are modified according to a temporally asymmetric Hebbian learning rule: synapses that are activated slightly before the cell fires are strengthened whereas those that are activated slightly after are weakened [2, 3]. Information regarding the postsynaptic activity of the cell is conveyed back to the dendritic locations of synapses by back-propagating action potentials from the soma. In this paper, we explore the hypothesis that recurrent excitation subserves the function of prediction and generation of temporal sequences in neocortical circuits [4, 5, 6]. We show "This research was supported by the Sloan Foundation and Howard Hughes Medical Institute. Predictive Sequence Learning in Recurrent Neocortical Circuits 165 that a temporal difference based learning rule for prediction applied to backpropagating action potentials reproduces the experimentally observed phenomenon of asymmetric Hebbian plasticity. We then show that such a learning mechanism can be used to learn temporal sequences and the property of direction selectivity emerges as a consequence of learning to predict moving stimuli. Space-time response plots of model neurons are shown to be similar to those of direction selective cells in alert macaque VI. 2 TEMPORALLY ASYMMETRIC HEBBIAN PLASTICITY AND TEMPORAL DIFFERENCE LEARNING To accurately predict input sequences, the recurrent excitatory connections in a network need to be adjusted such that the appropriate set of neurons are activated at each time step. This can be achieved by using a "temporal-difference" (TD) learning rule [5, 7]. In this paradigm of synaptic plasticity, an activated synapse is strengthened or weakened based on whether the difference between two temporally-separated predictions is positive or negative. This minimizes the errors in prediction by ensuring that the prediction generated by the neuron after synaptic modification is closer to the desired value than before (see [7] for more details). In order to ascertain whether temporally-asymmetric Hebbian learning in cortical neurons can be interpreted as a fonn of temporal-difference learning, we used a two-compartment model of a cortical neuron consisting of a dendrite and a soma-axon compartment. The compartmental model was based on a previous study that demonstrated the ability of such a model to reproduce a range of cortical response properties [8]. The presence of voItageactivated sodium channels in the dendrite allowed back-propagation of action potentials from the soma into the dendrite. To study plasticity, excitatory postsynaptic potentials (EPSPs) were elicited at different time delays with respect to postsynaptic spiking by presynaptic activation of a single excitatory synapse located on the dendrite. Synaptic currents were calculated using a kinetic model of synaptic transmission with model parameters fitted to whole-cell recorded AMPA currents (see [9] for more details). Synaptic plasticity was simulated by incrementing or decrementing the value for maximal synaptic conductance by an amount proportional to the temporal-difference in the postsynaptic membrane potential at time instants t + ~t and t ~t for presynaptic activation at time t. The delay parameter ~t was set to 5 ms to yield results consistent with previous physiological experiments [2]. Presynaptic input to the model neuron was paired with postsynaptic spiking by injecting a depolarizing current pulse (10 ms, 200 pA) into the soma. Changes in synaptic efficacy were monitored by applying a test stimulus before and after pairing, and recording the EPSP evoked by the test stimulus. Figure I A shows the results of pairings in which the postsynaptic spike was triggered 5 ms after and 5 ms before the onset of the EPSP respectively. While the peak EPSP amplitude was increased 58.5% in the former case, it was decreased 49.4% in the latter case, qualitatively similar to experimental observations [2]. The critical window for synaptic modifications in the model depends on the parameter ~t as well as the shape ofthe back-propagating action potential. This window of plasticity was examined by varying the time interval between presynaptic stimulation and postsynaptic spiking (with ~t = 5 ms). As shown in Figure IB, changes in synaptic efficacy exhibited a highly asymmetric dependence on spike timing similar to physiological data [2]. Potentiation was observed for EPSPs that occurred between I and 12 ms before the postsynaptic spike, with maximal potentiation at 6 ms. Maximal depression was observed for EPSPs occurring 6 ms after the peak of the postsynaptic spike and this depression gradually decreased, approaching zero for delays greater than 10 ms. As in rat neocortical neurons, Xenopus tectal neurons, and cultured hippocampal neurons (see [2]), a narrow transition zone (roughly 3 ms in the model) separated the potentiation and depression windows. 166 R. P. N. Rao and T. J. Sejnowski A I~< IS ~ B ~ , befote ~ ~ 150 15m. ISma ~ :~ ----r---, ____ 100 -----" ]: I . "-! i .... -"'.J ~ i i OO~ ___ J _ __ Jl__ _ ___ .5 -50 IS after I I~ ~ afler: ---.J ~ : ~ < c: , IS I ISms CU -100 :~ I>m. , .c --'1 _____ ------r---- U I I -40 before pairing -20 o 20 40 • & SI S2 Time of Synaptic Input (ms) Figure l: Synaptic Plasticity in a Model Neocortical Neuron. (A) (Left Panel) EPSP in the model neuron evoked by a presynaptic spike (S 1) at an excitatory synapse ("before"). Pairing this presynaptic spike with postsynaptic spiking after a 5 ms delay ("pairing") induces long-term potentiation ("after"). (Right Panel) If presynaptic stimulation (S2) occurs 5 ms after postsynaptic firing. the synapse is weakened resulting in a corresponding decrease in peak EPSP amplitude. (B) Critical window for synaptic plasticity obtained by varying the delay between pre- and postsynaptic spiking (negative delays refer to presynaptic before postsynaptic spiking). 3 RESULTS 3.1 Learning Sequences using Temporally Asymmetric Hebbian Plasticity To see how a network of model neurons can learn sequences using the learning mechanism described above, consider the simplest case of two excitatory neurons N 1 and N2 connected to each other, receiving inputs from two separate input neurons 11 and 12 (Figure 2A). Suppose input neuron 11 fires before input neuron 12, causing neuron Nl to fire (Figure 2B). The spike from Nl results in a sub-threshold EPSP in N2 due to the synapse S2. If input arrives from 12 any time between land 12 ms after this EPSP and the temporal summation of these two EPSPs causes N2 to fire, the synapse S2 will be strengthened. The synapse S l, on the other hand, will be weakened because the EPSP due to N2 arrives a few milliseconds after Nl has fired. Thus, on a subsequent trial, when input 11 causes neuron Nl to fire, Nl in turn causes N2 to fire several milliseconds before input 12 occurs due to the potentiation of the recurrent synapse S2 in previous trial(s) (Figure 2C). Input neuron 12 can thus be inhibited by the predictive feedback from N2 just before the occurrence of imminent input activity (marked by an asterisk in Figure 2C). This inhibition prevents input 12 from further exciting N2. Similarly, a positive feedback loop between neurons Nl and N2 is avoided because the synapse S 1 was weakened in previous trial(s) (see arrows in Figures 2B and 2C). Figure 2D depicts the process of potentiation and depression of the two synapses as a function of the number of exposures to the 11-12 input sequence. The decrease in latency of the predictive spike elicited in N2 with respect to the timing of input 12 is shown in Figure 2E. Notice that before learning, the spike occurs 3.2 ms after the occurrence of the input whereas after learning, it occurs 7.7 ms before the input. 3.2 Emergence of Direction Selectivity In a second set of simulations, we used a network of recurrently connected excitatory neurons as shown in Figure 3A receiving retinotopic sensory input consisting of moving pulses of excitation (8 ms pulse of excitation at each neuron) in the rightward and leftward directions. The task of the network was to predict the sensory input by learning appropriate recurrent connections such that a given neuron in the network starts firing several milliseconds before the arrival of its input pulse of excitation. The network was comprised of two parallel chains of neurons with mutual inhibition (dark arrows) between corresponding pairs of neurons along the two chains. The network was initialized such that within a chain, a given Predictive Sequence Learning in Recurrent Neocortical Circuits 167 A SI S2 Excitatory Neuron N2 Input Neuron 11 r. Input Neuron 12 Input I Input 2 B C Before Learning After Learning NI 11 II ~j~ I ~ N2 L ~ ~ 15 illS ~~ < 15 illS 12 1 12 ~ D 0.03 o o E 4 6 Synapse S2 66 °0000000000 0 000000 10 20 30 40 Time (number of trials) V) ••• • § 2 .. . . . .. .. ~ ~ 0·· .. .:: -2 .~ "E d:: -4 ... 0 ;>, -6 u " ~ -8 j 0 ... .... 10 20 30 40 Time (number of trials) Figure 2: Learning to Predict using Temporally Asymmetric Hebbian Learning. (A) Network of two model neurons Nl and N2 recurrently connected via excitatory synapses SI and S2, with input neurons 11 and 12. Nl and N2 inhibit the input neurons via inhibitory interneurons (darkened circles). (B) Network activity elicited by the sequence 11 followed by 12. (C) Network activity for the same sequence after 40 trials of learning. Due to strengthening of recurrent synapse S2. recurrent excitation from Nl now causes N2 to fire several ms before the expected arrival of input 12 (dashed line). allowing it to inhibit 12 (asterisk). Synapse SI has been weakened. preventing re-excitation of Nl (downward arrows show decrease in EPSP). (D) Potentiation and depression of synapses S 1 and S2 respectively during the course of learning. Synaptic strength was defined as maximal synaptic conductance in the kinetic model of synaptic transmission [9]. (E) Latency of predictive spike in N2 during the course of learning measured with respect to the time of input spike in 12 (dotted line). excitatory neuron received both excitation and inhibition from its predecessors and successors (Figure 3B). Excitatory and inhibitory synaptic currents were calculated using kinetic models of synaptic transmission based on properties of AMPA and GABAA receptors as determined from whole-cell recordings [9]. Maximum conductances for all synapses were initialized to small positive values (dotted lines in Figure 3C) with a slight asymmetry in the recurrent excitatory connections for breaking symmetry between the two chains. The network was exposed alternately to leftward and rightward moving stimuli for a total of 100 trials. The excitatory connections (labeled 'EXC' in Figure 3B) were modified according to the asymmetric Hebbian learning rule in Figure IB while the excitatory connections onto the inhibitory interneuron (labeled 'INH') were modified according to an asymmetric anti-Hebbian learning rule that reversed the polarity of the rule in Figure lB. The synaptic conductances learned by two neurons (marked NI and N2 in Figure 3A) located at corresponding positions in the two chains after 100 trials of exposure to the moving stimuli are shown in Figure 3C (solid line). Initially, for rightward motion, the slight asymmetry in 168 A c --- Input Stimulus (Rightward)f---Neuron NI EXC Synapse Number Neuron N2 Synapse Number R. P N. Rao and T. J. Sejnowski B Recurrent Excitatory Connections (EXC) l -4 -3 -2 -I () 2 Recurrent Inhibitory Connections (INH) -4 -3 -2 -l () D Neuron NI (Right-Selective) 2 Neuron N2 (Left-Selective) 4 4 Rightward I11II111111 ~~ ~~ Motion I11II111111 LLflward -.PJUL. Motion Figure 3: Direction Selectivity in the Model. (A) A model network consisting of two chains of recurrently connected neurons receiving retinotopic inputs_ A given neuron receives recurrent excita-tiorrand recurrent inhibition (white-headed arrows) as well as inhibition (dark-headed arrows) from its counterpart in the other chain_ (B) Recurrent connections to a given neuron (labeled '0') arise from 4 preceding and 4 succeeding neurons in its chain. Inhibition at a given neuron is mediated via a GAB Aergic interneuron (darkened circle). (C) Synaptic strength of recurrent excitatory (EXC) and inhibitory (IN H) connections to neurons Nt and N2 before (dotted lines) and after learning (solid lines). Synapses were adapted during 100 trials of exposure to alternating leftward and rightward moving stimuli. (D) Responses of neurons Nt and N2 to rightward and leftward moving stimuli_ As a result of learning, neuron N 1 has become selective for rightward motion (as have other neurons in the same chain) while neuron N2 has become selective for leftward motion_ In the preferred direction, each neuron starts firing several milliseconds before the actual input arrives at its soma (marked by an asterisk) due to recurrent excitation from preceding neurons_ The dark triangle represents the start of input stimulation in the network. the initial excitatory connections of neuron Nl allows it to fire slightly earlier than neuron N2 thereby inhibiting neuron N2. Additionally, since the EPSPs from neurons lying on the left of Nt occur before Nl fires, the excitatory synapses from these neurons are strengthened while the excitatory synapses from these same neurons to the inhibitory interneuron are weakened according to the two learning rules mentioned above. On the other hand, the excitatory synapses from neurons lying on the right side ofNl are weakened while inhibitory connections are strengthened since the EPSPs due to these connections occur after Nl has fired. The synapses on neuron N2 and its associated interneuron remain unaltered since there is no postsynaptic firing (due to inhibition by Nl) and hence no back-propagating action potentials in the dendrite. As shown in Figure 3C, after lOO trials, the excitatory and inhibitory connections to neuron Nl exhibit a marked asymmetry, with excitation originating from neurons on the left and inhibition from neurons on the right. Neuron N2 exhibits the opposite pattern of connectivity. As expected, neuron Nl was found to be selective for rightward motion while neuron N2 was selective for leftward motion (Figure 3D). Moreover, when stimulus motion is in the preferred direction, each neuron starts firing several milliseconds before the time of arrival of the input stimulus at its soma (marked by an asterisk) due to recurrent excitation from preceding neurons. Conversely, motion in the nonpreferred direction triggers recurrent inhibition from preceding neurons as well as inhibition Predictive Sequence Learning in Recurrent Neocortical Circuits 169 Monkey Data Model 5~ __ ~'~·~·u'~'~"_'~~ ____ ~'= _____ ·_· . . ...• ' .. ~ •... - ... p.!i! , .. '. . _~_~_~~ ~ .• :dc' ............ n. tH1· n ... --.... ~ 'Cd .... -,.. . . . . rftr_-4 sO' rtn •• h IL •• ~3• ; g j, ~~.~ : ·jHT!: %§§: ~ r ~.; "::n:; ;,::= :: : :: Q) ..... em 6ft. 1'b'1 ' , Q. en 1 •• 0.. -",., ~ + row +.+' •• _. d h, d "'d =fn h 'ft. ......... 'h,tr te'hz •• d + ~ • • hzc+ no • ....... ;::I:; ~ . _. _·h.e ~I~ ________ ~ C\I 6 r-' 1·~ N 0 50 Stimylus lime (rLonds) stimulus time (ms) ....... ----100 Figure 4: Comparison of Monkey and Model Space-Time Response Plots. (Left) Sequence of PSTHs obtained by flashing optimally oriented bars at 20 positions across the 50 -wide receptive field (RF) of a complex cell in alert monkey V 1 (from [11)). The cell's preferred direction is from the part of the RF represented at the bottom towards the top. Flash duration = 56 ms; inter-stimulus delay = 100 ms; 75 stimulus presentations. (Right) PSTHs obtained from a model neuron after stimulating the chain of neurons at 20 positions to the left and right side of the given neuron. Lower PSTHs represent stimulations on the preferred side while upper PSTHs represent stimulations on the null side. from the active neuron in the corresponding position in the other chain. Thus, the learned pattern of connectivity allows the direction selective neurons comprising the two chains in the network to conjointly code for and predict the moving input stimulus in each direction. The average firing rate of neurons in the network for the preferred direction was 75.7 Hz, which is in the range of cortical firing rates for moving bar stimuli. Assuming a 200 /-tm separation between excitatory model neurons in each chain and utilizing known values for the cortical magnification factor in monkey striate cortex, one can estimate the preferred stimulus velocity of model neurons to be 3.1 ° Is in the fovea and 27.9° Is in the periphery (at an eccentricity of 8°). Both of these values fall within the range of monkey striate cortical velocity preferences [11]. The model predicts that the neuroanatomical connections for a direction selective neuron should exhibit a pattern of asymmetrical excitation and inhibition similar to Figure 3C. A recent study of direction selective cells in awake monkey VI found excitation on the preferred side of the receptive field and inhibition on the null side consistent with the pattern of connections learned by the model [11]. For comparison with this experimental data, spontaneous background activity in the model was generated by incorporating Poisson-distributed random excitatory and inhibitory alpha synapses on the dendrite of each model neuron. Post stimulus time histograms (PSTHs) and space-time response plots were obtained by flashing optimally oriented bar stimuli at random positions in the cell's activating region. As shown in Figure 4, there is good qualitative agreement between the response plot for a complex cell and that for the model. Both space-time plots show a progressive shortening of response onset time and an increase in response transiency going in the preferred direction: in the model, this is due to recurrent excitation from progressively closer cells on the preferred side. Firing is reduced to below background rates 40-60 ms after stimulus onset in the upper part of the plots: in the model, this is due to recurrent inhibition from cells on the null side. The response transiency and shortening of response time course appears as a slant in the space-time maps, which can be related to the neuron's velocity sensitivity [11]. 170 R. P. N. Rao and T. J. Sejnowski 4 CONCLUSIONS Our results show that a network of recurrently connected neurons endowed with a temporaldifference based asymmetric Hebbian learning mechanism can learn a predictive model of its spatiotemporal inputs. When exposed to moving stimuli, neurons in a simulated network learned to fire several milliseconds before the expected arrival of an input stimulus and developed direction selectivity as a consequence of learning. The model predicts that a direction selective neuron should start responding several milliseconds before the preferred stimulus enters its retinal input dendritic field (such predictive neural activity has recently been reported in retinal ganglion cells [10)). Temporally asymmetric Hebbian learning has previously been suggested as a possible mechanism for sequence learning in the hippocampus [4] and as an explanation for the asymmetric expansion of hippocampal place fields during route learning [12]. Some of these theories require relatively long temporal windows of synaptic plasticity (on the order of several hundreds of milliseconds) [4] while others have utilized temporal windows in the millisecond range for coincidence detection [3]. Sequence learning in our model is based on a window of plasticity in the 10 to 15 ms range which is roughly consistent with recent physiological observations [2] (see also [13)). The idea that prediction and sequence learning may constitute an important goal of the neocortex has previously been suggested in the context of statistical and information theoretic models of cortical processing [4, 5,6]. Our biophysical simulations suggest a possible implementation of such models in cortical circuitry. Given the universality ofthe problem of encoding and generating temporal sequences in both sensory and motor domains, the hypothesis of predictive sequence learning in recurrent neocortical circuits may help provide a unifying principle for studying cortical structure and function. References [1] R. 1. Douglas et al., Science 269, 981 (1995); H. Suarez et aI., 1. Neurosci. 15,6700 (1995); R. Maex and G. A. Orban, 1. Neurophysiol. 75, 1515 (1996); P. Mineiro and D. Zipser, Neural Comput. 10, 353 (1998); F. S. Chance et aI., Nature Neuroscience 2, 277 (1999). [2] H. Markram et al., Science 275, 213 (1997); W. B. Levy and O. Steward, Neuroscience 8, 791 (1983); D. Debanne et aI., Proc. Natl. Acad. Sci. U.S.A. 91, 1148 (1994); L. I. Zhang et aI., Nature 395, 37 (1998); G. Q. Bi and M. M. Poo, 1. Neurosci. 18, 10464 (1998). [3] w. Gerstner et al., Nature 383, 76 (1996); R. Kempter et al., in Advances in Neural Info. Proc. Systems 11, M. S. Kearns, S. A. Solla and D. A. Cohn, Eds. (MIT Press, Cambridge, MA, 1999), pp. 125-131. [4] L. F. Abbott and K. I. Blum, Cereb. Cortex 6, 406 (1996); W. Gerstner and L. F. Abbott, 1. Comput. Neurosci. 4, 79 (1997); A. A. Minai and W. B. Levy, in Proceedings of the 1993 World Congress on Neural Networks II, 505 (1993). [5] P. R. Montague and T. J. Sejnowski, Learning and Memory 1, 1 (1994); P. R. Montague et al., Nature 377, 725 (1995); w. Schultz et aI., Science 275, 1593 (1997). [6] R. P. N. Rao and D. H. Ballard, Neural Computation 9, 721 (1997); R. P. N. Rao and D. H. Ballard, Nature Neuroscience 2, 79 (1999); H. Barlow, Perception 27, 885 (1998). [7] R. S. Sutton, Machine Learning 3, 9 (1988); R. S. Sutton and A. G. Barto, in Learning and Computational Neuroscience: Foundations of Adaptive Networks, M. Gabriel and J. W. Moore, editors (MIT Press, Cambridge, MA, 1990). [8] Z. F. Mainen and T. 1. Sejnowski, Nature 382, 363 (1996). [9] A. Destexhe et al., in Methods in NeurolUll Modeling, C. Koch and I. Segev, editors, (MIT Press, Cambridge, MA, 1998). [10] M.1. Berry et al., Nature 398,334 (1999). [11] M. S. Livingstone, Neuron 20, 509 (1998). [12] M. R. Mehta et aI., Proc. Natl. Acad. Sci. U.S.A. 94,8918 (1997). [13] L. F. Abbott and S. Song, in Advances in Neural Info. Proc. Systems 1 J, M. S. Keams, S. A. Solla and D. A. Cohn, Eds. (MIT Press, Cambridge, MA, 1999), pp. 69-75.
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Differentiating Functions of the Jacobian with Respect to the Weights Gary William Flake NEC Research Institute 4 Independence Way Princeton, NJ 08540 jiake@research.nj.nec.com Barak A. Pearlmutter Dept of Computer Science, FEC 313 University of New Mexico Albuquerque, NM 87131 bap@cs.unm.edu Abstract For many problems, the correct behavior of a model depends not only on its input-output mapping but also on properties of its Jacobian matrix, the matrix of partial derivatives of the model's outputs with respect to its inputs. We introduce the J-prop algorithm, an efficient general method for computing the exact partial derivatives of a variety of simple functions of the Jacobian of a model with respect to its free parameters. The algorithm applies to any parametrized feedforward model, including nonlinear regression, multilayer perceptrons, and radial basis function networks. 1 Introduction Let f (x, w) be an n input, m output, twice differentiable feedforward model parameterized by an input vector, x, and a weight vector w. Its Jacobian matrix is defined as [ ~ ()xl J= : aim aXI ~l a~" = df(x, w) . aim dx ax" The algorithm we introduce can be used to optimize functions of the form or 1 2 Ev(w) = 211Jv - bll (1) (2) where u, v, a, and b are user-defined constants. Our algorithm, which we call J-prop, can be used to calculate the exact value of both a Eu / aw or a Ev / aw in 0 (1) times the time required to calculate the normal gradient. Thus, I-prop is suitable for training models to have specific first derivatives, or for implementing several other well-known algorithms such as Double Backpropagation [1] and Tangent Prop [2]. Clearly, being able to optimize Equations 1 and 2 is useful; however, we suspect that the formalism which we use to derive our algorithm is actually more interesting because it allows us to modify J-prop to easily be applicable to a wide-variety of model types and 436 G. W. Flake and B. A. Pear/mutter objective functions. As such, we spend a fair portion of this paper describing the mathematical framework from which we later build J-prop. This paper is divided into four more sections. Section 2 contains background information and motivation for why optimizing the properties of the Jacobian is an important problem. Section 3 introduces our formalism and contains the derivation of the J-prop algorithm. Section 4 contains a brief numerical example of J-prop. And, finally, Section 5 describes further work and gives our conclusions. 2 Background and motivation Previous work concerning the modeling of an unknown function and its derivatives can be divided into works that are descriptive or prescriptive. Perhaps the best known descriptive result is due to White et al. [3,4], who show that given noise-free data, a multilayer perceptron (MLP) can approximate the higher derivatives of an unknown function in the limit as the number of training points goes to infinity. The difficulty with applying this result is the strong requirements on the amount and integrity of the training data; requirements which are rarely met in practice. This problem was specifically demonstrated by Principe, Rathie and Kuo [5] and Deco and Schiirmann [6], who showed that using noisy training data from chaotic systems can lead to models that are accurate in the input-output sense, but inaccurate in their estimates of quantities related to the Jacobian of the unknown system, such as the largest Lyapunov exponent and the correlation dimension. MLPs are particularly problematic because large weights can lead to saturation at a particular sigmoidal neuron which, in tum, results in extremely large first derivatives at the neuron when evaluated near the center of the sigmoid transition. Several methods to combat this type of over-fitting have been proposed. One of the earliest methods, weight decay [7], uses a penalty term on the magnitude of the weights. Weight decay is arguably optimal for models in which the output is linear in the weights because minimizing the magnitude of the weights is equivalent to minimizing the magnitude of the model's first derivatives. However, in the nonlinear case, weight decay can have suboptimal performance [1] because large (or small) weights do not always correspond to having large (or small) first derivatives. The Double Backpropagation algorithm [1] adds an additional penalty term to the error function equal to II a E / ax 112. Training on this function results in a form of regularization that is in many ways an elegant combination of weight decay and training with noise: it is strictly analytic (unlike training with noise) but it explicitly penalizes large first derivatives ofthe model (unlike weight decay). Double Backpropagation can be seen as a special case of J-prop, the algorithm derived in this paper. As to the general problem of coercing the first derivatives of a model to specific values, Simard, et at., [2] introduced the Tangent Prop algorithm, which was used to train MLPs for optical character recognition to be insensitive to small affine transformations in the character space. Tangent Prop can also be considered a special case of J-prop. 3 Derivation We now define a formalism under which J-prop can be easily derived. The method is very similar to a technique introduced by Pearlmutter [8] for calculating the product of the Hessian of an MLP and an arbitrary vector. However, where Pearlmutter used differential operators applied to a model's weight space, we use differential operators defined with respect to a model's input space. Our entire derivation is presented in five steps. First, we will define an auxiliary error Differentiating Functions of the Jacobian 437 function that has a few useful mathematical properties that simplify the derivation. Next, we will define a special differential operator that can be applied to both the auxiliary error function, and its gradient with respect to the weights. We will then see that the result of applying the differential operator to the gradient of the auxiliary error function is equivalent to analytically calculating the derivatives required to optimize Equations 1 and 2. We then show an example of the technique applied to an MLP. Finally, in the last step, the complete algorithm is presented. To avoid confusion, when referring to generic data-driven models, the model will always be expressed as a vector function y = f (x, w), where x refers to the model input and w refers to a vector of all of the tunable parameters of the model. In this way, we can talk about models while ignoring the mechanics of how the models work internally. Complementary to the generic vector notation, the notation for an MLP uses only scalar symbols; however, these symbols must refer to internal variable of the model (e.g., neuron thresholds, net inputs, weights, etc.), which can lead to some ambiguity. To be clear, when using vector notation, the input and output of an MLP will always be denoted by x and y, respectively, and the collection of all of the weights (including biases) map to the vector w. However, when using scalar arithmetic, the scalar notation for MLPs will apply. 3.1 Auxiliary error function Our auxiliary error function, E, is defined as T E(x, w) = u f(x, w). (3) Note that we never actually optimize with respect E; we define it only because it has the property that aE/ax = u T J, which will be useful to the derivation shortly. Note that a E / ax appears in the Taylor expansion of E about a point in input space: -T E(x + Ax, w) = E(x, w) + ~! Ax + 0 (1IAXI12) . (4) Thus, while holding the weights, w, fixed and letting Ax be a perturbation of the input, x, Equation 4 characterizes how small changes in the input of the model change the value of the auxiliary error function. Be setting Ax = rv, with v being an arbitrary vector and r being a small value, we can rearrange Equation 4 into the form: ~ [E(x+rv,w) -E(x,w)] +O(r) 1 [-] lim E(x + rv, w) - E(x, w) r~O r a I aE(x + rv,w) . r r=O (5) This final expression will allow us to define the differential operator in the next subsection. 3.2 Differential operator Let h(x, w) be an arbitrary twice differentiable function. We define the differentiable operator (6) 438 G. W Flake and B. A. Pearlmutter which has the property that Rv{E(x, w)} = u T Jv. Being a differential operator, Rv{-} obeys all of the standard rules for differentiation: Rv{c} Rv{ c· h(x, w)} Rv{h(x,w) + g(x,w)} Rv{h(x, w) . g(x, w)} Rv{h(g(x, w), w)} Rv{!h(X,W) } o = c· Rv{h(x, w)} Rv{ h(x, w)} + Rv{g(x, w)} Rv{h(x,w)}· g(x,w) + h(x,w)· Rv{g(x,w)} h'(g(x,w))· Rv{g(x,w)} d dt Rv{h(x, w)} The operator also yields the identity Rv{ x} = v. 3.3 Equivalence We will now see that the result of calculating Rv{ a E / aw} can be used to calculate both aEu/aw and aEv/aw. Note that Equations 3-5 all assume that both u and v are independent of x and w. To calculate aEu/aw and aEv/aw, we will actually set u or v to a value that depends on both x and w; however, the derivation still works because our choices are explicitly made in such a way that the chain rule of differentiation is not supposed to be applied to these terms. Hence, the correct analytical solution is obtained despite the dependence. To optimize with respect to Equation 1, we use: a 1 II T 112 (au T J) T T { a E } aw'2 J u - a = ~ (J u - a) = Rv aw ' (7) with v = (JT U - a). To optimize with respect to Equation 2, we use: ~! IIJv - bll 2 = (Jv _ b)T (aJv) = Rv{ aE} , aw2 aw aw (8) with u = (Jv - b). 3.4 Method applied to MLPs We are now ready to see how this technique can be applied to a specific type of model. Consider an MLP with L + 1 layers of nodes defined by the equations: y~ = g(x~) (9) N/ Xl t L 1-1 I ri Yj Wij i' (10) j In these equations, superscripts denote the layer number (starting at 0), subscripts index over terms in a particular layer, and NI is the number of input nodes in layer l. Thus, y~ is the output of neuron i at node layer l, and xi is the net input coming into the same neuron. Moreover, yf is an output of the entire MLP while y? is an input going into the MLP. The feedback equations calculated with respect to E are: aE ayf (11) Differentiating Functions of the Jacobian 8E 8y~ 8E 8x' t 8E 8w!j 8E 8()! J 8E 8xl ' t 439 (12) (13) (14) (15) where the Ui term is a component in the vector u from Equation 1. Applying the Rv {·} operator to the feedforward equations yields: Rv{Y?} Rv{yD Rv{ x~} g'(x~)Rv{ xD (for 1 > 0) N/ L Rv{y~-l } W~j' j (16) (17) (18) where the Vi term is a component in the vector v from Equation 2. As the final step, we apply the Rv{·} operator to the feedback equations, which yields: Rv{:~ } Rv{ :~} Rv{ :~} Rv{ :!,} Rv{ ::;} 3.5 Complete algorithm o (19) (20) (21) (22) (23) Implementing this algorithm is nearly as simple as implementing normal gradient descent. For each type of variable that is used in an MLP (net input, neuron output, weights, thresholds, partial derivatives, etc.), we require that an extra variable be allocated to hold the result of applying the Rv{·} operator to the original variable. With this change in place, the complete algorithm to compute 8Eu /8w is as follows: • Set u and a to the user specified vectors from Equation 1. • Set the MLP inputs to the value of x that J is to be evaluated at. • Perform a normal feedforward pass using Equations 9 and 10. • Set 8E/8yf to Ui. 440 G. W. Flake and B. A. Pearlmutter (a) (b) Figure 1: Learning only the derivative: showing (a) poor approximation of the function with (b) excellent approximation of the derivative. • Perform the feedback pass with Equations 11-15. Note that values in the aEjay? terms are now equal to JT U. • Set v to (JT u - a) • Perform a Rv{ .} forward pass with Equations 16-18. • Set the Rv{ 8Ej8yf} terms to O. • Perform a Rv{·} backward pass with Equations 19-23. After the last step, the values in the Rv{ 8E j 8w!j} and Rv{ 8 E j aeD terms contain the required result. It is important to note that the time complexity of the "J·forward" and "J. backward" calculations are nearly identical to the typical output and gradient evaluations (i.e., the "forward" and "backward" passes) of the models used. A similar technique can be used for calculating 8Evj8w. The main difference is that the Rv{ . } forward pass is performed between the normal forward and backward passes because u can only be determined after the Rv{ f (z, w)} has been calculated. 4 Experimental results To demonstrate the effectiveness and generality of the J-prop algorithm, we have implemented it on top of an existing neural network library [9] in such a way that the algorithm can be used on a large number of architectures, including MLPs, radial basis function net· works, and higher order networks. We trained an MLP with ten hidden tanh nodes on 100 points with conjugate gradient. The training exemplars consisted of inputs in [-1, 1] and a target derivative from 3 cos( 3x) + 5cos(lOx). Our unknown function (which the MLP never sees data from) is sin(3x) + l sin(lOx). The model quickly converges to a solution in approximately 100 iterations. Figure 1 shows the performance of the MLP. Having never seen data from the unknown function, the MLP yields a poor approximation of the function, but a very accurate approximation of the function's derivative. We could have trained on both outputs and derivatives, but our goal was to illustrate that J·prop can target derivatives alone. Differentiating Functions of the Jacobian 441 5 Conclusions We have introduced a general method for calculating the weight gradient of functions of the Jacobian matrix of feedforward nonlinear systems. The method can be easily applied to most nonlinear models in common use today. The resulting algorithm, J-prop, can be easily modified to minimize functionals from several application domains [10]. Some possible uses include: targeting known first derivatives, implementing Tangent Prop and Double Backpropagation, enforcing identical VO sensitivities in auto-encoders, deflating the largest eigenvalue and minimizing all eigenvalue bounds, optimizing the determinant for blind source separation, and building nonlinear controllers. While some special cases of the J-prop algorithm have already been studied, a great deal is unknown about how optimization of the Jacobian changes the overall optimization problem. Some anecdotal evidence seems to imply that optimization of the Jacobian can lead to better generalization and faster training. It remains to be seen if J-prop used on a nonlinear extension of linear methods will lead to superior solutions. Acknowledgements We thank Frans Coetzee, Yannis Kevrekidis, Joe O'Ruanaidh, Lucas Parra, Scott Rickard, Justinian Rosca, and Patrice Simard for helpful discussions. GWF would also like to thank Eric Baum and the NEC Research Institute for funding the time to write up these results. References [1] H. Drucker and Y. Le Cun. Improving generalization performance using double backpropagation. IEEE Transactions on Neural Networks, 3(6), November 1992. [2] P. Simard, B. Victorri, Y. Le Cun, and J. Denker. Tangent prop-A formalism for specifying selected invariances in an adaptive network. In John E. Moody, Steve J. Hanson, and Richard P. Lippmann, editors, Advances in Neural Information Processing Systems, volume 4, pages 895-903. Morgan Kaufmann Publishers, Inc., 1992. [3] H. White and A. R. Gallant. On learning the derivatives of an unknown mapping with multilayer feedforward networks. In Halbert White, editor, Artificial Neural Networks, chapter 12, pages 206-223. Blackwell, Cambridge, Mass., 1992. [4] H. White, K. Hornik, and M. Stinchcombe. Universal approximation of an unknown mapping and its derivative. In Halbert White, editor, Artificial Neural Networks, chapter 6, pages 55-77. Blackwell, Cambridge, Mass., 1992. [5] J. Principe, A. Rathie, and J. Kuo. Prediction of chaotic time series with neural networks and the issues of dynamic modeling. Bifurcations and Chaos, 2(4), 1992. [6] G. Deco and B. Schiirmann. Dynamic modeling of chaotic time series. In Russell Greiner, Thomas Petsche, and Stephen Jose Hanson, editors, Computational Learning Theory and Natural Learning Systems, volume IV of Making Learning Systems Practical, chapter 9, pages 137-153. The MIT Press, Cambridge, Mass., 1997. [7] G. E. Hinton. Learning distributed representations of concepts. In Proc. Eigth Annual Con! Cognitive Science Society, pages 1-12, Hillsdale, NJ, 1986. Erlbaum. [8] Barak A. Pearlmutter. Fast exact multiplication by the Hessian. Neural Computation, 6(1):147-160,1994. [9] G. W. Flake. Industrial strength modeling tools. Submitted to NIPS 99, 1999. [10] G. W. Flake and B. A. Pearl mutter. Optimizing properties of the Jacobian of nonlinear feedforward systems. In preperation, 1999.
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Effects of Spatial and Temporal Contiguity on the Acquisition of Spatial Information Thea B. Ghiselli-Crippa and Paul W. Munro Department of Information Science and Telecommunications University of Pittsburgh Pittsburgh, PA 15260 tbgst@sis.pitt.edu, munro@sis.pitt.edu Abstract Spatial information comes in two forms: direct spatial information (for example, retinal position) and indirect temporal contiguity information, since objects encountered sequentially are in general spatially close. The acquisition of spatial information by a neural network is investigated here. Given a spatial layout of several objects, networks are trained on a prediction task. Networks using temporal sequences with no direct spatial information are found to develop internal representations that show distances correlated with distances in the external layout. The influence of spatial information is analyzed by providing direct spatial information to the system during training that is either consistent with the layout or inconsistent with it. This approach allows examination of the relative contributions of spatial and temporal contiguity. 1 Introduction Spatial information is acquired by a process of exploration that is fundamentally temporal, whether it be on a small scale, such as scanning a picture, or on a larger one, such as physically navigating through a building, a neighborhood, or a city. Continuous scanning of an environment causes locations that are spatially close to have a tendency to occur in temporal proximity to one another. Thus, a temporal associative mechanism (such as a Hebb rule) can be used in conjunction with continuous exploration to capture the spatial structure of the environment [1]. However, the actual process of building a cognitive map need not rely solely on temporal associations, since some spatial information is encoded in the sensory array (position on the retina and proprioceptive feedback). Laboratory studies show different types of interaction between the relative contributions of temporal and spatial contiguities to the formation of an internal representation of space. While Clayton and Habibi's [2] series of recognition priming experiments indicates that priming is controlled only by temporal associations, in the work of McNamara et al. [3] priming in recognition is observed only when space and time are both contiguous. In addition, Curiel and Radvansky's [4] work shows that the effects of spatial and temporal contiguity depend on whether location or identity information is emphasized during learning. Moreover, other experiments ([3]) also show how the effects clearly depend on the task and can be quite different if an explicitly spatial task is used (e.g., additive effects in location judgments). 18 labels labels labels labels T. B. Ghiselli-Crippa and P W. Munro coordinates labels (A coeff.) labels coordinates (B coeff.) Figure 1: Network architectures: temporal-only network (left); spatio-temporal network with spatial units part of the input representation (center); spatio-temporal network with spatial units part of the output representation (right). 2 Network architectures The goal of the work presented in this paper is to study the structure of the internal representations that emerge from the integration of temporal and spatial associations. An encoder-like network architecture is used (see Figure 1), with a set of N input units and a set of N output units representing N nodes on a 2-dimensional graph. A set of H units is used for the hidden layer. To include space in the learning process, additional spatial units are included in the network architecture. These units provide a representation of the spatial information directly available during the learning/scanning process. In the simulations described in this paper, two units are used and are chosen to represent the (x, y) coordinates of the nodes in the graph. The spatial units can be included as part of the input representation or as part of the output representation (see Figure 1, center and right panels): both choices are used in the experiments, to investigate whether the spatial information could better benefit training as an input or as an output [5]. In the second case, the relative contribution of the spatial information can be directly manipulated by introducing weighting factors in the cost function being minimized. A two-term cost function is used, with a cross-entropy term for the N label units and a squared error term for the 2 coordinate units, ri indicates the actual output of unit i and ti its desired output. The relative influence of the spatial information is controlled by the coefficients A and B. 3 Learning tasks The left panel of Figure 2 shows an example of the type of layout used; the effective layout used in the study consists of N = 28 nodes. For each node, a set of neighboring nodes is defined, chosen on the basis of how an observer might scan the layout to learn the node labels and their (spatial) relationships; in Figure 2, the neighborhood relationships are represented by lines connecting neighboring nodes. From any node in the layout, the only allowed transitions are those to a neighbor, thus defining the set of node pairs used to train the network (66 pairs out of C(28, 2) = 378 possible pairs). In addition, the probability of occurrence of a particular transition is computed as a function of the distance to the corresponding neighbor. It is then possible to generate a sequence of visits to the network nodes, aimed at replicating the scanning process of a human observer studying the layout. Spatiotemporal Contiguity Effects on Spatial Information Acquisition 19 knife cup coin eraser eraser button Figure 2: Example of a layout (left) and its permuted version (right). Links represent allowed transitions. A larger layout of 28 units was used in the simulations. The basic learning task is similar to the grammar learning task of Servan-Schreiber et al. [6] and to the neighborhood mapping task described in [1] and is used to associate each of the N nodes on the graph and its (x, y) coordinates with the probability distribution of the transitions to its neighboring nodes. The mapping can be learned directly, by associating each node with the probability distribution of the transitions to all its neighbors: in this case, batch learning is used as the method of choice for learning the mapping. On the other hand, the mapping can be learned indirectly, by associating each node with itself and one of its neighbors, with online learning being the method of choice in this case; the neighbor chosen at each iteration is defined by the sequence of visits generated on the basis of the transition probabilities. Batch learning was chosen because it generally converges more smoothly and more quickly than online learning and gives qualitatively similar results. While the task and network architecture described in [1] allowed only for temporal association learning, in this study both temporal and spatial associations are learned simultaneously, thanks to the presence of the spatial units. However, the temporalonly (T-only) case, which has no spatial units, is included in the simulations performed for this study, to provide a benchmark for the evaluation of the results obtained with the spatio-temporal (S-T) networks. The task described above allows the network to learn neighborhood relationships for which spatial and temporal associations provide consistent information, that is, nodes experienced contiguously in time (as defined by the sequence) are also contiguous in space (being spatial neighbors). To tease apart the relative contributions of space and time, the task is kept the same, but the data employed for training the network is modified: the same layout is used to generate the temporal sequence, but the x , y coordinates of the nodes are randomly permuted (see right panel of Figure 2). If the permuted layout is then scanned following the same sequence of node visits used in the original version, the net effect is that the temporal associations remain the same, but the spatial associations change so that temporally neighboring nodes can now be spatially close or distant: the spatial associations are no longer consistent with the temporal associations. As Figure 4 illustrates, the training pairs (filled circles) all correspond to short distances in the original layout, but can have a distance anywhere in the allowable range in the permuted layout. Since the temporal and spatial distances were consistent in the original layout, the original spatial distance can be used as an indicator of temporal distance and Figure 4 can be interpreted as a plot of temporal distance vs. spatial distance for the permuted layout. The simulations described in the following include three experimental conditions: temporal only (no direct spatial information available); space and time consistent (the spatial coordinates and the temporal sequence are from the same layout); space and time inconsistent (the spatial coordinates and the temporal sequence are from different layouts). 20 T. B. Ghise/li-Crippa and P. W. Munro Hidden unit representations are compared using Euclidean distance (cosine and inner product measures give consistent results); the internal representation distances are also used to compute their correlation with Euclidean distances between nodes in the layout (original and permuted). The correlations increase with the number of hidden units for values of H between 5 and 10 and then gradually taper off for values greater than 10. The results presented in the remainder of the paper all pertain to networks trained with H = 20 and with hidden units using a tanh transfer function; all the results pertaining to S-T networks refer to networks with 2 spatial output units and cost function coefficients A = 0.625 and B = 6.25. 4 Results Figure 3 provides a combined view of the results from all three experiments. The left panel illustrates the evolution of the correlation between internal representation distances and layout (original and permuted) distances. The right panel shows the distributions of the correlations at the end of training (1000 epochs). The first general result is that, when spatial information is available and consistent with the temporal information (original layout), the correlation between hidden unit distances and layout distances is consistently better than the correlation obtained in the case of temporal associations alone. The second general result is that, when spatial information is available but not consistent with the temporal information (permuted layout), the correlation between hidden unit distances and original layout distances (which represent temporal distances) is similar to that obtained in the case of temporal associations alone, except for the initial transient. When the correlation is computed with respect to the permuted layout distances, its value peaks early during training and then decreases rapidly, to reach an asymptotic value well below the other three cases. This behavior is illustrated in the box plots in the right panel of Figure 3, which report the distribution of correlation values at the end of training. 4.1 Temporal-only vs. spatio-temporal As a first step in this study, the effects of adding spatial information to the basic temporal associations used to train the network can be examined. Since the learning task is the same for both the T-only and the S-T networks except for the absence or presence of spatial information during training, the differences observed can be attributed to the additional spatial information available to the S-T networks. The higher correlation between internal representation distances and original layout distances obtained when spatial information is 0 ~ ., ., 0 S and T CO"Isistent 0 i:i T-o" ~ ~ .. Sand T InCOnsistent .. -===s: 8 0 0 ii (corr with T distance) ... • 8 " " 0 0 ........... S and T Ir'ICOOSlStent E:2 '" (corr. Wflh S distance) N --'----' ci 0 0 0 0 0 200 400 600 800 1000 SandT T-only SandT SandT con_atent Inconsistent ineon.stant Ollnber 01 epochs (corr " th T ast ) (corr wth 5 dst ) Figure 3: Evolution of correlation during training (0 - 1000 epochs) (left). Distributions of correlations at the end of training (1000 epochs) (right). Spatiotemporal Contiguity Effects on Spatial Information Acquisition 21 N 0 ., 0 ", '" E 0 ~ ... 0 N 0 0 0 00 02 04 08 1 0 12 Figure 4: Distances in the original layout (x) vs_ distances in the permuted layout (y)_ The 66 training pairs are identified by filled circles_ dHU = 0.6 + 3.4dT + 0.3ds - 2.1(dT)2 + 0.4(d S )2 - 0.4dT ds 2 5 15 05 15 " Figure 5: Similarities (Euclidean distances) between internal representations developed by a S-T network (after 300 epochs)_ Figure 4 projects the data points onto the x, y plane_ available (see Figure 3) is apparent also when the evolution of the internal representations is examined_ As Figure 6 illustrates, the presence of spatial information results in better generalization for the pattern pairs outside the training set While the distances between training pairs are mapped to similar distances in hidden unit space for both the T-only and the S-T networks, the T-only network tends to cluster the non-training pairs into a narrow band of distances in hidden unit space. In the case of the S-T network instead, the hidden unit distances between non-training pairs are spread out over a wider range and tend to reflect the original layout distances. 4.2 Permuted layout As described above, with the permuted layout it is possible to decouple the spatial and temporal contributions and therefore study the effects of each. A comprehensive view of the results at a particular point during training (300 epochs) is presented in Figure 5, where the x, y plane represents temporal distance vs. spatial distance (see also Figure 4) and the z axis represents the similarity between hidden unit representations. The figure also includes a quadratic regression surface fitted to the data points. The coefficients in the equation of the surface provide a quantitative measure of the relative contributions of spatial (ds) and temporal distances (dT ) to the similarity between hidden unit representations (dHU ): (2) In general, after the transient observed in early training (see Figure 3), the largest and most significant coefficients are found for dT and (dT?, indicating a stronger dependence of dHU on temporal distance than on spatial distance. The results illustrated in Figure 5 represent the situation at a particular point during training (300 epochs). Similar plots can be generated for different points during training, to study the evolution of the internal representations. A different view of the evolution process is provided by Figure 7, in which the data points are projected onto the x,Z plane (top panel) and the y,z plane (bottom panel) at four different times during training. In the top panel, 14 22 T. B. Ghiselli-Crippa and P W Munro ::: ::: ~ ~ ~ ~. ~ .~. _ 0 ~ ,.. ~ _ 0 • N ,.. • N ~ ~ , 0 ~ ~ ~ 00 ~ ~, . , ~ ~ ~ ~ ~ ~ ~ , ' ',~-, ',' ~ -... ~ , ~ ~ ~ ~ ~ ~ 00 02 " 06 O. " 12 00 02 .. 06 .. " 12 00 02 .. 06 .. " " 00 02 .. .. .. " 12 "_d "_d "_d " _d ::: ::: ::: ::: ~ ~ ~ roo ~ ~ ~ o : s , e , rIP 0 00 :; ~ ~' ;; ~ _ 0 <P f/Po DO _ 0 ,.~,o 0 • , .' • N .. . • N ~ ~ tP DO 0 ~ : ~ : ~ . , ~ , .. ~ ~, ~ , " i ~ ',' ~ .~ ~ ~ , , ~ ~ .:. ~ .I' ~ ~ ~ g ~ 00 02 O. os .. 10 12 00 02 " 06 .. " 12 00 02 " .. .. 10 12 00 02 " O. o. " 12 "-' "-' "-' "-' Figure 6: Internal representation distances vs. original layout distances: S-T network (top) vs. T-only network (bottom). The training pairs are identified by filled circles. The presence of spatial information results in better generalization for the pairs outside the training set. the internal representation distances are plotted as a function of temporal distance (i.e., the spatial distance from the original layout), while in the bottom panel they are plotted as a function of spatial distance (from the permuted layout). The higher asymptotic correlation between internal representation distances and temporal distances, as opposed to spatial distances (see Figure 3), is apparent also from the examination of the evolutionary plots, which show an asymptotic behavior with respect to temporal distances (see Figure 7, top panel) very similar to the T-only case (see Figure 6, bottom panel). 5 Discussion The first general conclusion that can be drawn from the examination of the results described in the previous section is that, when the spatial information is available and consistent with the temporal information (original layout), the similarity structure of the hidden unit representations is closer to the structure of the original layout than that obtained by using temporal associations alone. The second general conclusion is that, when the spatial information is available but not consistent with the temporal information (permuted layout), the similarity structure of the hidden unit representations seems to correspond to temporal more than spatial proximity. Figures 5 and 7 both indicate that temporal associations take precedence over spatial associations. This result is in agreement with the results described in [1], showing how temporal associations (plus some high-level constraints) significantly contribute to the internal representation of global spatial information. However, spatial information certainly is very beneficial to the (temporal) acquisition of a layout, as proven by the results obtained with the S-T network vs. the T-only network. In terms of the model presented in this paper, the results illustrated in Figures 5 and 7 can be compared with the experimental data reported for recognition priming ([2], [3], [4]), with distance between internal representations corresponding to reaction time. The results of our model indicate that distances in both the spatially far and spatially close condition appear to be consistently shorter for the training pairs (temporally close) than for the nontraining pairs (temporally distant), highlighting a strong temporal effect consistent with the data reported in [2] and [4] (for spatially far pairs) and in [3] (only for the spatially close Spatiotemporal Contiguity Effects on Spatial Information Acquisition ~ ; ~-' ~. ~~. . ~ 0_ Sl ........... .. ri 0 0 ~'--_____ -.J 00 0 2 O. 0 1 01 10 12 In_d (T} ~ L..-____ -.l 00 02 0" 01 01 '0 12 In_d (S) j!I!A " ...... . ,. 0 lfIiiIo '0' ~ ,. ~ ~ • 110 0 \ .. 00 02 O. 01 01 10 1 2 In_d(TI 00 0.2 o. ot 01 10 12 .. _d(S) 00 02 0.4 01 01 10 12 l'I_d (T) 0.0 02 04 0 8 all 1 0 1 2 ... u:I (S) 0 0 02 O. 01 oa 10 12 In_den 23 ~l.-_____ -.J 00 02 O. 06 oa 10 12 !rUi (S) Figure 7: Internal representation distances vs. temporal distances (top) and vs. spatial distances (bottom) for a S-T network (permuted layout). The training pairs are identified by filled circles. The asymptotic behavior with respect to temporal distances (top panel) is similar to the T-only condition. The bottom panel indicates a weak dependence on spatial distances. case). For the training pairs (temporally close), slightly shorter distances are obtained for spatially close pairs vs. spatially far pairs; this result does not provide support for the experimental data reported in either [3] (strong spatial effect) or [2] (no spatial effect). For the non-training pairs (temporally distant), long distances are found throughout, with no strong dependence on spatial distance; this effect is consistent with all the reported experimental data. Further simulations and statistical analyses are necessary for a more conclusive comparison with the experimental data. References [1] Ghiselli-Crippa, TB. & Munro, P.w. (1994). Emergence of global structure from local associations. In J.D. Cowan, G. Tesauro, & J. Alspector (Eds.), Advances in Neural Information Processing Systems 6, pp. 1101-1108. San Francisco, CA: Morgan Kaufmann. [2] Clayton, K.N. & Habibi, A. (1991). The contribution of temporal contiguity to the spatial priming effect. Journal of Experimental Psychology: Learning. Memory. and Cognition 17:263-271. [3] McNamara, TP., Halpin. J.A. & Hardy, J.K. (1992). Spatial and temporal contributions to the structure of spatial memory. Journal of Experimental Psychology: Learning. Memory. and Cognition 18:555-564. [4] Curiel, J.M. & Radvansky, G.A. (1998). Mental organization of maps. Journal of Experimental Psychology: Learning. Memory. and Cognition 24:202-214. [5] Caruana, R. & de Sa, VR. (1997). Promoting poor features to supervisors: Some inputs work better as outputs. In M.e. Mozer, M.I. Jordan, & T Petsche (Eds.), Advances in Neural Information Processing Systems 9, pp. 389-395. Cambridge, MA: MIT Press. [6] Servan-Schreiber, D., Cleeremans, A. & McClelland, J.L. (1989). Learning sequential structure in simple recurrent networks. In D.S. Touretzky (Ed.), Advances in Neural Information Processing Systems 1, pp. 643-652. San Mateo, CA: Morgan Kaufmann.
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Agglomerative Information Bottleneck Noam Slonim Naftali Tishby* Institute of Computer Science and Center for Neural Computation The Hebrew University Jerusalem, 91904 Israel email: {noamm.tishby}(Qcs.huji.ac.il Abstract We introduce a novel distributional clustering algorithm that maximizes the mutual information per cluster between data and given categories. This algorithm can be considered as a bottom up hard version of the recently introduced "Information Bottleneck Method". The algorithm is compared with the top-down soft version of the information bottleneck method and a relationship between the hard and soft results is established. We demonstrate the algorithm on the 20 Newsgroups data set. For a subset of two newsgroups we achieve compression by 3 orders of magnitudes loosing only 10% of the original mutual information. 1 Introduction The problem of self-organization of the members of a set X based on the similarity of the conditional distributions of the members of another set, Y, {p(Ylx)}, was first introduced in [8] and was termed "distributional clustering" . This question was recently shown in [9] to be a special case of a much more fundamental problem: What are the features of the variable X that are relevant for the prediction of another, relevance, variable Y? This general problem was shown to have a natural in~ormation theoretic formulation: Find a compressed representation of the variable X, denoted X, such that the mutual information between X and Y, I (X; Y), is as high as possible, under a constraint on the mutual information between X and X, I (X; X). Surprisingly, this variational problem yields an exact self-consistent equations for the conditional distributions p(ylx), p(xlx), and p(x). This constrained information optimization problem was called in [9] The Information Bottleneck Method. The original approach to the solution of the resulting equations, used already in [8], was based on an analogy with the "deterministic annealing" approach to clustering (see [7]). This is a top-down hierarchical algorithm that starts from a single cluster and undergoes a cascade of cluster splits which are determined stochastically (as phase transitions) into a "soft" (fuzzy) tree of clusters. In this paper we propose an alternative approach to the information bottleneck 618 N Slonim and N Tishby problem, based on a greedy bottom-up merging. It has several advantages over the top-down method. It is fully deterministic, yielding (initially) "hard clusters", for any desired number of clusters. It gives higher mutual information per-cluster than the deterministic annealing algorithm and it can be considered as the hard (zero temperature) limit of deterministic annealing, for any prescribed number of clusters. Furthermore, using the bottleneck self-consistent equations one can "soften" the resulting hard clusters and recover the deterministic annealing solutions without the need to identify the cluster splits, which is rather tricky. The main disadvantage of this method is computational, since it starts from the limit of a cluster per each member of the set X. 1.1 The information bottleneck method The mutual information between the random variables X and Y is the symmetric functional of their joint distribution, I(X;Y) = L p(x,y) log ( ~~~'~\) = L p(x)p(ylx) log (P(Y(lx))) . xEX,yEY P P Y xEX,yEY P Y (1) The objective of the information bottleneck method is to extract a compact representation of the variable X, denoted here by X, with minimal loss of mutual information to another, relevance, variable Y. More specifically, we want to find a (possibly stochastic) map, p(xlx), that minimizes the (lossy) coding length of X via X, I(Xi X), under a constraint on the mutual information to the relevance variable I(Xi Y). In other words, we want to find an efficient representation of the variable X, X, such that the predictions of Y from X through X will be as close as possible to the direct prediction of Y from X. As shown in [9], by introducing a positive Lagrange multiplier 13 to enforce the mutual information constraint, the problem amounts to minimization of the Lagrangian: £[P(xlx)] = I(Xi X) - f3I(Xi Y) , (2) with respect to p(xlx), subject to the Markov condition X -t X -t Y and normalization. This minimization yields directly the following self-consistent equations for the map p(xlx), as well as for p(Ylx) and p(x): { p(xlx) = i(~~;) exp (-f3DKL [P(Ylx)lIp(Ylx)]) p(Ylx) = 2:xp(Ylx)p(xlx)~ p(x) = 2:x p(xlx)p(x) (3) where Z(f3, x) is a normalization function. The functional DKL[Pllq] == 2:y p(y) log ~f~~ is the Kulback-Liebler divergence [3J, which emerges here from the variational principle. These equations can be solved by iterations that are proved to converge for any finite value of 13 (see [9]). The Lagrange multiplier 13 has the natural interpretation of inverse temperature, which suggests deterministic annealing [7] to explore the hierarchy of solutions in X, an approach taken already in [8J . The variational principle, Eq.(2), determines also the shape of the annealing process, since by changing 13 the mutual informations Ix == I(X; X) and Iy == I(Y; X) vary such that My = 13-1 Mx . (4) Agglomerative Information Bottleneck 619 Thus the optimal curve, which is analogous to the rate distortion function in information theory [3], follows a strictly concave curve in the (Ix,Iy) plane, called the information plane. Deterministic annealing, at fixed number of clusters, follows such a concave curve as well, but this curve is suboptimal beyond a certain critical value of f3. Another interpretation of the bottleneck principle comes from the relation between the mutual information and Bayes classification error. This error is bounded above and below (see [6]) by an important information theoretic measure of the class conditional distributions p(XIYi), called the Jensen-Shannon divergence. This measure plays an important role in our context. The Jensen-Shannon divergence of M class distributions, Pi(X), each with a prior 7ri, 1 ~ i ~ M, is defined as, [6,4]. M M JS7r(Pl,P2, ···,PM] == H[L 7riPi(X)] - L 7riH [Pi(X)] , (5) i=l i=l where H[P(x)] is Shannon's entropy, H[P(x)] = - Ex p(x) logp(x). The convexity of the entropy and Jensen inequality guarantees the non-negativity of the JSdivergence. 1.2 The hard clustering limit For any finite cardinality of the representation IXI == m the limit f3 -+ 00 of the Eqs.(3) induces a hard partition of X into m disjoint subsets. In this limit each member x E X belongs only to the subset x E X for which p(Ylx) has the smallest DKL[P(ylx)lIp(ylx)] and the probabilistic map p(xlx) obtains the limit values 0 and 1 only. In this paper we focus on a bottom up agglomerative algorithm for generating "good" hard partitions of X. We denote an m-partition of X, i.e. X with cardinality m, also by Zm = {Zl,Z2, ... ,Zm}, in which casep(x) =p(Zi). We say that Zm is an optimal m-partition (not necessarily unique) of X if for every other m-partition of X, Z:n, I(Zm; Y) ~ I(Z:n; Y). Starting from the trivial N-partition, with N = lXI, we seek a sequence of merges into coarser and coarser partitions that are as close as possible to optimal. It is easy to verify that in the f3 -+ 00 limit Eqs.(3) for the m-partition distributions are simplified as follows. Let x == Z = {XI,X2, ... ,xl zl} , Xi E X denote a specific component (Le. cluster) of the partition Zm, then 1 {I if x E Z p(zlx) = 0 th . Vx E X o erWlse p(ylz) = plz) El~l P(Xi, y) Vy E Y p(z) = El~l P(Xi) (6) Using these distributions one can easily evaluate the mutual information between Zm and Y, I(Zm; Y), and between Zm and X, I(Zm; X), using Eq.(l). Once any hard partition, or hard clustering, is obtained one can apply "reverse annealing" and "soften" the clusters by decreasing f3 in the self-consistent equations, Eqs.( 3). Using this procedure we in fact recover the stochastic map, p(xlx), from the hard partition without the need to identify the cluster splits. We demonstrate this reverse deterministic annealing procedure in the last section. 620 N. Slonim and N. Tishby 1.3 Relation to other work A similar agglomerative procedure, without the information theoretic framework and analysis, was recently used in [1] for text categorization on the 20 newsgroup corpus. Another approach that stems from the distributional clustering algorithm was given in [5] for clustering dyadic data. An earlier application of mutual information for semantic clustering of words was used in [2]. 2 The agglomerative information bottleneck algorithm The algorithm starts with the trivial partition into N = IXI clusters or components, with each component contains exactly one element of X. At each step we merge several components of the current partition into a single new component in a way that locally minimizes the loss of mutual information leX; Y) = l(Zm; Y). Let Zm be the current m-partition of X and Zm denote the new m-partition of X after the merge of several components of Zm. Obviously, m < m. Let {Zl, Z2, ... , zd ~ Zm denote the set of components to be merged, and Zk E Zm the new component that is generated by the merge, so m = m - k + 1. To evaluate the reduction in the mutual information l(Zm; Y) due to this merge one needs the distributions that define the new m-partition, which are determined as follows. For every Z E Zm, Z f:. Zk, its probability distributions (p(z),p(ylz),p{zlx» remains equal to its distributions in Zm. For the new component, Zk E Zm, we define, { p(Zk) = L~=l P(Zi) p(yIZk) = P(~Ic) ~~=l P(Zi, y) \ly E Y . (zlx) = {1 1f x E ~i for some 1 ~ ~ ~ k \Ix X p 0 otherw1se E (7) It is easy to verify that Zm is indeed a valid m-partition with proper probability distributions. Using the same notations, for every merge we define the additional quantities: • The merge prior distribution: defined by ilk == (71"1,71"2, ... , 7I"k), where 7I"i is the prior probability of Zi in the merged subset, i.e. 7I"i == :t::). • The Y -information decrease: the decrease in the mutual information l(X; Y) due to a single merge, cHy(Zl' ""Zk) == l(Zm; Y) - l(Zm; Y) • The X-information decrease: the decrease in the mutual information l(X, X) due to a single merge, cHx (Z1, Z2, ... , Zk) == l(Zm; X) - l(Zm; X) Our algorithm is a greedy procedure, where in each step we perform "the best possible merge" , i.e. merge the components {Z1, ... , zd of the current m-partition which minimize cHy(Z1, ... , Zk). Since cHy(Zl, ... , Zk) can only increase with k (corollary 2), for a greedy procedure it is enough to check only the possible merging of pairs of components of the current m-partition. Another advantage of merging only pairs is that in this way we go through all the possible cardinalities of Z = X, from N to 1. For a given m-partition Zm = {Z1,Z2, ... ,Zm} there are m(~-1) possible pairs to merge. To find "the best possible merge" one must evaluate the reduction of information cHy(Zi, Zj) = l(Zm; Y) - l(Zm-1; Y) for every pair in Zm , which is O(m . WI) operations for every pair. However, using proposition 1 we know that cHy(Zi, Zj) = (P(Zi) + p(Zj» . JSn 2 (P(YIZi),p(Ylzj», so the reduction in the mutual Agglomerative Information Bottleneck 621 information due to the merge of Zi and Zj can be evaluated directly (looking only at this pair) in O(IYI) operations, a reduction of a factor of m in time complexity (for every merge). Input: Empirical probability matrix p(x,y), N = IX\, M = IYI Output: Zm : m-partition of X into m clusters, for every 1 ::; m ::; N Initialization: • Construct Z == X - For i = 1...N * Zi = {x;} * P(Zi) = p(Xi) * p(YIZi) = p(YIXi) for every y E Y * p(zlxj) = 1 if j = i and 0 otherwise - Z={Zl, ... ,ZN} • for every i, j = 1.. .N, i < j, calculate di,j = (p(Zi) +p(Zj))' JSn2[P(ylzi),p(ylzj)] (every di,j points to the corresponding couple in Z) Loop: • For t = 1... (N - 1) Find {a,.B} = argmini,j {di,j } (if there are several minima choose arbitrarily between them) Merge {z"" Zj3} => z : * p(z) = p(z",) + p(Zj3) * p(ylz) = r>li) (p(z""y) +p(zj3,y)) for every y E Y * p(zlx) = 1 if x E z'" U Zj3 and 0 otherwise, for every x E X - Update Z = {Z - {z"" z,q}} U{z} (Z is now a new (N - t)-partition of X with N - t clusters) - Update di ,j costs and pointers w.r.t. z (only for couples contained z'" or Zj3). • End For Figure 1: Pseudo-code of the algorithm. 3 Discussion The algorithm is non-parametric, it is a simple greedy procedure, that depends only on the input empirical joint distribution of X and Y. The output of the algorithm is the hierarchy of all m-partitions Zm of X for m = N, (N - 1), ... ,2,1. Moreover, unlike most other clustering heuristics, it has a built in measure of efficiency even for sub-optimal solutions, namely, the mutual information I(Zm; Y) which bounds the Bayes classification error. The quality measure of the obtained Zm partition is the fraction of the mutual information between X and Y that Zm captures. This is given by the curve II Z,; -: vs. m = 1 Zm I. We found that empirically this , curve was concave. If this is always true the decrease in the mutual information at every step, given by 8(m) == I(Z7n;Y)(-:~7n-l;Y) can only increase with decreasing m. Therefore, if at some point 8(m) becomes relatively high it is an indication that we have reached a value of m with "meaningful" partition or clusters. Further 622 N. Slonim and N. Tishby merging results in substantial loss of information and thus significant reduction in the performance of the clusters as features. However, since the computational cost of the final (low m) part of the procedure is very low we can just as well complete the merging to a single cluster. :;:~ :::"0.5 >~ 0.2 0 .4 0 .6 0.8 I(Z;X) / H(X) 0 .8 >" 1 06 • >" ... ~O . 4 -.6.+ 0.2 ING100 o~' ----;-;;;;;;-----"3000:=--= 121 %~--~ '0~--~2~0----~ 30~---4~0----~~~ IZI Figure 2: On the left figure the results of the agglomerative algorithm are shown in the "information plane", normalized I(Z; Y) vs. normalized I(Z; X) for the NGlOOO dataset. It is compared to the soft version of the information bottleneck via "reverse annealing" for IZI = 2,5, 10, 20, 100 (the smooth curves on the left). For IZI = 20, 100 the annealing curve is connected to the starting point by a dotted line. In this plane the hard algorithm is clearly inferior to the soft one. On the right-hand side: I(Zm, Y) of the agglomerative algorithm is plotted vs. the cardinality of the partition m for three subsets of the newsgroup dataset. To compare the performance over the different data cardinalities we normalize I(Zm ; Y) by the value of I(Zso ; Y) , thus forcing all three curves to start (and end) at the same points. The predictive information on the newsgroup for NGlOOO and NGIOO is very similar, while for the dichotomy dataset, 2ng, a much better prediction is possible at the same IZI, as can be expected for dichotomies. The inset presents the full curve of the normalized I(Z; Y) vs. IZI for NGIOO data for comparison. In this plane the hard partitions are superior to the soft ones. 4 Application To evaluate the ideas and the algorithm we apply it to several subsets of the 20Newsgroups dataset, collected by Ken Lang using 20, 000 articles evenly distributed among 20 UseNet discussion groups (see [1]). We replaced every digit by a single character and by another to mark non-alphanumeric characters. Following this preprocessing, the first dataset contained the 530 strings that appeared more then 1000 times in the data. This dataset is referred as NG1000. Similarly, all the strings that appeared more then 100 times constitutes the NG100 dataset and it contains 5148 different strings. To evaluate also a dichotomy data we used a corpus consisting of only two discussion groups out of the 20Newsgroups with similar topics: alt. atheism and talk. religion. misc. Using the same pre-processing, and removing strings that occur less then 10 times, the resulting "lexicon" contained 5765 different strings. We refer to this dataset as 2ng. We plot the results of our algorithm on these three data sets in two different planes. First, the normalized information ;g ~~~ vs. the size of partition of X (number of clusters), IZI. The greedy procedure directly tries to maximize J(Z; Y) for a given IZI, as can be seen by the strong concavity of these curves (figure 2, right). Indeed the procedure is able to maintain a high percentage of the relevant mutual information of the original data, while reducing the dimensionality of the "features" , Agglomerative Information Bottleneck 623 IZI, by several orders of magnitude. On the right hand-side of figure 2 we present a comparison between the efficiency of the procedure for the three datasets. The two-class data, consisting of 5765 different strings, is compressed by two orders of magnitude, into 50 clusters, almost without loosing any of the mutual information about the news groups (the decrease in I(Xi Y) is about 0.1%). Compression by three orders of magnitude, into 6 clusters, maintains about 90% of the original mutual information. Similar results, even though less striking, are obtained when Y contain all 20 newsgroups. The NG100 dataset was compressed from 5148 strings to 515 clusters, keeping 86% of the mutual information, and into 50 clusters keeping about 70% of the information. About the same compression efficiency was obtained for the NG1000 dataset. The relationship between the soft and hard clustering is demonstrated in the Information plane, i.e., the normalized mutual information values, :ti ;;~~ vs. Ik(~)' In this plane, the soft procedure is optimal since it is a direct maximization of I(Z; Y) while constraining I(Zi X). While the hard partition is suboptimal in this plane, as confirmed empirically, it provides an excellent starting point for reverse annealing. In figure 2 we present the results of the agglomerative procedure for NG1000 in the information plane, together with the reverse annealing for different values of IZI. As predicted by the theory, the annealing curves merge at various critical values of f3 into the globally optimal curve, which correspond to the "rate distortion function" for the information bottleneck problem. With the reverse annealing ("heating") procedure there is no need to identify the cluster splits as required in the original annealing ("cooling") procedure. As can be seen, the "phase diagram" is much better recovered by this procedure, suggesting a combination of agglomerative clustering and reverse annealing as the ultimate algorithm for this problem. References [1] L. D. Baker and A. K. McCallum. Distributional Clustering of Words for Text Classification In ACM SIGIR 98, 1998. [2] P. F. Brown, P.V. deSouza, R.L. Mercer, V.J. DellaPietra, and J.C. Lai. Class-based n-gram models of natural language. In Computational Linguistics, 18( 4}:467-479, 1992. [3] T. M. Cover and J. A. Thomas. Elements of Information Theory. John Wiley & Sons, New York, 1991. [4] R. EI-Yaniv, S. Fine, and N. Tishby. Agnostic classification of Markovian sequences. In Advances in Neural Information Processing (NIPS'97) , 1998. [5] T. Hofmann, J. Puzicha, and M. Jordan. Learning from dyadic data. In Advances in Neural Information Processing (NIPS'98), 1999. [6] J. Lin. Divergence Measures Based on the Shannon Entropy. IEEE Transactions on Information theory, 37(1}:145- 151, 1991. [7] K. Rose. Deterministic Annealing for Clustering, Compression, Classification, Regression, and Related Optimization Problems. Proceedings of the IEEE, 86(11}:2210- 2239, 1998. [8] F.C. Pereira, N. Tishby, and L. Lee. Distributional clustering of English words. In 30th Annual Meeting of the Association for Computational Linguistics, Columbus, Ohio, pages 183-190, 1993. [9] N. Tishby, W. Bialek, and F. C. Pereira. The information bottleneck method: Extracting relevant information from concurrent data. Yet unpublished manuscript, NEC Research Institute TR, 1998.
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Spike-based learning rules and stabilization of persistent neural activity Xiaohui Xie and H. Sebastian Seung Dept. of Brain & Cog. Sci., MIT, Cambridge, MA 02139 {xhxie, seung}@mit.edu Abstract We analyze the conditions under which synaptic learning rules based on action potential timing can be approximated by learning rules based on firing rates. In particular, we consider a form of plasticity in which synapses depress when a presynaptic spike is followed by a postsynaptic spike, and potentiate with the opposite temporal ordering. Such differential anti-Hebbian plasticity can be approximated under certain conditions by a learning rule that depends on the time derivative of the postsynaptic firing rate. Such a learning rule acts to stabilize persistent neural activity patterns in recurrent neural networks. 1 INTRODUCTION Recent experiments have demonstrated types of synaptic plasticity that depend on the temporal ordering of presynaptic and postsynaptic spiking. At cortical [ I] and hippocampal[2] synapses, long-term potentiation is induced by repeated pairing of a presynaptic spike and a succeeding postsynaptic spike, while long-term depression results when the order is reversed. The dependence of the change in synaptic strength on the difference l:!..t = tpost tpre between postsynaptic and presynaptic spike times has been measured quantitatively. This pairing function, sketched in Figure A o~i =~=::====: B 0L-.i _:3/_-----' o t - t post pre ~re 11111111111 111111111111111111 post 11111111111 11111111 •• 11. t .:oLl ____ \:;J __ ~ ° 1000 2000 time (ms) Figure I: (A) Pairing function for differential Hebbian learning. The change in synaptic strength is plotted versus the time difference between postsynaptic and presynaptic spikes. (B) Pairing function for differential anti-Hebbian learning. (C) Differential antiHebbian learning is driven by changes in firing rates. The synaptic learning rule of Eq. (l) is applied to two Poisson spike trains. The synaptic strength remains roughly constant in time, except when the postsynaptic rate changes. lA, has positive and negative lobes correspond to potentiation and depression. and a width of tens of milliseconds. We will refer to synaptic plasticity associated with this pairing function as differential Hebbian plasticity-Hebbian because the conditions for 200 X Xie and H. S. Seung potentiation are as predicted by Hebb[3], and differential because it is driven by the difference between the opposing processes of potentiation and depression. The pairing function of Figure IA is not characteristic of all synapses. For example, an opposite temporal dependence has been observed at electrosensory lobe synapses of electric fish[4]. As shown in Figure IB, these synapses depress when a presynaptic spike is followed by a postsynaptic one, and potentiate when the order is reversed. We will refer to this as differential anti-Hebbian plasticity. According to these experiments, the maximum ranges of the differential Hebbian and antiHebbian pairing functions are roughly 20 and 40 ms, respectively. This is fairly short, and seems more compatible with descriptions of neural activity based on spike timing rather than instantaneous firing rates[5, 6]. In fact, we will show that there are some conditions under which spike-based learning rules can be approximated by rate-based learning rules. Other people have also studied the relationship between spike-based and rate-based learning rules[7, 8]. The pairing functions of Figures IA and IB lead to rate-based learning rules like those traditionally used in neural networks, except that they depend on temporal derivatives of firing rates as well as firing rates themselves. We will argue that the differential antiHebbian learning rule of Figure IB could be a general mechanism for tuning the strength of positive feedback in networks that maintain a short-term memory of an analog variable in persistent neural activity. A number of recurrent network models have been proposed to explain memory-related neural activity in motor [9] and prefrontal [ 10] cortical areas, as well as the head direction system [11] and oculomotor integrator[ 12, 13, 14]. All of these models require precise tuning of synaptic strengths in order to maintain continuously variable levels of persistent activity. As a simple illustration of tuning by differential antiHebbian learning, a model of persistent activity maintained by an integrate-and-fire neuron with an excitatory autapse is studied. 2 SPIKE-BASED LEARNING RULE Pairing functions like those of Figure 1 have been measured using repeated pairing of a single presynaptic spike with a single postsynaptic spike. Quantitative measurements of synaptic changes due to more complex patterns of spiking activity have not yet been done. We will assume a simple model in which the synaptic change due to arbitrary spike trains is the sum of contributions from all possible pairings of presynaptic with postsynaptic spikes. The model is unlikely to be an exact description of real synapses, but could turn out to be approximately valid. We will write the spike train of the ith neuron as a series of Dirac delta functions, Si (t) = Ln <5(t - Tr), where Tr is the nth spike time of the ith neuron. The synaptic weight from neuron j to i at time t is denoted by Wij (t). Then the change in synaptic weight induced by presynaptic spikes occurring in the time interval [0, Tj is modeled as Wij(T + >.) - Wij(>') = [T dtj foo dti f(ti - tj)Si(ti) Sj(tj) (1) io -00 Each presynaptic spike is paired with all postsynaptic spikes produced before and after. For each pairing, the synaptic weight is changed by an amount depending on the pairing function f. The pairing function is assumed to be nonzero inside the interval [-T, Tj, and zero outside. We will refer to T as the pairing range. According to our model, each presynaptic spike results in induction of plasticity only after a latency>.. Accordingly, the arguments T + >. and >. of Wij on the left hand side of the equation are shifted relative to the limits T and 0 of the integral on the right hand side. We Spike-based Learning and Stabilization of Persistent Neural Activity 201 will assume that the latency>. is greater than the pairing range T, so that Wi} at any time is only influenced by presynaptic and postsynaptic spikes that happened before that time, and therefore the learning rule is causal. 3 RELATION TO RATE-BASED LEARNING RULES The learning rule of Eq. (1) is driven by correlations between presynaptic and postsynaptic activities. This dependence can be made explicit by making the change of variables u = ti - t j in Eq. (I), which yields Wij(T + >.) - Wij (>.) = iTT duf(u)Cij(u) (2) where we have defined the cross-correlation Cij(u) = !aT dt Si(t + u) Sj(t) . (3) and made use of the fact that f vanishes outside the interval [-T, T]. Our immediate goal is to relate Eq. (2) to learning rules that are based on the cross-correlation between firing rates, Crre(u) = !aT dt Vi(t + u) Vj(t) (4) There are a number of ways of defining instantaneous firing rates. Sometimes they are computed by averaging over repeated presentations of a stimulus. In other situations, they are defined by temporal filtering of spike trains. The following discussion is general, and should apply to these and other definitions of firing rates. The "rate correlation" is commonly subtracted from the total correlation to obtain the "spike correlation" C:rke = Cij - Cijate. To derive a rate-based approximation to the learning rule (2), we rewrite it as Wij(T + >.) - Wij(>') = iTT du f(u)Cijate(u) + iTT du f(u)C:rke (u) (5) and simply neglect the second term. Shortly we will discuss the conditions under which this is a good approximation. But first we derive another form for the first term by applying the approximation Vi(t + u) ~ Vi(t) + UVi(t) to obtain j T duf(u)Crre(u) ~ iT dt[fiovi(t) + 131Vi(t)]VJ (t) -T 0 (6) where we define (7) This approximation is good when firing rates vary slowly compared to the pairing range T . The learning rule depends on the postsynaptic rate through fio Vi + 131 Vi . When the first term dominates the second, then the learning rule is the conventional one based on correlations between firing rates, and the sign of fio determines whether the rule is Hebbian or anti-Hebbian. In the remainder of the paper, we will discuss the more novel case where 130 = O. This holds for the pairing functions shown in Figures lA and IB, which have positive and negative lobes with areas that exactly cancel in the definition of 130. Then the dependence on 202 X Xie and H. S. Seung postsynaptic activity is purely on the time derivative of the firing rate. Differential Hebbian learning corresponds to /31 > 0 (Figure IA), while differential anti-Hebbian learning leads to /31 < 0 (Figure IB). To summarize the /30 = 0 case, the synaptic changes due to rate correlations are approximated by Wij ex: -ViVj (diff. anti-Hebbian) (8) for slowly varying rates. These formulas imply that a constant postsynaptic firing rate causes no net change in synaptic strength. Instead, changes in rate are required to induce synaptic plasticity. To illustrate this point, Figure lC shows the result of applying differential anti-Hebbian learning to two spike trains. The presynaptic spike train was generated by a 50 Hz Poisson process, while the postsynaptic spike train was generated by an inhomogeneous Poisson process with rate that shifted from 50 Hz to 200 Hz at 1 sec. Before and after the shift, the synaptic strength fluctuates but remains roughly constant. But the upward shift in firing rate causes a downward shift in synaptic strength, in accord with the sign of the differential anti-Hebbian rule in Eq. (8). The rate-based approximation works well for this example, because the second term of Eq. (5) is not so important. Let us return to the issue of the general conditions under which this term can be neglected. With Poisson spike trains, the spike correlations C:Pike (u) are zero in the limit T -7 00, but for finite T they fluctuate about zero. The integr~l over u in the second term of (5) dampens these fluctuations. The amount of dampening depends on the pairing range T, which sets the limits of integration. In Figure 1 C we used a relatively long pairing range of 100 ms, which made the fluctuations small even for small T. On the other hand, if T were short, the fluctuations would be small only for large T_ Averaging over large T is relevant when the amplitUde of f is small, so that the rate of learning is slow. In this case, it takes a long time for significant synaptic changes to accumulate, so that plasticity is effectively driven by integrating over long time periods T in Eq. (l). In the brain, nonvanishing spike correlations are sometimes observed even in the T -7 00 limit, unlike with Poisson spike trains. These correlations are often roughly symmetric about zero, in which case they should produce little plasticity if the pairing functions are antisymmetric as in Figures lA and lB. On the other hand, if the spike correlations are asymmetric, they could lead to substantial effects[6]. 4 EFFECTS ON RECURRENT NETWORK DYNAMICS The learning rules of Eq. (8) depend on both presynaptic and postsynaptic rates, like learning rules conventionally used in neural networks. They have the special feature that they depend on time derivatives, which has computational consequences for recurrent neural networks of the form Xi + Xi = L Wiju(Xj) + bi j (9) Such classical neural network equations can be derived from more biophysically realistic models using the method of averaging[ 15] or a mean field approximation[ 16]. The firing rate of neuron j is conventionally identified with Vj = u(Xj). The cost function E( {Xi}; {Wij}) = ~ Li v; quantifies the amount of drift in firing rate at the point Xl , ... , X N in the state space of the network. If we consider Vi to be a function of Xi and Wij defined by (9), then the gradient ofthe cost function with respect to Wij is given by BE / BWij = U' (Xi)ViVj. Assuming that U is a monotonically increasing function so that u' (xd > 0, it follows that the differential Hebbian update of (8) increases the cost function, Spike-based Learning and Stabilization of Persistent Neural Activity 203 and hence increases the magnitude of the drift velocity. In contrast, the differential antiHebbian update decreases the drift velocity. This suggests that the differential anti-Hebbian update could be useful for creating fixed points of the network dynamics (9). 5 PERSISTENT ACTIVITY IN A SPIKING AUTAPSE MODEL The preceding arguments about drift velocity were based on approximate rate-based descriptions of learning and network dynamics. It is important to implement spike-based learning in a spiking network dynamics, to check that our approximations are valid. INHIBITORY BURST • Therefore we have numerically simulated the simple recurrent circuit of integrate-and-fire neurons shown in Figure 2. The core of the circuit is the "memory neuron," which makes an excitatory autapse onto itself. It also receives synaptic input from three input neurons: a tonic neuron, an excitatory burst neuron, and an inhibitory burst neuron. It is known that this circuit can store a shortterm memory of an analog variable in persistent activity, if the strengths of the autapse and tonic synapse are precisely tuned[ 17]. Here we show that this tuning can be accomplished by the spikebased learning rule of Eq. (1), with a differential anti-Hebbian pairing function like that of Figure 1 B. Figure 2: Circuit diagram for autapse model The memory neuron is described by the equations C dV m dt dr Tsyn dt + r = n (10) (1) where V is the membrane potential. When V reaches V'thres, a spike is considered to have occurred, and V is reset to Vreset. Each spike at time Tn causes a jump in the synaptic activation r of size CY.r/Tsyn, after which r decays exponentially with time constant Tsyn until the next spike. The synaptic conductances of the memory neuron are given by (12) The term W r is recurrent excitation from the autapse, where W is the strength of the autapse. The synaptic activations ro, r +, and r _ of the tonic, excitatory burst, and inhibitory burst neurons are governed by equations like (10) and (1), with a few differences. These neurons have no synaptic input; their firing patterns are instead determined by applied currents lapp,o, lapp,+ and lapp,_. The tonic neuron has a constant applied current, which makes it fire repetitively at roughly 20 Hz (Figure 3). For the excitatory and inhibitory burst neurons the applied current is normally zero, except for brief 100 ms current pulses that cause bursts of action potentials. As shown in Figure 3, if the synaptic strengths W and Wo are arbitrarily set before learning, the burst neurons cause only transient changes in the firing rate of the memory neuron. After applying the spike-based learning rule (1) to tune both W and Wo, the memory 204 X Xie and H. S. Seung 111111111111 I ~IIIIIIIII I IUIIIIIIII I /untuned 111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 I I~ ____ ~I~ ____ ~ ______ =-____ _ I I " tuned 1IIIIIIIIIIIIIIIIIIIIIIIIIIINIIIIII'.tl 1 sec 1111111111'"111111111111111111111111111 Figure 3: Untuned and tuned autapse activity. The middle three traces are the membrane potentials of the three input neurons in Figure 2 (spikes are drawn at the reset times of the integrate-and-fire neurons). Before learning, the activity of the memory neuron is not persistent, as shown in the top trace. After the spike-based learning rule (1) is applied to the synaptic weights Wand Wo, then the burst inputs cause persistent changes in activity. em = 1 nF, gL = 0.025 J-lS, VL = -70 mY, VE = 0 mY, VI = -70 mY, vthres = -52 mY, Vreset = -59 mY, a s = 1, Tsyn = 100 ms, Iapp,o = 0.5203 nA, I app,± = 0 or 0.95 nA, Tsyn,O = 100 ms, Tsyn,+ = Tsyn,- = 5 ms, W+ = 0.1, W_ = 0.05. neuron is able to maintain persistent activity. During the interburst intervals (from A after one burst until A before the next), we made synaptic changes using the differential antiHebbian pairing function f(t) = -Asin(7l'tjT) for spike time differences in the range [-T, T] with A = 1.5 X 10-4 and T=A=120 ms. The resulting increase in persistence time can be seen in Figure 4A, along with the values of the synaptic weights versus time. To quantify the performance of the system at maintaining persistent activity, we determined the relationship between dv / dt and v using a long sequence of interburst intervals, where v was defined as the reciprocal of the interspike interval. If Wand Wo are fixed at optimally tuned values, there is still a residual drift, as shown in Figure 4B. But if these parameters are allowed to adapt continuously, even after good tuning has been achieved, then the residual drift is even smaller in magnitude. This is because the learning rule tweaks the synaptic weights during each interburst interval, reducing the drift for that particular firing rate. Autapse learning is driven by the autocorrelation of the spike train, rather than a crosscorrelation. The peak in the autocorrelogram at zero lag has no effect, since the pairing function is zero at the origin. Since the autocorrelation is zero for small time lags, we used a fairly large pairing range in our simulations. In a recurrent network of many neurons, a shorter pairing range would suffice, as the cross-correlation does not vanish near zero. 6 DISCUSSION We have shown that differential anti-Hebbian learning can tune a recurrent circuit to maintain persistent neural activity. This behavior can be understood by reducing the spike-based learning rule (l) to the rate-based learning rules ofEqs. (6) and (8). The rate-based approximations are good if two conditions are satisfied. First, the pairing range must be large, or the rate of learning must be slow. Second, spike synchrony must be weak, or have little effect on learning due to the shape of the pairing function. The differential anti-Hebbian pairing function results in a learning rule that uses -Vi as a negative feedback signal to reduce the amount of drift in firing rate, as illustrated by our simulations of an integrate-and-fire neuron with an excitatory autapse. More generally, the learning rule could be relevant for tuning the strength of positive feedback in networks that maintain a short-term memory of an analog variable in persistent neural activity. Spike-based Learning and Stabilization of Persistent Neural Activity 205 A B 250 c ' 6 I 200 0.395 W WO 0.16 1 4 ~150 2 .~~ ~ 0385 0.12 1 ~ ~r· ~~~ i! :I: 0 §100 0 10 20 ~-2 "me Is) "" I ~~ -4f '0 50 -at 00 ! -8' i 5 10 15 20 25 20 40 60 80 100 tlme(s) rate (Hzl Figure 4: Tuning the autapse. (A) The persistence time of activity increases as the weights Wand Wo are tuned. Each transition is driven by pseudorandom bursts of input (B) Systematic relationship between drift dv/dt in firing rate and v, as measured from a long sequence of interburst intervals. If the weights are continuously fine-tuned ('*') the drift is less than with fixed well-tuned weights ('0'). For example, the learning rule could be used to improve the robustness of the oculomotor integrator[12, 13, 14] and head direction system[l1] to mistuning of parameters. In deriving the differential forms of the learning rules in (8), we assumed that the areas under the positive and negative lobes of the pairing function are equal, so that the integral defining 130 vanishes. In reality, this cancellation might not be exact. Then the ratio of 131 and 130 would limit the persistence time that can be achieved by the learning rule. Both the oculomotor integrator and the head direction system are also able to integrate vestibular inputs to produce changes in activity patterns. The problem of finding generalizations of the present learning rules that train networks to integrate is still open. References [1] H. Markram, J. Lubke, M. Frotscher, and B. Sakmann. Science, 275(5297):213-5, 1997. [2] G. Q. Bi and M. M. Poo. 1 Neurosci, 18(24):10464-72,1998. [3] D. O. Hebb. Organization of behavior. Wiley, New York, 1949. [4] C. C. Bell, V. Z. Han, Y. Sugawara, and K. Grant. Nature, 387(6630):278-81 , 1997. [5] w. Gerstner, R. Kempter, 1. L. van Hemmen, and H. Wagner. Nature, 383(6595):76-81, 1996. [6] L. F. Abbott and S. Song. Adv. Neural Info. Proc. Syst., 11, 1999. [7] P. D. Roberts. 1. Comput. Neurosci., 7:235-246, 1999. [8] R. Kempter, W. Gerstner, and J. L. van Hemmen. Phys. Rev. E, 59(4):4498-4514, 1999. [9] A. P. Georgopoulos, M. Taira, and A. Lukashin. Science, 260:47-52, 1993. [10] M. Camperi and X. J. Wang. 1 Comput Neurosci, 5(4):383-405, 1998. [11] K. Zhang. 1. Neurosci., 16:2112-2126, 1996. [12] S. C. Cannon, D. A. Robinson, and S. Shamma. Bio!. Cybern., 49:127-136,1983. [13] H. S. Seung. Proc. Nat!. A cad. Sci. USA, 93:13339-13344, 1996. [14] H. S. Seung, D. D. Lee, B. Y. Reis, and D. W. Tank. Neuron, 2000. [15] B. Ermentrout. Neural Comput., 6:679-695, 1994. [16] O. Shriki, D. Hansel, and H. Sompolinsky. Soc. Neurosci. Abstr., 24:143, 1998. [17] H. S. Seung, D. D. Lee, B. Y. Reis, and D. W. Tank. 1. Comput. Neurosci., 2000. PART III THEORY
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Nonlinear Discriminant Analysis using Kernel Functions Volker Roth & Volker Steinhage University of Bonn, Institut of Computer Science III Romerstrasse 164, D-53117 Bonn, Germany {roth, steinhag}@cs.uni-bonn.de Abstract Fishers linear discriminant analysis (LDA) is a classical multivariate technique both for dimension reduction and classification. The data vectors are transformed into a low dimensional subspace such that the class centroids are spread out as much as possible. In this subspace LDA works as a simple prototype classifier with linear decision boundaries. However, in many applications the linear boundaries do not adequately separate the classes. We present a nonlinear generalization of discriminant analysis that uses the kernel trick of representing dot products by kernel functions. The presented algorithm allows a simple formulation of the EM-algorithm in terms of kernel functions which leads to a unique concept for unsupervised mixture analysis, supervised discriminant analysis and semi-supervised discriminant analysis with partially unlabelled observations in feature spaces. 1 Introduction Classical linear discriminant analysis (LDA) projects N data vectors that belong to c different classes into a (c - 1)-dimensional space in such way that the ratio of between group scatter SB and within group scatter Sw is maximized [1]. LDA formally consists of an eigenvalue decomposition of Sv) S B leading to the so called canonical variates which contain the whole class specific information in a (c - I)-dimensional subspace. The canonical variates can be ordered by decreasing eigenvalue size indicating that the first variates contain the major part of the information. As a consequence, this procedure allows low dimensional representations and therefore a visualization of the data. Besides from interpreting LDA only as a technique for dimensionality reduction, it can also be seen as a multi-class classification method: the set of linear discriminant functions define a partition of the projected space into regions that are identified with class membership. A new observation x is assigned to the class with centroid closest to x in the projected space. To overcome the limitation of only linear decision functions some attempts have been made to incorporate nonlinearity into the classical algorithm. HASTIE et al. [2] introduced the so called model of Flexible Discriminant Analysis: LDA is reformulated in the framework of linear regression estimation and a generalization of this method is given by using nonlinear regression techniques. The proposed regression techniques implement the idea of using nonlinear mappings to transform the input data into a new space in which again a linear regression is performed. In real world Nonlinear Discriminant Analysis Using Kernel Functions 569 applications this approach has to deal with numerical problems due to the dimensional explosion resulting from nonlinear mappings. In the recent years approaches that avoid such explicit mappings by using kernel functions have become popular. The main idea is to construct algorithms that only afford dot products of pattern vectors which can be computed efficiently in high-dimensional spaces. Examples of this type of algorithms are the Support Vector Machine [3] and Kernel Principal Component Analysis [4]. In this paper we show that it is possible to formulate classical linear regression and therefore also linear discriminant analysis exclusively in terms of dot products. Therefore, kernel methods can be used to construct a nonlinear variant of discriminant analysis. We call this technique Kernel Discriminant Analysis (KDA). Contrary to a similar approach that has been published recently [5J, our algorithm is a real multi-class classifier and inherits from classical LDA the convenient property of data visualization. 2 Review of Linear Discriminant Analysis Under the assumption of the data being centered (i.e. Ei Xi = 0) the scatter matrices S B and Sw are defined by SB = ""~ ~ ""nj (x(j)) (x~))T (1) L...,;J=1 nj L...,;l,m=l I Sw = ""c ""nj (x(j) _ ~ ""nj XU)) (x(j) _ ~ ""nj x(j)) T (2) L...,;j=l L...,;l=l I nj L...,;l=l I I nj L...,;m=l m where nj is the number of patterns x~j) that belong to class j. LDA chooses a transformation matrix V that maximizes the objective function J(V) = IVTSB VI. (3) IVTSwVI The columns of an optimal V are the generalized eigenvectors that correspond to the nonzero eigenvalues in SBVi = Ai SWVi' In [6J and [7J we have shown, that the standard LDA algorithm can be restated exclusively in terms of dot products of input vectors. The final equation is an eigenvalue equation in terms of dot product matrices which are of size N x N. Since the solution of high-dimensional generalized eigenvalue equations may cause numerical problems (N may be large in real world applications), we present an improved algorithm that reformulates discriminant analysis as a regression problem. Moreover, this version allows a simple implementation of the EM-algorithm in feature spaces. 3 Linear regression analysis In this section we give a brief review of linear regression analysis which we use as "building block" for LDA. The task of linear regression analysis is to approximate the regression function by a linear function r(x) = E(YIX = x) ~ c + xT f'. (4) on the basis of a sample (YI, Xl), ... ,(Y N , x N ). Let now y denote the vector (YI, ... ,YN)T and X denote the data matrix which rows are the input vectors. U sing a quadratic loss function, the optimal parameters c and f' are chosen to minimize the average squared residual ASR = N-Illy - c IN + Xf'112 + f'Tnf'. (5) IN denotes a N-vector of ones, n denotes a ridge-type penalty matrix n = €I which penalizes the coefficients of f'. Assuming the data beirig centered, i.e E~l Xi = 0, the parameters of the regression function are given by: N c = N- 1 "". Yi =: /-Ly , f' = (XT X + €I)-l XT y. (6) L...,;t=l 570 V. Roth and V. Steinhage 4 LDA by optimal scoring In this section the LDA problem is linked to linear regression using the framework of penalized optimal scoring. We give an overview over the detailed derivation in [2] and [8]. Considering again the problem with c classes and N data vectors, the class-memberships are represented by a categorical response variable 9 with c levels. It is useful to code the n responses in terms of the indicator matrix Z: Zi,j = 1, if the i-th data vectJr belongs to class j, and 0 otherwise. The point of optimal scoring is to turn categorical variables into quantitative ones by assigning scores to classes: the score vector 9 assigns the real number 9j to the j-th level of g. The vector Z9 then represents a vector of scored training data and is regressed onto the data matrix X. The simultaneous estimation of scores and regression coefficients constitutes the optimal scoring problem: minimize the criterion ASR(9, (3) = N- 1 [IIZ9 - X{311 2 + {3TO{3] (7) under the constraint ~ IIZ9UZ = 1. According to (6), for a given score 9 the minimizing {3 is given by {3os = (XT X + 0)-1 xT Z(J, (8) and the partially minimized criteri9n becomes: minASR(9,{3) = 1- N-19TZ™(0)Z9, (3 (9) where M(O) = X(XTX +O)-IXT denotes the regularized hat or smoother matrix. Minimizing of (9) under the constraint ~ IIZ9W = 1 can be performed by the following procedure: 1. Choose an initial matrix 8 0 satisfying the constraint N- 18'{; ZT Z80 = I and set 8 0 = Z80 ~ 2. Run a multi-response regression of 80 onto X: 80 = M(0)80 = XB, where B is the ma.1rix of regression coefficients. 3. Eigenanalyze 8 0T 8 0 to obtain the optimal scores, and update the matrix of regression coefficients: B* = BW, with W being the matrix of eigenvectors. It can be shown, that the final matrix B* is, up to a diagonal scale matrix, equivalent to the matrix of LDA-vectors, see [8]. 5 Ridge regression using only dot products The penalty matrix 0 in (5) assures that the penalized d x d covariance matrix i: = XT X + d is a symmetric nonsingular matrix. Therefore, it has d eigenvectors ei with accomplished positive eigenvalues Ii such that the following equations hold: - -1 "d 1 T ~ = 6 -eie· i=1 Ii t (10) The first equation implies that the first 1 leading eigenvectors ei with eigenvalues Ii > € have an expansion in terms of the input vectors. Note that 1 is the number of nonzero eigenvalues of the unpenalized covariance matrix X T X. Together with (6), it follows for the general case, when the dimensionality d may extend l, that {3 can be written as the sum of two terms: an expansion in terms of the vectors Xi with coefficients ai and a similar expansion in terms of the remaining eigenvectors: I:N I:d T I:d {3 = . aixi + . ~jej = X a + . ~jej , t=1 )=1+1 )=1+1 (11) with a = (a1 ... an) T. However, the last term can be dropped, since every eigenvector ej, j = 1 + 1, ... ,d is orthogonal to every vector Xi and does not influence the value of the regression function (4). The problem of penalized linear regression can therefore be stated as minimizing Nonlinear Discriminant Analysis Using Kernel Functions 571 ASR(a) = N- 1 [Ily - XXT al1 2 + aTXOXTaJ. (12) A stationary vector a is determined by a = (XXT + O)-ly. (13) Let now the dot product matrix K be defined by Kij = xT Xj and let for a given test point (Xl) the dot product vector kl be defined by kl = XXI . With this notation the regression function of a test point (xL) reads T( )-1 r(Xl) = /-Ly + kl K + €I y . (14) This equation requires only dot products and we can apply the kernel trick. The final equation (14), up to the constant term /-Ly , has also been found by SAUNDERS et al., [9J. They restated ridge regression in dual variables and optimized the resulting criterion function with a lagrange multiplier technique. Note that our derivation, which is a direct generalization of the standard linear regression formalism, leads in a natural way to a class of more general regression functions including the constant term. 6 LDA using only dot products Setting f3 = XT a as in (11) and using the notation of section 5, for a given score o the optimal vector a is given by: aas = (XXT + 0)-1 ZO. (15) Analogous to (9), the partially minimized criterion becomes: min ASR(O, a) = 1 - N- 10T ZT M(O)ZO, ex with M(O) = XXT(XXT + 0)-1 = K(K + €I)-l. (16) To minimize (16) under the constraint tv IIZOW = 1 the procedure described in section 4 can be used when M(O) is substituted by M(O). The matrix Y which rows are the input vectors projected onto the column vectors of B* is given by: Y = XB* = K(K + €I)-l Z8oW. (17) Note that again the dot product matrix K is all that is needed to calculate Y. 7 The kernel trick The main idea of constructing nonlinear algorithms is to apply the linear methods not in the space of observations but in a feature space F that is related to the former by a nonlinear mapping ¢ : RN ---+ F, X ---+ ¢(x) . Assuming that the mapped data are centered in F, i.e. L~=l ¢(Xi) = 0, the presented algorithms remain formally unchanged if the dot product matrix K is computed in F: Kij = (¢(Xi) . ¢(Xj)). As shown in [4], this assumption can be dropped by writing ¢ instead of the mapping ¢: ¢(Xi) := ¢(Xi) ~ L~=l ¢(Xi). Computation of dot products in feature spaces can be done efficiently by using kernel functions k(xi, Xj) [3]: For some choices of k there exists a mapping ¢ into some feature space F such that k acts as a dot product in F. Among possible kernel functions there are e.g. Radial Basis Function (RBF) kernels of the form k(x,y) = exp(-llx - YW/c). 8 The EM-algorithm in feature spaces LDA can be derived as the maximum likelihood method for normal populations with different means and common covariance matrix ~ (see [11]). Coding the class membership of the observations in the matrix Z as in section 4, LDA maximizes the (complete data) log-likelihood function 572 V. Roth and V. Steinhage This concept can be generalized for the case that only the group membership of Nc < N observations is known ([14], p.679): the EM-algorithm provides a convenient method for maximizing the likelihood function with missing data: E-step: set Pki = Prob(xi E class k) { Zik' if the class membership of Xi has been observed Pki = 1Tk ¢>" (z.) h· '" () [1/2( )T~-l ( )] Lk=l 1Tk¢>"(Z;) ' ot erWlse, 'f'k Xi ex exp Xi ILk L.; Xi ILk M-step: set 1 N 'irk = N LPki' i=l The idea behind this approach is that even an unclassified observation can be used for estimation if it is given a proper weight according to its posterior probability for class membership. The M-step can be seen as weighted mean and covariance maximum likelihood estimates in a weighted and augmented problem: we augment the data by replicating the N observations c times, with the l-th such replication having observation weights Plio The maximization of the likelihood function can be achieved via a weighted and augmented LDA. It turns out that it is not necessary to explicitly replicate the observations and run a standard LDA: the optimal scoring version of LDA described in section 4 allows an implicit solution of the augmented problem that still uses only N observations. Instead of using a response indicator matrix Z, one uses a blurred response Matrix Z, whose rows consist of the current class probabilities for each observation. At each M-step this Z is used in a multiple linear regression followed by an eigen-decomposition. A detailed derivation is given in [11]. Since we have shown that the optimal scoring problem can be solved in feature spaces using kernel functions this is also the case for the whole EM-algorithm: the E-step requires only differences in Mahalonobis distances which are supplied by KDA. After iterated application of the E- and M-step an observation is classified to the class k with highest probability Pk. This leads to a unique framework for pure mixture analysis (Nc = 0), pure discriminant analysis (Nc = N) and the semisupervised models of discriminant analysis with partially unclassified observations (0 < Nc < N) in feature spaces. 9 Experiments Waveform data: We illustrate KDA on a popular simulated example, taken from [10], pA9-55 and used in [2, 11]. It is a three class problem with 21 variables. The learning set consisted of 100 observations per class. The test set was of size 1000. The results are given in table 1. Table 1: Results for waveform data. The values are averages over 10 simulations. The 4 entries above the line are taken from [11]. QDA: quadratic discriminant analysis, FDA: flexible discriminant analysis, MDA: mixture discriminant analysis. Technique Training Error [%] Test Error [%] LDA 12.1(0.6) 19.1(0.6) QDA 3.9(OA) 20.5(0.6) FDA (best model parameters) 10.0(0.6) 19.1(0.6) MDA (best model parameters) 13.9{0.5) 15.5(0.5) KDA (RBF kernel, (7 = 2, € = 1.5) 10.7(0.6) 14.1(0.7) Nonlinear Discriminant Analysis Using Kernel Functions 573 The Bayes risk for the problem is about 14% [10]. KDA outperforms the other nonlinear versions of discriminant analysis and reaches the Bayes rate within the error bounds, indicating that one cannot expect significant further improvement using other classifiers. Figure 1 demonstrates the data visualization property of KDA. Since for a 3 class problem the dimensionality of the projected space equals 2, the data can be visualized without any loss of information. In the left plot one can see the projected learn data and the class centroids, the right plot shows the test data and again the class centroids of the learning set. Figure 1: Data visualization with KDA. Left: learn set, right: test set To demonstrate the effect of using unlabeled data for classification we repeated the experiment with waveform data using only 20 labeled observations per class. We compared the the classification results on a test set of size 300 using only the labeled data (error rate E 1 ) with the results of the EM-model which considers the test data as incomplete measurements during an iterative maximization of the likelihood function (error rate E2). Using a RBF kernel (0" = 250), we obtained the following mean error rates over 20 simulations: El = 30.5(3.6)%, E2 = 17.1(2.7)%. The classification performance could be drastically improved when including the unlabelled data into the learning process. Object recognition: We tested KDA on the MPI Chair Database l . It consists of 89 regular spaced views form the upper viewing hemisphere of 25 different classes of chairs as a training set and 100 random views of each class as a test set. The available images are downscaled to 16 x 16 pixels. We did not use the additional 4 edge detection patterns for each view. Classification results for several classifiers are given in table 2. KDA poly. kernel 2.1 For a comparison of the computational performance we also trained the SVM-light implementation (V 2.0) on the data, [13]. In this experiment with 25 classes the KDA algorithm showed to be Significantly faster than the SVM: using the RBFkernel, KDA was 3 times faster, with the polynomial kernel KDA was 20 times faster than SVM-light. 10 Discussion In this paper we present a nonlinear version of classical linear discriminant analysis. The main idea is to map the input vectors into a high- or even infinite dimensional feature space and to apply LDA in this enlarged space. Restating LDA in a way that only dot products of input vectors are needed makes it possible to use kernel representations of dot products. This overcomes numerical problems in high-dimensional IThe database is available via ftp:/ /ftp.mpik-tueb.mpg.de/pub/chair_dataset/ 574 V. Roth and V. Steinhage feature spaces. We studied the classification performance of the KDA classifier on simulated waveform data and on the MPI chair database that has been widely used for benchmarking in the literature. For medium size problems, especially if the number of classes is high, the KDA algorithm showed to be significantly faster than a SVM while leading to the same classification performance. From classical LDA the presented algorithm inherits the convenient property of data visualization, since it allows low dimensional views of the data vectors. This makes an intuitive interpretation possible, which is helpful in many practical applications. The presented KDA algorithm can be used as the maximization step in an EM algorithm in feature spaces. This allows to include unlabeled observation into the learning process which can improve classification results. Studying the performance of KDA for other classification problems as well as a theoretical comparison of the optimization criteria used in the KDA- and SVM-algorithm will be subject of future work. Acknowledgements This work was supported by Deutsche Forschungsgemeinschaft, DFG. We heavily profitted from discussions with Armin B. Cremers, John Held and Lothar Hermes. References [1] R. Duda and P. Hart, Pattern Classification and Scene Analysis. Wiley & Sons, 1973. [2] T. Hastie, R. Tibshirani, and A. Buja, "Flexible discriminant analysis by optimal scoring," JASA, vol. 89, pp. 1255-1270, 1994. [3] V. N. Vapnik, Statistical learning theory. Wiley & Sons, 1998. [4] B. Sch6lkopf, A. Smola, and K.-R. Muller, "Nonlinear component analysis as a kernel eigenvalue problem," Neural Computation, vol. 10, no. 5, pp. 1299-1319, 1998. [5] S. Mika, G. Ratsch, J. Weston, B. Sch6lkopf, and K.-R. Miiller, "Fisher discriminant analysis with kernels," in Neural Networks for Signal Processing IX (Y.-H. Hu, J. Larsen, E. Wilson, and S. Douglas, eds.), pp. 41-48, IEEE, 1999. [6] V. Roth and V. Steinhage, "Nonlinear discriminant analysis using kernel functions," Tech. Rep. IAI-TR-99-7, Department of Computer Science III, Bonn University, 1999. [7] V. Roth, A. Pogoda, V. Steinhage, and S. Schroder, "Pattern recognition combining feature- and pixel-based classification within a real world application," in Mustererkennung 1999 (W. Forstner, J. Buhmann, A. Faber, and P. Faber, eds.), Informatik aktuell, pp. 120-129, 21. DAGM Symposium, Bonn, Springer, 1999. [8] T. Hastie, A. Buja, and R. Tibshirani, "Penalized discriminant analysis," AnnStat, vol. 23, pp. 73-102, 1995. [9] S. Saunders, A. Gammermann, and V. Vovk, "Ridge regression learning algorithm in dual variables," tech. rep., Royal Holloway, University of London, 1998. [10] L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone, Classification and Regression Trees. Monterey, CA: Wadsworth and Brooks/Cole, 1984. [11] T. Hastie and R. Tibshirani, "Discriminant analysis by gaussian mixtures," JRSSB, vol. 58, pp. 158-176, 1996. [12] B. Scholkopf, Support Vector Learning. PhD thesis, 1997. R. Oldenbourg Verlag, Munich. [13] T. Joachims, "Making large-scale svm learning practical," in Advances in Kernel Methods - Support Vector Learning (B. Scholkopf, C. Burges, and A. Smola, eds.), MIT Press, 1999. [14] B. Flury, A First Course in Multivariate Statistics. Springer, 1997.
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Support Vector Method for Multivariate Density Estimation Vladimir N. Vapnik Royal Halloway College and AT &T Labs, 100 Schultz Dr. Red Bank, NJ 07701 vlad@research.att.com Abstract Sayan Mukherjee CBCL, MIT E25-201 Cambridge, MA 02142 sayan@ai.mit.edu A new method for multivariate density estimation is developed based on the Support Vector Method (SVM) solution of inverse ill-posed problems. The solution has the form of a mixture of densities. This method with Gaussian kernels compared favorably to both Parzen's method and the Gaussian Mixture Model method. For synthetic data we achieve more accurate estimates for densities of 2, 6, 12, and 40 dimensions. 1 Introduction The problem of multivariate density estimation is important for many applications, in particular, for speech recognition [1] [7]. When the unknown density belongs to a parametric set satisfying certain conditions one can estimate it using the maximum likelihood (ML) method. Often these conditions are too restrictive. Therefore, non-parametric methods were proposed. The most popular of these, Parzen's method [5], uses the following estimate given data Xl, ... , Xl: (1) where K'Yl(t - Xi) is a smooth function such that J K'Yl(t - xi)dt = 1. Under some conditions on "Yl and K'Yl (t - Xi), Parzen's method converges with a fast asymptotic rate. An example of such a function is a Gaussian with one free parameter "Y; (the width) The structure of the Parzen estimator is too complex: the number of terms in (1) is equal to the number of observations (which can be hundreds of thousands). 660 V. N. Vapnik and S. Mukherjee Researchers believe that for practical problems densities can be approximated by a mixture with few elements (Gaussians for Gaussian Mixture Models (GMM)). Therefore, the following parametric density model was introduced m P(x, a,~) = 2: (}:iP(X, ai, ~i)' i=l m (}: ~ 0, 2: (}:i = 1, (3) i=l where P(x, ai, ~i) are Gaussians with different vectors ai and different diagonal covariance matrices ~i; (}:i is the proportion of the i-th Gaussian in the mixture. It is known [9] that for general forms of Gaussian mixtures the ML estimate does not exist. To use the ML method two values are specified: a lower bound on diagonal elements of the covariance matrix and an upper bound on the number of mixture elements. Under these constraints one can estimate the mixture parameters using the EM algorithm. This solution, however, is based on predefined parameters. In this article we use an SVM approach to obtain an estimate in the form of a mixture of densities. The approach has no free parameters. In our experiments it performs better than the GMM method. 2 Density estimation is an ill-posed problem A density p(t) is defined as the solution of the equation i~ p(t) dt = F(x), (4) where F(x) is the probability distribution function. Estimating a density from data involves solving equation (4) on a given set of densities when the distribution function F(x) is unknown but a random i.i.d. sample Xl, ... , Xe is given. The empirical distribution function Fe(x) is a good approximation of the actual distribution, where O( u) is the step-function. In the univariate case, for sufficiently large l the distribution of the supremum error between F(x) and Ft(x) is given by the Kolmogorov-Smirnov distribution P{sup IF(x) - Fe(x)1 < c/Vi} = 1- 22:( _1)k-1 exp{ -2c2k2 }. x (5) k=l Hence, the problem of density estimation can be restated as solving equation (4) but replacing the distribution function F(x) with the empirical distribution function Fe(x) which converges to the true one with the (fast) rate 0(1)' for univariate and multivariate cases. The problem of solving the linear operator equation Ap = F with approximation Ft(x) is ill-posed. In the 1960's methods were proposed for solving ill-posed problems using approximations Ft converging to F as l increases. The idea of these methods was to Support Vector Method for Multivariate Density Estimation 661 introduce a regularizing functional O(P) (a semi-continuous, positive functional for which O(p) ~ c, c > 0 is a compactum) and define the solution Pt which is a trade-off between O(p) and IIAp - Ftll. The following two methods which are asymptotically equivalent [11] were proposed by Tikhonov [8] and Phillips [6] min [IIAp - Ftll 2 + ItO(P)] , It > 0, It -+ 0, p minO(p) s.t. IIAp - Fill < et, et > 0, et -+ O. p (6) (7) For the stochastic case it can be shown for both methods that if Ft(x) converges in probability to F(x) and Ii -+ 0 then for sufficiently large f and arbitrary 1/ and J.L the following inequality holds [10] [9] [3] (8) where f > fO(1/, J.L) and PEl (p,Pi), PE2(F, Fi ) are metrics in the spaces p and F. Since Fi(X) -+ F(x) in probability with the rate O( ~), from equation (8) it follows that if Ii > O( ~) the solutions of equation (4) are consistent. 3 Choice of regularization parameters For the deterministic case the residual method [2] can be used to set the regularization parameters (,i in (6) and ei in (7)) by setting the parameter (,i or ei) such that Pi satisfies the following IIApi - Fill = IIF(x) - Fi(X) II = Ui, (9) where Ui is the known accuracy of approximation of F(x) by Fi(X). We use this idea for the stochastic case. The Kolmogorov-Smirnov distribution is used to set Ui, Ui = c/...[i, where c corresponds to an appropriate quantile. For the multivariate case one can either evaluate the appropriate quantile analytically [4] or by simulations. The density estimation problem can be solved using either regularization method (6) or (7). Using method (6) with a L2 norm in image space F and regularization functional O(p) = (Tp, Tp) where T is a convolution operator, one obtains Parzen's method [10] [9] with kernels defined by operator T 4 SVM for density estimation We apply the SVM technique to equation (7) for density estimation. We use the C norm in (7) and solve equation (4) in a set of functions belonging to a Reproducing Kernel Hilbert Space (RKHS). We use the regularization functional O(P) = IIplit = (p,p)1i. (10) A RKHS can be defined by a positive definite kernel K (x, y) and an inner product (f, g)1i in Hilbert space 1-l such that (f(x),K(x,Y))1i = f(y) "If E 1-l. (11) 662 V. N. Vapnik and S. Mukherjee Note that any positive definite function K(x,y) has an expansion 00 K(x,y) = LAi<!)i(x)<Pi(Y) (12) i=l where Ai and <Pi(X) are eigenvalues and eigenfunctions of K(x, y). Consider the set of functions 00 f(x,c) = LCi<Pi(X) i=l and the inner product 00 * ** (f( *) f( **) ""' ci Ci X,C , X,C = ~~. i=l t Kernel (12), inner product (14), and set (13) define a RKHS and For functions from a RKHS the functional (10) has the form 00 2 ""' C· O(p) = ~ At. , i=l t (13) (14) (15) where Ai is the i-th eigenvalue of the kernel K(x, y). The choice of the kernel defines smoothness properties on the solution. To use method (7) to solve for the density in equation (4) in a RKHS with a solution satisfying condition (9) we minimize O(p) = (p,p)ll subject to the constraint We look for a solution of equation (4) with the form l p(t) = L fiiKl'l (Xi, t). (16) i=l Accounting for (16) and (11) minimizing (10) is equivalent to minimizing l O(p,p) = (p,p)ll = L fiifijKl'l(Xi,Xj) (17) i,j=l Support Vector Method for Multivariate Density Estimation 663 subject to constraints (18) i f3i ~ 0, I:f3i = 1. (19) i=1 This optimization problem is closely related to the SV regression problem with an O"i-insensitive zone [9]. It can be solved using the standard SVM technique. Generally, only a few of the f3i will be nonzero, the Xi corresponding to these f3i are called support vectors. Note that kernel (2) has width parameter 'Yi. We call the value of this parameter admissible if it satisfies constraint (18) (the solution satisfies condition (9)). The admissible set 'Ymin :::; 'Yi :::; 'Ymax is not empty since for Parzen's method (which also has form (16)) such a value does exist. Among the 'Yi in this admissible set we select the one for which O(P) is smallest or the number of support vectors is minimum. Choosing other kernels (for example Laplacians) one can estimate densities using non-Gaussian mixture models which for some problems are more appropriate [1]. 5 Experiments Several trials of estimates constructed from sampling distributions were examined. Boxplots were made of the L1 (p) norm over the trials. The horizontal lines of the boxplot indicate the 5%, 25%, 50%, 75%, and 95% quantiles of the error distribution. For the SVM method we set O"i = c/V'i, where c = .36, .41, .936, and 1.75 for two, six, twelve and forty dimensions. For Parzen's method 'Yi was selected using a leave-one out procedure. The GMM method uses the EM algorithm and sets all parameters except n, the upper bound on the number of terms in the mixture [7]. Figure (1) shows plots of the SVM estimate using a Gaussian kernel and the GMM estimate when 60 points were drawn form a mixture of a Gaussian and Laplacian in two dimensions. Figure (2a) shows four boxplots of estimating a density defined by a mixture of two Laplacians in a two dimensional space using 200 observations. Each boxplot shows outcomes of 100 trials: for the SVM method, Parzen's method, and the GMM method with parameters n = 2, and n = 4. Figure (2c) shows the distribution of the number of terms for the SVM method. Figure (2b) shows boxplots of estimating a density defined by the mixture of four Gaussians in a six dimensional space using 600 observations. Each boxplot shows outcomes of 50 trials: for the SVM method, Parzen's method, and the G MM method with parameters n = 4, and n = 8. Figure (2c) shows the distribution of the number of terms for the SVM method. Figure (3a) shows boxplots of outcomes of estimating a density defined by the mixture of four Gaussians and four Laplacians in a twelve dimensional space using 664 V. N. Vapnik and S. Mukherjee 400 observations. Each boxplot shows outcomes of 50 trials: for the SVM method, Parzen's method, and the GMM method with parameter n = 8. Figure (3c) shows the distribution of the number of terms for the SVM method. Figure (3b) shows boxplots of outcomes of estimating a density defined by the mixture of four Gaussians and four Laplacians in a forty dimensional space using 480 observations. Each box-plot shows outcomes of 50 trials: for the SVM method, Parzen's method, and the GMM method with parameter n = 8. Figure (3c) shows the distribution of the number of terms for the SVM method. 6 Summary A method for multivariate density estimation based on the SVM technique for solving ill-posed problems is introduced. This method has a form of a mixture of densities. The estimate in general has only a few terms. In experiments on synthetic data this method is more accurate than the GMM method. References [1] S. Basu and C.A. Micchelli. Parametric density estimation for the classification of acoustic feature vectors in speech recognition. In Nonlinear Modeling, Advanced Black-Box Techniques. Kluwer Publishers, 1998. [2] V.A. Morozov. Methods for solving incorrectly posed problems. Springer-Verlag, Berlin, 1984. [3] S. Mukherjee and V. Vapnik. Multivariate density estimation: An svm approach. AI Memo 1653, Massachusetts Institute of Technology, 1999. [4] S. Paramasamy. On multivariate kolmogorov-smirnov distribution. Statistics fj Probability Letters, 15:140-155, 1992. [5] E. Parzen. On estimation of a probability density function and mode. Ann. Math. Statis. , 33:1065-1076, 1962. [6] D.L. Phillips. A technique for the numerical solution of integral equations of the first kind. J.Assoc. Comput. Machinery, 9:84- 97, 1962. [7] D. Reynolds and R. Rose. Robust text-independent speaker identification using gaussian mixture speaker models. IEEE Trans on Speech and Audio Processing, 3(1):1-27, 1995. [8] A. N. Tikhonov. Solution of incorrectly formulated problems and the regularization method. Soviet Math. Dokl., 4:1035-1038, 1963. [9] V. N. Vapnik. Statistical learning theory. J. Wiley, 1998. [10] V.N. Vapnik and A.R. Stefanyuk. Nonparametric methods for restoring probability densities. Avtomatika i Telemekhanika, (8):38-52, 1978. [11] V.V. Vasin. Relationship of several variational methods for the approximate solution of ill-posed problems. Math Notes, 7:161- 166, 1970. Support Vector Method/or Multivariate Density Estimation 665 , . -r ~ ,··-r· .~ . (a) (b) (c) Figure 1: (a) The true distribution (b) the GMM case with 4 mixtures (c) the Parzen case (d) the SVM case for 60 points. 0.16 X 10-3 --,2.1 0.14 --,-0.12 , 0 --r 1.9 ~ e 0.1 8 I .S ~, ~ :::; 15 ~O.08 8 1.7 ~ ~ B z 0.06 El9 1.6 , 0.04 + ~ 1.5 $ 1.4 -+~ 0.02 SVM Parzen GMM2 GMM4 SVM Parzen GMM4 GMMS 2-"'" 6 dirnll'l!ilQ(ls (a) (b) (c) Figure 2: ( a) Boxplots of the L1 (P) error for the mixture of two Laplacians in two dimensions for the SVM method, Parzen's method, and the GMM method with 2 and 4 Gaussians. (b) Boxplots of the L 1(P) error for mixture of four Gaussians in six dimensions with the SVM method, Parzen's method, and the GMM method with 4 mixtures. (c) Boxplots of distribution of the number of terms for the SVM method for the two and six dimensional cases. 0.16r-~---------' 0.14 0.12 g 0.1 & ~O .08 0.08 0.04 SVM Parzen GMMS (a) 0.12 0.1 ~0 .08 ~ ~O.oe 0.04 0.02 B 25 -- SVM Parzen GMMS 12 Dim 40 Dim (b) (c) Figure 3: (a) Boxplots of the Ll (p) error for the mixture of four Laplacians and four Gaussians in twelve dimensions for the SVM method, Parzen's method, and the GMM method with 8 Gaussians. (b) Boxplots of the L1 (P) error for the mixture of four Laplacians and four Gaussians in forty dimensions for the SVM method, Parzen's method, and the GMM method with 8 Gaussians. (c) Boxplots of distribution of the number of terms for the SVM method for the twelve and forty dimensional cases.
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On input selection with reversible jump Markov chain Monte Carlo sampling Peter Sykacek Austrian Research Institute for Artificial Intelligence (OFAI) Schottengasse 3, A-10lO Vienna, Austria peter@ai. univie. ac. at Abstract In this paper we will treat input selection for a radial basis function (RBF) like classifier within a Bayesian framework. We approximate the a-posteriori distribution over both model coefficients and input subsets by samples drawn with Gibbs updates and reversible jump moves. Using some public datasets, we compare the classification accuracy of the method with a conventional ARD scheme. These datasets are also used to infer the a-posteriori probabilities of different input subsets. 1 Introduction Methods that aim to determine relevance of inputs have always interested researchers in various communities. Classical feature subset selection techniques, as reviewed in [1], use search algorithms and evaluation criteria to determine one optimal subset. Although these approaches can improve classification accuracy, they do not explore different equally probable subsets. Automatic relevance determination (ARD) is another approach which determines relevance of inputs. ARD is due to [6] who uses Bayesian techniques, where hierarchical priors penalize irrelevant inputs. Our approach is also "Bayesian": Relevance of inputs is measured by a probability distribution over all possible feature subsets. This probability measure is determined by the Bayesian evidence of the corresponding models. The general idea was already used in [7] for variable selection in linear regression mo.dels. Though our interest is different as we select inputs for a nonlinear classification model. We want an approximation of the true distribution over all different subsets. As the number of subsets grows exponentially with the total number of inputs, we can not calculate Bayesian model evidence directly. We need a method that samples efficiently across different dimensional parameter spaces. The most general method that can do this is the reversible jump Markov chain Monte Carlo sampler (reversible jump Me) recently proposed in [4]. The approach was successfully applied by [8] to determine a probability distribution in a mixture density model with variable number of kernels and in [5] to sample from the posterior of RBF regression networks with variable number of kernels. A Markov chain that switches between different input subsets is useful for two tasks: Counting how often a particular subset was visited gives us a relevance measure of the corresponding inputs; For classification, we approximate On Input Selection with Reversible Jump MCMC 639 the integral over input sets and coefficients by summation over samples from the Markov chain. The next sections will show how to implement such a reversible jump MC and apply the proposed algorithm to classification and input evaluation using some public datasets. Though the approach could not improve the MLP-ARD scheme from [6] in terms of classification accuracy, we still think that it is interesting: We can assess the importance of different feature subsets which is different than importance of single features as estimated by ARD. 2 Methods The classifier used in this paper is a RBF like model. Inference is performed within a Bayesian framework. When conditioning on one set of inputs, the posterior over model parameters is already multimodal. Therefore we resort to Markov chain Monte Carlo (MCMC) -sampling techniques to approximate the desired posterior over both model coefficients and feature subsets. In the next subsections we will propose an appropriate architecture for the classifier and a hybrid sampler for model inference. This hybrid sampler consists of two parts: We use Gibbs updates ([2]) to sample when conditioning on a particular set of inputs and reversible jump moves that carry out dimension switching updates. 2.1 The classifier I~ order to allow input relevance determination by Bayesian model selection, the classifier needs at least one coefficient that is associated with each input: Roughly speaking, the probability of each model is proportional to the likelihood of the most probable coefficients, weighted by their posterior width divided by their prior width. The first factor always increases when using more coefficients (or input features). The second will decrease the more inputs we use and together this gives a peak for the most probable model. A classifier that satisfies these constraints is the so called classification in the sampling paradigm. We model class conditional densities and together with class priors express posterior probabilities for classes. In neural network literature this approach was first proposed in [10). We use a model that allows for overlapping class conditional densities: D K p(~lk) = L WkdP(~I~) , p(~) = L PkP(~lk) (1) d=l k=l Using Pk for the J{ class priors and p(~lk) for the class conditional densities, (1) expresses posterior probabj,Jities for classes as P(kl~) = PkP(~lk)/p(~). We choose the component densities, p(~IcI> d), to be Gaussian with restricted parametrisation: Each kernel is a multivariate normal distribution with a mean and a diagonal covariance matrix. For all Gaussian kernels together, we get 2 * D * I parameters, with I denoting the current input dimension and D denoting the number of kernels. Apart from kernel coefficients, cI>d , (1) has D coefficients per class, Wkd, indicating the prior kernel allocation probabilities and J{ class priors. Model (1) allows to treat labels of patterns as missing data and use labeled as well as unlabeled data for model inference. In this case training is carried out using the likelihood of observing inputs and targets: p(T, X18) = rrr;=lrr;::=lPkPk(~nk Ifu)rr~=lp(bnI8) , (2) where T denotes labeled and X unlabeled training data. In (2) 8 k are all coefficients the k-th class conditional density depends on. We further use 8 for all model 640 P. Sykacek coefficients together, nk as number of samples belonging to class k and m as index for unlabeled samples. To make Gibbs updates possible, we further introduce two latent allocation variables. The first one, d, indicates the kernel number each sample was generated from, the second one is the unobserved class label c, introduced for unlabeled data. Typical approaches for training models like (1), e.g. [3] and [9], use the EM algorithm, which is closely related to the Gibbs sampler introduce in the next subsection. 2.2 Fixed dimension sampling In this subsection we will formulate Gibbs updates for sampling from the posterior when conditioning on a fixed set of inputs. In order to allow sampling from the full conditional, we have to choose priors over coefficients from their conjugate family: • Each component mean, !!!d, is given a Gaussian prior: !!!d '" Nd({di). • The inverse variance of input i and kernel d gets a Gamma prior: u;;l '" r( a, ,Bi). • All d variances of input i have a common hyperparameter, ,Bi, that has itself a Gamma hyperprior: ,Bi ,...., r(g, hi). • The mixing coefficients, ~, get a Dirichlet prior: ~ '" 1J (6w , ... , 6w ). • Class priors, P, also get a Dirichlet prior: P '" 1J(6p , ... ,6p). The quantitative settings are similar to those used in [8]: Values for a are between 1 and 2, g is usually between 0.2 and 1 and hi is typically between 1/ Rr and 10/ Rr, with Ri denoting the i'th input range. The mean gets a Gaussian prior centered at the midpoint, e, with diagonal inverse covariance matrix ~, with "'ii = 1/ Rr. The prior counts dw and 6p are set to 1 to give the corresponding probabilities non-informative proper Dirichlet priors. The Gibbs sampler uses updates from the full conditional distributions in (3). For notational convenience we use ~ for the parameters that determine class conditional densities. We use m as index over unlabeled data and Cm as latent class label. The index for all data is n, dn are the latent kernel allocations and nd the number of samples allocated by the d-th component. One distribution does not occur in the prior specification. That is Mn(l, ... ) which is a multinomial-one distribution. Finally we need some counters: ml ... mK are the counts per class and mlk .. mDk count kernel allocations of class-k-patterns. The full conditional of the d-th kernel variances and the hyper parameter ,Bi contain i as index of the input dimension. There we express each u;J separately. In the expression of the d-th kernel mean, I On Input Selection with Reversible Jump MCMC illd, we use .lGt to denote the entire covariance matrix. p(~J .. ) p(~I···) p(PI···) p(illdl···) ( { PkP(~mlfu) }) Mn 1, I:k PkP(~mlfu)' k = l..K Mn (1, { WtndP(~nl~) ,d= l..D}) I:, Wt,.dP(~nl~) r (9 + Da. hi + ;; ud,! ) 1) (ow + mlk, ... ,ow + mDk) 1) (op + ml, ... , op + mK) N ((nd~l + ~)-l(ndVdl~ + ~S), (ndVd 1 + ~)-l) r (a + ~d, f3i + ~ L (~n,i -llid,i)2) i£,. Vnld,.=d 2.3 Moving between different input subsets 641 (3) The core part of this sampler are reversible jump updates, where we move between different feature subsets. The probability of a feature subset will be determined by the corresponding Bayesian model evidence and by an additional prior over number of inputs. In accordance with [7J, we use the truncated Poisson prior: p(I) = 1/ ( Ijax ) c ~~ , where c is a constant and Imax the total nr. of inputs. Reversible jump updates are generalizations of conventional Metropolis-Hastings updates, where moves are bijections (x, u) H (x', u'). For a thorough treatment we refer to [4J. In order to switch subsets efficiently, we will use two different types of moves. The first consist of a step where we add one input chosen at random and a matching step that removes one randomly chosen input. A second move exchanges two inputs which allows "tunneling" through low likelihood areas. Adding an input, we have to increase the dimension of all kernel means and diagonal covariances. These coefficients are drawn from their priors. In addition the move proposes new allocation probabilities in a semi deterministic way. Assuming the ordering, Wk,d ~ Wk,d+1: op Beta(ba , bb + 1) Vd ~ D/2 { W~'D+l-d = Wk,D+l-d + Wk ,dOp w~ , d = wk,d(1 - op) (4) The matching step proposes removing a randomly chosen input. Removing corresponding kernel coefficients is again combined with a semi deterministic proposal of new allocation probabilities, which is exactly symmetric to the proposal in (4). 642 P. Sykacek Table 1: Summary of experiments Data avg(#) max(#) RBF (%,n a ) MLP (%,nb) Ionosphere 4.3 9 (91.5,11) 95.5,4 Pima 4 7 (78.9,111 79.8,8 Wine 4.4 8 (100, 01 96.8,2 We accept births with probability: n, min( 1, lh. rt x p(;(;/) G, J2,; r g exp ( - 05 ;" (I'd - <d)' ) x (~':) f g (.,.~ -')"-1 exp( -Ii' "'~-') x dm / (I + 1) x 1 bm/(Imax - I) (~, V27i) D TID exp ( -0.5Ih(J.l~ - ed)2) 1 x D ). (5) (Ii;)) TID (0"~-2)a-l exp( _(3'0"~-2) The first line in (5) are the likelihood and prior ratio. The prior ratio results from the difference in input dimension, which affects the kernel means and the prior over number of inputs. The first term of the proposal ratio is from proposing to add or remove one input. The second term is the proposal density of the additional kernel components which cancels with the corresponding term in the prior ratio. Due to symmetry of the proposal (4) and its reverse in a death move, there is no contribution from changing allocation probabilities. Death moves are accepted with probability ad = l/ab. The second type of move is an exchange move. We select a new input and one from the model inputs and propose new mean coefficients. This gives the following acceptance probability: min( 1, lh. ratio x (*,J2;) D TID exp ( -0.5"Jb(J.ld - ed)2) (~, J2;) D ITD exp ( -0.5 Ih(J.ld - ed)2) cm/ I TID N(J.ldl···) ) x---:-:--'-----..,..x cm/(Imax - I) TID N(J.l~I···) . (6) The first line of (6) are again likelihood and prior ratio. For exchange moves, the prior ratio is just the ratio from different values in the kernel means. The first term in the proposal ratio is from proposing to exchange an input. The second term is the proposal density of new kernel mean components. The last part is from proposing new allocation probabilities. 3 Experiments Although the method can be used with labeled and unlabeled data, the following experiments were performed using only labeled data. For all experiments we set a = 2 and 9 = 0.2. The first two data sets are from the VCI repositoryl. We use 1 Available at http://www.ics.uci.edu/ mlearn/MLRepository.html. On Input Selection with Reversible Jump MCMC 643 the Ionosphere data which has 33 inputs, 175 training and 176 test samples. For this experiment we use 6 kernels and set h = 0.5. The second data is the wine recognition data which provides 13 inputs, 62 training and 63 test samples. For this data, we use 3 kernels and set h = 0.28. The third experiment is performed with the Pima data provided by B. D. Ripley2. For this one we use 3 kernels and set h = 0.16. For all experiments we draw 15000 samples from the posterior over coefficients and input subsets. We discard the first 5000 samples as burn in and use the rest for predictions. Classification accuracy, is compared with an MLP classifier using R. Neals hybrid Monte Carlo sampling with ARD priors on inputs. These experiments use 25 hidden units. Table 1 contains further details: avg( #) is the average and max(#) the maximal number of inputs used by the hybrid sampler; RBF (%, na) is the classification accuracy of the hybrid sampler and the number of errors it made that were not made by the ARD-MLP; MLP(%, nb) is the same for the ARD-MLP. We compare classifiers by testing (na, nb) against the null hypothesis that this is an observation from a Binomial Bn(na +nb, 0.5) distribution. This reveals that neither difference is significant. Although we could not improve classification accuracy on these data, this does not really matter because ARD methods usually lead to high generalization accuracy and we can compete. The real benefit from using the hybrid sampler is that we can infer probabilities telling us how much different subsets contribute to an explanation of the target variables. Figure 3 shows the occurrence probabilities of feature subsets and features. Note that table 1 has also details about how many features were used in these problems. Especially the results from Ionosphere data are interesting as on average we use only 4.3 out of 33 input features. For ionosphere and wine data the Markov chain visits about 500 different input subsets within 10000 samples. For the Pima data the number is about 60 and an order of magnitude smaller. 4 Discussion In this paper we have discussed a hybrid sampler that uses Gibbs updates and reversible jump moves to approximate the a-posteriori distribution over parameters and input subsets in nonlinear classification problems. The classification accuracy of the method could compete with R. Neals MLP-ARD implementation. However the real advantage of the method is that it provides us with a relevance measure of feature subsets. This allows to infer the optimal number of inputs and how many different explanations the data provides. Acknow ledgements I want to thank several people for having used resources they provide: I have used R.Neals hybrid Markov chain sampler for the MLP experiments; The data used for the experiments were obtained form the University at Irvine repository and from B. D. Ripley. Furthermore I want to express gratitude to the anonymous reviewers for their comments and to J.F.G. de Freitas for useful discussions during the conference. This work was done in the framework of the research project GZ 607.519/2V /B/9/98 "Verbesserung der Biosignalverarbeitung durch Beruecksichtigung von Unsicherheit und Konfidenz", funded by the Austrian federal ministry of science and transport (BMWV). 2 Available at http://www.stats.ox.ac.uk 644 10 5 Probabil~les of input subsets-Ionosphere 9.1% :[7.14) 8.6%: [3. 4) 7.9%: [4. 7) 4.6% : [4. 15) o j, I. j 1. J o 100 200 300 400 500 Probabilities of input subsets-Pima 20r-----~--------------TO 15 10 5 17.3% : [2. 7j 16.1%:[2. 9.8% : [2. 5) 8.6% : [2. 5. 7) °0L-~~~2~0~~~~40~~~-6~0 Probabilities of input subsets - Wine 10r---~--~--~----~--~ 5 9.5°,,": [1. 13) 5.2%: [7. 13) 3,4°,,": [ 13) 3.1%: [1.12.13) 100 200 300 400 500 Probabilities of inputs - Ionosphere 30r-----~------------~_, 20 10 10 20 30 Probabilities of inputs - Pima 40r---~~--------~-----' 30 20 10 o~--~~--~----~----~ o 246 Probabilities of inputs - Wine 8 3Or-------~-------------, 20 10 0U-------~------~----~ o 5 10 P. Sykacek Figure 1: Probabilities of inputs and input subsets measuring their relevance. References [1] P. A. Devijver and J. V. Kittler. Pattern Recognition. A Statistical Approach. PrenticeHall, Englewood Cliffs, NJ, 1982. [2] S. Geman and D. Geman. Stochastic relaxation, gibbs distributions and the bayesian restoration of images. IEEE Trans. Pattn. Anal. Mach. Intel., 6:721-741, 1984. [3] Z. Ghahramani, M.1. Jordan Supervised Learning from Incomplete Data via an EM Approach In Cowan J.D., et al.(eds.), Advances in Neural Information Processing Systems 6, Morgan Kaufmann, Los Altos/Palo Alto/San Francisco, pp.120-127, 1994. [4] P. J. Green. Reversible jump markov chain monte carlo computation and bayesian model determination. Biometrika, 82:711-732, 1995. [5] C. C. Holmes and B. K. Mallick. Bayesian radial basis functions of variable dimension. Neural Computation, 10:1217-1234, 1998. [6] R. M. Neal. Bayesian Learning for Neural Networks. Springer, New York, 1986. [7] D. B. Phillips and A. F. M. Smith. Bayesian model comparison via jump diffusioons. In W.R. Gilks, S. Richardson, and D.J. Spiegelhalter, editors, Markov Chain Monte Carlo in Practice, pages 215-239, London, 1996. Chapman & Hall. [8] 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. [9] M. Stensmo, T.J. Sejnowski A Mixture Model System for Medical and Machine Diagnosis In Tesauro G., et al.(eds.), Advances in Neural Information Processing System 7, MIT Press, Cambridge/Boston/London, pp.1077-1084, 1995. [10] H. G. C. Traven A neural network approach to statistical pattern classification by "semi parametric" estimation of probability density functions IEEE Trans. Neur. Net., 2:366-377, 1991.
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Uniqueness of the SVM Solution Christopher J .C. Burges Advanced Technologies, Bell Laboratories, Lucent Technologies Holmdel, New Jersey burges@iucent.com David J. Crisp Centre for Sensor Signal and Information Processing, Deptartment of Electrical Engineering, University of Adelaide, South Australia dcrisp@eleceng.adelaide.edu.au Abstract We give necessary and sufficient conditions for uniqueness of the support vector solution for the problems of pattern recognition and regression estimation, for a general class of cost functions. We show that if the solution is not unique, all support vectors are necessarily at bound, and we give some simple examples of non-unique solutions. We note that uniqueness of the primal (dual) solution does not necessarily imply uniqueness of the dual (primal) solution. We show how to compute the threshold b when the solution is unique, but when all support vectors are at bound, in which case the usual method for determining b does not work. 1 Introduction Support vector machines (SVMs) have attracted wide interest as a means to implement structural risk minimization for the problems of classification and regression estimation. The fact that training an SVM amounts to solving a convex quadratic programming problem means that the solution found is global, and that if it is not unique, then the set of global solutions is itself convex; furthermore, if the objective function is strictly convex, the solution is guaranteed to be unique [1]1. For quadratic programming problems, convexity of the objective function is equivalent to positive semi-definiteness of the Hessian, and strict convexity, to positive definiteness [1]. For reference, we summarize the basic uniqueness result in the following theorem, the proof of which can be found in [1]: Theorem 1: The solution to a convex programming problem, for which the objective function is strictly convex, is unique. Positive definiteness of the Hessian implies strict convexity of the objective function. Note that in general strict convexity of the objective function does not neccesarily imply positive definiteness of the Hessian. Furthermore, the solution can still be unique, even if the objective function is loosely convex (we will use the term "loosely convex" to mean convex but not strictly convex). Thus the question of uniqueness IThis is in contrast with the case of neural nets, where local minima of the objective function can occur. 224 C. J. C. Burges and D. J. Crisp for a convex programming problem for which the objective function is loosely convex is one that must be examined on a case by case basis. In this paper we will give necessary and sufficient conditions for the support vector solution to be unique, even when the objective function is loosely convex, for both the clasification and regression cases, and for a general class of cost function. One of the central features of the support vector method is the implicit mapping ~ of the data Z E Rn to some feature space F, which is accomplished by replacing dot products between data points Zi, Zj, wherever they occur in the train and test algorithms, with a symmetric function K (Zi' Zj ), which is itself an inner product in F [2]: K(Zi' Zj) = (~(Zi)' ~(Zj» = (Xi, Xj), where we denote the mapped points in F by X = ~(z). In order for this to hold the kernel function K must satisfy Mercer's positivity condition [3]. The algorithms then amount to constructing an optimal separating hyperplane in F, in the pattern recognition case, or fitting the data to a linear regression tube (with a suitable choice of loss function [4]) in the regression estimation case. Below, without loss of generality, we will work in the space F, whose dimension we denote by dF. The conditions we will find for non-uniqueness of the solution will not depend explicitly on F or ~. Most approaches to solving the support vector training problem employ the Wolfe dual, which we describe below. By uniqueness of the primal (dual) solution, we mean uniqueness of the set of primal (dual) variables at the solution. Notice that strict convexity of the primal objective function does not imply strict convexity of the dual objective function. For example, for the optimal hyperplane problem (the problem of finding the maximal separating hyperplane in input space, for the case of separable data), the primal objective function is strictly convex, but the dual objective function will be loosely convex whenever the number of training points exceeds the dimension of the data in input space. In that case, the dual Hessian H will necessarily be positive semidefinite, since H (or a submatrix of H, for the cases in which the cost function also contributes to the (block-diagonal) Hessian) is a Gram matrix of the training data, and some rows of the matrix will then necessarily be linearly dependent [5]2. In the cases of support vector pattern recognition and regression estimation studied below, one of four cases can occur: (1) both primal and dual solutions are unique; (2) the primal solution is unique while the dual solution is not; (3) the dual is unique but the primal is not; (4) both solutions are not unique. Case (2) occurs when the unique primal solution has more than one expansion in terms of the dual variables. We will give an example of case (3) below. It is easy to construct trivial examples where case (1) holds, and based on the discussion below, it will be clear how to construct examples of (4). However, since the geometrical motivation and interpretation of SVMs rests on the primal variables, the theorems given below address uniqueness of the primal solution3• 2 The Case of Pattern Recognition We consider a slightly generalized form of the problem given in [6], namely to minimize the objective function F = (1/2) IIwl12 + L Ci~f (1) 2Recall that a Gram matrix is a matrix whose ij'th element has the form (Xi,Xj) for some inner product (,), where Xi is an element of a vector space, and that the rank of a Gram matrix is the maximum number of linearly independent vectors Xi that appear in it [6]. 3Due to space constraints some proofs and other details will be omitted. Complete details will be given elsewhere. Uniqueness of the SVM Solution with constants p E [1,00), Gi > 0, subject to constraints: Yi(W . Xi + b) > 1 ~i' i = 1,,,,,1 C; > 0 i = 1 ... 1 ,:>. -' " 225 (2) (3) where W is the vector of weights, b a scalar threshold, ~i are positive slack variables which are introduced to handle the case of nonseparable data, the Yi are the polarities of the training samples (Yi E {± I} ), Xi are the images of training samples in the space F by the mapping ~, the Gi determine how much errors are penalized (here we have allowed each pattern to have its own penalty), and the index i labels the 1 training patterns. The goal is then to find the values of the primal variables {w, b, ~i} that solve this problem. Most workers choose p = 1, since this results in a particularly simple dual formulation, but the problem is convex for any p 2: 1. We will not go into further details on support vector classification algorithms themselves here, but refer the interested reader to [3], [7]. Note that, at the solution, b is determined from w and ~i by the Karush Kuhn Tucker (KKT) conditions (see below), but we include it in the definition of a solution for convenience. Note that Theorem 1 gives an immediate proof that the solution to the optimal hyperplane problem is unique, since there the objective function is just (1/2)lIwI1 2 , which is strictly convex, and the constraints (Eq. (2) with the ~ variables removed) are linear inequality constraints which therefore define a convex set4. For the discussion below we will need the dual formulation of this problem, for the case p = 1. It takes the following form: minimize ~ L-ijG:iG:jYiYj(Xi,Xj) L-iG:i subject to constraints: TJi > 0, G:i 2: 0 Gi G:i + TJi LG:iYi 0 (4) (5) (6) and where the solution takes the form w = L-i G:iYiXi, and the KKT conditions, which are satisfied at the solution, are TJi~i = 0, G:i (Yi (w . Xi + b) - 1 + ~i) = 0, where TJi are Lagrange multipliers to enforce positivity of the ~i' and G:i are Lagrange multipliers to enforce the constraint (2). The TJi can be implicitly encapsulated in the condition 0 ~ ai :::; Gi , but we retain them to emphasize that the above equations imply that whenever ~i =/; 0, we must have ai = Gi . Note that, for a given solution, a support vector is defined to be any point Xi for which G:i > O. Now suppose we have some solution to the problem (1), (2), (3). Let Nl denote the set {i : Yi = 1, W · Xi + b < I}, N2 the set {i : Yi = -1, W· Xi + b > -I}, N3 the set {i : Yi = 1, W· Xi + b = I}, N4 the set {i : Yi = -1, W· Xi + b = -I}, Ns the set {i : Yi = 1, W· Xi + b > I}, and N6 the set {i : Yi = -1, W· Xi + b < -I}. Then we have the following theorem: Theorem 2: The solution to the soft-margin problem, (1), (2) and (3), is unique for p > 1. For p = 1, the solution is not unique if and only if at least one of the following two conditions holds: (7) (8) iENl UN3 iEN2 Furthermore, whenever the solution is not unique, all solutions share the same w, and any support vector Xi has Lagrange multiplier satisfying ai = Gi , and when (7) 4This is of course not a new result: see for example [3]. 226 C. 1. C. Burges and D. 1. Crisp holds, then N3 contains no support vectors, and when (8) holds, then N4 contains no support vectors. Proof: For the case p > 1, the objective function F is strictly convex, since a sum of strictly convex functions is a strictly convex function, and since the function g( v) = vP , v E lR+ is strictly convex for p > 1. FUrthermore the constraints define a convex set, since any set of simultaneous linear inequality constraints defines a convex set. Hence by Theorem 1 the solution is unique. For the case p = 1, define Z to be that dF + i-component vector with Zi = Wi, i = 1, ... ,dF, and Zi = ~i' i = dF + 1", . ,dF + t. In terms of the variables z, the problem is still a convex programming problem, and hence has the property that any solution is a global solution. Suppose that we have two solutions, Zl and Z2' Then we can form the family of solutions Zt, where Zt == (1 - t)ZI + tZ2, and since the solutions are global, we have F(zd = F(Z2) = F(zt). By expanding F(zt) - F(zt} = 0 in terms of Zl and Z2 and differentiating twice with respect to t we find that WI = W2. Now given wand b, the ~i are completely determined by the KKT conditions. Thus the solution is not unique if and only if b is not unique. Define 0 == min {miniENl ~i' miniEN6 (-1 - W • Xi - b)}, and suppose that condition (7) holds. Then a different solution {w', b', e} is given by w' = w, b' = b + 0, and ~~ = ~i - 0, Vi E N 1 , ~~ = ~i + 0, Vi E N2 uN4 , all other ~i = 0, since by construction F then remains the same, and the constraints (2), (3) are satisfied by the primed variables. Similarly, suppose that condition (8) holds. Define 0 == min{miniEN2~i,miniEN5(w·xi+b-l)}. Then a different solution {w',b',e} is given by w' = w, b' = b - 0, and ~~ = ~i - 0, Vi E N2 , ~: = ~i + 0, Vi E NI U N3 , all other ~i = 0, since again by construction F is unchanged and the constraints are still met. Thus the given conditions are sufficient for the solution to be nonunique. To show necessity, assume that the solution is not unique: then by the above argument, the solutions must differ by their values of b. Given a particular solution b, suppose that b + 0, 0 > 0 is also a solution. Since the set of solutions is itself convex, then b + 0' will also correspond to a solution for all 0' : 0 ~ 0' ~ O. Given some b' = b + 0', we can use the KKT conditions to compute all the ei, and we can choose 0' sufficiently small so that no ~i' i E N6 that was previously zero becomes nonzero. Then we find that in order that F remain the same, condition (7) must hold. If b - 0, 0 > 0 is a solution, similar reasoning shows that condition (8) must hold. To show the final statement of the theorem, we use the equality constraint (6), together with the fact that, from the KKT conditions, all support vectors Xi with indices in NI uN2 satisfy (Xi = Ci • Substituting (6) in (7) then gives L:N3 (Xi + L:N4 (Ci (Xi) = 0 which implies the result, since all (Xi are non-negative. Similarly, substituting (6) in (8) gives L:,M (Ci (Xi) + L:.Af. (Xi = 0 which again . l' h 1 3 4 Imp les t e resu t. 0 Corollary: For any solution which is not unique, letting S denote the set of indices of the corresponding set of support vectors, then we must have L:iES CiYi = O. FUrthermore, if the number of data points is finite, then for at least one of the family of solutions, all support vectors have corresponding ~i i= O. Note that it follows from the corollary that if the Ci are chosen such that there exists no subset r of the train data such that L:iET CiYi = 0, then the solution is guaranteed to be unique, even if p = 1. FUrthermore this can be done by choosing all the Ci very close to some central value C, although the resulting solution can depend sensitively on the values chosen (see the example immediately below). Finally, note that if all Ci are equal, the theorem shows that a necessary condition for the solution to be non-unique is that the negative and positive polarity support vectors be equal in number. Uniqueness of the SVM Solution 227 A simple example of a non-unique solution, for the case p = 1, is given by a train set in one dimension with just two examples, {Xl = 1, YI = I} and {xz = -1, Yz = -11' with GI = Cz == C. It is straightforward to show analytically that for G 2: 2' the solution is unique, with w = 1, 6 = 6 = b = 0, and marginS equal to 2, while for C < ! there is a family of solutions, with -1 + 2C ::; b ::; 1 - 2C and 6 = 1- b - 2C, 6 = 1 + b - 2G, and margin l/C. The case G < ! corresponds to Case (3) in Section (1) (dual unique but primal not), since the dual variables are uniquely specified by a = C. Note also that this family of solutions also satisfies the condition that any solution is smoothly deformable into another solution [7J. If GI > Cz, the solution becomes unique, and is quite different from the unique solution found when Gz > CI . When the G's are not equal, one can interpret what happens in terms of the mechanical analogy [8J, with the central separating hyperplane sliding away from the point that exerts the higher force, until that point lies on the edge of the margin region. Note that if the solution is not unique, the possible values of b fall on an interval of the real line: in this case a suitable choice would be one that minimizes an estimate of the Bayes error, where the SVM output densities are modeled using a validation set6 . Alternatively, requiring continuity with the cases p > 1, so that one would choose that value of b that would result by considering the family of solutions generated by different choices of p, and taking the limit from above of p -t 1, would again result in a unique solution. 3 The Case of Regression Estimation 7 Here one has a set of l pairs {xI,Yd,{xz,yz},···,{XI,YI}, {Xi E :F,Yi E R}, and the goal is to estimate the unknown functional dependence j of the Y on the X, where the function j is assumed to be related to the measurements {Xi,Yi} by Yi = j(Xi) +ni, and where ni represents noise. For details we refer the reader to [3], [9]. Again we generalize the original formulation [10], as follows: for some choice of positive error penalties Gi, and for positive €i, minimize I F = ~ Ilwllz + 2)Gi~f + C;(~np) i=l with constant p E [1, 00), subject to constraints Yi - w . Xi - b < €i + ~i W • Xi + b - Yi < €i + ~; ~;*) > 0 (9) (10) (11) (12) where we have adopted the notation ~;*) == {~i ' ~;} [9J. This formulation results in an "€ insensitive" loss function, that is, there is no penalty (~}*) = 0) associated with point Xi if IYi - w . Xi - bl ::; €i. Now let {3, {3* be the Lagrange multipliers introduced to enforce the constraints (10), (11). The dual then gives 2: {3i = 2: {3;, 0::; {3i ::; Gi , 0::; {3; ::; G;, (13) 5The margin is defined to be the distance between the two hyperplanes corresponding to equality in Eq. (2), namely 2/lIwll, and the margin region is defined to be the set of points between the two hyperplanes. 6This method was used to estimate b under similar circumstances in [8]. 7The notation in this section only coincides with that used in section 2 where convenient. 228 C. J. C. Burges and D. J. Crisp which we will need below. For this formulation, we have the following Theorem 3: For a given solution, define !(Xi, Yi) == Yi W • Xi - b, and define Nl to be the set of indices {i : !(Xi, Yi) > fi}, N2 the set {i : !(Xi, Yi) = fd, N3 the set {i : !(Xi,Yi) = -fi}, and N4 the set {i : !(Xi,Yi) < -fi}. Then the solution to (9) - (12) is unique for p > 1, and for p = 1 it is not unique if and only if at least one of the following two conditions holds: L Ci LC; (14) iENIUN2 iEN4 L C'! , LCi (15) iEN3UN4 iENl Furthermore, whenever the solution is not unique, all solutions share the same w, and all support vectors are at bound (that iss, either f3i = Ci or f3i = Cn, and when (14) holds, then N3 contains no support vectors, and when (15) holds, then N2 contains no support vectors. The theorem shows that in the non-unique case one will only be able to move the tube (and get another solution) if one does not change its normal w. A trivial example of a non-unique solution is when all the data fits inside the f-tube with room to spare, in which case for all the solutions, the normal to the f-tubes always lies along the Y direction. Another example is when all Ci are equal, all data falls outside the tube, and there are the same number of points above the tube as below it. 4 Computing b when all SV s are at Bound The threshold b in Eqs. (2), (10) and (11) is usually determined from that subset of the constraint equations which become equalities at the solution and for which the corresponding Lagrange multipliers are not at bound. However, it may be that at the solution, this subset is empty. In this section we consider the situation where the solution is unique, where we have solved the optimization problem and therefore know the values of all Lagrange multipliers, and hence know also w, and where we wish to find the unique value of b for this solution. Since the ~~.) are known once b is fixed, we can find b by finding that value which both minimizes the cost term in the primal Lagrangian, and which satisfies all the constraint equations. Let us consider the pattern recognition case first. Let S+ (S_) denote the set of indices of positive (negative) polarity support vectors. Also let V+ (V_) denote the set of indices of positive (negative) vectors which are not support vectors. It is straightforward to show that if 2:iES_ Ci > 2:iES+ Ci, then b = max {maxiES_ (-1 W • Xi), maxiEV+ (1 W • Xi)}, while if 2:iES_ Ci < 2:iES+ Ci, then b = min {miniEs+ (1 W • Xi), miniEv_ (-1 W • Xi)}' Furthermore, if 2:iES_ Ci = 2:iES+ Ci, and if the solution is unique, then these two values coincide. In the regression case, let us denote by S the set of indices of all support vectors, S its complement, SI the set of indices for which f3i = Ci, and S2 the set of indices for which f3i = C;, so that S = SI U S2 (note SI n S2 = 0). Then if 2:iES2 C; > 2:iESl Ci, the desired value of b is b = max{m~Es(Yi - W· Xi + fi), maxiES(Yi - W· Xi - fi)} while if 2:iES2 C; < 2:iESl Ci, then b min {miniEs(Yi W • Xi - fi), miniES(Yi - W· Xi + fi)}' 8Recall that if Ei > 0, then {3i{3; = O. Uniqueness of the SVM Solution Again, if the solution is unique, and if also l:iES2 c; two values coincide. 5 Discussion 229 We have shown that non-uniqueness of the SVM solution will be the exception rather than the rule: it will occur only when one can rigidly parallel transport the margin region without changing the total cost. If non-unique solutions are encountered, other techniques for finding the threshold, such as minimizing the Bayes error arising from a model of the SVM posteriors [8], will be needed. The method of proof in the above theorems is straightforward, and should be extendable to similar algorithms, for example Mangasarian's Generalized SVM [11]. In fact one can extend this result to any problem whose objective function consists of a sum of strictly convex and loosely convex functions: for example, it follows immediately that for the case of the lI-SVM pattern recognition and regression estimation algorithms [12], with arbitrary convex costs, the value of the normal w will always be unique. Acknowledgments C. Burges wishes to thank W. Keasler, V. Lawrence and C. Nohl of Lucent Technologies for their support. References [1] R. Fletcher. Practical Methods of Optimization. John Wiley and Sons, Inc., 2nd edition, 1987. [2] B. E. Boser, I. M. Guyon, and V .Vapnik. A training algorithm for optimal margin classifiers. In Fifth Annual Workshop on Computational Learning Theory, Pittsburgh, 1992. ACM. [3] V. Vapnik. Statistical Learning Theory. John Wiley and Sons, Inc., New York, 1998. [4] A.J. Smola and B. Scholkopf. On a kernel-based method for pattern recognition, regression, approximation and operator inversion. Algorithmica, 22:211 - 231, 1998. [5] Roger A. Horn and Charles R. Johnson. Matrix Analysis. Cambridge University Press, 1985. [6] C. Cortes and V. Vapnik. Support vector networks. Machine Learning, 20:273-297, 1995. [7] C.J.C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2}:121-167, 1998. [8] C. J. C. Burges and B. Scholkopf. Improving the accuracy and speed of support vector learning machines. In M. Mozer, M. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems 9, pages 375-381, Cambridge, MA, 1997. MIT Press. [9] A. Smola and B. Scholkopf. A tutorial on support vector regression. Statistics and Computing, 1998. In press: also, COLT Technical Report TR-1998-030. [10] V. Vapnik, S. Golowich, and A. Smola. Support vector method for function approximation, regression estimation, and signal processing. Advances in Neural Information Processing Systems, 9:281-287, 1996. [11] O.L. Mangarasian. Generalized support vector machines, mathematical programming technical report 98-14. Technical report, University of Wisconsin, October 1998. [12] B. Scholkopf, A. Smola, R. Williamson and P. Bartlett, New Support Vector Algorithms, NeuroCOLT2 NC2-TR-1998-031, 1998.
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The Parallel Problems Server: an Interactive Tool for Large Scale Machine Learning Charles Lee Isbell, Jr. isbell @research.att.com AT&T Labs 180 Park Avenue Room A255 Florham Park, NJ 07932-0971 Parry Husbands PIRHusbands@lbl.gov Lawrence Berkeley National LaboratorylNERSC 1 Cyclotron Road, MS 50F Berkeley, CA 94720 Abstract Imagine that you wish to classify data consisting of tens of thousands of examples residing in a twenty thousand dimensional space. How can one apply standard machine learning algorithms? We describe the Parallel Problems Server (PPServer) and MATLAB*P. In tandem they allow users of networked computers to work transparently on large data sets from within Matlab. This work is motivated by the desire to bring the many benefits of scientific computing algorithms and computational power to machine learning researchers. We demonstrate the usefulness of the system on a number of tasks. For example, we perform independent components analysis on very large text corpora consisting of tens of thousands of documents, making minimal changes to the original Bell and Sejnowski Matlab source (Bell and Sejnowski, 1995). Applying ML techniques to data previously beyond their reach leads to interesting analyses of both data and algorithms. 1 Introduction Real-world data sets are extremely large by the standards of the machine learning community. In text retrieval, for example, we often wish to process collections consisting of tens or hundreds of thousands of documents and easily as many different words. Naturally, we would like to apply machine learning techniques to this problem; however, the sheer size of the data makes this difficult. This paper describes the Parallel Problems Server (PPServer) and MATLAB *P. The PPServer is a "linear algebra server" that executes distributed memory algorithms on large data sets. Together with MATLAB*P, users can manipulate large data sets within Matlab transparently. This system brings the efficiency and power of highly-optimized parallel computation to researchers using networked machines but maintain the many benefits of interactive environments. We demonstrate the usefulness of the PPServer on a number of tasks. For example, we perform independent components analysis on very large text corpora consisting of tens of thousands of documents with minimal changes to the original Bell and Sejnowski Matlab source (Bell and Sejnowski, 1995). Applying ML techniques to datasets previously beyond 704 Libraries Computational & Interface Routines C. L. Isbell, Jr. and P. Husbands Machine) MatlabS Local Variables ml ... [ ··· .. ········l m, ... [ ............ . m, ... [ ....... . m, ... [ ............ . Machin~, Figure 1: Use of the PPServer by Matlab is almost completely transparent. PPServer variables are tied to the PPServer itself while Matlab maintains handles to the data. Using Matlab's object system, functions using PPServer variables invoke PPServer commands implicitly. their reach, we discover interesting analyses of both data and algorithms. 2 The Parallel Problems Server The Parallel Problems Server (PPServer) is the foundation of this work. The PPServer is a realization of a novel client-server model for computation on very large matrices. It is compatible with any Unix-like platform supporting the Message Passing Interface (MPI) library (Gropp, Lusk and Skjellum, 1994). MPI is the standard for multi-processor communication and is the most portable way for writing parallel code. The PPServer implements functions for creating and removing distributed matrices, loading and storing them from/to disk using a portable format, and performing elementary matrix operations. Matrices are two-dimensional single or double precision arrays created on the PPServer itself (functions are provided for transferring matrix sections to and from a client). The PPServer supports both dense and sparse matrices. The PPServer communicates with clients using a simple request-response protocol. A client requests an action by issuing a command with the appropriate arguments, the server executes that command, and then notifies the client that the command is complete. The PPServer is directly extensible via compiled libraries called packages. The PPServer implements a robust protocol for communicating with packages. Clients (and other packages) can load and remove packages on-the-fty, as well as execute commands within packages. Package programmers have direct access to information about the PPServer and its matrices. Each package represents its own namespace, defining a set of visible function names. This supports data encapsulation and allows users to hide a subset of functions in one package by loading another that defines the same function names. Finally, packages support common parallel idioms (eg applying a function to every element of a matrix), making it easier to add common functionality. All but a few PPServer commands are implemented in packages, including basic matrix operations. Many highly-optimized public libraries have been realized as packages using appropriate wrapper functions. These packages include ScaLAPACK (Blackford et a1., 1997), S3L (Sun's optimized version of ScaLAPACK), PARPACK (Maschhoff and Sorensen, 1996), and PETSc (PETSc, ). Large Scale Machine Learning Using The Parallel Problems Server 705 1 function H=hilb(n) 2 J l:nj 3 J = J (ones (n, 1) , : ) j 4 I J' j 5 E = ones (n, n) j 6 H = E. / (I+J-1) j Figure 2: Matlab code for producing Hilbert matrices. When n is influenced by P, each of the constructors creates a PPServer object instead of a Matlab object. 3 MATLAB*P By directly using the PPServer's client communication interface, it is possible for other applications to use the PPServer's functionality. We have implemented a client interface for Matlab, called MA1LAB*P. MATLAB*P is a collection of Matlab 5 objects, Matlab m-files (Matlab's scripting language) and Matlab MEX programs (Matlab's extemallanguage API) that allows for the transparent integration of Matlab as a front end for the Parallel Problems Server. The choice of Matlab was influenced by several factors. It is the de facto standard for scientific computing, enjoying wide use in industry and academia. In the machine learning community, for example, algorithms are often written as Matlab scripts and made freely available. In the scientific computing community, algorithms are often first prototyped in Matlab before being optimized for languages such as Fortran. We endeavor to make interaction with the PPServer as transparent as possible for the user. In principle, a typical Matlab user should never have to make explicit calls to the PPServer. Further, current Matlab programs should not have to be rewritten to take advantage of the PPServer. Space does not permit a complete discussion of MA1LAB*P (we refer the reader to (Husbands and Isbell, 1999)); however, we will briefly discuss how to use prewritten Matlab scripts without modification. This is accomplished through the simple but innovative P notation. We use Matlab 5's object oriented features to create PPServer objects automatically. P is a special object we introduce in Matlab that acts just like the integer 1. A user typing a=ones (1000*P, 1000) or b=rand( 1000, 1000*P) obtains two 1000-by-lOOO matrices distributed in parallel. The reader can guess the use of P here: it indicates that a is distributed by rows and b by columns. To a user, a and b are matrices, but within Matlab, they are handles to special distributed types that exist on the PPServer. Any further references to these variables (e.g. via such commands as eig, svd, inv, *, +, -) are recognized as a call to the PPServer rather than as a traditional Matlab command. Figure 2 shows the code for Matlab's built in function hilb. The call hilb (n) produces the n x n Hilbert matrix (Hij = i+}-l)' When n is influenced by P, a parallel array results: • J=l: n in line 2 creates the PPServer vector 1,2, · . . , n and places a handle to it in J. Note that this behavior does not interfere with the semantics of for loops (for i=l: n) as Matlab assigns to i the value of each column of 1: n: the numbers 1,2, ... , n. • ones (n, 1) in line 3 produces a PPServer matrix. • Emulation of Matlab's indexing functions results in the correct execution of line 3. 706 C. L. Isbell, Jr. and P. Husbands • Overloading of ' (the transpose operator) executes line 4 on the PPServer. • In line 5, E is generated on the PPServer because ofthe overloading of ones. • Overloading elementary matrix operations makes H a PPServer matrix (line 6). The Parallel Problems Server and MA1LAB *p have been tested extensively on a variety of platforms. They currently run on Cray supercomputers! , clusters of symmetric multiprocessors from Sun Microsystems and DEC as well as on clusters of networked Intel PCs. The PPServer has also been tested with other clients, including Common LISP. Although computational performance varies depending upon the platform, it is clear that the system provides distinct computational advantages. Communication overhead (in our experiments, roughly two milliseconds per PPServer command) is negligible compared to the computational and space advantage afforded by transparent access to highly-optimized linear algebra algorithms. 4 Applications in Text Retrieval In this section we demonstrate the efficacy of the PPServer on real-world machine learning problems. In particular we explore the use of the PPServer and MA1LAB*P in the text retrieval domain. The task in text retrieval is to find the subset of a collection of documents relevant to a user's information request. Standard approaches are based on the Vector Space Model (VSM). A document is a vector where each dimension is a count of the occurrence of a different word. A collection of documents is a matrix, D, where each column is a document vector di. The similarity between tw~ documents is their inner product, Jf d j. Queries are just like documents, so the relevance of documents to a query, q, is DT q. Typical small collections contain a thousand vectors in a ten thousand dimensional space, while large collections may contain 500,000 vectors residing in hundreds of thousands of dimensions. Clearly, well-understood standard machine learning techniques may exhibit unpredictable behavior under such circumstances, or simply may not scale at all. Classically, ML-like approaches try to construct a set of linear operators which extract the underlying "topic" structure of documents. Documents and queries are projected into that new (usually smaller) space before being compared using the inner product. The large matrix support in MA1LAB*P enables us to use matrix decomposition techniques for extracting linear operators easily. We have explored several different algorithms(Isbell and Viola, 1998). Below, we discuss two standard algorithms to demonstrate how the PPServer allows us to perform interesting analysis on large datasets. 4.1 Latent Semantic Indexing Latent Semantic Indexing (LSI) (Deerwester et al., 1990) constructs -a smaller document matrix by using the Singular Value Decomposition (SVD): D = U SVT . U contains the eigenvectors of the co-occurrence matrix while the diagonal elements of S (referred to as singular values) contain the square roots of their corresponding eigenValUes. The eigenvectors with the largest eigenvalues capture the axes of largest variation in the data. LSI projects documents onto the k-dimensional subspace spanned by the first k columns of U (denoted Uk) so thatthe documents are now: V[ = S;;lUk. Queries are similarly projected. Thus, the document-query scores for LSI can be obtained with simple Matlab code: 1 Although there is no Matlab for the Cray, we are still able to use it to "execute" Matlab code -in parallel. Large Scale Machine Learning Using The Parallel Problems Server '" .. .. IG ------------ - ---. , , 707 Figure 3: The first 200 singular values of a collection of about 500,000 documents and 200,000 terms, and singular values for half of that collection. Computation for on the full collection took only 62 minutes using 32 processors on a Cray TIE. D=dsparse('term-doc'); %D SPARSE reads a sparse matrix Q=dsparse('queries'); [U,S,VJ=svds(D,k); % compute the k-SVD of D sc=getlsiscores(U,S,V,Q); % computes v*(l/s)*u'*q The scores that are returned can then be combined with relevance judgements to obtain precision/recall curves that are displayed in Matlab: r=dsparse('judgements'); [pr,reJ=precisionrecall(sc,r); plot (re ( , @' ) , pr ( , @' ) ) ; In addition to evaluating the performance of various techniques, we can also explore characteristics of the data itself. For example, many implementations of LSI on large collections use only a subset of the documents for computational reasons. This leads one to question how the SVD is affected. Figure 3 shows the first singular values for one large collection as well as for a random half of that collection. It shows that the shape of the curves are remarkably similar (as they are for the other half). This suggests that we can derive a projection matrix from just half of the collection. An evaluation of this technique can easily be performed using our system. Prernlinary experiments show nearly identical retrieval performance. 4.2 What are the Independent Components of Documents? Independent components analysis (ICA)(Bell and Sejnowski, 1995) also recovers linear projections from data. Unlike LSI, which finds principal components, ICA finds axes that are statistically independent. ICA's biggest success is probably its application to the blind source separation or cocktail party problem. In this problem, one observes the output of a number of microphones. Each microphone is assumed to be recording a linear mixture of a number of unknown sources. The task is to recover the original sources. There is a natural embedding of text retrieval within this framework. The words that are observed are like microphone signals, and underlying ''topics'' are the source signals that give rise to them. Figure 4 shows a typical distribution of words projected along axes found by ICA.2 Most words have a value close to zero. The histogram shows only the words large positive or 2These results are from a collection containing transcripts of White House press releases from 1993. There are 1585 documents and 18,675 distinct words. 708 africa apartheid I -1 -0.75 anc transition mandela continent elite ethiopia C. L. Isbell, Jr. and P. Husbands saharan 0.5 0.75 Figure 4: Distribution of words with large magnitude an ICA axis from White House text. negative values. One group of words is made up of highly-related terms; namely, "africa," "apartheid," and "mandela." The other group of words are not directly related, but each cooccurs with different individual words in the first group. For example, "saharan" and "africa" occur together many times, but not in the context of apartheid and South Africa; rather, in documents concerning US policy toward Africa in general. As it so happens, "saharan" acts as a discriminating word for these SUbtopics. As observed in (Isbell and Viola, 1998), it appears that ICA is finding a set of words, S, that selects for related documents, H, along with another set of words, T, whose elements do not select for H, but co-occur with elements of S. Intuitively, S selects for documents in a general subject area, and T removes a specific subset of those documents, leaving a small set of highly related documents. This suggests a straightforward algorithm to achieve the same goal directly. This local clustering approach is similar to an unsupervised version of Rocchio with Query Zoning (Singhal, 1997). Further analysis of ICA on similar collections reveals other interesting behavior on large datasets. For example, it is known that ICA will attempt to find an unmixing matrix that is full rank. This is in conflict with the notion that these collections actually reside in a much smaller subspace. We have found in our experiments with ICA that some axes are highly kurtotic while others produce gaussian-like distributions. We conjecture that any axis that results in a gaussian-like distribution will be split arbitrarily among all "empty" axes. For all intents and purposes, these axes are uninformative. This provides an automatic noisereduction technique for ICA when applied to large datasets. For the purposes of comparison, Figure 5 illustrates the performance of several algorithms (including ICA and various clustering techniques) on articles from the Wall Street Journal.3 5 Discussion We have shown that MATLAB *p enables portable, high-performance interactive supercomputing using the Parallel Problems Server, a powerful mechanism for writing and accessing optimized algorithms. Further, the client communication protocol makes it possible to implement transparent integration with sufficiently powerful clients, such as Matlab 5. With such a tool, researchers can now use Matlab as something more than just a way for prototyping algorithms and working on small problems. MATLAB*P makes it possible to interactively operate on and visualize large data sets. We have demonstrated this last claim by using the PPServer system to apply ML techniques to large datasets, allowing for analyses of both data and algorithms. MATLAB*P has also been used to implement versions of Diverse Density(Maron, 1998), MIMIC(DeBonet, Isbell and Viola, 1996), and gradient descent. 3The WSJ collection contains 42,652 documents and 89,757 words Large Scale Machine Learning Using The Parallel Problems Server 709 .71~~~-~;:=======il l.SI 0.0 --~_a.... - - ....... DocuMnIIt_ a....... _ . teA T .... """, Recall Figure 5: A comparison of different algorithms on the Wall Street Journal References Bell, A. and Sejnowski, T. (1995). An infonnation-maximizaton approach to blind source separation and blind deconvolution. Neural Computation, 7:1129-1159. Blackfor<~, L. S., Choi, J., Cleary, A, D' Azevedo, E., Demmel, J., Dhilon, I., Dongarra, 1., Hammarling, S., Henry, G., Petitet, A, Stanley, K, Walker, D., and Whaley, R. (1997). ScaLAPACK Users' Guide. http://www.netlib.orglscalapacklsluglscalapack..slug.htrnl. DeBonet, J., Isbell, C., and Viola, P. (1996). Mimic: Finding optima by estimating probability densities. In Advances in Neural Information Processing Systems. Deerwester, S., Dumais, S. T., Landauer, T. K, Furnas, G. w., and Harshman, R. A (1990). Indexing by latent semantic analysis. Journal of the Society for Information Science, 41(6):391-407. Frakes, W. B. and Baeza-Yates, R., editors (1992). Information Retrieval: Data Structures and Algorithms. Prentice-Hall. Gropp, W., Lusk, E., and Skjellum, A (1994). Using MPI: Portable Parallel Programming with the Message-Passing Interface. The MIT Press. Husbands, P. and Isbell, C. (1999). MITMatlab: A tool for interactive supercomputing. In Proceedings of the Ninth SIAM Conference on Parallel Processingfor Scientific Computing. Isbell, C. and Viola, P. (1998). Restructuring sparse high dimensional data for effective retrieval. In Advances in Neural Information Processing Systems. Kwok, K L. (1996). A new method of weighting query tenns for ad-hoc retrieval. In Proceedings of the 19th ACMlSIGIR Conference,pages 187-195. Maron, O. (1998). A framework for multiple-instance learning. In Advances in Neural Information Processing Systems. Maschhoff, K J. and Sorensen, D. C. (1996). A Portable Implementation of ARPACK for Distributed Memory Parallel Computers. In Preliminary Proceedings of the Copper Mountain Conference on Iterative Methods. O'Brien, G. W. (1994). Infonnation management tools for updating an svd-encoded indexing scheme. Technical Report UT-CS-94-259, University of Tennessee. PETSc. The Portable, Extensible Toolkit for Scientific Computation. http://www.mcs.anl.gov/home/group/petsc.htrnl. PPServer. The Parallel Problems Server Web Page. http://www.ai.mit.edulprojects/ppserver. Sahami, M., Hearst, M., and Saund, E. (1996). Applying the multiple cause mixture model to text categorization. In Proceedings of the 13th International Machine Learning Conference. Singhal, A (1997). Learning routing queries in a query zone. 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Search for Information Bearing Components in Speech Howard Hua Yang and Hynek Hermansky Department of Electrical and Computer Engineering Oregon Graduate Institute of Science and Technology 20000 NW, Walker Rd., Beaverton, OR97006, USA {hyang,hynek}@ece.ogi.edu, FAX:503 7481406 Abstract In this paper, we use mutual information to characterize the distributions of phonetic and speaker/channel information in a timefrequency space. The mutual information (MI) between the phonetic label and one feature, and the joint mutual information (JMI) between the phonetic label and two or three features are estimated. The Miller's bias formulas for entropy and mutual information estimates are extended to include higher order terms. The MI and the JMI for speaker/channel recognition are also estimated. The results are complementary to those for phonetic classification. Our results show how the phonetic information is locally spread and how the speaker/channel information is globally spread in time and frequency. 1 Introduction Speech signals typically carry information about number of target sources such as linguistic message, speaker identity, and environment in which the speech was produced. In most realistic applications of speech technology, only one or a few information targets are important. For example, one may be interested in identifying the message in the signal regardless of the speaker or the environments in which the speech was produced, or the identification of the speaker is needed regardless of the words the targeted speaker is saying. Thus, not all components of the signal may be equally relevant for a decoding of the targeted information in the signal. The speech research community has at its disposal rather large speech databases which are mainly used for training and testing automatic speech recognition (ASR) systems. There have been relatively few efforts to date to use such databases for deriving reusable knowledge about speech and speech communication processes which could be used for improvements of ASR technology. In this paper we apply information-theoretic approaches to study a large hand-labeled data set of fluent speech to learn about the information structure of the speech signal including the distribution of speech information in frequency and in time. Based on the labeled data set, we analyze the relevancy of the features for phonetic 804 H. H. YangandH Hermansky classifications and speaker/channel variability. The features in this data set are labeled with respect to underlying phonetic classes and files from which the features come from. The phoneme labels relate to the linguistic message in the signal, and the file labels carry the information about speakers and communication channels (each file contains speech of a single speaker transmitted through one telephone channel). Thus, phoneme and file labels are two target variables for statistical inference. The phoneme labels take 19 different values corresponding to 19 broad phoneme categories in the OGI Stories database [2]. The file labels take different values representing different speakers in the OGI Stories database. The relevancy of a set of features is measured by the joint mutual information (JMI) between the features and a target variable. The phoneme target variable represents in our case the linguistic message. The file target variable represents both different speakers and different telephone channels. The joint mutual information between a target variable and the features quantifies the relevancy of the features for that target variable. Mutual information measure the statistical dependence between random variables. Morris et al (1993) used mutual information to find the critical points of information for classifying French Vowel-Plosive-Vowel utterances. Bilmes(1998) showed recently that the information appears to be spread over relatively long temporal spans. While Bilmes used mutual information between two variables on nonlabeled data to reveal the mutual dependencies between the components of the spectral energies in time and frequency, we focused on joint mutual information between the phoneme labels or file labels and one, two or three feature variables in the time-frequency plane[7, 6] and used this concept to gain insight into how information about phonemes and speaker/channel variability is distributed in the time-frequency plane. 2 Data Set and Preprocessing The data set used in this paper is 3-hour phonetically labeled telephone speech, a subset of the English portion (Stories) ofthe OGI multi-lingual database [2] containing approximately 50 seconds of extemporaneous speech from each of 210 different speakers. The speech data is labeled by a variable Y taking 19 values representing 19 most often occurring phoneme categories. The average phoneme duration is about 65 ms and the number of phoneme instances is 6542l. Acoustic features X (fk, t) for the experiments are derived from a short-time analysis of the speech signal with a 20 ms analysis window (Hamming) at the frame t advanced in 10 ms steps. The logarithmic energy at a frequency fk is computed from the squared magnitude FFT using a critical-band spaced (log-like in the frequency variable) weighting function in a manner similar to that of the computation of Perceptual Linear Prediction coefficients [3]. In particular, the 5-th, 8-th and 12th bands are centered around 0.5, 1 and 2 kHz respectively. Each feature X(fk, t) is labeled by a phoneme label YP(t) and a file label Y J (t). We use mutual information to measure the relevancy of X(/k, t - d) across all frequencies fk and in a context window - D ::; d ::; + D for the phoneme classification and the speaker/channel identification. 3 Estimation of MI and Bias Correction In this paper, we only consider the mutual information (MI) between discrete random variables. The phoneme label and the file label are discrete random variables. Search for Information Bearing Components in Speech 805 However, the feature variables are bounded continuous variables. To obtain the quantized features, we divide the maximum range of the observed features into cells of equal volume so that we can use histogram to estimate mutual information defined by " p(x, y) I(Xi Y) = L...Jp(x, y) log2 (x) ( )' x,y P P Y If X and Yare jointly Gaussian, then I(Xi Y) = -~ In(1 - p2) where p is the correlation coefficient between X and Y. However, for speech data the feature variables are generally non-Gaussian and target variables are categorical type variables. Correlations involving a categorical variable are meaningless. The MI can also be written as I(XiY) = H(X) + H(Y) - H(X, Y) = H(Y) - H(YIX) = H(X) - H(XIY) (1) where H (Y IX) is a conditional entropy defined by H(YIX) = - L:p(x) L:p(Ylx) log2P(ylx). x y The two equations in (1) mean that the MI is the uncertainty reduction about Y give X or the uncertainty reduction about X give Y. Based on the histogram, H(X) is estimated by H(X) = - L: ni log2 ni . n n ~ where ni is the number of data points in the i-th cell and n is the data size. And I(X i Y) is estimated by i(Xi Y) = H(X) + H(Y) - H(X, Y). Miller(1954)[4] has shown that H(X) is an underestimate of H(X) and i(Xi Y) is an overestimate of I (X i Y) . The biases are A r - 1 1 E[H(X)] - H(X) = - 2In(2)n + O( n2 ) (2) E[i(X;Y)]-I(X;Y) = (r-l)(c-l) +O(~) 2In(2)n n2 (3) where rand c are the number of cells for X and Y respectively. Interestingly, the first order terms in (2) and (3) do not depend on the probability distribution. After using these formulas to correct the'estimates, the new estimates have the same variances as the old estimates but with reduced biases. However, these formulas break down when rand n are of the same order. Extending Miller's approach, we find a high order correction for the bias. Let {pd be the probability distribution of X, then E[H(X)] - H(X) r-l 1 = - 2In(2)n + 6In(2)n2 (S( {Pi}) - 3r + 2) 1 1 --(S({pd) - 1) + 0(-) 4n3 n 4 (4) 806 H. H. Yang and H. Hermansky The last two terms in the bias (4) depend on the unknown probabilities {pd. In practice they are approximated by the relative frequency estimates. Similarly, we can find the bias formulas of the high order terms O(nl:2) and O(n\) for the MI estimate. When X is evenly distributed, Pi = 1/r, so S( {pd) = r2 and , r-1 1 2 1 2 1 E[H(X)]- H(X) = - 2ln(2)n + 6ln(2)n2 (r - 3r + 2) - 4n3 (r -1) + O(n4). Theoretically S( {Pi}) has no upper bound when one of the probabilities is close to zero. However, in practice it is hard to collect a sample to estimate a very small probability. For this reason, we assume that Pi is either zero or greater than 6/r where 6 > 0 is a small constant does not depend on nor r . Under this assumption S( {pd) ::; r2/6 and the amplitUde of the last term in (4) is less than 4!3 (r2 /6 - 1) . 4 MI in Speech for Phonetic Classification The three hour telephone speech in the OGI database gives us a sample size greater than 1 million, n = 1050000. To estimate the mutual information between three features and a target variable, we need to estimate the entropy H(Xl' X 2, X3, Y). Take B = 20 as the number of bins for each feature variable and C = 19 is the number of phoneme categories. Then the total number of cells is r = B3 * C. After a constant adjustment, assuming 6 = 1, the bias is O( :2) = 6ln(12)n2 (r2 - 3r + 2) = 0.005(bits). It is shown in Fig. 1(a) that X(/4,t) and X(/5,t) are most relevant features for phonetic classification. From Fig. 1(b), at 5 Bark the MI spread around the current frame is 200 ms. Given one feature Xl, the information gain due to the second feature is the difference I(Xl,X2;Y)- I(Xl;Y) = I(X2;YIXd where I(X2; YIXd is called the information gain of X 2 given Xl. It is a conditional mutual information defined by It is shown in Fig. 1(c)-(d) that given X(/5, t) across different bands the maximum information gain is achieved by X(/g, t), and within 5 Bark band the maximum information gain is achieved by X (/5 , t - 5). The mutual informations I(X(/4' t), X(/k, t + d); Y) for k = 1, ... ,15, k ¥ 4, and d = ±1, ... ,±1O, the information gain from the second feature in the vicinity of the first one, are shown in Fig. 2. The asymmetric distribution of the MI around the neighborhood (/5, d = 0) indicates that the phonetic information is spread asymmetrically through time but localized in about 200 ms around the current frame. Based on our data set, we have H(Y) = 3.96 (bits). The JMI for three frequency features and three temporal features are shown in Fig. 1{e)-{f). Based on these estimates, the three frequency features give 28% reduction in uncertainty about Y while the three temporal features give 19% reduction. Search for Information Bearing Components in Speech 807 0.55 0.5 0.45 0.4 :II .Iii 0 .35 :iii! 0.3 0 .25 0 .2 0,'5 0 6 9 12 15 earl< (a) 0 .9 0 .85 0.8 0 . 75 :II .S; 0.7 :iii 0 .85 0.8 0.55 0.5 0 3 8 9 12 15 Bark (C) 1.2 1 . 1 :II .Iii 0.9 :iii 0 . 8 0.7 o .eO~-~3--~B,.------:9:----:,:'::2---:', S· ea'" (e) 0 .5 0.4 :II 0.3 .S; :iii! 0 .2 ) 0 . 1 ~ 0 -400 200 0 200 400 tim_ shift In rna (b) 0.9 O .BS 0 .8 0 .75 :II .'" 07 :iii ~~ 0 .65 0 .6 0 .55 0 .5 -400 -200 0 200 400 time shift In rna (d) 1 . 2 1 . 1 :II .& 0.9 :iii 0 . 8 o.71=--------..,. ---------------~~0~0~---:2~0~0--~0-----2~0~0~-~400 tl",. shirt In rna (f) Figure 1: (a) MIs of individual features in different bands. (b) MIs of individual feature at 5 Bark with different lOms-frame shifts. (c) JMIs of two features: at 5 Bark and in other bands. (d) JMIs of two features: current frame and shifted frames, both at 5 Bark. (e) JMIs of three features: at 5 Bark, 9 Bark and in other bands. The dashed line is th~ JMI level achieved by the two features X (15, t) and X (19, t). (f) JMIs of three features: current frame, 5th frame before current frame, and other shifted frames, all at 5 Bark. The dashed line is the JMI level achieved by X (15 , t) and X (15 , t - 5). The size of our data set is n = 1050000. Therefore, we can reliably estimate the joint 808 H H Yang and H Hermansky MI between three features and the phoneme label. However, to estimate the JMI for more than 3 features we have the problem of curse of dimensionality since for k features, r = Bk * C is exponential increasing. For example, when k = 4, B = 20, and C = 19, the second order bias is O(1/n2) = 2.02 (bits) which is too high to be ignored. To extend our approach beyond the current three-feature level, we need either to enlarge our data set or to find an alternative to the histogram based MI estimation. 0.95 0.9 0.85 ~ 0.8 .Ii 0.75 ~ 0.7 ~0.65 0.6 0.55 0.5 15 Bark o -100 . .. ... ..... ; 100 frame shill In ms Figure 2: The 3-D plot of joint mutual information around X(!4 , t). An asymmetric distribution is apparent especially around 4 Bark and 5 Bark. 1 .3 .---~-~--~-~----, 1 . 2 1 . 1 ~ 0 .9 .S; 2 0 .8 0 . 7 0 .6 0 .5 3 6 9 12 15 Bark (a) 1.5 .----~--~--~----, 1.4 1.3 1 .2 1.1 0.9 O . B 0 . 7 0 . 6 I~ '" '" , , 1.,t band I 15-th band ___ _ \ I \ I \ I I I I I I, ~ ;' , , ~~O~0~--~'0~0---0~--'~00~-~ 200 time shift In rna (b) Figure 3: (a) The MI between one frequency feature and the file label. (b) The JMI between two features and the file identity labels. 5 MI in Speech for Speaker/Channel Recognition The linguistic variability expressed by phoneme labels is not the only variability present in speech. We use the mutual information to evaluate relevance to other Search for Information Bearing Components in Speech 809 sources of variabilities such as speaker/channel variability. Taking the file label as a target variable, we estimated the mutual information for one and two features. It is shown in Fig. 3(a) that the most relevant features are in the very low frequency channels, which in our case of telephone speech carry only very little speech information. Fig. 3(b) shows that the second most relevant feature for speaker/channel recognition is at least 150 ms apart from the first most relevant feature. These results suggest that the information about the speaker and the communication channel is not localized in time. These results are complementary to the results for phonetic classification shown in Fig. 1 (a) and (d) . 6 CONCLUSIONS Our results have shown that the information theoretic analysis of labeled speech data is feasible and useful for obtaining reusable knowledge about speech/channel variabilities. The joint mutual information of two features for phonetic classification is asymmetric around the current frame. We also estimated the joint mutual information between the phoneme labels and three feature variables. The uncertainty about the phonetic classification is reduced by adding more features. The maximum uncertainty reductions due to three frequency features and three temporal features are 28% and 19% respectively. The mutual informations of one and two features for speaker/channel recognition are estimated. The results show that the most relevant features are in the very low frequency bands. At 1 Bark and 5 Bark, the second most relevant temporal feature for speaker/channel recognition is at least 150 ms apart from the first most relevant feature. These results suggest that the information about the speaker and the communication channel is not localized in time. These results are complementary to the results for phonetic classification for which the mutual information is generally localized with some time spread. References [1] J. A. Bilmes. Maximum mutual information based reduction strategies for crosscorrelation based joint distribution modeling. In ICASSP98 , pages 469-472, April 1998. [2] R. Cole, M. Fanty, M. Noel, and T . Lander. Telephone speech corpus development at CSLU. In ICSLP, pages 1815-1818, Yokohama, Sept. 1994. [3] H. Hermansky. Perceptual linear predictive (PLP) analysis of speech. 1. Acoust. Soc. Am., 87(4):1738-1752, April 1990. [4] G. A. Miller. Note on the bias of information estimates. In H. Quastler, editor, Information Theory and Psychology , pages 95-100. The Free Press, Illinois, 1954. [5] Andrew Morris, Jean-Luc Schwartz, and Pierre Escudier. An information theoretical investigation into the distribution of phonetic information across the auditory spectogram. Computer Speech fj Language, 7(2):121-136, April 1993. [6] H. H. Yang, S. Van Vuuren, , S. Sharma, and H. Hermansky. Relevancy of timefrequency features for phonetic classification and speaker-channel recognition. Accepted by Speech Communication, 1999. [7] H. H. Yang, S. Van Vuuren, and H. Hermansky. Relevancy of time-frequency features for phonetic classification measured by mutual information. In ICASSP99, pages 1:225-228, Phoenix, March 1999. PART VII VISUAL PROCESSING
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An Oscillatory Correlation Framework for Computational Auditory Scene Analysis GuyJ.Brown Department of Computer Science University of Sheffield Regent Court, 211 Portobello Street, Sheffield S 1 4DP, UK Email: g.brown@dcs.shefac.uk DeLiang L. Wang Department of Computer and Information Science and Centre for Cognitive Science The Ohio State University Columbus, OH 43210-1277, USA Email: dwang@cis.ohio-state.edu Abstract A neural model is described which uses oscillatory correlation to segregate speech from interfering sound sources. The core of the model is a two-layer neural oscillator network. A sound stream is represented by a synchronized population of oscillators, and different streams are represented by desynchronized oscillator populations. The model has been evaluated using a corpus of speech mixed with interfering sounds, and produces an improvement in signal-to-noise ratio for every mixture. 1 Introduction Speech is seldom heard in isolation: usually, it is mixed with other environmental sounds. Hence, the auditory system must parse the acoustic mixture reaching the ears in order to retrieve a description of each sound source, a process termed auditory scene analysis (ASA) [2]. Conceptually, ASA may be regarded as a two-stage process. The first stage (which we term 'segmentation') decomposes the acoustic stimulus into a collection of sensory elements. In the second stage ('grouping'), elements that are likely to have arisen from the same environmental event are combined into a perceptual structure called a stream. Streams may be further interpreted by higher-level cognitive processes. Recently, there has been a growing interest in the development of computational systems that mimic ASA [4], [1], [5]. Such computational auditory scene analysis (CASA) systems are inspired by auditory function but do not model it closely; rather, they employ symbolic search or high-level inference engines. Although the performance of these systems is encouraging, they are no match for the abilities of a human listener; also, they tend to be complex and computationally intensive. In short, CASA currently remains an unsolved problem for real-time applications such as automatic speech recognition. Given that human listeners can segregate concurrent sounds with apparent ease, computational systems that are more closely modelled on the neurobiological mechanisms of hearing may offer a performance advantage over existing CAS A systems. This observation - together with a desire to understand the neurobiological basis of ASA - has led some investigators to propose neural network models of ASA. Most recently, Brown and Wang [3] have given an account of concurrent vowel separation based on oscillatory correlation. In this framework, oscillators that represent a perceptual stream are synchronized (phase locked with zero phase lag), and are desynchronized from oscillators that represent different streams [8]. Evidence for the oscillatory correlation theory comes from neurobiological studies which report synchronised oscillations in the auditory, visual and olfactory cortices (see [10] for a review). 748 G. J. Brown and D. L. Wang In this paper, we propose a neural network model that uses oscillatory correlation as the underlying neural mechanism for ASA; streams are formed by synchronizing oscillators in a two-dimensional time-frequency network. The model is evaluated on a task that involves the separation of two time-varying sounds. It therefore extends our previous study [3], which only considered the segregation of vowel sounds with static spectra. 2 Model description The input to the model consists of a mixture of speech and an interfering sound source, sampled at a rate of 16 kHz with 16 bit resolution. This input signal is processed in four stages described below (see [10] for a detailed account). 2.1 Peripheral auditory processing Peripheral auditory frequency selectivity is modelled using a bank of 128 gammatone filters with center frequencies equally distributed on the equivalent rectangular bandwidth (ERB) scale between 80 Hz and 5 kHz [1]. Subsequently, the output of each filter is processed by a model of inner hair cell function. The output of the hair cell model is a probabilistic representation of auditory nerve firing activity. 2.2 Mid-level auditory representations Mechanisms similar to those underlying pitch perception can contribute to the perceptual separation of sounds that have different fundamental frequencies (FOs) [3]. Accordingly, the second stage of the model extracts periodicity information from the simulated auditory nerve firing patterns. This is achieved by computing a running autocorrelation of the auditory nerve activity in each channel, forming a representation known as a correlogram [1], [5]. At time step j, the autocorrelation A(iJ,'t) for channel i with time lag 't is given by: K-I A(i, j,'t) = I. r(i,j-k)r(i,j-k-'t)w(k) (1) k=O Here, r is the output of the hair cell model and w is a rectangular window of width K time steps. We use K = 320, corresponding to a window width of 20 ms. The autocorrelation lag 't is computed in L steps of the sampling period between 0 and L-1 ; we use L = 201, corresponding to a maximum delay of 12.5 ms. Equation (1) is computed for M time frames, taken at 10 ms intervals (i.e., at intervals of 160 steps of the time indexj). For periodic sounds, a characteristic 'spine' appears in the correlogram which is centered on the lag corresponding to the stimulus period (Figure 1A). This pitch-related structure can be emphasized by forming a 'pooled' correlogram s(j,'t), which exhibits a prominent peak at the delay corresponding to perceived pitch: N s(j, 't) = I. A (i, j, 't) (2) i = I It is also possible to extract harmonics and formants from the correlogram, since frequency channels that are excited by the same acoustic component share a similar pattern of periodicity. Bands of coherent periodicity can be identified by cross-correlating adjacent correlogram channels; regions of high correlation indicate a harmonic or formant [1]. The cross-correlation C(iJ) between channels i and i+ 1 at time frame j is defined as: L-I C(i,j) = IL.A(i,j, 't)A(i+l,j, 't) (l~i~N-l) (3) t=O Here, A(i, j , 't) is the autocorrelation function of (1) which has been normalized to have zero mean and unity variance. A typical cross-correlation function is shown in Figure 1A. Oscillatory Correlation for CAS A 749 2.3 Neural oscillator network: overview Segmentation and grouping take place within a two-layer oscillator network (Figure IB). The basic unit of the network is a single oscillator, which is defined as a reciprocally connected excitatory variable x and inhibitory variable y [7]. Since each layer of the network takes the form of a time-frequency grid, we index each oscillator according to its frequency channel (i) and time frame (j): Xij = 3xij-xt+2-Yij+lij+Sij+P Yij = £(y(1 + tanh(xi/~» - Yij) (4a) (4b) Here, Ii} represents external input to the oscillator, Si} denotes the coupling from other oscillators in the network, c, 'Y and ~ are parameters, and p is the amplitude of a Gaussian noise term. If coupling and noise are ignored and Ii} is held constant, (4) defines a relaxation oscillator with two time scales. The x-nullcline, i.e. Xii' = 0, is a cubic function and the y-nullcline is a sigmoid function. If Ii" > 0, the two nul clines intersect only at a point along the middle branch of the cubic with ~ chosen small. In this case, the oscillator exhibits a stable limit cycle for small values of c, and is referred to as enabled. The limit cycle alternates between silent and active phases of near steady-state behaviour. Compared to motion within each phase, the alternation between phases takes place rapidly, and is referred to as jumping. If Ii" < 0, the two nullclines intersect at a stable fixed point. In this case, no oscillation occurs. Hence, oscillations in (4) are stimulus-dependent. 2.4 Neural oscillator network: segment layer In the first layer of the network, segments are formed - blocks of synchronised oscillators that trace the evolution of an acoustic component through time and frequency. The first layer is a two-dimensional time-frequency grid of oscillators with a global inhibitor (see Figure IB). The coupling term Sij in (4a) is defined as Sij = ~ Wij,k/H(xk/-ex )- WzH(z-ez) (5) kl E N(i, j) where H is the Heaviside function (i.e., H(x) = I for x ~ 0, and zero otherwise), Wij,kl is the connection weight from an oscillator (iJ) to an oscillator (k,/) and N(iJ) is the four nearest neighbors of (iJ). The threshold ex is chosen so that an oscillator has no influence on its A 5000 B 'N ::z:: ';:: 2741 u <= I!.l '" i3" 1457 d:: ... ~ I!.l U Q) <= <= ~ ..c: U i i , j , I n.D 2.5 5.0 7.5 10.0 12.5 Autocorrelation Lag (ms) Figure I: A. Correlogram of a mixture of speech and trill telephone, taken 450 ms after the start of the stimulus. The pooled correlogram is shown in the bottom panel, and the crosscorrelation function is shown on the right. B. Structure of the two-layer oscillator network. 750 G. J. Brown and D. L. Wang neighbors unless it is in the active phase. The weight of neighboring connections along the time axis is uniformly set to 1. The connection weight between an oscillator (iJ) and its vertical neighbor (i+lJ) is set to 1 if C(iJ) exceeds a threshold Se; otherwise it is set to O. Wz is the weight of inhibition from the global inhibitor z, defined as (6) where <roo = 1 if xi} 2:: Sz for at least one oscillator (iJ), and <roo = 0 otherwise. Hence Sz is a threshold. If <roo = 1, z ~ 1. Small segments may form which do not correspond to perceptually significant acoustic components. In order to remove these noisy fragments, we introduce a lateral potential Pi} for oscillator (iJ), defined as [11]: Pij = (1 - Pij)H[ L.. H(xkl - ex) - epJ - £Pij (7) kleNp(i,j) Here, Sp is a threshold. Nf(iJ) is called the potential neighborhood of (iJ), which is chosen to be (iJ-l) and (iJ+l). I both neighbors of (iJ) are active, Pi} approaches 1 on a fast time scale; otherwise, Pij relaxes to 0 on a slow time scale determined by c. The lateral potential plays its role by gating the input to an oscillator. More specifically, we replace (4a) with iij = 3xij-x:j +2-Yij+ lijH(pij-e) +Sij+P (4a') With Pij initialized to 1, it follows that Pij will drop below the threshold S unless the oscillator (iJ) receives excitation from its entire potential neighborhood. Given our choice of neighborhood in (5), this implies that a segment must extend for at least three consecutive time frames. Oscillators that are stimulated but cannot maintain a high potential are relegated to a discontiguous 'background' of noisy activity. An oscillator (iJ) is stimulated if its corresponding input lij > O. Oscillators are stimulated only if the energy in their corresponding correlogram channel exceeds a threshold Sa. It is evident from (1) that the energy in a correlogram channel i at time j corresponds to A(iJ,O); thus we set Ii} = 0.2 if A(iJ,O) > Sa' and Iij = -5 otherwise. Figure 2A shows the segmentation of a mixture of speech and trill telephone. The network was simulated by the LEGION algorithm [8], producing 94 segments (each represented by a distinct gray level) plus the background (shown in black). For convenience we show all segments together in Figure 2A, but each actually arises during a unique time interval. B 5000 g 2741 >. <.) c: <) ~ 1457 ~ !:S 1: 729 <) U 03 c: c: 315 ~ ..c: U 80 0.0 Time (seconds) 1.5 Time (seconds) Figure 2: A. Segments formed by the first layer of the network for a mixture of speech and trill telephone. B. Categorization of segments according to FO. Gray pixels represent the set P, and white pixels represent regions that do not agree with the FO. Oscillatory Correlation for CASA 751 2.5 Neural oscillator network: grouping layer The second layer is a two-dimensional network of laterally coupled oscillators without global inhibition. Oscillators in this layer are stimulated if the corresponding oscillator in the first layer is stimulated and does not form part of the background. Initially, all oscillators have the same phase, implying that all segments from the first layer are allocated to the same stream. This initialization is consistent with psychophysical evidence suggesting that perceptual fusion is the default state of auditory organisation [2]. In the second layer, an oscillator has the same form as in (4), except that Xu is changed to: iii = 3xij - x~ + 2 - Yij + Ii) 1 + !1H(Pij - a)] + Sij + P (4a") Here, Jl is a small positive parameter; this implies that an oscillator with a high lateral potential gets a slightly higher external input. We choose NpCiJ) and aR so that oscillators which correspond to the longest segment from the first layer are the first to jump to the active phase. The longest segment is identified by using the mechanism described in [9]. The coupling term in (4a") consists of two types of coupling: e v Sij = Sij + Sij (8) Here, S;j represents mutual excitation between oscillators within each segment. We set S~ = 4 if the active oscillators from the same segment occupy more than half of the length of the segment; otherwise S~j = 0.1 if there is at least one active oscillator from the same segment. The coupling term S; denotes vertical connections between oscillators corresponding to different frequency channels and different segments, but within the same time frame. At each time frame, an FO is estimated from the pooled correlogram (2) and this is used to classify frequency channels into two categories: a set of channels, P, that are consistent with the FO, and a set of channels that are not (Figure 2B). Given the delay 'tm at which the largest peak occurs in the pooled correlogram, for each channel i at time frame j, i E P if AU, j, 'tm )/ A(i, j, 0) > ad (9) Since AUJ,O) is the energy in correlogram channel i at time j, (9) amounts to classification on the basis of an energy threshold. We use ad = 0.95. The delay 'tm can be found by using a winner-take-all network, although for simplicity we currently apply a maximum selector. A 5IMM) N :r: '-' 2741 >. u c: Q) ;:::l g' 1457 .. u.. .... Q) C 729 Q) U Q) c: ; 315 ..c: U 80 Time (seconds) N :r: '-' 2741 >. u c: Q) ;:::l go 1457 ~ .... Q) c Q) U "0 c: § ..c: U Time (seconds) Figure 3: A. Snapshot showing the activity of the second layer shortly after the start of simulation. Active oscillators (white pixels) correspond to the speech stream. B. Another snapshot, taken shortly after A. Active oscillators correspond to the telephone stream. 752 G. J. Brown and D. L. Wang The FO classification process operates on channels, rather than segments. As a result, channels within the same segment at a particular time frame may be allocated to different FO categories. Since segments cannot be decomposed, we enforce a rule that all channels of the same frame within each segment must belong to the same FO category as that of the majority of channels. After this conformational step, vertical connections are fonned such that, at each time frame, two oscillators of different segments have mutual excitatory links if the two corresponding channels belong to the same FO category; otherwise they have mutual inhibitory links. S~ is set to -O.S if (iJ) receives an input from its inhibitory links; similarly, s~ is set to O.S if (iJ) receives an input from its vertical excitatory links. At present, our model has no mechanism for grouping segments that do not overlap in time. Accordingly, we limit operation of the second layer to the time span of the longest segment. After fonning lateral connections and trimming by the longest segment, the network is numerically solved using the singular limit method [6]. Figure 3 shows the response of the second layer to the mixture of speech and trill telephone. The figure shows two snapshots of the second layer, where a white pixel indicates an active oscillator and a black pixel indicates a silent oscillator. The network quickly forms two synchronous blocks, which desynchronize from each other. Figure 3A shows a snapshot taken when the oscillator block (stream) corresponding to the segregated speech is in the active phase; Figure 3B shows a subsequent snapshot when the oscillator block corresponding to the trill telephone is in the active phase. Hence, the activity in this layer of the network embodies the result of ASA; the components of an acoustic mixture have been separated using FO infonnation and represented by oscillatory correlation. 2.6 Resynthesis The last stage of the model is a resynthesis path. Phase-corrected output from the gammatone filterbank is divided into 20 ms sections, overlapping by 10 ms and windowed with a raised cosine. A weighting is then applied to each section, which is unity if the corresponding oscillator is in its active phase, and zero otherwise. The weighted filter outputs are summed across all channels to yield a resynthesized wavefonn. A 70 '"'""' 60 c:Q "0 '-' .9 50 <;:; .... 0 40 til ·0 s= 30 I B I e;; 20 s= tlO . r;; 10 s= «I 0 :; 0 -10 r-fI t- , F ::1 • ·n, ~ I f .i~ I I •• .J1l • • I NO Nl N2 N3 N4 N5 N6 N7 N8 N9 Intrusion type B 100 90 '"'""' ~ 80 '-' "0 0 70 .... 0 > 0 60 u ~ 50 » f:.Il 40 o s= 0 ..c: 30 u 0 0 20 0.. CIl IO 0 ",.-,,n I"·' F Irr-~ :;-'\ ~ 'i ' ", m fi t H< itllE "1 ',' ill J "f Bi t'fl J U:, f' j: ,d NO Nl N2 N3 N4 N5 N6 N7 N8 N9 Intrusion type Figure 4: A. SNR before (black bar) and after (grey bar) separation by the model. Results are shown for voiced speech mixed with ten intrusions (NO = 1 kHz tone; Nl = random noise; N2 = noise bursts; N3 = 'cocktail party' noise; N4 = rock music; NS = siren; N6 = trill telephone; N7 = female speech; N8 = male speech; N9 = female speech). B. Percentage of speech energy recovered from each mixture after separation by the model.
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Manifold Stochastic Dynamics for Bayesian Learning Mark Zlochin Department of Computer Science Technion - Israel Institute of Technology Technion City, Haifa 32000, Israel zmark@cs.technion.ac.il YoramBaram Department of Computer Science Technion - Israel Institute of Technology Technion City, Haifa 32000, Israel baram@cs.technion.ac.il Abstract We propose a new Markov Chain Monte Carlo algorithm which is a generalization of the stochastic dynamics method. The algorithm performs exploration of the state space using its intrinsic geometric structure, facilitating efficient sampling of complex distributions. Applied to Bayesian learning in neural networks, our algorithm was found to perform at least as well as the best state-of-the-art method while consuming considerably less time. 1 Introduction In the Bayesian framework predictions are made by integrating the function of interest over the posterior parameter distribution, the lattt~r being the normalized product of the prior distribution and the likelihood. Since in most problems the integrals are too complex to be calculated analytically, approximations are needed. Early works in Bayesian learning for nonlinear models [Buntineand Weigend 1991, MacKay 1992] used Gaussian approximations to the posterior parameter distribution. However, the Gaussian approximation may be poor, especially for complex models, because of the multi-modal character of the posterior distribution. Hybrid Monte Carlo (HMC) [Duane et al. 1987] introduced to the neural network community by [Neal 1996], deals more successfully with multi-modal distributions but is very time consuming. One of the main causes of HMC inefficiency is the anisotropic character of the posterior distribution - the density changes rapidly in some directions while remaining almost constant in others. We present a novel algorithm which overcomes the above problem by using the intrinsic geometrical structure of the model space. 2 Hybrid Monte Carlo Markov Chain Monte Carlo (MCMC) [Gilks et al. 1996] approximates the value E[a] = / a(O)Q(O)dO Manifold Stochastic Dynamics for Bayesian Learning 695 by the mean 1 IV a = N L a(O(t») t=l where e(l) , ... , O(N) are successive states of the ergodic Markov chain with invariant distribution Q(8) . In addition to ergodicity and invariance of Q(O) another quality we would like the Markov chain to have is rapid exploration of the state space. While the first two qualities are rather easily attained, achieving rapid exploration of the state space is often nontrivial. A state-of-the-art MCMC method, capable of sampling from complex distributions, is Hybrid Monte Carlo [Duane et al. 1987]. The algorithm is expressed in terms of sampling from canonical distribution for the state, q, of a "physical" system, defined in terms of the energy function E( q) I: P(q) ex exp(-E(q)) (1) To allow the use of dynamical methods, a "momentum" variable, p, is introduced , with the same dimensionality as q. The canonical distribution over the "phase space" is defined to be: P(q,p) ex exp(-H(q,p)) (2) where H(q ,p) = E(q) + K(p) is the "Hamiltonian", which represents the total energy. f{ (p) is the "kinetic energy" due to momentum, defined as n 2 K (p) = '" J!.L ~2m' i=l l (3) where pi , i = 1, . . . , n are the momentum components and m i is the "mass" associated with i'th component, so that different components can be given different weight. Sampling from the canonical distribution can be done using stochastic dynamics method [Andersen 1980], in which the task is split into two sub tasks - sampling uniformly from values of q and p with a fixed total energy, H(q ,p), and sampling states with different values of H. The first task is done by simulating the Hamiltonian dynamics of the system: dqi BH Pi =+dT BPi m j Different energy levels are obtained by occasional stochastic Gibbs sampling [Geman and Geman 1984] of the momentum. Since q and p are independent, p may be updated without reference to q by drawing a value with probability density proportional to exp( - K (p)), which, in the case of (3), can be easily done, since the Pi'S have independent Gaussian distributions. In practice, Hamiltonian dynamics cannot be simulated exactly, but can be approximated by some discretization using finite time steps. One common approximation is leapfrog discretization [Neal 1996], In the hybrid Monte Carlo method stochastic dynamic transitions are used to generate candidate states for the Metropolis algorithm [Metropolis et al. 1953]. This eliminates certain 1 Note that any probability density that is nowhere zero can be put in this form, by simply defining E( q) = - log P( q) - log Z, for any convenient Z). 696 M Zlochin and Y. Baram drawbacks of the stochastic dynamics such as systematic errors due to leapfrog discretization, since Metropolis algorithm ensures that every transition keeps canonical distribution invariant. However, the empirical comparison between the uncorrected stochastic dynamics and the HMC in application to Bayesian learning in neural networks [Neal 1996] showed that with appropriate discretization stepsize there is no notable difference between the two methods. A modification proposed in [Horowitz 1991] instead of Gibbs sampling of momentum, is to replace p each time by p. cos (0) + ( . sin( 0), where 0 is a small angle and ( is distributed according to N(O, 1). While keeping canonical distribution invariant, this scheme, called momentum persistence, improves the rate of exploration. 3 Riemannian geometry A Riemannian manifold [Amari 1997] is a set e ~ R n equipped with a metric tensor G which is a positive semidefinite matrix defining the inner product between infinitesimal increments as: < dOl, d02 >= doT . G . d02 Let us denote entries of G by Gi,j and entries of G- l by Gi,j. This inner product naturally gives us the norm II dO IIb=< dO, dO >= dOT. G . dO. The Jeffrey prior over e is defined by the density function: 11" ( 0) ex: JiG(ijI where I . I denotes determinant. 3.1 Hamiltonian dynamics over a manifold For Riemannian manifold the dynamics take a more general form than the one described in section 2. If the metric tensor is G and all masses are set to one then the Hamiltonian is given by: 1 H(q,p) = E(q) + 2pT . G- l . P (4) The dynamics are governed by the following set of differential equations [Chavel 1993]: where r~ , k are the Christoffel symbols given by: r i. k =! ~Gi,m(OGm,k + oGm,j _ OGj,k) J, 2 ~ oqj Oqk oqm and q = ~: is related to p by q = G-lp. Manifold Stochastic Dynamics for Bayesian Learning 697 3.2 Riemannian geometry of functions In regression the log-likelihood is proportional to the empirical error, which is simply the Euclidean distance between the target point, t, and candidate function evaluated over the sample. Therefore, the most natural distance measure between the models is the Euclidean seminorm : I d(Ol,{;2)2 =11 hi - !(Plir= L(f(Xi,01) - !(Xi,02)f (5) i=1 The resulting metric tensor is: I G = L{Y'e!(xi,O). Y'd(Xi,Of} = JT . J (6) i=1 where V' e denotes gradient and J = [(] ~~~ d] is the Jacobian matrix. J 3.3 Bayesian geometry A Bayesian approach would suggest the inclusion of prior assumptions about the parameters in the manifold geometry. If, for example, a priori 0 "" N (0, 1/ a), then the log-posterior can be written as: I n 10gp(Olx) = P L(f(Xi , OI) - t)2 + a L(Ok - 0)2 i=l k=1 where P is inverse noise variance. Therefore, the natural metric in the model space is I n d(01, ( 2)2 = P L(f(Xi, ( 1) - !(Xi, (2))2 + a L(O.! - Ok)2 i=l with the metric tensor: "T " GB=p·G+a·I=J .J where j is the "extended Jacobian": j"j = { where &i,j is the Kroneker's delta. i < I i > I k=1 (7) (8) Note, that as a -+ 0, GB -+ PG, hence as the prior becomes vaguer we approach a nonBayesian paradigm. If, on the other hand, a -+ 00 or P . G -+ 0, the Bayesian geometry approaches the Euclidean geometry ofthe parameter space. These are the qualities that we would like the Bayesian geometry to have - if the prior is "strong" in comparison to the likelihood, the exact form of G should be of little importance. The definitions above can be applied to any log-concave prior distribution with the inverse Hessian of the log-prior, (V'V' logp( 0)) -1, replacing a I in (7). The framework is not restricted to regression. For a general distribution class it is natural to use Fisher information matrix, I, as a metric tensor [Amari 1997}. The Bayesian metric tensor then becomes: GB = I + (V'V'logp(O))-l (9) 698 M Zlochin and y. Baram 4 Manifold Stochastic Dynamics As mentioned before, the energy landscape in many regression problems is anisotropic. This degrades the performance of HMC in two aspects: • The dynamics may not be optimal for efficient exploration of the posterior distribution as suggested by the studies of Gaussian diffusions [Hwang et al. 1993]. • The resulting differential equations are stiff [Gear 1971], leading to large discretization errors, which in turn necessitates small time steps, implying that the computational burden is high. Both of these problems disappear if instead of the Euclidean Hamiltonian dynamics used in HMC we simulate dynamics over the manifold equipped with the metric tensor G B proposed in the previous section. In the context of regression from the definition G B = jT . j, we obtain an alternative • & d2q . . & equatIOn lor dT2 ,In a matnx lorm: 2 ' d q = -G- 1("V E + jT oj q) dT2 B dT (10) In the canonical distribution P(q,p) ex: exp(-H(q,p)) the conditional distribution of p given q is a zero-mean Gaussian with the covariance matrix G B (q) and the marginal distribution over q is proportional to exp( -E(q))1r(q). This is equivalent to mUltiplying the prior by the Jeffrey prior2. The sampling from the canonical distribution is two-fold: • Simulate the Hamiltonian dynamics (3.1) for one time-step using leapfrog discretisation. • Replace p using momentum persistence. Unlike the HMC case, the momentum perturbation (is distributed according to N(O, GB). The actual weights mUltiplying the matrices I and G in (7) may be chosen to be different from the specified a and /3, so as to improve numerical stability. 5 Empirical comparison 5.1 Robot ann problem We compared the performance of the Manifold Stochastic Dynamics (MSD) algorithm with the standard HMC. The comparison was carried using MacKay's robot arm problem which is a common benchmark for Bayesian methods in neural networks [MacKay 1992, Neal 1996]. The robot arm problem is concerned with the mapping: YI = 2.0 cos Xl + 1.3 COS(XI + X2) + el, Y2 = 2.0 sin Xl + 1.3 sin(xi + X2) + e2 where el, e2 are independent Gaussian noise variables of standard deviation 0.05. The dataset used by Neal and Mackay contained 200 examples in the training set and 400 in the test set. 2In fact, since the actual prior over the weights is unknown, a truly Bayesian approach would be to use a non-informative prior such as 71"( q). In this paper we kept the modified prior which is the product of 7I"(q) and a zero-mean Gaussian. Manifold Stochastic Dynamics for Bayesian Learning 699 1.2 , 0.9 ...,...-- - - 0.8 --, , "0.7 , 0.8 ", , 0.6 0.6 0.5 0.4 0.4 0.3 0.2 0.2 0 0.1 0 -0.2 0 10 20 30 40 50 0 10 20 30 40 50 Figure 1: Average (over the 1 0 runs) autocorrelation of input-to-hidden (left) and hiddento-output (right) weights for HMC with 100 and 30 leapfrog steps per iteration and MSD with single leapfrog step per iteration. The horizontal axis gives the lags, measured in number of iterations. We used a neural network with two input units, one hidden layer containing 8 tanh units and two linear output units. The hyperparameter f3 was set to its correct value of 400 and 0" was chosen to be 1. 5.2 Algorithms We compared MSD with two versions of HMC - with 30 and with 100 leapfrog steps per iteration, henceforth referred to as HMC30 and HMCIOO. MSD was run with a single leapfrog step per iteration. In all three algorithms momentum was resampled using persistence with cos(O) = 0.95. A single iteration of HMC100 required about 4.8 . 106 floating point operations (flops), HMC30 required 1.4 . 106 flops and MSD required 0.5 . 106 flops. Hence the computationalload of MSD was about one third of that of HMC30 and 10 times lower than that of HMClOO. The discretization stepsize for HMC was chosen so as to keep the rejection rate below 5%. An equivalent criterion of average error in the Hamiltonian around 0.05 was used for the MSD. All three sampling algorithms were run 10 times, each time for 3000 iteration with the first 1000 samples discarded in order to allow the algorithms to reach the regions of high probability. 5.3 Results One appropriate measure for the rate of state space exploration is weights autocorrelation [Neal 1996]. As shown in Figure 1, the behavior of MSD was clearly superior to that of HMC. Another value of interest is the total squared error over the test set. The predictions for the test set were made as follows. A subsample of 100 parameter vectors waS generated by taking every twentieth sample vector starting from 1001 and on. The predicted value was 700 M. Zlochin and Y. Baram the average over the empirical function distribution of this sUbsample. The total squared errors, nonnalized with respect to the variance on the test cases, have the following statistics (over the 10 runs): average standard deviation HMC30 1.314 0.074 HMCI00 1.167 0.044 MSD 1.161 0.023 The average error ofHMC30 is high, indicating that the algorithm failed to reach the region of high probability. The errors of HMC 1 00 and MSD are comparable but the standard deviation for MSD is twice as low as that for HMC 1 00, meaning that the estimate obtained using MSD is more reliable. 6 Conclusion We have described a new algorithm for efficient sampling from complex distributions such as those appearing in Bayesian learning with non-linear models. The empirical comparison shows that our algorithm achieves results superior to the best achieved by existing algorithms in considerably smaller computation time. References [Amari 1997] Amari S., "Natural Gradient Works Efficiently in Learning", Neural Computation, vol. 10, pp.251-276. [Andersen 1980] Andersen H.e., "Molecular dynamics simulations at constant pressure and/or temperature", Journal of Chemical Physics, vol. 3,pp. 589-603. [Buntine and Weigend 1991] "Bayesian back-propagation", Complex systems, vol. 5, pp. 603-643. [Chavel 1993] Chavel I., Riemannian Geometry: A Modem Introduction, University Press, Cambridge. [Duane et al. 1987] "Hybrid Monte Carlo", Physics Letters B,vol. 195,pp. 216-222. [Gear 1971] Gear e.W., Numerical initial value problems in ordinary differential equations, Prentice Hall. [Geman and Geman 1984] Geman S.,Geman D., "Stochastic relaxation,Gibbs distributions and the Bayesian restoration of images", IEEE Trans.,PAMI6,721-741. [Gilks et al. 1996] Gilks W.R., Richardson S. and Spiegelhalter DJ., Markov Chain Monte Carlo in Practice, Chapman&Hall. [Hwang et al. 1993] Hwang, C.,-R, Hwang-Ma S.,-Y. and Shen. S.,-J., "Accelerating Gaussian diffusions", Ann. Appl. Prob. , vol. 3, 897-913. [Horowitz 1991] Horowitz A.M., "A generalized guided Monte Carlo algorithm", Physics Letters B" vol. 268, pp. 247-252. [MacKay 1992] MacKay D.le., Bayesian Methods for Adaptive Models, Ph.D. thesis, California Institute of Technology. [Metropolis et al. 1953] Metropolis N., Rosenbluth A.W., Rosenbluth M.N., Teller A.H. and Teller E., "Equation of State Calculations by Fast Computing Machines", Journal of Chemical Physics,vol.21,pp. 1087-1092. [Neal 1996] Neal, R.M., Bayesian Learn ing for Neural Networks, Springer 1996. PART V IMPLEMENTATION
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Bifurcation Analysis of a Silicon Neuron Girish N. Patel] , Gennady s. Cymbalyuk2,3, Ronald L. Calabrese2, and Stephen P. DeWeerth1 lSchool of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, Ga. 30332-0250 {girish.patel, steve.deweerth} @ece.gatech.edu 2Department of Biology Emory University 1510 Clifton Road, Atlanta, GA 30322 {gcym, rcalabre}@biology.emory.edu 3Institute of Mathematical Problems in Biology RAS Pushchino, Moscow Region, Russia 142292 (on leave) Abstract We have developed a VLSI silicon neuron and a corresponding mathematical model that is a two state-variable system. We describe the circuit implementation and compare the behaviors observed in the silicon neuron and the mathematical model. We also perform bifurcation analysis of the mathematical model by varying the externally applied current and show that the behaviors exhibited by the silicon neuron under corresponding conditions are in good agreement to those predicted by the bifurcation analysis. 1 Introduction The use of hardware models to understand dynamical behaviors in biological systems is an approach that has a long and fruitful history [1 ][2]. The implementation in silicon of oscillatory neural networks that model rhythmic motor-pattern generation in animals is one recent addition to these modeling efforts [3][4]. The oscillatory patterns generated by these systems result from intrinsic membrane properties of individual neurons and their synaptic interactions within the network [5]. As the complexity of these oscillatory silicon systems increases, effective mathematical analysis becomes increasingly more important to our understanding their behavior. However, the nonlinear dynamical behaviors of the model neurons and the large-scale interconnectivity among these neurons makes it very difficult to analyze theoretically the behavior of the resulting very large-scale integrated (VLSI) systems. Thus, it is important to first identify methods for modeling the model neurons that underlie these oscillatory systems. Several simplified neuronal models have been used in the mathematical simulations of pattern generating networks [6][7][8]. In this paper, we describe the implementation of a 732 G. N Patel, G. S. Cymbalyuk, R. L. Calabrese and S. P. DeWeerth two-state-variable silicon neuron that has been used effectively to develop oscillatory networks [9][10]. We then derive a mathematical model of this implementation and analyze the neuron and the model using nonlinear dynamical techniques including bifurcation analysis [11]. Finally, we compare the experimental data derived from the silicon neuron to that obtained from the mathematical model. 2 The silicon model neuron The schematic for our silicon model neuron is shown in Figure 1. This silicon neuron is inspired by the two-state, Morris-Lecar neuron model [12][ 13]. Transistor M I ' analogous to the voltage-gated calcium channel in the Morris-Lecar model, provides an instantaneous inward current that raises the membrane potential towards V High when the membrane is depolarized. Transistor M2 ' analogous to the voltage-gated potassium channel in the Morris-Lecar model, provides a delayed outward current that lowers the membrane potential toward V Low when the membrane is depolarized. V H and V L are analogous to the half-activation voltages for the inward and outward currents, respectively. The voltages across CI and C2 are the state variables representing the membrane potential, V, and the slow "activation" variable of the outward current, W, respectively. The W -nullcline represents its steady-state activation curve. Unlike the Morris-Lecar model, our silicon neuron model does not possess a leak current. Using current conservation at node V, the net current charging CI is given by (1) where iH and iL are the output currents of a differential pair circuit, and a p and aN describe the ohmic effects of transistors MJ and M2, respectively. The net current into C2 is given by (2) where ix is the output current of the OTA, and ~p and ~N account for ohmic effects of the pull-up and the pull-down transistors inside the OTA. VHigh OTA v w '------<>--- V Low Figure 1: Circuit diagram of the silicon neuron. The circuit incorporates analog building blocks including two differential pair circuits composed of a bias current, IBH, and transistors M4-Ms, and a bias current, IBL, and transistors M6-M7' and a single followerintegrator circuit composed of an operational transconductance amplifier (OTA), Xl in the configuration shown and a load capacitor, C2. The response of the follower-integrator circuit is similar to a first-order low-pass filter. Bifurcation Analysis of a Silicon Neuron 733 The output currents of the differential-pair and an OTA circuits, derived by using subthreshold transistor equations [2], are a Fenni function and a hyperbolic-tangent function, respectively [2]. Substituting these functions for iH, iL, and ix in (1) and (2) yields where K(V-YH) / UT K(W - YL) / UT . e e C 1 V = Iextap + IBH K(V _ YH) / UT a p - IBL K(W _ YL)/ UT aN l+e l+e I v - V High/ UT a p = -e YLow - V I UT aN = 1 - e - W / UT ~N = 1- e (3) (4) UT is the thennal voltage, V dd is the supply voltage, and K is a fabrication dependent parameter. The tenns a p and aN limit the range of V to within V High and V Low' and the terms ~p and ~N limit the range of W to within the supply rails (Vdd and Gnd). In order to compare the model to the experimental results, we needed to determine values for all of the model parameters. V Hi!\h' V Low' V H' V L ' and V dd were directly measured in experiments. The parameters IBH and IBL were measured by voltage-clamp experiments performed on the silicon neuron. At room temperature, UT ::::: 0.025 volts. The value of K ::::: 0.65 was estimated by measuring the slope of the steady-state activation curve of inward current. Because W was implemented as an inaccessible node, IT could only be estimated. Based on the circuit design, we can assume that the bias currents IT and IBH are of the same order of magnitude. We choose IT::::: 2.2 nA to fit the bifurcation diagram (see Figure 3). Cl and C2, which are assumed to be identical according to the physical design, are time scaling parameters in the model. We choose their values (Cl = C2 = 28 pF) to fit frequency dependence on lext (see Figure 4). 3 Bifurcation analysis The silicon neuron and the mathematical model! described by (3) demonstrate various dynamical behaviors under different parametric conditions. In particular, stable oscillations and steady-state equilibria are observed for different values of the externally applied current, I ext . We focused our analysis on the influence of I ext on the neuron behavior for two reasons: (i) it provides insight about effects of synaptic currents, and (ii) it allows comparison with neurophysiological experiments in which polarizing current is used as a primary control parameter. The main results of this work are presented as the comparison between the mathematical models and the experimental data represented as bifurcation diagrams and frequency dependencies. The null clines described by (3) and for lext = 32 nA are shown in Figure 2A. In the regime that we operate the circuit, the W -null cline is an almost-linear curve and the Vnullcline is an N-shaped curve. From (3), it can be seen that when IBH + lext > IBL the nullclines cross at (V, W)::::: (VHigh, VHigh) and the system has high voltage (about 5 volts) steady-state equilibrium. Similarly, for Iext close to zero, the system has one stable equilibrium point close to (V, W) ::::: (V Low' V Low). !The parameters used throughout the analyses of the model are V Low = 0 V , V High = 5 V, V L = V H = 2.5 V, I BH = 6.5 nA , I BL = 42 nA, IT = 2.2 nA , V dd = 5 V, Vt = 0.025 mV, and K = 0.65. 734 A B 2.85 2.8 2.75 ~ 2.7 (5 > .....,.. ~ 2.65 2.6 2.55 2.5 G. N Patel, G. S. Cymba/yule, R. L. Calabrese and S. P. De Weerth / / @ / W-nullcline V-nullcline I 2.45~ ______ ~ ______ ~ ______ ~ ______ ~ ______ ~1 o 3.2 3 2.8 In l2.6 > 2.4 2.2 2 0 5 10 2 3 V (volts) 15 20 time (msec) 4 5 25 30 35 Figure 2: Nullclines and trajectories in the model of the silicon neuron for lex! = 32 nA. The system exhibits a stable limit-cycle (filled circles), an unstable limit-cycle (unfilled circles), and stable equilibrium point. Unstable limit-cycle separates the basins of attraction of the stable limit-cycle and stable equilibrium point. Thus, trajectories initiated within the area bounded by the unstable limit-cycle approach the stable equilibrium point (solid line in A's inset, and "x's" in B). Trajectories initiated outside the unstable limitcycle approach the stable limit-cycle. In A, the inset shows an expansion at the intersection of the V - and W -nullclines. Bifurcation Analysis of a Silicon Neuron 735 A Experimental data 5 ........................... ~ ",>0< • x • • • x 4 • • ~ .. 003 • ~ .• 0 x xxxx •• x 2:>t#* xxxx >2 x I • • 1 .. • • • • • • 0 ••••••••••••••••••••••••••• x 0 10 20 30 40 50 lext (nAmps) Modeling data B 5 ....................... -. ( . 4 • • • • • 003 ,J .0 rt---------; > ->2 • 1 • • . ~ 0 '-....................... 0 10 20 30 40 50 I ext (nAmps) Figure 3: Bifurcation diagrams of the hardware implementation (A) and of the mathematical model (B) under variation of the externally applied current. In A, the steadystate equilibrium potential of V is denoted by "x"s. The maximum and minimum values of V during stable oscillations are denoted by the filled circles. In B, the stable and unstable equilibrium points are denoted by the solid and dashed curve, respectively, and the minimum and maximum values of the stable and unstable oscillations are denoted by the filled and unfilled circles, respectively. In B, limit-cycle oscillations appear and disappear via sub-critical Andronov-Hopf bifurcations. The bifurcation diagram (B) was computed with the LOCBIF program [14]. 736 G. N. Patel, G. S. Cymbalyuk, R. L. Calabrese and S. P. De Weerth A Experimental data 100~--~--~----~~ 80 • N • ~ >. 60 • • g • .~~.. ~ 40 • ••• .., t:T ... ~ . LL 20 o~--~----~------~ o 10 20 30 lext (nAmps) B 100 - 80 N ::c >. 60 0 c Q) :l 40 t:T Q) ... LL 20 0 0 Modeling data • • • • • • ~. • ••• • •• . ~ 10 20 30 lext (nAmps) Figure 4: Frequency dependence of the silicon neuron (A) and the mathematical model (B) on the externally applied current. For moderate values of lext ([1 nA,34 nA)), the stable and unstable equilibrium points are close to (V, W) ::::: (V H' V L) (Figure 3). In experiments in which lext was varied, we observed a hard loss of the stability of the steady-state equilibrium and a transition into oscillations at lext = 7.2 nA (I ext = 27.5 nA). In the mathematical model, at the critical value of lext = 7.7 nA (lext = 27.8 nA), an unstable limit cycle appears via a subcritical Andronov-Hopf bifurcation. This unstable limit cycle merges with the stable limit cycle at the fold bifurcation at lext = 3.4 nA (lext = 32.1 nA). Similarly, in the experiments, we observed hard loss of stability of oscillations at lext = 2.0 nA (I ext = 32.8 nA). Thus, the system demonstrates hysteresis. For example. when lext = 20 nA the silicon neuron has only one stable regime, namely, stable oscillations. Then if external current is slowly increased to lext = 32.8 nA. the form of oscillations changes. At this critical value of the current, the oscillations suddenly lose stability, and only steady-state equilibrium is stable. Now, when the external current is reduced, the steady-state equilibrium is observed at the values of the current where oscillations were previously exhibited. Thus, within the ranges of externally applied currents (2.0,7.2) and (27.5,32.8), oscillations and a steady-state equilibrium are stable regimes as shown in Figure 2. 4 Discussion We have developed a two-state silicon neuron and a mathematical model that describes the behavior of this neuron. We have shown experimentally and verified mathematically that this silicon neuron has three regions of operation under the variation of its external current (one of its parameters). We also perform bifurcation analysis of the mathematical model by varying the externally applied current and show that the behaviors exhibited by the silicon neuron under corresponding conditions are in good agreement to those predicted by the bifurcation analysis. This analysis and comparison to experiment is an important step toward our understanding of a variety of oscillatory hardware networks that we and others are developing. The Bifurcation Analysis of a Silicon Neuron 737 model facilitates an understanding of the neurons that the hardware alone does not provide. In particular for this neuron, the model allows us to determine the location of the unstable fixed points and the types of bifurcations that are exhibited. In higher-order systems, we expect that the model will provide us insight about observed behaviors and complex bifurcations in the phase space. The good matching between the model and the experimental data described in this paper gives us some confidence that future analysis efforts will prove fruitful. Acknowledgments S. DeWeerth and G. Patel are funded by NSF grant IBN-95 II 721 , G.S. Cymbalyuk is supported by Russian Foundation of Fundamental Research grant 99-04-49112, R.L. Calabrese and G.S. Cymbalyuk are supported by NIH grants NS24072 and NS34975. References [1] Van Der Pol, B (1939) Biological rhythms considered as relaxation oscillations In H. Bremmer and c.J. Bouwkamp (eds) Selected Scientific Papers, Vol 2, North Holland Pub. Co., 1960. [2] Mead, C.A. Analog VLSI and Neural Systems. Addison-Wesley, Reading, MA, 1989. [3] Simoni, M.E, Patel, G.N., DeWeerth, S.P., & Calabrese, RL. Analog VLSI model of the leech heartbeat elemental oscillator. Sixth Annual Computational Neuroscience Meeting, 1997. in Big Sky, Montana. [4] DeWeerth, S., Patel, G., Schimmel, D., Simoni, M. and Calabrese, R (1997). In Proceedings of the Seventeenth Conference on Advanced Research in VLSI, RB. Brown and A.T. Ishii (eds), Los Alamitos, CA: IEEE Computer Society, 182-200. [5] Marder, E. & Calabrese, RL. (1996) Principles of rhythmic motor pattern generation. Physiological Reviews 76 (3): 687-717. [6] Kopell, N. & Ermentrout, B. (1988) Coupled oscillators and the design of central pattern generators. Mathematical biosciences 90: 87-109. [7] Skinner, EK., Turrigiano, G.G., & Marder, E. (1993) Frequency and burst duration in oscillating neurons and two-cell networks. Biological Cybernetics 69: 375-383. [8] Skinner, EK., Gramoll, S., Calabrese, R.L., Kopell, N. & Marder, E. (1994) Frequency control in biological half-center oscillators. In EH. Eeckman (ed.), Computation in neurons and neural systems, pp. 223-228, Boston: Kluwer Academic Publishers. [9] Patel, G. Holleman, J., DeWeerth, S. Analog VLSI model of intersegmental coordination with nearest-neighbor coupling. In , 1997. [10] Patel, G. A neuromorphic architecture for modelling intersegmental coordination. Ph.D. dissertation, Georgia Institute of Technology, 1999. [11] J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Applied Mathematical Sciences, 42. Springer-Verlag, New York, New York, Heidelberg, Berlin, 1983. [12] Morris, C. and Lecar, H. (1981) Voltage oscillations in the barnacle giant muscle fiber. Biophys. J, 35: 193-213. [13] Rinzel, J. & Ermentrout, G.B. (1989) Analysis of Neural Excitability and Oscillations. In C. Koch and I. Segev (eds) Methods in Neuronal Modeling from Synapses to Networks. MIT press, Cambridge, MA. [14] Khibnik, A. I. , Kuznetsov, Yu.A., Levitin, v.v., Nikolaev, E.V. (1993) Continuation techniques and interactive software for bifurcation analysis of ODEs and iterated maps. Physica D 62 (1-4): 360-367.
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Speech Modelling Using Subspace and EM Techniques Gavin Smith Cambridge University Engineering Department Cambridge CB2 1PZ England gas1 oo3@eng.cam.ac.uk Joao FG de Freitas Computer Science Division 487 Soda Hall UC Berkeley CA 94720-1776, USA. jfgf@cs.berkeley.edu 1 Mahesan Niranjan Computer Science Sheffield University Sheffield. S 1 4DP England m.niranjan@dcs.shef.ac.uk Abstract Tony Robinson Cambridge University Engineering Department Cambridge CB2 IPZ England ajr@eng.cam.ac.uk The speech waveform can be modelled as a piecewise-stationary linear stochastic state space system, and its parameters can be estimated using an expectation-maximisation (EM) algorithm. One problem is the initialisation of the EM algorithm. Standard initialisation schemes can lead to poor formant trajectories. But these trajectories however are important for vowel intelligibility. The aim of this paper is to investigate the suitability of subspace identification methods to initialise EM. The paper compares the subspace state space system identification (4SID) method with the EM algorithm. The 4SID and EM methods are similar in that they both estimate a state sequence (but using Kalman filters and Kalman smoothers respectively), and then estimate parameters (but using least-squares and maximum likelihood respectively). The similarity of 4SID and EM motivates the use of 4SID to initialise EM. Also, 4SID is non-iterative and requires no initialisation, whereas EM is iterative and requires initialisation. However 4SID is sub-optimal compared to EM in a probabilistic sense. During experiments on real speech, 4SID methods compare favourably with conventional initialisation techniques. They produce smoother formant trajectories, have greater frequency resolution, and produce higher likelihoods. 1 Work done while in Cambridge Engineering Dept., UK. Speech Modelling Using Subspace and EM Techniques 797 1 Introduction This paper models speech using a stochastic state space model, where model parameters are estimated using the expectation-maximisation (EM) technique. One problem is the initialisation of the EM algorithm. Standard initialisation schemes can lead to poor formant trajectories. These trajectories are however important for vowel intelligibility. This paper investigates the suitability of subspace state space system identification (4SIO) techniques [10,11], which are popular in system identification, for EM initialisation. Speech is split into fixed-length, overlapping frames. Overlap encourages temporally smoother parameter transitions between frames. Oue to the slow non-stationary behaviour of speech, each frame of speech is assumed quasi-stationary and represented as a linear time-invariant stochastic state space (SS) model. Xt+l = AXt + Wt Yt CXt + Vt (1) (2) The system order is p. Xt E ~p X l is the state vector. A E ~p x p and C E ~l x p are system parameters. The output Yt E ~ is the speech signal at the microphone. Process and observation noises are modelled as white zero-mean Gaussian stationary noises Wt E ~p x l f"V N(O, Q) and Vt E ~ f"V N(O, R) respectively. The problem definition is to estimate parameters e = (A, c, Q, R) from speech Yt only. The structure of the paper is as follows. The theory section describes EM and 4SIO applied to the parameter estimation of the above SS model. The similarity of 4SIO and EM motivates the use of 4SID to initialise EM. Experiments on real speech then compare 4SIO with more conventional initialisation methods. The discussion then compares 4SIO with EM. 2 Theory 2.1 The Expectation-Maximisation (EM) Technique Given a sequence of N observations Yl:N of a signal such as speech, the maximum likelihood estimate for the parameters is 9ML = arg maxep(Yl:N!e) . EM breaks the maximisation of this potentially difficult likelihood function down into an iterative maximisation of a simpler likelihood function, generating a new estimate ek each iteration. Rewriting P(Yl:N!e) in terms of a hidden state sequence Xl:N, and taking expectations over P(Xl:N!Yl:N, ek) 10gp(Yl:N!e) 10gp(Yl:N!e) = 10gp(Xl:N,yl:N!e) -logp(Xl:N!Yl:N,e) Ek[logp(Xl:N, Yl:N!e)] - E k[logp(Xl :N!Yl:N, e)] (3) (4) Iterative maximisation of the first expectation in equation 4 guarantees an increase in 10gp(Yl:N!e). (5) This converges to a local or global maximum depending on the initial parameter estimate eo. Refer to [8] for more details. EM can thus be applied to the stochastic state space 798 G. Smith, J. F. G. d. Freitas, T. Robinson and M Niranjan model of equations 1 and 2 to determine optimal parameters e. An explanation is given in [3]. The EM algorithm applied to the SS system consists of two stages per iteration. Firstly, given current parameter estimates, states are estimated using a Kalman smoother. Secondly, given these states, new parameters are estimated by maximising the expected log likelihood function. We employ the Rauch-Tung-Striebel formulation of the Kalman smoother [2]. 2.2 The State-Space Model Equations 1 and 2 can be cast in block matrix form and are termed the state sequence and block output equations respectively [10]. Note that the use of blocking and fixed-length signals applies restrictions to the general model in section 1. i > P is the block size. Xi+I,i+j YI!i AiXl,j + arWI!i riXI,j + HrWI!i + VI!i (6) (7) Xi+I,i+j is a state sequence matrix; its columns are the state vectors from time (i + 1) to (i+j). XI,j is similarly defined. Y W is a Hankel matrix of outputs from time 1 to (i+j-1). W and V are similarly defined. a i is a reversed extended controllability-type matrix, r i is the extended observability matrix and Hi is a Toeplitz matrix. These are all defined below where IPxp is an identity matrix. XI,j def [Xl X2 X3 ••. Xj] [ci, 1 r·~ ~ a w def [Ai- I Ai- 2 ... I] ~ [ y, Y2 Yj 1 [ c1-, :J Y ~f Y2 Y3 Yj+1 H~~f 0 l!i : , Yi Yi+l Yi+j-l C A sequence of outputs can be separated into two block output equations containing past and future outputs denoted with subscriptsp and! respectively. With Yp dg YI!i, Y, dg de, de, Y i+1!2i and similarly for W and V, and Xp = XI,j and X, = Xi+I,i+j, past and future are related by the equations AiXp +arWp riXp + HiWp + Vp rix, + HiW, + V, 2.3 Subspace State Space System Identification (4SID) Techniques (8) (9) (10) Comments throughout this section on 4SIO are largely taken from the work of Van Overschee and Oe Moor [10]. 4SIO methods are related to instrumental variable (IV) methods [11]. 4SIO algorithms are composed of two stages. Stage one involves the low-rank approximation and estimation of the extended observability matrix directly from the output Speech Modelling Using Subspace and EM Techniques 799 data. For example, consider the future output block equation 10. Y, undergoes an orthogonal projection onto the row space ofY p' This is denoted by Y, /'J p = Y, YJ (Y p YJ) ty p, where t is the Moore-Penrose inverse. r iX, /'Jp + HfW, /'J p + V, /'Jp rix,/'Jp (11) Stage two involves estimation of system parameters. The singular value decomposition of Y, /'Jp allows the observability and state sequence matrices to be estimated to within a similarity transform from the column and row spaces respectively. From these two matrices, system parameters (A, c, Q, R) can be determined by least-squares. There are two interesting comments. Firstly, the orthogonal projection from stage one coincides with a minimum error between true data Y, and its linear prediction from Y p in the Frobenius norm. Greater flexibility is obtained by weighting the projection with matrices WI and W 2 and analysing this: WI (YJi'J p )W2 • 4SID and IV methods differ with respect to these weighting matrices. Weighting is similar to prefiltering the observations prior to analysis to preferentially weight some frequency domain, as is common in identification theory [6]. Secondly, the state estimates from stage two can be considered as outputs from a parallel bank of Kalman filters, each one estimating a state from the previous i observations, and initialised using zero conditions. The particular subspace algorithm and software used in this paper is the sto-pos algorithm as detailed in [10]. Although this algorithm introduces a small bias into some of the parameter estimates, it guarantees positive realness of the covariance sequence, which in turn guarantees the definition of a forward innovations model. 3 Experiments Experiments are conducted on the phrase "in arithmetic", spoken by an adult male. The speech waveform is obtained from the Eurom 0 database [4] and sampled at 16 kHz. The speech waveform is divided into fixed-length, overlapping frames, the mean is subtracted and then a hamming window is applied. Frames are 15 ms in duration, shifted 7.5 ms each frame. Speech is modelled as detailed in section 1. All models are order 8. A frame is assumed silent and no analysis done when the mean energy per sample is less than an empirically defined threshold. For the EM algorithm, a modified version of the software in [3] is used. The initial state vector and covariance matrix are set to zero and identity respectively, and 50 iterations are applied. Q is updated by taking its diagonal only in the M-step for numerical stability (see [3]). In these experiments, three schemes are compared at initialising parameters for the EM algorithm, that is the estimation of 9 0 . These schemes are compared in terms of their formant trajectories relative to the spectrogram and their likelihoods. The three schemes are • 4SID. This is the subspace method in section 2.3 with block size 16 . • ARMA. This estimates 9 0 using the customised Matlab armax function!, which models the speech waveform as an autoregressive moving average (ARMA) process, with order 8 polynomials. I armax minimises a robustified quadratic prediction error criterion using an iterative GaussNewton algorithm, initialised using a four-stage least-squares instrumental variables algorithm [7]. 800 G. Smith, J F. G. d. Freitas, T. Robinson and M Niranjan • AR(l). This uses a simplistic method, and models the speech waveform as a first order autoregressive (AR) process with some randomness introduced into the estimation. It still initialises all parameters fully2. Results are shown in Figures 1 and 2. Figure 1 shows the speech waveform, spectrogram and formant trajectories for EM with all three initialisation schemes. Here formant frequencies are derived from the phase of the positive phase eigenvalues of A after 50 iterations of EM. Comparison with the spectrogram shows that for this order 8 model, 4SID-EM produces best formant trajectories. Figure 2 shows mean average plots of likelihood against EM iteration number for each initialisation scheme. 4SID-EM gives greater likelihoods than ARMA-EM and AR(l)-EM. The difference in formant trajectories between subspaceEM and ARMA-EM despite the high likelihoods, demonstrates the multi-modality of the likelihood function. For AR(l)-EM, a few frames were not estimated due to numerical instability. 4 Discussion Both the 4SID and EM algorithms employ similar methodologies: states are first estimated using a Kalman device, and then these states are used to estimate system parameters according to similar criteria. However in EM, states are estimated using past, present and future observations with a Kalman smoother; system parameters are then estimated using maximum likelihood (ML). Whereas in 4SID, states are estimated using the previous i observations only with non-steady state Kalman filters. System parameters are then estimated using least-squares (LS) subject to a positive realness constraint for the covariance sequence. Refer also to [5] for a similar comparison. 4SID algorithms are sub-optimal for three reasons. Firstly, states are estimated using only partial observations sequences. Secondly, the LS criterion is only an approximation to the ML criterion. Thirdly, the positive realness constraint introduces bias. A positive realness constraint is necessary due to a finite amount of data and any lacking in the SS model. For this reason, 4SID methods are used to initialise rather than replace EM in these experiments. 4SID methods also have some advantages. Firstly, they are linear and non-iterative, and do not suffer from the disadvantages typical of iterative algorithms (including EM) such as sensitivity to initial conditions, convergence to local minima, and the definition of convergence criteria. Secondly, they require little prior parameterisation except the definition of the system order, which can be determined in situ from observation of the singular values of the orthogonal projection. Thirdly, the use of the SVD gives numerical robustness to the algorithms. Fourthly, they have higher frequency resolution than prediction error minimisation methods such as ARMA and AR [1]. 5 Conclusions 4SID methods can be used to initialise EM giving better formant tracks, higher likelihoods and better frequency resolution than more conventional initialisation methods. In the future we hope to compare 4SID methods with EM in a principled probabilistic manner, investigate weighting matrices further, and apply these methods to speech enhancement. Further work is done by Smith et al. in [9], and similar work done by Grivel et aI. in [5]. Acknowledgements We are grateful for the use of 4SID software supplied with [10] and the EM software of 2Presented in the software in [3], this method is best used when the dimensions of the state space and observations are the same. Speech Modelling Using Subspace and EM Techniques 801 le4 <I) "0 .5 0 Q.. E t.s -le4 0 0.1 0.3 time / s 0.5 0.7 8 N ::I: ....: ....... 4 0" <I) ..::: 0 0 0.1 0.3 0.5 0.7 8 N tH::I: + ....: ....... 0" 4 <I) ..::: 8 N 1+14::I: ....: ~ ....... 0" <I) ..::: + 80 00 8 + ++ + +t ++ + N ::I: + + + + ....: ....... 4 ? 0" +1 + ~ + 0 1 Figure 1: (a) Time waveform and (b) spectrogram for "in arithmetic". Formant trajectories are estimated using EM and a SS model initialised with three different schemes: (d) 4SID, (e) ARMA and (t) AR(l). 802 G. Smith. J. F. G. d. Freitas. T. Robinson and M. Niranjan Zoubin Ghahramani [3]. Gavin Smith is supported by the Schiff Foundation, Cambridge University. At the time of writing, Nando de Freitas was supported by two University of the Witwatersrand Merit Scholarships, a Foundation for Research Development Scholarship (South Africa), an ORS award and a Trinity College External Research Studentship (Cambridge). ·1400 ·1700 ,,' 2 10 iteration nuLr so Figure 2: Likelihood convergence plots for EM and the SS model initialised with 4SID [- -], ARMA [-] and AR(I) [-.] for the experiments in Figure 1. Plots are the mean average over all frames where analysed. 6 References [1] Arun, K.S. & Kung, S.Y. (1990) Balanced Approximation of Stochastic Systems. SIAM Journal on Matrix Analysis and Applications, vol. 11, no. 1, pp. 42--68. [2] Gelb, A. ed., (1974) Applied Optimal Estimation. Cambridge, MA: MIT Press. [3] Ghahramani, Z. & Hinton, G. (1996) Parameter Estimation for Linear Dynamical Systems, Tech. rep. CRG-TR-96-2, Dept. of Computer Science, Univ. of Toronto. Software at www.gatsby.ucl.ac.ukrzoubinlsoJtware.html. [4] Grice, M. & Barry, W. (1989) Multi-lingual Speech Input/Output: Assessment, Methodology and Standardization, Tech. rep., University College, London, ESPRIT Project 1541 (SAM), extension phase final report. [5] Grivel, E., Gabrea, M. & Najim, M. (1999) Subspace State Space Model Identification For Speech Enhancement, Paper 1622, ICASSP'99. [6] Ljung, L. (1987) System Identification: Theory for the User. Englewood Cliffs, NJ: Prentice-Hall, Inc. [7] Ljung, L. (1991) System Identification Toolbox For Use With MatLab. 24 Prime Park Way, Natrick, MA, USA: The MathWorks, Inc. [8] McLachlan, G.J. & Krishnan, T. (1997) The EM Algorithm and Extensions. John Wiley and Sons Inc. [9] Smith, G.A. & Robinson, A.J. & Niranjan, M. (2000) A Comparison Between the EM and Subspace Algorithms for the Time-Invariant Linear Dynamical System. Tech. rep. CUEDIF-INFENGffR.366, Engineering Dept., Cambridge Univ., UK. [10] Van Overschee, P. & De Moor, B. (1996) Subspace Identification for Linear Systems: Theory, Implementation, Applications. Dordrecht, Netherlands: Kluwer Academic Publishers. [11] Viberg, M. & Wahlberg, B. & Ottersten, B. (1997) Analysis of State Space System Identification Methods Based on Instrumental Variables and Subspace Fitting. Automatica, vol. 33, no. 9, pp. 1603-1616.
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Optimal Kernel Shapes for Local Linear Regression Dirk Ormoneit Trevor Hastie Department of Statistics Stanford University Stanford, CA 94305-4065 ormoneit@stat.stanjord.edu Abstract Local linear regression performs very well in many low-dimensional forecasting problems. In high-dimensional spaces, its performance typically decays due to the well-known "curse-of-dimensionality". A possible way to approach this problem is by varying the "shape" of the weighting kernel. In this work we suggest a new, data-driven method to estimating the optimal kernel shape. Experiments using an artificially generated data set and data from the UC Irvine repository show the benefits of kernel shaping. 1 Introduction Local linear regression has attracted considerable attention in both statistical and machine learning literature as a flexible tool for nonparametric regression analysis [Cle79, FG96, AMS97]. Like most statistical smoothing approaches, local modeling suffers from the so-called "curse-of-dimensionality", the well-known fact that the proportion of the training data that lie in a fixed-radius neighborhood of a point decreases to zero at an exponential rate with increasing dimension of the input space. Due to this problem, the bandwidth of a weighting kernel must be chosen very big so as to contain a reasonable sample fraction. As a result, the estimates produced are typically highly biased. One possible way to reduce the bias of local linear estimates is to vary the "shape" of the weighting kernel. In this work, we suggest a method for estimating the optimal kernel shape using the training data. For this purpose, we parameterize the kernel in terms of a suitable "shape matrix" , L, and minimize the mean squared forecasting error with respect to L. For such an approach to be meaningful, the "size" of the weighting kernel must be constrained during the minimization to avoid overfitting. We propose a new, entropy-based measure of the kernel size as a constraint. By analogy to the nearest neighbor approach to bandwidth selection [FG96], the suggested measure is adaptive with regard to the local data density. In addition, it leads to an efficient gradient descent algorithm for the computation of the optimal kernel shape. Experiments using an artificially generated data set and data from the UC Irvine repository show that kernel shaping can improve the performance of local linear estimates substantially. The remainder of this work is organized as follows. In Section 2 we briefly review Optimal Kernel Shapes for Local Linear Regression 541 local linear models and introduce our notation. In Section 3 we formulate an objective function for kernel shaping, and in Section 4 we discuss entropic neighborhoods. Section 5 describes our experimental results and Section 6 presents conclusions. 2 Local Linear Models Consider a nonlinear regression problem where a continuous response y E JR is to be predicted based on a d-dimensional predictor x E JRd. Let D == {(Xt, Yt), t = 1, . .. ,T} denote a set of training data. To estimate the conditional expectation f(xo) == E[ylxo], we consider the local linear expansion f(x) ~ 0:0 + (x - xo),/3o in the neighborhood of Xo. In detail, we minimize the weighted least squares criterion T C(o:,/3;xo) == ~)Yt - 0: - (Xt - xo)'/3)2k(xt,xo) (1) t=1 to determine estimates of the parameters 0:0 and /30. Here k(xt, xo) is a non-negative weighting kernel that assigns more weight to residuals in the neighborhood of Xo than to residuals distant from Xo. In multivariate problems, a standard way of defining k(xt, xo) is by applying a univariate, non-negative "mother kernel" </>(z) to the distance measure Ilxt - xolln == J(Xt - xo)'O(Xt - xo): k(xt, xo) == : (1lxt - xolln) . ES=1 </> (1lxs - xolln) (2) Here 0 is a positive definite d x d matrix determining the relative importance assigned to different directions of the input space. For example, if </>(z) is a standard normal density, k(xt, xo) is a normalized multivariate Gaussian with mean Xo and covariance matrix 0- 1 . Note that k(xt, xo) is normalized so as to satisfy E;=1 k(xt, xo) = 1. Even though this restriction is not relevant directly with regard to the estimation of 0:0 and /30, it will be needed in our discussion of entropic neighborhoods in Section 4. Using the shorthand notation i(xo, 0) == (&0, ~b)" the solution of the minimization problem (1) may be written conveniently as i(xo,O) = (X'W X)-1 X'WY, (3) where X is the T x (d + 1) design matrix with rows (1, x~ - xb)" Y is the vector of response values, and W is a TxT diagonal matrix with entries Wt,t = k(xt, xo). The resulting local linear fit at Xo using the inverse covariance matrix 0 is simply !(xo; 0) == &0. Obviously, !(xo; 0) depends on 0 through the definition of the weighting kernel (2). In the discussion below, our focus is on choices of 0 that lead to favorable estimates of the unknown function value f(xo). 3 Kernel Shaping The local linear estimates resulting from different choices of 0 vary considerably in practice. A common strategy is to choose 0 proportional to the inverse sample covariance matrix. The remaining problem of finding the optimal scaling factor is equivalent to the problem of bandwidth selection in univariate smoothing [FG96, BBB99]. For example, the bandwidth is frequently chosen as a function of the distance between Xo and its kth nearest neighbor in practical applications [FG96]. In this paper, we take a different viewpoint and argue that optimizing the "shape" 542 D. Onnoneit and T. Hastie of the weighting kernel is at least as important as optimizing the bandwidth. More specifically, for a fixed "volume" of the weighting kernel, the bias of the estimate can be reduced drastically by shrinking the kernel in directions of large nonlinear variation of f (x), and stretching it in directions of small nonlinear variation. This idea is illustrated using the example shown in Figure 1. The plotted function is sigmoidal along an index vector K, and constant in directions orthogonal to K,. Therefore, a "shaped" weighting kernel is shrunk in the direction K, and stretched orthogonally to K" minimizing the exposure of the kernel to the nonlinear variation. Figure 1: Left: Example of a single index model of the form y = g(X'K) with K = (1,1) and g(z) = tanh(3z). Right: The contours of g(z) are straight lines orthogonal to K. To distinguish formally the metric and the bandwidth of the weighting kernel, we rewrite 0 as follows: 0== A' (LL' + I). (4) Here A corresponds to the inverse bandwidth, and L may be interpreted as a metricor shape-matrix. Below we suggest an algorithm which is designed to minimize the bias with respect to the kernel metric. Clearly, for such an approach to be meaningful, we need to restrict the "volume" of the weighting kernel; otherwise, the bias of the estimate could be minimized trivially by choosing a zero bandwidth. For example, we might define A contingent on L so as to satisfy 101 = c for some constant c. A serious disadvantage of this idea is that, by contrast to the nearest neighbor approach, 101 is independent of the design. As a more appropriate alternative, we define A in terms of a measure of the number of neighboring observations. In detail, we fix the volume of k(xt, xo) in terms of the "entropy" of the weighting kernel. Then, we choose A so as to satisfy the resulting entropy constraint. Given this definition of the bandwidth, we determine the metric of k (Xt, xo) by minimizing the mean squared prediction error: T C(L; D) == I)Yt - f(Xt; 0»2 (5) t=l with respect to L. In this way, we obtain an approximation of the optimal kernel shape because the expectation of C(L; D) differs from the bias only by a variance term which is independent of L. Details of the entropic neighborhood criterion and of the numerical minimization procedure are described next. 4 Entropic Neighborhoods We mentioned previously that, for a given shape matrix L, we choose the bandwidth parameter A in (4) so as to fulfill a volume constraint on the weighting kernel. For this purpose, we interpret the kernel weights k(xt, xo) as probabilities. In particular, Optimal Kernel Shapes for Local Linear Regression 543 as k(Xt, xo) > 0 and Et k(xt, xo) = 1 by definition (2), we can formulate the local entropy of k(xt, xo): T H(O) == - I: k(xt, xo) log k(xt, xo). (6) t=l The entropy of a probability distribution is typically thought of as a measure of uncertainty. In the context of the weighting kernel k(xt, xo), H(O) can be used as a smooth measure of the "size" of the neighborhood that is used for averaging. To see this, note that in the extreme case where equal weights are placed on all observations in D, the entropy is maximized. At the other extreme, if the single nearest neighbor of Xo is assigned the entire weight of one, the entropy attains its minimum value zero. Thus, fixing the entropy at a constant value c is similar to fixing the number k in the nearest neighbor approach. Besides justifying (6), the correspondence between k and c can also be used to derive a more intuitive volume parameter than the entropy level c. We specify c in terms of a hypothetical weighting kernel that places equal weight on the k nearest neighbors of Xo and zero weight on the remaining observations. Note that the entropy of this hypothetical kernel is log k. Thus, it is natural to characterize the size of an entropic neighborhood in terms of k, and then to determine A by numerically solving the nonlinear equation system (for details, see [OH99]) H(O) = logk. (7) More precisely, we report the number of neighbors in terms of the equivalent sample fraction p == kiT to further intuition. This idea is illustrated in Figure 2 using a one- and a two-dimensional example. The equivalent sample fractions are p = 30% and p = 50%, respectively. Note that in both cases the weighting kernel is wider in regions with few observations, and narrower in regions with many observations. As a consequence, the number of observations within contours of equal weighting remains approximately constant across the input space. " ': . . . ,. 0·:··:··."···---------. . . .,', . ~', .~,' .,'. ': . 0.2 . 03 04 0.1 O. ot 1 Figure 2: Left: Univariate weighting kernel k(-, xo) evaluated at Xo = 0.3 and Xo = 0.7 based on a sample data set of 100 observations (indicated by the bars at the bottom). Right: Multivariate weighting kernel k(·, xo) based on a sample data set of 200 observations. The two ellipsoids correspond to 95% contours of a weighting kernel evaluated at (0.3,0.3)' and (0.6,0.6)' . To summarize, we define the value of A by fixing the equivalent sample fraction parameter p, and subsequently minimize the prediction error on the training set with respect to the shape matrix L. Note that we allow for the possibility that L may be of reduced rank I :::; d as a means of controlling the number of free parameters. As a minimization procedure, we use a variant of gradient descent that 544 D. Ormoneit and T. Hastie accounts for the entropy constraint. In particular, our algorithm relies on the fact that (7) is differentiable with respect to L. Due to space limitations, the interested reader is referred to [OH99] for a formal derivation of the involved gradients and for a detailed description of the optimization procedure. 5 Experiments In this section we compare kernel shaping to standard local linear regression using a fixed spherical kernel in two examples. First, we evaluate the performance using a simple toy problem which allows us to estimate confidence intervals for the prediction accuracy using Monte Carlo simulation. Second, we investigate a data set from the machine learning data base at UC Irvine [BKM98]. 5.1 Mexican Hat Function In our first example, we employ Monte Carlo simulation to evaluate the performance of kernel shaping in a five-dimensional regression problem. For this purpose, 20 sets of 500 data points each are generated independently according to the model y = coS(SJxI + x~) . exp( -(xi + x~)). (8) Here the predictor variables Xl, ... ,X5 are drawn according to a five-dimensional standard normal distribution. Note that, even though the regression is carried out in a five-dimensional predictor space, y is really only a function of the variables Xl and X2 . In particular, as dimensions two through five do not contribute any information with regard to the value of y, kernel shaping should effectively discard these variables. Note also that there is no noise in this example. Figure 3: Left: "True" Mexican hat function. Middle: Local linear estimate using a spherical kernel (p = 2%). Right: Local linear estimate using kernel shaping (p = 2%) . Both estimates are based on a training set consisting of 500 data points. Figure 3 shows a plot of the true function, the spherical estimate, and the estimate using kernel shaping as functions of Xl and X2. The true function has the familiar "Mexican hat" shape, which is recovered by the estimates to different degrees. We evaluate the local linear estimates for values of the equivalent neighborhood fraction parameter p in the range from 1% to 15%. Note that, to warrant a fair comparison, we used the entropic neighborhood also to determine the bandwith of the spherical estimate. For each value of p, 20 models are estimated using the 20 artificially generated training sets, and subsequently their performance is evaluated on the training set and on the test set of 31 x 31 grid points shown in Figure 3. The shape matrix L has maximal rank 1 = 5 in this experiment. Our results for local linear regression using the spherical kernel and kernel shaping are summarized in Table 1. Performance is measured in terms of the mean R2-value of the 20 models, and standard deviations are reported in parenthesis. Optimal Kernel Shapes for Local Linear Regression Algorithm spherical kernel spherical kernel spherical kernel spherical kernel spherical kernel kernel shaping kernel shaping kernel shaping kernel shaping p=l% p=2% p=5% p = 10% p= 20% p= 1% p=2% p=5% p= 15% Training R2 0.961 (0.005) 0.871 (0.014) 0.680 (0.029) 0.507 (0.038) 0.341 (0.039) 0.995 (0.001) 0.984 (0.002) 0.923 (0.009) 0.628 (0.035) 0.215 (0.126) 0.293 (0.082) 0.265 (0.043) 0.213 (0.030) 0.164 (0.021) 0.882 (0.024) 0.909 (0.017) 0.836 (0.023) 0.517 (0.035) 545 Table 1: Performances in the toy problem. The results for kernel shaping were obtained using 200 gradient descent steps with step size a = 0.2. The results in Table 1 indicate that the optimal performance on the test set is obtained using the parameter values p = 2% both for kernel shaping (R2 = 0.909) and for the spherical kernel (R2 = 0.293). Given the large difference between the R2 values, we conclude that kernel shaping clearly outperforms the spherical kernel on this data set. ---Figure 4: The eigenvectors of the estimate of n obtained on the first of 20 training sets. The graphs are ordered from left to right by increasing eigenvalues (decreasing extension of the kernel in that direction): 0.76,0.76, 0.76, 33.24, 34.88. Finally, Figure 4 shows the eigenvectors of the optimized n on the first of the 20 training sets. The eigenvectors are arranged according to the size of the corresponding eigenvalues. Note that the two rightmost eigenvectors, which correspond to the directions of minimum kernel extension, span exactly the Xl -x2-space where the true function lives. The kernel is stretched in the remaining directions, effectively discarding nonlinear contributions from X3, X4, and X5' 5.2 Abalone Database The task in our second example is to predict the age of abalone based on several measurements. More specifically, the response variable is obtained by counting the number of rings in the shell in a time-consuming procedure. Preferably, the age of the abalone could be predicted from alternative measurements that may be obtained more easily. In the data set, eight candidate measurements including sex, dimensions, and various weights are reported along with the number of rings of the abalone as predictor variables. We normalize these variables to zero mean and unit variance prior to estimation. Overall, the data set consists of 4177 observations. To prevent possible artifacts resulting from the order of the data records, we randomly draw 2784 observations as a training set and use the remaining 1393 observations as a test set. Our results are summarized in Table 2 using various settings for the rank l, the equivalent fraction parameter p, and the gradient descent step size a. The optimal choice for p is 20% both for kernel shaping (R2 = 0.582) and for the spherical kernel (R2 = 0.572). Note that the performance improvement due to kernel shaping is negligible in this experiment. 546 Kernel spherical kernel spherical kernel spherical kernel spherical kernel spherical kernel spherical kernel kernel shaping kernel shaping kernel shaping kernel shaping kernel shaping kernel shaping p = 0.05 p = 0.10 P = 0.20 P = 0.50 p = 0.70 P = 0.90 l - 5, p - 0.20, a = 0.5 l = 5, p = 0.20, a = 0.2 l = 2, P = 0.10, a = 0.2 l = 2, P = 0.20, a = 0.2 l = 2, P = 0.50, a = 0.2 l = 2, p = 0.20, a = 0.5 D. Ormoneit and T. Hastie Training R2 0.752 0.686 0.639 0.595 0.581 0.568 0.705 0.698 0.729 0.663 0.603 0.669 0.543 0.564 0.572 0.565 0.552 0.533 0.575 0.577 0.574 0.582 0.571 0.582 Table 2: Results using the Abalone database after 200 gradient descent steps. 6 Conclusions We introduced a data-driven method to improve the performance of local linear estimates in high dimensions by optimizing the shape of the weighting kernel. In our experiments we found that kernel shaping clearly outperformed local linear regression using a spherical kernel in a five-dimensional toy example, and led to a small performance improvement in a second, real-world example. To explain the results of the second experiment, we note that kernel shaping aims at exploiting global structure in the data. Thus, the absence of a larger performance improvement may suggest simply that no corresponding structure prevails in that data set. That is, even though optimal kernel shapes exist locally, they may vary accross the predictor space so that they cannot be approximated by any particular global shape. Preliminary experiments using a localized variant of kernel shaping did not lead to significant performance improvements in our experiments. Acknowledgments The work of Dirk Ormoneit was supported by a grant of the Deutsche Forschungsgemeinschaft (DFG) as part of its post-doctoral program. Trevor Hastie was partially supported by NSF grant DMS-9803645 and NIH grant ROI-CA-72028-01. Carrie Grimes pointed us to misleading formulations in earlier drafts of this work. References [AMS97] C. G. Atkeson, A. W. Moore, and S. Schaal. Locally weighted learning. Artificial Intelligence Review, 11:11-73, 1997. [BBB99] M. Birattari, G. Bontempi, and H. Bersini. Lazy learning meets the recursive least squares algorithm. In M. J. Kearns, S. A. Solla, and D. A. Cohn, editors, Advances in Neural Information Processing Systems 11. The MIT Press, 1999. [BKM98] C. Blake, E. Koegh, and C. J. Merz. UCI Repository of machine learning databases. http://vvw.ics.uci.edu/-mlearn/MLRepository.html. [Cle79] W. S. Cleveland. Robust locally weighted regression and smoothing scatterplots. Journal of the American Statistical Association, 74:829-836, 1979. [FG96] J. Fan and 1. Gijbels. Local Polynomial Modelling and Its Applications. Chapman & Hall, 1996. [OH99] D. Ormoneit and T . Hastie. Optimal kernel shapes for local linear regression. Tech. report 1999-11, Department of Statistics, Stanford University, 1999.
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Robust Neural Network Regression for Offline and Online Learning Thomas Briegel* Siemens AG, Corporate Technology D-81730 Munich, Germany thomas.briegel@mchp.siemens.de Volker Tresp Siemens AG, Corporate Technology D-81730 Munich, Germany volker.tresp@mchp.siemens.de Abstract We replace the commonly used Gaussian noise model in nonlinear regression by a more flexible noise model based on the Student-tdistribution. The degrees of freedom of the t-distribution can be chosen such that as special cases either the Gaussian distribution or the Cauchy distribution are realized. The latter is commonly used in robust regression. Since the t-distribution can be interpreted as being an infinite mixture of Gaussians, parameters and hyperparameters such as the degrees of freedom of the t-distribution can be learned from the data based on an EM-learning algorithm. We show that modeling using the t-distribution leads to improved predictors on real world data sets. In particular, if outliers are present, the t-distribution is superior to the Gaussian noise model. In effect, by adapting the degrees of freedom, the system can "learn" to distinguish between outliers and non-outliers. Especially for online learning tasks, one is interested in avoiding inappropriate weight changes due to measurement outliers to maintain stable online learning capability. We show experimentally that using the t-distribution as a noise model leads to stable online learning algorithms and outperforms state-of-the art online learning methods like the extended Kalman filter algorithm. 1 INTRODUCTION A commonly used assumption in nonlinear regression is that targets are disturbed by independent additive Gaussian noise. Although one can derive the Gaussian noise assumption based on a maximum entropy approach, the main reason for this assumption is practicability: under the Gaussian noise assumption the maximum likelihood parameter estimate can simply be found by minimization of the squared error. Despite its common use it is far from clear that the Gaussian noise assumption is a good choice for many practical problems. A reasonable approach therefore would be a noise distribution which contains the Gaussian as a special case but which has a tunable parameter that allows for more flexible distributions. In this paper we use the Student-t-distribution as a noise model which contains two free parameters - the degrees of freedom 1/ and a width parameter (72. A nice feature of the t-distribution is that if the degrees of freedom 1/ approach infinity, we recover the Gaussian noise model. If 1/ < 00 we obtain distributions which are more heavy-tailed than the Gaussian distribution including the Cauchy noise model with 1/ = 1. The latter *Now with McKinsey & Company, Inc. 408 T. Briegel and V. Tresp is commonly used for robust regression. The first goal of this paper is to investigate if the additional free parameters, e.g. v, lead to better generalization performance for real world data sets if compared to the Gaussian noise assumption with v = 00. The most common reason why researchers depart from the Gaussian noise assumption is the presence of outliers. Outliers are errors which occur with low probability and which are not generated by the data-generation process that is subject to identification. The general problem is that a few (maybe even one) outliers of high leverage are sufficient to throw the standard Gaussian error estimators completely off-track (Rousseeuw & Leroy, 1987). In the second set of experiments we therefore compare how the generalization performance is affected by outliers, both for the Gaussian noise assumption and for the t-distribution assumption. Dealing with outliers is often of critical importance for online learning tasks. Online learning is of great interest in many applications exhibiting non-stationary behavior like tracking, signal and image processing, or navigation and fault detection (see, for instance the NIPS*98 Sequential Learning Workshop). Here one is interested in avoiding inappropriate weight chances due to measurement outliers to maintain stable online learning capability. Outliers might result in highly fluctuating weights and possible even instability when estimating the neural network weight vector online using a Gaussian error assumption. State-of-the art online algorithms like the extended Kalman filter, for instance, are known to be nonrobust against such outliers (Meinhold & Singpurwalla, 1989) since they are based on a Gaussian output error assumption. The paper is organized as follows. In Section 2 we adopt a probabilistic view to outlier detection by taking as a heavy-tailed observation error density the Student-t-distribution which can be derived from an infinite mixture of Gaussians approach. In our work we use the multi-layer perceptron (MLP) as nonlinear model. In Section 3 we derive an EM algorithm for estimating the MLP weight vector and the hyperparameters offline. Employing a state-space representation to model the MLP's weight evolution in time we extend the batch algorithm of Section 3 to the online learning case (Section 4). The application of the computationally efficient Fisher scoring algorithm leads to posterior mode weight updates and an online EM-type algorithm for approximate maximum likelihood (ML) estimation of the hyperparameters. In in the last two sections (Section 5 and Section 6) we present experiments and conclusions, respectively. 2 THE t-DENSITY AS A ROBUST ERROR DENSITY We assume a nonlinear regression model where for the t-th data point the noisy target Yt E R is generated as (1) and Xt E Rk is a k-dimensional known input vector. g(.;Wt) denotes a neural network model characterized by weight vector Wt E Rn , in our case a multi-layer perceptron (MLP). In the offline case the weight vector Wt is assumed to be a fixed unknown constant vector, i.e. Wt == w. Furthermore, we assume that Vt is uncorrelated noise with density Pv, (. ). In the offline case, we assume Pv, ( .) to be independent of t, i.e. Pv, (.) == Pv (.). In the following we assume that Pv (.) is a Student-t-density with v degrees of freedom with r(!±l) z2-~ Pv(z)=T(zI0-2,v)= y'7W2qv)(1+-2-) , v,o->O. (2) 01W 2" 0V It is immediately apparent that for v = 1 we recover the heavy-tailed Cauchy density. What is not so obvious is that for v -t 00 we obtain a Gaussian density. For the derivation of the EM-learning rules in the next section it is important to note that the t-denstiy can be thought of as being an infinite mixture of Gaussians of the form (3) Robust Neural Network Regression for Offline and Online Learning 3 2 -1 ._-._._ ..... _ .. __ ...... _ •• _" 2 ... ~,:,. - ..... ~ -...... -... -......... ;:~,' -3 , -e -4 -2 ° z , ". v-T5 . ,'; .. ,..................... ..~ .... ~ . ;,-_. __ .-......... ::-: .... . - .-.2 4 " Boeton Houaing ct.,_ with addtttve oult .... 1.2r-~-~--"-.--~----..." 1.1 liJO.9 0.8 0 .7 0.5'----:---5~-7;;IO:-----:15~---;20~--::;!25 nu_ Of 0UIIIen ('K.] 409 Figure 1: Left: ¢(.)-functions for the Gaussian density (dashed) and t-densities with II = 1,4,15 degrees of freedom. Right: MSE on Boston Housing data test set for additive outliers. The dashed line shows results using a Gaussian error measure and the continuous line shows the results using the Student-t-distribution as error measure. where T(zI0'2, II) is the Student-t-density with II degrees of freedom and width parameter 0'2, N(zIO, 0'2/U) is a Gaussian density with center 0 and variance 0'2/U and U '" X~/II where X~ is a Chi-square distribution with II degrees of freedom evaluated at U > O. To compare different noise models it is useful to evaluate the "¢-function" defined as (Huber,1964) ¢(z) = -ologpv(z)/oz (4) i.e. the negative score-function of the noise density. In the case of i.i.d. samples the ¢function reflects the influence of a single measurement on the resulting estimator. Assuming Gaussian measurement errors Pv(z) = N(zIO,0'2) we derive ¢(z) = z/0'2 which means that for Izl -+ 00 a single outlier z can have an infinite leverage on the estimator. In contrast, for constructing robust estimators West (1981) states that large outliers should not have any influence on the estimator, i.e. ¢(z) -+ 0 for Izl -+ 00. Figure 1 (left) shows ¢(z) for different II for the Student-t-distribution. It can be seen that the degrees of freedom II determine how much weight outliers obtain in influencing the regression. In particular, for finite II, the influence of outliers with Izl -+ 00 approaches zero. 3 ROBUST OFFLINE REGRESSION As stated in Equation (3), the t-density can be thought of as being generated as an infinite mixture of Gaussians. Maximum likelihood adaptation of parameters and hyperparameters can therefore be performed using an EM algorithm (Lange et al., 1989). For the t-th sample, a complete data point would consist of the triple (Xt, Yt, Ut) of which only the first two are known and Ut is missing. In the E-step we estimate for every data point indexed by t (5) where at = E[ut IYt, Xt] is the expected value of the unknown Ut given the available data (Xt,Yt) andwhere«5t = (Yt - g(Xti W0 1d»)2 /0'2,old. In the M-step the weights wand the hyperparameters 0'2 and II are optimized using T argm~n{LOt(Yt - g(XtiW»)2} (6) t=l 410 T. Briegel and V. Tresp T ~ L at [(Yt - g(Xt; wnew))2] (7) t=l { Tv v v argm:x T log 2" - Tlog{f( 2")} T T +( ~ - 1) L (3t - ~ L at} t=l t=l (8) where (9) with the Digamma function DG(z) = 8f(z)/8z. Note that the M-step for v is a onedimensional nonlinear optimization problem. Also note that the M-steps for the weights in the MLP reduce to a weighted least squares regression problem in which outliers tend to be weighted down. The exception of course is the Gaussian case with v ~ 00 in which all terms obtain equal weight. 4 ROBUST ONLINE REGRESSION For robust online regression, we assume that the model Equation (1) is still valid but that w can change over time, i.e. w = Wt. In particular we assume that Wt follows a first order random walk with normally distributed increments, i.e. (10) and where Wo is normally distributed with center ao and covariance Qo. Clearly, due to the nonlinear nature of 9 and due to the fact that the noise process is non-Gaussian, a fully Bayesian online algorithm which for the linear case with Gaussian noise can be realized using the Kalman filter is clearly infeasible. On the other hand, if we consider data 'D = {Xt, yt}f=l' the negative log-posterior -logp(WTI'D) of the parameter sequence WT = (wJ, ... , w~) T is up to a normalizing constant -logp(WTI'D) ex: and can be used as the appropriate cost function to derive the posterior mode estimate W¥AP for the weight sequence. The two differences to the presentation in the last section are that first, Wt is allowed to change over time and that second, penalty terms, stemming from the prior and the transition density, are included. The penalty terms are penalizing roughness of the weight sequence leading to smooth weight estimates. A suitable way to determine a stationary point of -logp(WTI'D), the posterior mode estimate of WT , is to apply Fisher scoring. With the current estimate WT1d we get a better estimate wTew = wTld +171' for the unknown weight sequence WT where 'Y is the solution of (12) with the negative score function S(WT) = -8logp(WT1'D)/8WT and the expected information matrix S(WT) = E[82 10gp(WTI'D)/8WT8WT ]. By applying the ideas given in Fahrmeir & Kaufmann (1991) to robust neural network regression it turns out that solving (12), i.e. to compute the inverse of the expected information matrix, can be performed by Robust Neural Network Regression/or Offline and Online Learning 411 Cholesky decomposition in one forward and backward pass through the set of data 'D. Note that the expected information matrix is a positive definite block-tridiagonal matrix. The forward-backward steps have to be iterated to obtain the posterior mode estimate W.pAP for WT. For online posterior mode smoothing, it is of interest to smooth backwards after each filter step t. If Fisher scoring steps are applied sequentially for t = 1,2, ... , then the posterior mode smoother at time-step t - 1, wl~~~P = (W~t-l"'" wi-I lt-l) T together with the step-one predictor Wtlt-l = Wt-I lt-l is a reasonable starting value for obtaining the posterior mode smoother WtMAP at time t. One can reduce the computational load by limiting the backward pass to a sliding time window, e.g. the last Tt time steps, which is reasonable in non-stationary environments for online purposes. Furthermore, if we use the underlying assumption that in most cases a new measurement Yt should not change estimates too drastically then a single Fisher scoring step often suffices to obtain the new posterior mode estimate at time t. The resulting single Fisher scoring step algorithm with lookback parameter Tt has in fact just one additional line of code involving simple matrix manipulations compared to online Kalman smoothing and is given here in pseudo-code. Details about the algorithm and a full description can be found in Briegel & Tresp (1999). Online single Fisher scoring step algorithm (pseudo-code) for t = 1,2, ... repeat the following four steps: • Evaluate the step-one predictor Wt lt-l. • Perform the forward recursions for s = t - Tt, ... , t. • New data point (Xt, yd arrives: evaluate the corrector step Wtlt. • Perform the backward smoothing recursions ws-Ilt for s = t, ... , t - Tt. For the adaptation of the parameters in the t-distribution, we apply results from Fahrmeir & Kunstler (1999) to our nonlinear assumptions and use an online EM-type algorithm for approximate maximum likelihood estimation of the h yperparameters lit and (7F. We assume the scale factors (7F and the degrees of freedom lit being fixed quantities in a certain time window of length ft, e.g. (7F = (72, lit = 11, t E {t - ft, t}. For deriving online EM update equations we treat the weight sequence Wt together with the mixing variables Ut as missing. By linear Taylor series expansion of g(.; w s ) about the Fisher scoring solutions Wslt and by approximating posterior expectations E[ws I'D] with posterior modes Wslt, S E {t - ft, t} and posterior covariances cov[ws I'D] with curvatures :Eslt = E[( Ws - Wslt) (ws - Wslt) T I'D] in the E-step, a somewhat lengthy derivation results in approximate maximum likelihood update rules for (72 and 11 similar to those given in Section 3. Details about the online EM-type algorithm can be found in Briegel & Tresp (1999). 5 EXPERIMENTS 1. Experiment: Real World Data Sets. In the first experiment we tested if the Studentt-distribution is a useful error measure for real-world data sets. In training, the Studentt-distribution was used and both, the degrees of freedom 11 and the width parameter (72 were adapted using the EM update rules from Section 3. Each experiment was repeated 50 times with different divisions into training and test data. As a comparison we trained the neural networks to minimize the squared error cost function (including an optimized weight decay term). On the test data set we evaluated the performance using a squared error cost function. Table 1 provides some experimental parameters and gives the test set performance based on the 50 repetitions of the experiments. The additional explained variance is defined as [in percent] 100 x (1 - MSPE, IMSPEN) where MSPE, is the mean squared prediction error using the t-distribution and MSPEN is the mean squared prediction error using the Gaussian error measure. Furthermore we supply the standard 412 T. Briegel and V. Tresp Table I: Experimental parameters and test set performance on real world data sets. Data Set I # Inputs/Hidden I Training I Test I Add.Exp.Var. [%] I Std. [%] I Boston Housing (13/6) 400 106 4.2 0.93 Sunspot (1217) 221 47 5.3 0.67 Fraser River (1217) 600 334 5.4 0.75 error based on the 50 experiments. In all three experiments the networks optimized with the t-distribution as noise model were 4-5% better than the networks optimized using the Gaussian as noise model and in all experiments the improvements were significant based on the paired t-test with a significance level of 1 %. The results show clearly that the additional free parameter in the Student-t-distribution does not lead to overfitting but is used in a sensible way by the system to value down the influence of extreme target values. Figure 2 shows the normal probability plots. Clearly visible is the derivation from the Gaussian distribution for extreme target values. We also like to remark that we did not apply any preselection process in choosing the particular data sets which indicates that non-Gaussian noise seems to be the rule rather than the exception for real world data sets. :: 0.99 0." ~095 1090 1075 i oso i0 25 1010 ; 005 002 00\ 0003 000'''._0, ~--=" -<>5o-----O0 ---,07' -',---' ,e~"""MWlgWllllh~ .. -8fTttdenllly Figure 2: Normal probability plots of the three training data sets after learning with the Gaussian error measure. The dashed line show the expected normal probabilities. The plots show clearly that the residuals follow a more heavy-tailed distribution than the normal distribution. 2. Experiment: Outliers. In the second experiment we wanted to test how our approach deals with outliers which are artificially added to the data set. We started with the Boston housing data set and divided it into training and test data. We then randomly selected a subset of the training data set (between 0.5% and 25%) and added to the targets a uniformly generated real number in the interval [-5,5]. Figure I (right) shows the mean squared error on the test set for different percentages of added outliers. The error bars are derived from 20 repetitions of the experiment with different divisions into training and test set. It is apparent that the approach using the t-distribution is consistently better than the network which was trained based on a Gaussian noise assumption. 3. Experiment: Online Learning. In the third experiment we examined the use of the t-distribution in online learning. Data were generated from a nonlinear map y = 0.6X2 + bsin(6x) - 1 where b = -0.75, -0.4, -0.1,0.25 for the first, second, third and fourth set of 150 data points, respectively. Gaussian noise with variance 0.2 was added and for training, a MLP with 4 hidden units was used. In the first experiment we compare the performance of the EKF algorithm with our single Fisher scoring step algorithm. Figure 3 (left) shows that our algorithm converges faster to the correct map and also handles the transition in the model (parameter b) much better than the EKE In the second experiment with a probability of 10% outliers uniformly drawn from the interval [-5,5] were added to the targets. Figure 3 (middle) shows that the single Fisher scoring step algorithm using the Robust Neural Network Regression/or Offline and Online Learning 413 t-distribution is consistently better than the same algorithm using a Gaussian noise model and the EKE The two plots on the right in Figure 3 compare the nonlinear maps learned after 150 and 600 time steps, respectively. w ~ . ! IO~O~-C'00::---=200~-=311l;--::::""'--5;:;;1Il-----:! 1Dl r",. 1~' 10 0 '. 100 2CIO 300 400 5CXl all r",. 0.5 Figure 3: Left & Middle: Online MSE over each of the 4 sets of training data. On the left we compare extended Kalman filtering (EKF) (dashed) with the single Fisher scoring step algorithm with Tt = 10 (GFS-lO) (continuous) for additive Gaussian noise. The second figure shows EKF (dashed-dotted), Fisher scoring with Gaussian error noise (GFS1 0) (dashed) and t-distributed error noise (TFS-l 0) (continuous), respectively for data with additive outliers. Right: True map (continuous), EKF learned map (dashed-dotted) and TFS-I0 map (dashed) after T = 150 and T = 600 (data sets with additive outliers). 6 CONCLUSIONS We have introduced the Student-t-distribution to replace the standard Gaussian noise assumption in nonlinear regression. Learning is based on an EM algorithm which estimates both the scaling parameters and the degrees of freedom of the t-distribution. Our results show that using the Student-t-distribution as noise model leads to 4-5% better test errors than using the Gaussian noise assumption on real world data set. This result seems to indicate that non-Gaussian noise is the rule rather than the exception and that extreme target values should in general be weighted down. Dealing with outliers is particularly important for online tasks in which outliers can lead to instability in the adaptation process. We introduced a new online learning algorithm using the t-distribution which leads to better and more stable results if compared to the extended Kalman filter. References Briegel, T. and Tresp, V. (1999) Dynamic Neural Regression Models, Discussion Paper, Seminar flir Statistik, Ludwig Maximilians Universitat Milnchen. de Freitas, N., Doucet, A. and Niranjan, M. (1998) Sequential Inference and Learning, NIPS*98 Workshop, Breckenridge, CO. Fahrmeir, L. and Kaufmann, H. (1991) On Kalman Filtering, Posterior Mode Estimation and Fisher Scoring in Dynamic Exponential Family Regression, Metrika 38, pp. 37-60. Fahrmeir, L. and Kilnstler, R. (1999) Penalized Likelihood smoothing in robust state space models, Metrika 49, pp. 173-191. Huber, p.r. (1964) Robust Estimation of Location Parameter, Annals of Mathematical Statistics 35, pp.73-101. Lange, K., Little, L., Taylor, J. (989) Robust Statistical Modeling Using the t-Distribution, JASA 84, pp. 881-8%. Meinhold, R. and SingpurwaIla, N. (1989) Robustification of Kalman Filter Models, JASA 84, pp. 470-496. Rousseeuw, P. and Leroy, A. (1987) Robust Regression and Outlier Detection, John Wiley & Sons. West, M. (1981) Robust Sequential Approximate Bayesian Estimation, JRSS B 43, pp. 157-166.
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Leveraged Vector Machines Yoram Singer Hebrew University singer@cs.huji.ac.il Abstract We describe an iterative algorithm for building vector machines used in classification tasks. The algorithm builds on ideas from support vector machines, boosting, and generalized additive models. The algorithm can be used with various continuously differential functions that bound the discrete (0-1) classification loss and is very simple to implement. We test the proposed algorithm with two different loss functions on synthetic and natural data. We also describe a norm-penalized version of the algorithm for the exponential loss function used in AdaBoost. The performance of the algorithm on natural data is comparable to support vector machines while typically its running time is shorter than of SVM. 1 Introduction Support vector machines (SVM) [1, 13] and boosting [10, 3, 4, 11] are highly popular and effective methods for constructing linear classifiers. The theoretical basis for SVMs stems from Vapnik's seminal on learning and generalization [12] and has proved to be of great practical usage. The first boosting algorithms [10, 3], on the other hand, were developed to answer certain fundamental questions about PAC-learnability [6]. While mathematically beautiful, these algorithms were rather impractical. Later, Freund and Schapire [4] developed the AdaBoost algorithm, which proved to be a practically useful meta-learning algorithm. AdaBoost works by making repeated calls to a weak learner. On each call the weak learner generates a single weak hypothesis, and these weak hypotheses are combined into an ensemble called strong hypothesis. Recently, Schapire and Singer [11] studied a simple generalization of AdaBoost in which a weak-hypothesis can assign a real-valued confidence to each prediction. Even more recently, Friedman, Hastie, and Tibshirani [5] presented an alternative view of boosting from a statistical point of view and also described a new family of algorithms for constructing generalized additive models of base learners in a similar fashion to AdaBoost. The work of Friedman, Hastie, and Tibshirani generated lots of attention and motivated research in classification algorithms that employ various loss functions [8, 7]. In this work we combine ideas from the research mentioned above and devise an alternative approach to construct vector machines for classification. As in SVM, the base predictors that we use are Mercer kernels. The value of a kernel evaluated at an input pattern, i.e., the dot-product between two instances embedded in a high-dimensional space, is viewed as a real-valued prediction. We describe a simple extension to additive models in which the prediction of a base-learner is a linear transformation of a given kernel. We then describe an iterative algorithm that greedily adds kernels. We derive our algorithm using the exponentialloss function used in AdaBoost and the loss function used by Friedman, Hastie, and Tibshirani [5] in "LogitBoost". For brevity we call the resulting classifiers boosted vector machines (BVM) and logistic vector machines (LVM). We would like to note in passing Leveraged Vector Machines 611 that the resulting algorithms are not boosting algorithms in the PAC sense. For instance, the weak-Iearnability assumption that the weak-learner can always find a weak-hypothesis is violated. We therefore adopt the terminology used in [2] and call the resulting classifiers leveraged vector machines. The leveraging procedure we give adopts the chunking technique from SVM. After presenting the basic leveraging algorithms we compare their performance with SVM on synthetic data. The experimental results show that the leveraged vector machines achieve similar performance to SVM and often the resulting vector machines are smaller than the ones obtained by SVM. The experiments also demonstrate that BVM is especially sensitive to (malicious) label noise while LVM seems to be more insensitve. We also describe a simple norm-penalized extension of BVM that provides a partial solution to overfitting in the presence of noise. Finally, we give results of experiments performed with natural data from the DCI repository and conclude. 2 Preliminaries Let S = ((Xl, yd, ... ,(xm, Ym)) be a sequence of training examples where each instance Xi belongs to a domain or instance space X, and each label Yi is in {-I, +1}. (The methods described in this paper to build vector machines and SVMs can be extended to solve multiclass problems using, for instance, error correcting output coding. Such methods are beyond the scope of this paper and will be discussed elsewhere). For convenience, we will use iii to denote (Yi + 1) /2 E {O, I}. As is boosting, we assume access to a weak or base learning algorithm which accepts as input a weighted sequence of training examples S. Given such input, the weak learner computes a weak (or base) hypothesis h. In general, h has the form h : X -+ ~. We interpret the sign of h(x) as the predicted label (-1 or + 1) to be assigned to instance X, and the magnitude Ih(x)1 as the "confidence" in this prediction. To build vector machines we use the notion of confidence-rated predictions. take for base hypotheses sample-based Mercer kernels [13], and define the confidence (i.e., the magnitude of prediction) of a base learner to be the value of its dot-product with another instance. The sign of the prediction is set to be the label of the corresponding instance. Formally, for each base hypothesis h there exist (Xj,Yj) E S such that h(x) = YjK(Xj, x) and K(u, v) defines an inner product in a feature space: K(u, v) = 2:::~1 ak'lfJk (U)'Ij;k (v). We denote the function induced by an instance label pair (Xj, Yj) with a kernel K by </>j (x) = yjK (Xj, x). Our goal is to find a classifier f(x), called a strong hypothesis in the context of boosting algorithms, ofthe form f(x) = 2::::=1 atht(x) + /3, such that the signs of the predictions of the classifier should agree, as much as possible, with the labels of the training instances. The leverage algorithm we describe maintains a distribution Dover {I, ... , m}, i.e., over the indices of S. This distribution is simply a vector of non-negative weights, one weight per example and is an exponential function of the classifier f which is built incrementally, 1 m D(i) = Z exp (-Yd(Xi)) where Z = L exp (-Yd(Xi)) . i=l (1) For a random function 9 of the input instances and the labels, we denote the sample expectation of 9 according to D by ED(g) = 2::::1 D(i)g(Xi, Yi). We also use this notation to denote the expectation of matrices of random functions. We will convert a confidencerated classifier f into a randomized predictor by using the soft-max function and denote it by P(Xi) where p exp (f(Xi)) (Xi) = exp (f(Xi)) + exp (- f(Xi)) 1 1 + exp (-2f(Xi)) . (2) 612 Y. Singer 3 The leveraging algorithm The basic procedure to construct leveraged vector machines builds on ideas from [11, 5] by extending the prediction to be a linear function of the base classifiers. The algorithm works in rounds, constructing a new classifier It from the previous one It-I by adding a new base hypothesis ht to the current classifier, It- Denoting by Dt and Pt+1 the distribution and probability given by Eqn. (1) and Eqn. (2) using It and It+l' the algorithm attempts to minimize either the exponential function that arise in AdaBoost: m m Z = 2: exp (-ydt(Xi)) = 2: exp (-Yi(ft-l (Xi) + atht(Xi) + f3t)) i=1 i=1 m '" 2: Dt(i) exp (-Yi(atht(Xi) + f3t)) , (3) i=1 or the logistic loss function: m (4) i=1 m i=1 m - 2: (fh log(Pt+1 (Xi)) + (1 - ih) log(1 - Pt+1 (Xi))) (5) i=1 We initialize lo(x) to be zero everywhere and run the procedure for a predefined number of rounds T. The final classifier is therefore IT(X) = '£'['=1 (atht(x) + f3t) = f3 + '£'['=1 atht(x) where f3 = '£t f3t . We would like to note parenthetically that it is possible to use other loss functions that bound the 0-1 (classification) loss (see for instance [8]). Here we focus on the above loss functions, Land Z. Fixing It-I and ht, these functions are convex in at and f3t which guarantees, under mild conditions (details omitted due to lack of space), the uniqueness of at and f3t . On each round we look for the current base hypothesis ht that will reduce the loss function (Z or L) the most. As discussed before, each input instance X j defines a function <Pj (x) and is a candidate for ht(x). In general, there is no close form solution for Eqn. (3) and (5) and finding a and f3 for each possible input instance is time consuming. We therefore use a quadratic approximation for the loss functions. Using the quadratic approximation, for each <Pj we can find a and f3 analytically and calculate the reduction in the loss function. Let \7 Z = (~~, ~~) T and \7 L = (~~, ~~) T be the column vectors of the partial derivatives of Z and L w.r.t a and f3 (fixing It-I and ht ). Similarly, let \72 Z and \72 L be the 2 x 2 matrices of second order derivatives of Z and L with respect to a and f3. Then, quadratic approximation yields that (a,f3)T = (\72Z)-1 \7Z and (a,f3)T = (\72L)-1 \7L. On each round t we maintain a distribution D t which is defined from It as given by Eqn. (l) and conditional class probability estimates Pt(Xi) as given by Eqn. (2). Solving the linear equation above for a and f3 for each possible instance is done by setting ht (x) = <Pj (x), we get for Z (6) and for L (7) Leveraged Vector Machines ~--- -....... . :~_------,7"--~~~'i====~ Figure 1: Comparison of the test error as a function of number of leveraging rounds when using full numerical search for a and f3, a "one-step" numerical search based on a quadratic approximation of the loss function, and a one-step search with chunking of the instances. 613 Note that the equations above share much in common and require, after pre-computing P(Xi), the same amount of computation time. After calculating the value of a and f3 for each instance (x j , Y j ), we simply evaluate the corresponding value of the loss function, choose the instance (Xj> , Yj» that attains the minimal loss, and set ht = <pj>. We then numerically search for the optimal value of a and f3 by iterating Eqn. (6) or Eqn. (7) and summing the values into at and f3t. We would like to note that typically two or three iterations suffice and we can save time by using the value of a and f3 found using the quadratic approximation without a full numerical search for the optimal value of a and f3. (See also Fig. 1.) We repeat this process for T rounds or until no instance can serve as a base hypothesis. We note that the same instance can be chosen more than once, although not in consecutive iterations, and typically only a small fraction of the instances is actually used in building f. Roughly speaking, these instances are the "support patterns" of the leveraged machines although they are not necessarily the geometric support patterns. As in SVMs, in order to make the search for a base hypothesis efficient we pre-compute and store K(x, x') for all pairs x i- x' from 8. Storing these values require 1812 space, which might be prohibited in large problems. To save space, we employ the idea of chunking used in SVM. We partition 8 into r blocks 81,82 , ..• ,Sr of about the same size. We divide the iterations into sub-groups such that all iterations belonging to the ith sub-group use and evaluate kernels based on instances from the ith block only_ When switching to a new block k we need to compute the values K(x, x') for x E 8 and x, E Sk. This division into blocks might be more expensive since we typically use each block of instances more than once. However, the storage of the kernel values can be done in place and we thus save a factor of r in memory requirements. In practice we found that chunking does not hurt the performance. In Fig. 1 we show the test error as a function of number of rounds when using (a) full numerical search to determine a and f3 on each round, (b) using the quadratic approximation ("one-step") to find a and f3, and (c) using quadratic approximation with chunking. The number of instances in the experiment is 1000, each block for chunking is of size 100, and we switch to a different block every 100 iterations. (Further description of the data is given in the next section.) In this example, after 10 iterations, there is virtually no difference in the performance of the different schemes. 4 Experiments with synthetic data In this section we describe experiments with synthetic data comparing different aspects of leveraged vector machines to SVMs. The original instance space is two dimensional where the positive class includes all points inside a circle of radius R, i.e., an instance (UI, U2) E 1R? is labeled +1 iff ui + u~ ~ R. The instances were picked at random according to a zero mean unit variance normal distribution and R was set such exactly half of the instances belong to the positive class. In all the experiments described in this section we generated 10 groups of training and test sets each of which includes 1000 train and test examples. Overall, there are 10,000 training examples and 10,000 test examples. The 614 .• ' J» - . ._\ .~ - • OM • ! , .. - _.'-I - I -D .. SVM i ~ I ! V . -•• i· ~ .. 5 • , ... Y. Singer Figure 2: Performance comparison of SVM and BVM as a function of the training data size (left), the dimension of the kernels (middle), and the number of redundant features. ' 0 ~ t ea t ot .. ~II 0 "" • I' ....... ~ • IMI ~ L'" - . . svu • •• /;I .. Figure 3: Train and test errors for SVM, LVM, and BVM as a function of the label noise. average variance of the estimates of the empirical errors across experiments is about 0.2%. For SVM we set the regularization parameter, C , to 100 and used 500 iterations to build leveraged machines. In all the experiments without noise the results for BVM and LVM were practically the same. We therefore only compare BVM to SVM in Fig. 2. Unless said otherwise we used polynomials of degree two as kernels: K(X,' x) = (x· x' + 1)2. Hence, the data is separable in the absence of noise. In the first experiment we tested the sensitivity to the number of training examples by omitting examples from the training data (without any modification to the test sets). On the left part of Fig. 2 we plot the test error as a function of the number of training examples. The test error of BVM is almost indistinguishable from the error of SVM and performance of both methods improves very fast as a function of training examples. Next, we compared the performance as a function of the dimension of polynomial constituting the kernel. We ran the algorithms with kernels of the form K(X,' x) = (x · x' + l)d for d = 2, ... ,8. The results are depicted in the middle plots of Fig. 2. Again, the performance of BVM and SVM is very close (note the small scale of the y axis for the test error in this experiment). To conclude the experiments with clean, realizable, data we checked the sensitivity to irrelevant features of the input. Each input instance (Ul' U2) was augmented with random elements U3," . ,Ul to form an input vector of dimension l. The right hand side graphs of Fig. 2 shows the test error as a function of 1 for 1 = 2, ... , 12. Once more we see that the performance of both algorithms is very similar. We next compared the performance of the algorithms in the presence of noise. We used kernels of dimension two and instances without redundant features. The label of each instance was flipped with probability E. We ran 15 sets of experiments, for € = 0.01, ... , 0.15. As before, each set included 10 runs each of which used 1000 training examples and 1000 test examples. In Fig. 3 we show the average training error (left), and the average test error (right), for each of the algorithms. It is apparent from the graphs that BVMs built based on the exponential loss are much more sensitive to noise than SVMs and LVMs, and their generalization error degrades significantly, even for low noise rates. The generalization error ofLVMs is, on the other hand, only slightly worse than the that of SVMs, although the Leveraged Vector Machines .~ -LW ----_. --~ ..... LW 615 ~ ~ - .. - .. --Figure 4: The training error, test error, and the cumulative Ll norm (L~'=l la~ I) as a function of the number of leveraging iterations for LVM,BVM, and PBVM. only algorithmic difference in constructing BVMs and LVMs is in the loss function. The fact that LVMs exhibit performance similar to SVM can be partially attributed to the fact that the asymptotic behavior of their loss functions is the same. 5 A norm-penalized version One of the problems with boosting and the corresponding leveraging algorithm with the exponential loss described here, is that it might increase the confidence on a few instances while misclassifying many other instances, albeit with a small confidence. This often happens on late rounds, during which the distribution D t (i) is concentrated on a few examples, and the leveraging algorithm typically assigns a large weight to a weak hypothesis that does not effect most of the instances. It is therefore desired to control the complexity of the leveraged classifiers by limiting the magnitude of base hypotheses' weights. Several methods have been proposed to limit the confidence of AdaBoost, using, for instance, regularization (e.g., [9]) or "smoothing" the predictions [11]. Here we propose a norm-penalized method for BVM that is very simple to implement and maintains the convexity properties of the objective function. Following the idea Cortes and Vapnik's of SVMs in the nonseparable case [1] we add the following penalization term: ,0 exp (L;=1 latlP) . Simple algebric manipulation implies that the objective function at the tth round for BVMs with the penalization term above is, m Zt = I: Dt(i) exp (-Yi(atht(xi) + f3t» +,t exp{latIP) . (8) i=l It is also easy to show that the penalty parameter should be updated after each round is: ,t = ,t-l exp(lat-lIP)/Zt-l. Since Zt < 1, unless there is no kernel function better than random, ,t typically increases as a function of t, forcing more and more the new weights to be small. Note that Eqn. (8) implies that the search for a base predictor ht and weights at, f3t on each round can still be done independently of previous rounds by maintaining the distribution D t and a single regularization value 't. The penalty term for p = 1 and p = 2 simply adds a diagonal term to the matrix of second order derivatives (Eqn. (6» and the algorithm follows the same line (details omitted). For brevity we call the norm-penalized leveraging procedure PBVM. In Fig. 4 we plot the test error (right), training error (middle), and Lt latl as functions of number of rounds for LVM, BVM, and PBVM with p = 1 ,0 = 0.01. The training set in this example was made small on purpose (200 examples) and was contaminated with 5% label noise. In this very small example both LVM and BVM overfit while PBVM stops increasing the weights and finds a reasonably good classifier. The plots demonstrate that the norm-penalized version can safeguard against overfitting by preventing the weights from growing arbitrarily large, and that the effect of the penalized version is very similar to early stopping. We would like 616 Y. Singer SVM LVM BVM RBVM SVM LVM BVM PBVM #Example DataSet & Size Size Size Size Error Error Error Error (Source) #Feature labor (UC!) 57 : 16 12.5 13.7 16.1 13.6 6.0 14.0 14.0 12.0 echocard. (uci) 74 : 12 7.8 13.0 12.6 12.4 8.6 5.7 10.0 10.0 bridges (uci) 102 : 7 27.2 20.2 18.5 17.9 15.0 15.0 23.0 14.0 hepati tis (uci) 155: 19 41.2 13.5 17.4 14.0 21.3 22.0 22.7 22.0 horse·colic (uci) 300: 23 122.0 13.0 13.0 13.0 14.7 14.7 14.7 13.2 liver (uci) 345 : 6 228.6 11.3 12.8 10.7 33.8 35.6 33.5 35.6 ionosphere (uci) 351: 34 63.4 58.9 67.9 59.1 13.7 13.1 16.9 13.7 vote (uci) 435 : 16 37.0 37.0 41.0 37.0 4.4 5.2 5.9 5.2 ticketl (att) 556 : 78 48.1 84.6 89.3 82.3 8.4 3.3 11.5 5.1 ticket2 (att) 556: 53 52.6 77.1 75.4 74.0 6.6 6.4 8.0 6.4 ticket3 (att) 556 : 61 46.1 76.2 77.8 73.3 6.9 4.9 7.6 6.7 bands (uci) 690: 39 265.5 78.2 76.4 75.6 32.8 33.2 34.3 33.3 breast-wisc (uci) 699: 9 49.3 26.5 24.4 24.0 3.5 3.6 4.1 4.1 pima (uci) 768 : 8 360.7 47.7 30.3 22.8 23.0 22.6 23.2 22.1 german (uci) 1000: 10 485.2 89.8 96.5 87.0 23.5 24.0 23.8 24.1 weather (uci) 1000: 35 562.0 52.0 52.0 52.0 25.9 25.4 25.4 25.4 network (att) 2600: 35 1031.0 42.0 43.0 42.0 24.8 21.2 23.5 21.2 splice (uci) 3190: 60 318.0 153.0 156.0 153.0 8.0 8.4 8.4 8.4 boa (att) 5000: 68 637.0 183.0 178.0 160.0 41.5 40.8 40.8 41.0 Table 1: Summary of results for a collection of binary classification problems. to note that we found experimentally that the norm-penalized version does compensate for incorrect estimates of a and fJ due to malicious label noise. The experimental results given in the next section show, however, that it does indeed help in preventing overfitting when the training set is small. 6 Experiments with natural data We compared the practical performance of leveraged vector machines with SVMs on a collection of nineteen dataset from the UCI machine learning repository and AT&T networking and marketing data. For SVM we set C = 100. We built each of the leveraged vector machines using 500 rounds. For PBVM we used again p = 1 and 'Yo = 0.0l. We used chunking in building the leveraged vector machines, dividing each training set into 10 blocks. For all the datasets, with the exception of "boa", we used lO-fold cross validation to calculate the test error. (The dataset "boa" has 5000 training examples and 6000 test examples.) The performance of SVM, LVM, and PBVM seem comparable. In fact, with the exception of a very few datasets the differences in error rates are not statistically significant. Of the three methods (SVM, PBVM, and LVM), LVM is the simplest to implement the time required to build an LVM is typically much shorter than that of an SVM. It is also worth noting that the size of leveraged machines is often smaller than the size of the corresponding SVM. Finally, it apparent that PBVMs frequently yield better results than BVMs, especially for small and medium size datasets. References [I] Corinna Cones and Vladimir Vapnik. Suppon-vector networks. Machine Learning, 20(3):273-297, September 1995. [2] N. Duffy and D. Helmbold. A geometric approach to leveraging weak learners. EuroCOLT '99. [3] Yoav Freund. Boosting a weak learning algorithm by majority. Information and Computation, 121(2):256-285, 1995. [4] Yoav Freund and Roben E. Schapire. A decision· theoretic generalization of on-line learning and an application to boosting. Journal of Computer and System Sciences, 55(1):119-139, August 1997. [5] J. Friedman, T. Hastie, and R. Tibshirani. Additive logistic regression: a statistical view of boosting. Tech. Repon, 1998. [6] Michael Kearns and Leslie G. Valiant. Cryptographic limitations on learning Boolean formulae and finite automata. Journal of the Associationfor Computing Machiner)" 41(1):67-95, January 1994. [7] John D. Laffeny. Additive models, boosting and inference for generalized divergences. In Proceedings of the Twelfth Annual Conference on Computational Learning Theor)" 1999. [8] L. Mason, J. Baxter. P. Banlett, and M. Frean. Doom II. Technical repon. Depa. of Sys. Eng. ANU 1999. [9] G. Rlitsch, T.Onoda. and K.-R. Miiller. Regularizing adaboost. In Advances in Neural Info. Processing Systems 12,1998. [10] Roben E. Schapire. The strength of weak learnability. Machine Learning, 5(2):197-227,1990. [II] Roben E. Schapire and Yoram Singer. Improved boosting algorithms using confidence-rated predictions. COLT'98. [12] V. N. Vapnik. Estimation of Dependences Based on Empirical Data. Springer-Verlag, 1982. [13] Vladimir N. Vapnik. The Nature of Statistical Learning Theor),. Springer, 1995.
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Neural Network Based Model Predictive Control Stephen Piche Pavilion Technologies Austin, TX 78758 spiche@pav.com Gene Boe Pavilion Technologies Austin, TX 78758 gboe@pav.com Jim Keeler Pavilion Technologies Austin, TX 78758 jkeeler@pav.com Doug Johnson Pavilion Technologies Austin, TX 78758 djohnson@pav.com Abstract Greg Martin Pavilion Technologies Austin, TX 78758 gmartin@pav.com Mark Gerules Pavilion Technologies Austin, TX 78758 mgerules@pav.com Model Predictive Control (MPC), a control algorithm which uses an optimizer to solve for the optimal control moves over a future time horizon based upon a model of the process, has become a standard control technique in the process industries over the past two decades. In most industrial applications, a linear dynamic model developed using empirical data is used even though the process itself is often nonlinear. Linear models have been used because of the difficulty in developing a generic nonlinear model from empirical data and the computational expense often involved in using nonlinear models. In this paper, we present a generic neural network based technique for developing nonlinear dynamic models from empirical data and show that these models can be efficiently used in a model predictive control framework. This nonlinear MPC based approach has been successfully implemented in a number of industrial applications in the refining, petrochemical, paper and food industries. Performance of the controller on a nonlinear industrial process, a polyethylene reactor, is presented. 1 Introduction Model predictive control has become the standard technique for supervisory control in the process industries with over 2,000 applications in the refining, petrochemicals, chemicals, pulp and paper, and food processing industries [1]. Model Predictive Control was developed in the late 70's and came into wide-spread use, particularly in the refining industry, in the 80's. The economic benefit of this approach to control has been documented [1,2]. 1030 s. Piche, J. Keeler, G. Martin, G. Boe, D. Johnson and M. Gerules Several factors have contributed to the wide-spread use of MPC in the process industries: 1. Multivariate Control: Industrial processes are typically coupled multipleinput multiple-output (MIMO) systems. MIMO control can be implemented using MPC. 2. Constraints: Constraints on the inputs and outputs of a process due to safety considerations are common in the process industries. These constraints can be integrated into the control calculation using MPC. 3. Sampling Period: Unlike systems in other industries such as automotive or aerospace, the open-loop settling times for many processes is on the order of hours rather than milliseconds. This slow settling time translates to sampling periods on the order of minutes. Because the sampling period is sufficiently long, the complex optimization calculations that are required to implement MPC can be solved at each sampling period. 4. Commercial Tools: Commercial tools that facilitate model development and controller implementation have allowed proliferation of MPC in the process industries. U nti! recently, industrial applications of MPC have relied upon linear dynamic models even though most processes are nonlinear. MPC based upon linear models is acceptable when the process operates at a single setpoint and the primary use of the controller is the rejection of disturbances. However, many chemical processes, including polymer reactors, do not operate at a single setpoint. These processes are often required to operate at different set points depending upon the grade of the product that is to be produced. Because these processes operate over the nonlinear range of the system, linear MPC often results in poor performance. To properly control these processes, a nonlinear model is needed in the MPC algorithm. This need for nonlinear models in MPC is well recognized. A number of researchers and commercial companies have developed both simulation and industrial applications using a variety of different technologies including both first principles and empirical approaches such as neural networks [3,4]. Although a variety of different models have been developed, they have not been practical for wide scale industrial application. On one hand, nonlinear models built using first principle techniques are expensive to develop and are specific to a process. Conversely, many empirically based nonlinear models are not appropriate for wide scale use because they require costly plant tests in multiple operating regions or because they are too computationally expensive to use in a real-time environment. This paper presents a nonlinear model that has been developed for wide scale industrial use. It is an empirical model based upon a neural network which is developed using plant test data from a single operating region and historical data from all regions. This is in contrast to the usual approach of using plant test data from multiple regions. This model has been used on over 50 industrial applications and was recognized in a recent survey paper on nonlinear MPC as the most widely used nonlinear MPC controller in the process industries[l]. Neural Network Based Model Predictive Control 1031 After providing a brief overview of model predictive control in the next section, we present details on the formulation of the nonlinear model. After describing the model, an industrial application is presented that validates the usefulness of the nonlinear model in an MPC algorithm. 2 Model Predictive Control Model predictive control is based upon solving an optimization problem for the control actions at each sampling interval. Using MPC, an optimizer computes future control actions that minimize the difference between a model of the process and desired performance over a time horizon (typically the time horizon is greater than the open-loop settling time of the process). For example, given a linear model of process, (1) where u(t) represents the input to the process, the optimizer may be used to minimize an objective function at time t, T J = 2)(Yt+i - Yt+i)2 + (Ut+i - Ut+i_l)2) (2) i=l where Yt is the desired set point for the output and T is the length of the time horizon. In addition to minimizing an objective function, the optimizer is used to observe a set of constraints. For example, it is common to place upper and lower bounds on the inputs as well as bounds on the rate of change of the input, U upper 2:: Ut+i 2:: Ul ower V 1:::; i :::; T AUupper 2:: Ut+i - Ut+i-l 2:: AUlower V 1:::; i :::; T (3) (4) where Uupper and Ulower are the upper and lower input bounds while AUupper and AUlower are the upper and lower rate of change bounds. After the trajectory of future control actions is computed, only the first value in the trajectory is sent as a setpoint to the actuators. The optimization calculation is re-run at each sampling interval using a model which has been updated using feedback. The form of the model, the objective function, the constraints and the type of optimizer have been active areas of research over the past two decades. A number of excellent survey papers on MPC cover these topics [1,2,4]. As discussed above, we have selected a MIMO nonlinear model which is presented in the next section. Although the objective function given above contains two terms (desired output and input move suppression), the objective function used in our implementation contains thirteen separate terms. (The details of the objective function are beyond the scope of this paper.) Our implementation uses the constraints given above in (3) and (4). Because we use nonlinear models, a nonlinear programming technique must be used to solve the optimization problem. We use LS-GRG which is a reduced gradient solver [5]. 1032 S. Piche, J. Keeler, G. Martin, G. Roe, D. Johnson and M Gerules 3 A Generic and Parsimonious Nonlinear Model For a nonlinear model to achieve wide-spread industrial use, the model must be parsimonious so that it can be efficiently used in an optimization problem. Furthermore, it must be developed from limited process data. As discussed below, the nonlinear model we use is composed of a combination of a nonlinear steady state model and a linear dynamic model which can be derived from available data. The method of combining the models results in a parsimonious nonlinear model. 3.1 Process data and component models The quantity and quality of available data ultimately determines the structure of an empirical model. In developing our models, the available data dictated the type of model that could be created. In the process industries, two types of data are available: 1. Historical data: The values of the inputs and outputs of most processes are saved at regular intervals to a data base. Furthermore, most processing companies retain historical data associated with their plant for several years. 2. Plant tests: Open-loop testing is a well accepted practice for determining the process dynamics for implementation ofMPC. However, open-loop testing in multiple operating regions is not well accepted and is impractical in most cases even if it were accepted. Most practitioners of MPC models have used plant test data and ignored historical data. Practitioners have ignored the historical data in the past because it was difficult to extract and preprocess the data, and build models. Historical data was also viewed as not useful because it was collected in closed-loop and therefore process dynamics could not be extracted in many cases. Using only the plant test data, the practitioner is limited to linear dynamic models. We chose to use the historical data because it can be used to create nonlinear steady state models of processes that operate at multiple setpoints. Combining the nonlinear steady state model with linear dynamic models from the plant test data provides a generic approach to developing nonlinear models. To easily facilitate the development of nonlinear models, a suite of tools has been developed for data extraction and preprocessing as well as model training. The nonlinear steady state models, Yss = NNss(u) (5) are implemented by a feedforward neural network and trained using variants of the backpropagation algorithm [6]. The developer has a great deal of flexibility in determining the architecture of the network including the ability to select which inputs affect which outputs. Finally, an algorithm for specifying bounds on the gain (Jacobian) of the model has recently been implemented [7]. Because of limited plant test data, the dynamic models are restricted to second order models with input time delay, Yt = -alYt-l a2Yt-2 + b1 Ut-d-l + b2Ut-d-2 (6) Neural Network Based Model Predictive Control 1033 The parameters of (6) are identified by minimizing the squared error between the model and the plant test data. To prevent a biased estimate of the parameters, the identification problem is solved using an optimizer because of the correlation in the model inputs [8]. Tools for selecting the identification regions and viewing the results are provided. 3.2 Combining the nonlinear steady state and dynamic models A variety of techniques exist for combining nonlinear steady state and linear dynamic models. The dynamic models can be used to either preprocess the inputs or postprocess the outputs of the steady state model. These models, referred to as Hammerstein and Weiner models respectively [8], contain a large number of parameters and are computationally expensive in an optimization problem when the model has many inputs and outputs. These models, when based upon neural networks, also extrapolate poorly. Gain scheduling is often used to combine nonlinear steady state models and linear dynamic models. Using a neural network steady state model, the gain at the current operating point, Ui, ayss gi = au I U=Ui (7) is used to update the gain of the linear dynamic model of (6), (8) where = b 1 + al + a2 19i b1 + b2 (9) b 1 + al + a2 2gi b1 + b2 (10) The difference equation is linearized about the point Ui and Yi = N N(Ui), thus, ~Y = Y - Yi and ~u = U Ui· To simplify the equations above, a single-input singleoutput (8180) system is used. Gain scheduling results in a parsimonious model that is efficient to use in the MPC optimization problem, however, because this model does not incorporate information about the gain over the entire trajectory, its use leads to suboptimal performance in the MPC algorithm. Our nonlinear model approach remedies this problem. By solving a steady state optimization problem whenever a setpoint change is made, it is possible to compute the final steady state values of the inputs, U f. Given the final steady state input values, the gain associated with the final steady state can be computed. For a 8180 system, this gain is given by (11) Using the initial and final gain associated with a setpoint change, the gain structure over the entire trajectory can be approximated. This two point gain scheduling overcomes the limitations of regular gain scheduling in MPC algorithms. 1034 s. Piche, J Keeler, G. Martin, G. Boe. D. Johnson and M Gerules Combining the initial and final gain with the linear dynamic model, a quadratic difference equation is derived for the overall nonlinear model, where = bi (1 + al + a2)(9f - 9i) (b1 + b2)(uf - ud b2 (1 + al + a2)(9f - 9d (b1 + b2)(uf - ud (13) (14) and VI and V2 are given by (9) and (10). Use of the gain at the final steady state introduces the last two terms of (12). This model allows the incorporation of gain information over the entire trajectory in the MPC algorithm. The gain at of (12) at Ui is 9i while at uf it is 9f. Between the two points, the gain is a linear combination of 9i and 9 f. For processes with large gain changes, such as polymer reactors, this can lead to dramatic improvements in MPC controller performance. An additional benefit of using the model of (12) is that we allow the user to bound the initial and final gain and thus control the amount of nonlinearity used in the model. For practitioners who are use to implementing MPC with linear models, using gain bounds allows them to transition from linear to nonlinear models. This ability to control the amount of nonlinearity used in the model has been important for acceptance of this new model in many applications. Finally, bounding the gains can be used to guarantee extrapolation performance of the model. The nonlinear model of (12) fits the criteria needed in order to allow wide spread use of nonlinear models for MPC. The model is based upon readily available data and has a parsimonious representation allowing models with many inputs and outputs to be efficiently used in the optimizer. Furthermore, it addresses the primary nonlinearity found in processes, that being the significant change in gain over the operating region. 4 Polymer Application The nonlinear model described above has been used in a wide-variety of industrial applications including Kamyr digesters (pUlp and paper), milk evaporators and dryers (food processing), toluene diamine purification (chemicals), polyethylene and polypropylene reactors (polymers) and a fluid catalytic cracking unit (refining). Highlights of one such application are given below. A MPC controller that uses the model described above has been applied to a Gas Phase High Density Polyethylene reactor at Chevron Chemical Co. in Cedar Bayou, Texas [9]. The process produces homopolymer and copolymer grades over a wide range of melt indices. It's average production rate per year is 230,000 tons. Optimal control of the process is difficult to achieve because the reactor is a highly coupled nonlinear MIMO system (7 inputs and 5 outputs). For example, a number of input-output pairs exhibit gains that varying by a factor of 10 or more over the operating region. In addition, grade changes are made every few days. During these transitions nonprime polymer is produced. Prior to commissioning these controllers, Neural Network Based Model Predictive Control 1035 these transitions took several hours to complete. Linear and gain scheduling based controller have been tried on similar reactors and have delivered limited success. The nonlinear model was constructed using only historical data. The nonlinear steady state model was trained upon historical data from a two year period. This data contained examples of all the products produced by the reactor. Accurate dynamic models were derived both from historical data and knowledge of the process, thus, no step tests were conducted on the process. Excellent performance of this controller has been reported [9]. A two-fold decrease in the variance of the primary quality variable (melt index) has been achieved. In addition, the average transition time has been decreased by 50%. Unscheduled shutdowns which occurred previously have been eliminated. Finally, the controller, which has been on-line for two years, has gained high operator acceptance. 5 Conclusion A generic and parsimonious nonlinear model which can be used in an MPC algorithm has been presented. The model is created by combining a nonlinear steady state model with a linear dynamic models. They are combined using a two-point gain scheduling technique. This nonlinear model has been used for control of a nonlinear MIMO polyethylene reactor at Chevron Chemical Co. The controller has also been used in 50 other applications in the refining, chemicals, food processing and pulp and paper industries. References [1] Qin, S.J. & Badgwell, T.A. (1997) An overview of industrial model predictive control technology. In J. Kantor, C. Garcia and B. Carnahan (eds.), Chemical Process Control AIChE Symposium Series, pp. 232-256. NY: AIChB. [2] Seborg, D.E. (1999) A perspective on advanced strategies for Process Control (Revisited). to appear in Pmc. of European Control Conf. Karlsruhe, Germany. [3] Qin, S.J. & Badgwell, T.A. (1998) An overview of nonlinear model predictive control applications. Pmc. IFAC Workshop on Nonlinear Model Predictive Control - Assessment and Future Directions, Ascona, Switzerland, June 3-5. [4] Meadow, E.S. & Rawlings, J.B. (1997) Model predictive control. In M. Hesnon and D. Seborg (eds.), Nonlinear Model Predictive Control, pp. 233-310. NJ: Prentice Hall. [5] Nash, S. & Sofer, A. (1996) Linear and Nonlinear Programming. NY: McGraw-Hill. [6] Rumelhart D.E, Hinton G.B. & Williams, R.J. (1986) Learning internal representations by error propagation. In D. Rumelhart and J. McClelland (eds.), Parallel Distributed Processing, pp. 318-362. Cambridge, MA: MIT Press. [7] Hartman, E. (2000) Training feedforward neural networks with gain constraints. To appear in Neural Computation. [8] Ljung, L. (1987) System Identification. NJ: Prentice Hall. [9] Goff S., Johnson D. & Gerules, M. (1998) Nonlinear control and optimization of a high density polyethylene reactor. Proc. Chemical Engineering Expo, Houston, June.
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Coastal Navigation with Mobile Robots Nicholas Roy and Sebastian Thrun School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 { nicholas. roy I sebastian. thrun } @cs.cmu.edu Abstract The problem that we address in this paper is how a mobile robot can plan in order to arrive at its goal with minimum uncertainty. Traditional motion planning algorithms often assume that a mobile robot can track its position reliably, however, in real world situations, reliable localization may not always be feasible. Partially Observable Markov Decision Processes (POMDPs) provide one way to maximize the certainty of reaching the goal state, but at the cost of computational intractability for large state spaces. The method we propose explicitly models the uncertainty of the robot's position as a state variable, and generates trajectories through the augmented pose-uncertainty space. By minimizing the positional uncertainty at the goal, the robot reduces the likelihood it becomes lost. We demonstrate experimentally that coastal navigation reduces the uncertainty at the goal, especially with degraded localization. 1 Introduction For an operational mobile robot, it is essential to prevent becoming lost. Early motion planners assumed that a robot would never be lost - that a robot could always know its position via dead reckoning without error [7]. This assumption proved to be untenable due to the small and inevitable inconsistencies in actual robot motion; robots that rely solely on dead reckoning for their position estimates lose their position quickly. Mobile robots now perform position tracking using a combination of sensor data and odometry [2, 10, 5]. However, the robot's ability to track its position can vary considerably with the robot's position in the environment. Some parts of the environment may lack good features for localization [11]. Other parts of the environment can have a large number of dynamic features (for example, people) that can mislead the localization system. Motion planners rarely, if ever, take the robot's position tracking ability into consideration. As the robot's localization suffers, the likelihood that the robot becomes lost increases, and as a consequence, the robot is less likely to complete the given trajectory. Most localization systems therefore compensate by adding environment-specific knowledge to the localization system, or by adding additional sensing capabilities to the robot, to guarantee that the robot can complete every possible path. In general, however, such alterations to the position tracking abilities of the robot have limitations, and an alternative scheme must be used to ensure that the robot can navigate with maximum reliability. The conventional planners represent one end of a spectrum of approaches (figure 1), in that a plan can be computed easily, but at the cost of not modelling localization performance. At opposite end of the spectrum is the Partially Observable Markov Decision Process 1044 Conventional Path Planner Tractable Not Robust N Roy and S. Thntn POMDP Intracwble Robust Figure 1: The continuum of possible approaches to the motion planning, from the robust but intractable POMDP, to the potentially failure-prone but real-time conventional planners. Coastal navigation lies in the middle of this spectrum. (POMDP). POMDPs in a sense are the brass ring of planning with uncertainty; a POMDP policy will make exactly the right kind of compromise between conventional optimality considerations and certainty of achieving the goal state. Many people have examined the use of POMDPs for mobile robot navigation [5, 6, 8]. However, computing a POMDP solution is computationally intractable (PSPACE-hard) for large state systems - a mobile robot operating in the real world often has millions of possible states. As a result, many of the mobile robot POMDP solutions have made simplifying assumptions about the world in order to reduce the state space size. Many of these assumptions do not scale to larger environments or robots. In contrast, our hypothesis is that only a small number of the dimensions of the uncertainty matter, and that we can augment the state with these dimensions to approximate a solution to the POMDP. The coastal navigation model developed in this paper represents a tradeoff between robust trajectories and computational tractability, and is inspired by traditional navigation of ships. Ships often use the coasts of continents for navigation in the absence of better tools such as GPS, since being close to the land allows sailors to determine with high accuracy where they are. The success of this method results from coast lines containing enough information in their structure for accurate localization. By navigating sufficiently close to areas of the map that have high information content, the likelihood of getting lost can be minimized. 2 Modelling Uncertainty The problem that we address in this paper is how a mobile robot can plan in order to arrive at its goal with minimum uncertainty. Throughout this discussion, we will be assuming a known map of the environment [9]. The position, x, of the robot is given as the location (x, y) and direction e, defined over a space X = (X, Y, 8). Our localization method is a grid-based implementation of Markov localization [3, 5]. This method represents the robot's belief in its current position using a 3-dimensional grid over X = (X, Y, 8), which allows for a discrete approximation of arbitrary probability distributions. The probability that the robot has a particular pose x is given by the probability p(x). State Augmentation We can extend the state of the robot from the 3-dimensional pose space to an augmented pose-uncertainty space. We can represent the uncertainty of the robot's positional distribution as the entropy, H(Px ) = -J p(x) log(p(x)) dx (1) x We therefore represent the state space of the robot as the tuple S (x,y,e,H(x,y,e)) (x, H(x)) State Transitions In order to construct a plan between two points in the environment, we need to be able to represent the effect of the robot's sensing and moving actions. The implementation of Markov localization provides the following equations for the tracking Coastal Navigation with Mobile Robots the robot's pose from x to x': p(x'lu) p(x'lz) J p(x'lx, u)p(x)dx x ap(zlx)p(x) 1045 (2) (3) These equations are taken from [3, 12], where equation (2) gives the prediction phase of localization (after motion u), and equation (3) gives the update phase of localization (after receiving observation z). a is a normalizing constant. We extend these equations to the fourth dimension as follows: 3 Planning p(slu) p(slz) (p(xlu),ll(p(xlu))) (p(xlz), ll(p(xlz))) (4) (5) Equations (4) and (5) provide a mechanism for tracking the robot's state, and in fact contain redundant information, since the extra state variable ll(x) is also contained in the probability distribution p(x). However, in order to make the planning problem tractable, we cannot in fact maintain the probabilistic sensing model. To do so would put the planning problem firmly in the domain ofPOMDPs, with the associated computational intractability. Instead, we make a simplifying assumption, that is, that the positional probability distribution of the robot can be represented at all times by a Gaussian centered at the mean x. This allows us to approximate the positional distribution with a single statistic, the entropy. In POMDP terms, we using the assumption of Gaussian distributions to compress the belief space to a single dimension. We can now represent the positional probability distribution completely with the vector s, since the width of the Gaussian is represented by the entropy ll(x). More importantly, the simplifying assumption allows us to track the state of the robot deterministically. Although the state transitions are stochastic (as in equation (4», the observations are not. At any point in time, the sensors identify the true state of the system, with some certainty given by II (p(xlz)). This allows us to compress the state transitions into a single rule: p(slu) (p(xlu),ll(p(xlu,z))) (6) The final position of the robot depends only on the motion command 1l and can be identified by sensing z. However, the uncertainty of the pose, ll(p(xlll, z)), is a function not only of the motion command but also the sensing. The simplifying assumption of Gaussian models is in general untenable for localization; however, we shall see that this assumption is sufficient for the purpose of motion planning. One final modification must be made to the state transition rule. In a perfect world, it would be possible to predict exactly what observation would be made. However, it is exactly the stochastic and noisy nature of real sensors that generates planning difficulty, yet the update rule (6) assumes that it is possible to predict measurement z at pose x. Deterministic prediction is not possible; however, it is possible to compute probabilities for sensor measurements, and thus generate an expected value for the entropy based on the probability distribution of observations Z, which leads to the final state transition rule: p(slu) (p(xlu), Ez[ll(p(xlu , z))]) (7) where Ez[ll(p(xlll, z))] represents the expected value of the entropy of the pose distribution over the space of possible sensor measurements. With the transition rule in equation (7), we can now compute the transition probabilities for any particular state using a model of the robot's motion, a model of the robot's sensor and a map of th~ environment. The probability p(xlu) is given by a model of the robot's motion, and can be easily precomputed for each action u. The expectation term Ez [ll] 1046 N. Roy and S. Thrun can also be precomputed for each possible state s. The precomputation of these transition probabilities is very time-intensive, because it requires simulating sensing at each state in the environment, and then computing the posterior distribution. However, as the precomputation is a one-time operation for the environment and robot, planning itself can be an online operation and is (in the limit) unaffected by the speed of computing the transition probabilities. 3.1 Computing Trajectories With the state update rule given in equation (7), we can now compute the optimal trajectory to a particular goal. We would in fact like to compute not just the optimal trajectory from the current robot position, but the optimal action from any position in the world. If the robot should deviate from the expected trajectory for any reason (such as error in the motion, or due to low-level control constraints), interests of efficiency suggest precomputing actions for continuing to the goal, rather than continually replanning as these contingencies arise. Note that the motion planning problem as we have now phrased it can be viewed as the problem of computing the optimal policy for a given problem. The Markovian, stochastic nature of the transitions, coupled with the need to compute the optimal policy for all states, suggests a value iteration approach. Value iteration attempts to find the policy that maximizes the long-term reward [1,4]. The problem becomes one of finding the value function, J(s) which assigns a value to each state. The optimal action at each state can then be easily computed by determining the expected value of each action at each state, from the neighboring values. We use a modified form of Bellman's equations to give the value of state J (s) and policy as N J(Si) m:x[R(sd + C(s, u) + L p(Sj lSi, u) . J(Sj)] (8) j=1 N argmax[R(si) + C(s, u) + L p(Sj lSi, u) . J(Sj)] (9) Il j=1 By iterating equation (8), the value function iteratively settles to a converged value over all states. Iteration stops when no state value changes above some threshold value. In the above equations, R(sd is the immediate reward at state si, p(Sj lSi , u) is the transition probability from state si to state Sj, and C(s, u) is the cost of taking action u at state s. Note that the form of the equations is undiscounted in the traditional sense, however, the additive cost term plays a similar role in that the system is penalized for policies that take longer trajectories. The cost in general is simply the distance of one step in the given direction u, although the cost of travel close to obstacles is higher, in order to create a safety margin around obstacles. The cost of an action that would cause a collision is infinite, preventing such actions from being used. The immediate reward is localized only at the goal pose. However, the goal pose has a range of possible values for the uncertainty, creating a set of goal states, g. In order to reward policies that arrive at a goal state with a lower uncertainty, the reward is scaled linearly with goal state uncertainty. R( xd = {~ - H (s) S (; 9 otherwise (10) By implementing the value iteration given in the equations (8) and (9) in a dynamic program, we can compute the value function in O( nkcrid where n is the number of states in the environment (number of positions x number of entropy levels) and kcrit is the number of iterations to convergence. With the value function computed, we can generate the optimal action for any state in O(a) time, where a is the number of actions out of each state. Coastal Navigation with Mobile Robots 1047 4 Experimental Results Figure 2 shows the mobile robot, Minerva, used for this research. Minerva is a RWI B-18, and senses using a 360 0 field of view laser range finder at 10 increments. Figure 2: Minerva, the B-18 mobile robot used for this research, and an example environment map, the Smithsonian National Museum of American History. The black areas are the walls and obstacles. Note the large sparse areas in the center of the environment. Also shown in figure 2 is an example environment,the Smithsonian National Museum of American History. Minerva was used to generate this map, and operated as a tour-guide in the museum for two weeks in the summer of 1998. This museum has many of the features that make localization difficult -large open spaces, and many dynamic obstacles (people) that can mislead the sensors. Startillg Positioll Start POSitiOIl ~ ~~ ~ .,.~ ~ . I . .-A. ~ (a) Conventional (b) Coastal (c) Sensor Map Figure 3: Two examples in the museum environment. The left trajectory is given by a conventional, shortest-path planner. The middle trajectory is given by the coastal navigation planner. The black areas correspond to obstacles, the dark grey areas correspond to regions where sensor infonnation is available, the light grey areas to regions where no sensor infonnation is available. Figure 3 shows the effect of different planners in the sample environment. Panel (a) shows the trajectory of a conventional, shortest distance planner. Note that the robot moves di1048 N. Roy and S. Thrun rectly towards the goal. Panel (b) shows the trajectory given by the coastal planner. In both examples, the robot moves towards an obstacle, and relocalizes once it is in sensor range of the obstacle, before moving towards the goal. These periodic relocalizations are essential for the robot to arrive at the goal with minimum positional uncertainty, and maximum reliability. Panel (c) shows the sensor map of the environment. The black areas show obstacles and walls, and the light grey areas are where no information is available to the sensors, because all environmental features are outside the range of the sensors. The dark grey areas indicate areas where the information gain from the sensors is not zero; the darker grey the area, the better the information gain from the sensors. 20 18 16 14 ~ 12 t;I " to ,., 8 e Jl 6 4 2 0 ·2 0 Positional Uncenainty at Goal Conventional Navigation -coastal Navigation ! ...... + .......... +- ......... .1 ............. I ............... J 0.5 I 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Figure 4: The performance of the coastal navigation algorithm compared to the coastal motion planner. The graph depicts the entropy of the position probability distribution against the range of the laser sensor. Note that the coastal navigation dramatically improves the certainty of the goal position with shorter range laser sensing. Figure 4 is a comparison of the average positional certainty (computed as entropy of the positional probability) of the robot at its goal position, compared to the range of the laser range sensor. As the range of the laser range gets shorter, the robot can see fewer and fewer environmental features - this is essentially a way of reducing the ability of the robot to localize itself. The upper line is the performance of a conventional shortest-distance path planner, and the lower line is the coastal planner. The coastal planner has a lower uncertainty for all ranges of the laser sensor, and is substantially lower at shorter ranges, confirming that the coastal navigation has the most effect when the localization is worst. 5 Conclusion In this paper, we have described a particular problem of motion planning - how to guarantee that a mobile robot can reach its goal with maximum reliability. Conventional motion planners do not typically plan according to the ability of the localization unit in different areas of the environment, and thus make no claims about the robustness of the generated trajectory. In contrast, POMDPs provide the correct solution to the problem of robust trajectories, however, computing the solution to a POMDP is intractable for the size of the state space for typical mobile robot environments. We propose a motion planner with an augmented state space that represents positional uncertainty explicitly as an extra dimension. The motion planner then plans through poseuncertainty space, to arrive at the goal pose with the lowest possible uncertainty. This can be seen to be an approximation to a POMDP where the multi-dimensional belief space is represented as a subset of statistics, in this case the entropy of the belief space. We have shown some experimental comparisons with a conventional motion planner. Not only did the coastal navigation generated trajectories that provided substantial improvement of the positional certainty at the goal compared to the conventional planner, but the improvement became more pronounced as the localization was degraded. Coastal Navigation with Mobile Robots 1049 The model presented here, however, is not complete. The entire methodology hinges upon the assumption that the robot's probability distribution can be adequately represented by the entropy of the distribution. This assumption is valid if the distribution is restricted to a uni-modal Gaussian, however, most Markov localization methods that are based on this assumption fail, because multi-modal, non-Gaussian positional distributions are quite common for moving robots. Nonetheless, it may be that multiple uncertainty statistics along multiple dimensions (e.g., x and y) may do a better job of capturing the uncertainty sufficiently. It is an question for future work as to how many statistics can capture the uncertainty of a mobile robot, and under what environmental conditions. Acknowledgments The authors gratefully acknowledge the advice and collaboration of Tom Mitchell throughout the development of this work. Wolfram Burgard and Dieter Fox played an instrumental role in the development of earlier versions of this work, and their involvement and discussion of this new model is much appreciated. This work was partially funded by the Fonds pour la Formation de Chercheurs et l' Aide a la Recherche (FCAR). References [1] R. Bellman. Dynamic Programming. Princeton University Press, NJ, 1957. [2] w. Burgard, D. Fox, D. Hennig, and T. Schmidt. Estimating the absolute position of a mobile robot using position probability grids. In AAAI, 1996. [3] D. Fox, W. Burgard, and S. Thrun. Active Markov localization for mobile robots. Robotics and Autonomous Systems, 25(3-4), 1998. [4] R. A. Howard. Dynamic Programming and Markov Processes. MIT, 1960. [5] L. Kaelbling, A. R. Cassandra, and J. A. Kurien. Acting under uncertainty: Discrete Bayesian models for mobile-robot navigation. In IROS, 1996. [6] S. Koenig and R. Simmons. The effect of representation and knowledge on goal-directed exploration with reinforcement learning algorithms. Machine Learning Journal, 22:227-250,1996. [7] J .-c. Latombe. Robot Motion Planning. Kluwer Academic Publishers, 1991. [8] S. Mahadevan and N. Khaleeli. Robust mobile robot navigation using partially-observable semi-Markov decision processes. 1999. [9] H. P. Moravec and A. Elfes. High resolution maps from wide angle sonar. In ICRA, 1985. [10] R. Sim and G. Dudek. Mobile robot localization from learned landmarks. In lROS, 1998. [11] H. Takeda, C. Facchinetti, and J.-c. Latombe. Planning the motions of mobile robot in a sensory uncertainty field. IEEE Trans. on Pattern Analysis and Machine Intelligence, 16(10), 1994. [12] S. Thrun, D. Fox, and W. Burgard. A probabilistic approach to concurrent mapping and localization for mobile robots. Machine Learning, 431,1998.
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Boosting with Multi-Way Branching in Decision Trees Yishay Mansour AT&T Labs-Research 180 Park Ave Florham Park NJ 07932 David McAllester {mansour, dmac }@research.att.com Abstract It is known that decision tree learning can be viewed as a form of boosting. However, existing boosting theorems for decision tree learning allow only binary-branching trees and the generalization to multi-branching trees is not immediate. Practical decision tree algorithms, such as CART and C4.5, implement a trade-off between the number of branches and the improvement in tree quality as measured by an index function. Here we give a boosting justification for a particular quantitative trade-off curve. Our main theorem states, in essence, that if we require an improvement proportional to the log of the number of branches then top-down greedy construction of decision trees remains an effective boosting algorithm. 1 Introduction Decision trees have been proved to be a very popular tool in experimental machine learning. Their popularity stems from two basic features they can be constructed quickly and they seem to achieve low error rates in practice. In some cases the time required for tree growth scales linearly with the sample size. Efficient tree construction allows for very large data sets. On the other hand, although there are known theoretical handicaps of the decision tree representations, it seem that in practice they achieve accuracy which is comparable to other learning paradigms such as neural networks. While decision tree learning algorithms are popular in practice it seems hard to quantify their success ,in a theoretical model. It is fairly easy to see that even if the target function can be described using a small decision tree, tree learning algorithms may fail to find a good approximation. Kearns and, Mansour [6] used the weak learning hypothesis to show that standard tree learning algorithms perform boosting. This provides a theoretical justification for decision tree learning similar Boosting with Multi-Way Branching in Decision Trees 301 to justifications that have been given for various other boosting algorithms, such as AdaBoost [4]. Most decision tree learning algorithms use a top-down growth process. Given a current tree the algorithm selects some leaf node and extends it to an internal node by assigning to it some "branching function" and adding a leaf to each possible output value of this branching function. The set of branching functions may differ from one algorithm to another, but most algorithms used in practice try to keep the set of branching functions fairly simple. For example, in C4.5 [7], each branching function depends on a single attribute. For categorical attributes, the branching is according to the attribute's value, while for continuous attributes it performs a comparison of the attribute with some constant. Of course such top-down tree growth can over-fit the data it is easy to construct a (large) tree whose error rate on the training data is zero. However, if the class of splitting functions has finite VC dimension then it is possible to prove that, with high confidence of the choice of the training data, for all trees T the true error rate of T is bounded by f(T) + 0 (JITI/m) where f(T) is the error rate of T on the training sample, ITI is the number of leaves of T, and m is the size of the training sample. Over-fitting can be avoided by requiring that top-down tree growth produce a small tree. In practice this is usually done by constructing a large tree and then pruning away some of its nodes. Here we take a slightly different approach. We assume a given target tree size s and consider the problem of constructing a tree T with ITI = sand f(T) as small as possible. We can avoid over-fitting by selecting a small target value for the tree size. A fundamental question in top-down tree growth is how to select the branching function when growing a given leaf. We can think of the target size as a "budget" . A four-way branch spends more of the tree size budget than does a two-way branch a four-way branch increases the tree size by roughly the same amount as two twoway branches. A sufficiently large branch would spend the entire tree size budget in a single step. Branches that spend more of the tree size budget should be required to achieve more progress than branches spending less ofthe budget. Naively, one would expect that the improvement should be required to be roughly linear in the number of new leaves introduced one should get a return proportional to the expense. However, a weak learning assumption and a target tree size define a nontrivial game between the learner and an adversary. The learner makes moves by selecting branching functions and the adversary makes moves by presenting options consistent with the weak learning hypothesis. We prove here that the learner achieve a better value in this game by selecting branches that get a return considerably smaller than the naive linear return. Our main theorem states, in essence, that the return need only be proportional to the log of the number of branches. 2 Preliminaries We assume a set X of instances and an unknown target function f mapping X to {O,l}. We assume a given "training set" S which is a set of pairs of the form (x, f(x)). We let 1l be a set of potential branching functions where each hE 1l is a function from X to a finite set Rh we allow different functions in 1l to have different ranges. We require that for any h E 1l we have IRhl ~ 2. An 1l-tree is 302 Y. Mansour and D. McAllester a tree where each internal node is labeled with an branching function h E 1i and has children corresponding to the elements of the set Rh. We define ITI to be the number ofleafnodes ofT. We let L(T) be the set ofleafnodes ofT. For a given tree T, leaf node f of T and sample S we write Sl to denote the subset of the sample S reaching leaf f. For f E T we define Pl to be the fraction of the sample reaching leaf f, i.e., ISll/ISI. We define ql to be the fraction of the pairs (x, f(x» in Sl for which f(x) = 1. The training error ofT, denoted i(T), is L:lEL(T)Plmin(ql, 1- ql). 3 The Weak Learning Hypothesis and Boosting Here, as in [6], we view top-down decision tree learning as a form of Boosting [8, 3]. Boosting describes a general class of iterative algorithms based on a weak learning hypothesis. The classical weak learning hypothesis applies to classes of Boolean functions. Let 1i2 be the subset of branching functions h E 1i with IRhl = 2. For c5 > ° the classical c5-weak learning hypothesis for 1i2 states that for any distribution on X there exists an hE 1i2 with PrD(h(x) f f(x)) ~ 1/2-c5. Algorithms designed to exploit this particular hypothesis for classes of Boolean functions have proved to be quite useful in practice [5]. Kearns and Mansour show [6] that the key to using the weak learning hypothesis for decision tree learning is the use of an index function I : [0, 1] ~ [0,1] where I(q) ~ 1, I(q) ~ min(q, (1- q)) and where I(T) is defined to be L:lEL(T) PlI(ql). Note that these conditions imply that i(T) ~ I(T). For any sample W let qw be the fraction of pairs (x, f(x)) E W such that f(x) = 1. For any h E 1i let Th be the decision tree consisting of a single internal node with branching function h plus a leaf for each member of IRh I. Let Iw (Th) denote the value of I(Th) as measured with respect to the sample W. Let ~ (W, h) denote I (qW ) - Iw (Th). The quantity ~(W, h) is the reduction in the index for sample W achieved by introducing a single branch. Also note that Pt~(Sl, h) is the reduction in I(T) when the leaf f is replaced by the branch h. Kearns and Mansour [6] prove the following lemma. Lemma 3.1 (Kearns & Mansour) Assuming the c5-weak learning hypothesis for 1i2, and taking I(q) to be 2Jq(1- q), we have that for any sample W there exists an h E 1i2 such that ~(W,h) ~ ~:I(qw). This lemma motivates the following definition. Definition 1 We say that 1i2 and I satisfies the "I-weak tree-growth hypothesis if for any sample W from X there exists an hE 1i2 such that ~(W, h) ~ "II(qw). Lemma 3.1 states, in essence, that the classical weak learning hypothesis implies the weak tree growth hypothesis for the index function I(q) = 2J q(l - q). Empirically, however, the weak tree growth hypothesis seems to hold for a variety of index functions that were already used for tree growth prior to the work of Kearns and Mansour. The Ginni index I(q) = 4q(1 - q) is used in CART [1] and the entropy I(q) = -q log q - (1- q) log(l- q) is used in C4.5 [7]. It has long been empirically observed that it is possible to make steady progress in reducing I(T) for these choices of I while it is difficult to make steady progress in reducing i(T). We now define a simple binary branching procedure. For a given training set S and target tree size s this algorithm grows a tree with ITI = s. In the algorithm Boosting with Multi-Way Branching in Decision Trees 303 o denotes the trivial tree whose root is a leaf node and Tl h denotes the result of , replacing the leaf l with the branching function h and a new leaf for each element of Rh. T=0 WHILE (ITI < s) DO l f- argmaxl 'ftl1(til) h f- argmaxhEl£:l~(Sl' h) T f- Tl,h; END-WHILE We now define e(n) to be the quantity TI~:ll(l-;). Note that e(n) ~ TI~:/ e-7 = ~ .. -l /" e--Y Wi"'l 1 S < e--Y Inn = n--Y. Theorem 3.2 (Kearns & Mansour) 1f1l2 and I satisfy the ,-weak tree growth hypothesis then the binary branching procedure produces a tree T with i(T) ~ I(T) ~ e(ITI) ~ ITI--Y· Proof: The proof is by induction on the number of iterations of the procedure. We have that 1(0) ~ 1 = e(l) so the initial tree immediately satisfies the condition. We now assume that the condition is satisfied by T at the begining of an iteration and prove that it remains satisfied by Tl,h at the end of the iteration. Since I(T) = LlET Ih1(til) we have that the leaf l selected by the procedure is such that Pl1(til) 2: II~)· By the ,-weak tree growth assumption the function h selected by the procedure has the property that ~(Sl, h) 2: ,1(ql). We now have that I(T) - I(Tl,h) = Pl~(Sl' h) 2: P1I1(til) 2: ,II?il " This implies that I(Tl,h) ~ I(T) - rh1(T) = (1- j;)I(T) ~ (1- rh)e(ITI) = e(ITI + 1) = e(ITl,hl). o 4 Statement of the Main Theorem We now construct a tree-growth algorithm that selects multi-way branching functions. As with many weak learning hypotheses, the ,-weak tree-growth hypothesis can be viewed as defining a game between the learner and an adversary. Given a tree T the adversary selects a set of branching functions allowed at each leaf of the tree subject to the constraint that at each leaf l the adversary must provide a binary branching function h with ~(Sl' h) 2: ,1(til). The learner then selects a leaf land a branching function h and replaces T by Tl,h. The adversary then again selects a new set of options for each leaf subject to the ,-weak tree growth hypothesis. The proof of theorem 3.2 implies that even when the adversary can reassign all options at every move there exists a learner strategy, the binary branching procedure, guaranteed to achieves a final error rate of ITI--Y. Of course the optimal play for the adversary in this game is to only provide a single binary option at each leaf. However, in practice the "adversary" will make mistakes and provide options to the learner which can be exploited to achieve even lower error rates. Our objective now is to construct a strategy for the learner which can exploit multi-way branches provided by the adversary. We first say that a branching function h is acceptable for tree T and target size 304 Y. Mansour and D. MeAl/ester s if either IRhl = 2 or ITI < e(IRh!)s"Y/(2IRh!). We also define g(k) to be the quantity (1 - e(k»/"Y. It should be noted that g(2) = 1. It should also be noted that e( k) '" e -'Y Ink and hence for "Y In k small we have e( k) '" 1 "Y In k and hence g(k) '" Ink. We now define the following multi-branch tree growth procedure. T=0 WHILE (ITI < s) DO l +- argm~ Ptl(qt) h +- argmaxhEll, h acceptable for T and s ~(St, h)/g(IRhl) T +- Tt,h; END-WHILE A run of the multi-branch tree growth procedure will be called "Y-boosting if at each iteration the branching function h selected has the property that ~(St, h) / g(lRh I) ~ "YI(qt). The "Y-weak tree growth hypothesis implies that ~(St,h)/g(IRhl) ~ "YI(qt)/g(2) = "YI(qt). Therefore, the "Y-weak tree growth hypothesis implies that every run of the multi-branch growth procedure is "Y-bootsing. But a run can be "Y-bootsing by exploiting mutli-way branches even when the "Y-weak tree growth hypothesis fails. The following is the main theorem of this paper. Theorem 4.1 1fT is produced by a "Y-boosting run of the multi-branch tree-growth procedure then leT) ~ e(ITI) ~ ITI-'Y· 5 Proof of Theorem 4.1 To prove the main theorem we need the concept of a visited weighted tree, or VWtree for short. A VW-tree is a tree in which each node m is assigned both a rational weight Wm E [0,1] and an integer visitation count Vm ~ 1. We now define the following VW tree growth procedure. In the procedure Tw is the tree consisting of a single root node with weight wand visitation count 1. The tree Tt.w1 ..... w/c is the result of inserting k new leaves below the leaf l where the ith new leaf has weight Wi and new leaves have visitation count 1. W +- any rational number in [0,1] T+-Tw FOR ANY NUMBER OF STEPS REPEAT THE FOLLOWING l +- argmaxt e(tI:~wl Vt +- Vt + 1 OPTIONALLY T +- Tt.Wl .... ,Wlll WITH WI + .. . Wtll ~ e(vt)wt We first prove an analog of theorem 3.2 for the above procedure. For a VW-tree T we define ITI to be LtEL(T) Vt and we define leT) to be LtEL(T) e( Vt)Wt. Lemma 5.1 The VW procedure maintains the invariant that leT) ~ e(ITI). Proof: The proof is by induction on the number of iterations of the algorithm. The result is immediate for the initial tree since eel) = 1. We now assume that leT) ~ e(IT!) at the start of an iteration and show that this remains true at the end of the iteration. Boosting with Multi-Way Branching in Decision Trees 305 We can associate each leaf l with Vt "subleaves" each of weight e(vt)wt/Vt. We have that ITI is the total number of these subleaves and I(T) is the total weight of these subleaves. Therefore there must exist a subleaf whose weight is at least I(T)/ITI. Hence there must exist a leaf l satisfying e(vt)wt/Vt 2': I(T)/ITI. Therefore this relation must hold of the leaf l selected by the procedure. Let T' be the tree resulting from incrementing Vt. We now have I(T) - I(T') = e(vt)wt- e(vt + l)wt = e(vt)wt- (1- ;;)e(vt)wt = ;;e(vt)wt 2': "/I~)' So we have I(T') ~ (1 - ffl )I(T) ~ (1 - ffl )e(ITI) = e(IT'I). Finally, if the procedure grows new leaves we have that the I(T) does not increase and that ITI remains the same and hence the invariant is maintained. 0 For any internal node m in a tree T let C(m) denote the set of nodes which are children of m. A VW-tree will be called locally-well-formed if for every internal node m we have that Vm = IC(m)l, that I:nEC(m) Wn ~ e(IC(m)l)wm . A VW-tree will be called globally-safe ifmaxtEL(T) e(vt)wt/Vt ~ millmEN(T) e(vt-1)wt/(vt-1) where N(T) denotes the set of internal nodes of T. Lemma 5.2 If T is a locally well-formed and globally safe VW-tree, then T is a possible output of the VW growth procedure and therefore I(T) ~ e(ITI). Proof: Since T is locally well formed we can use T as a "template" for making nondeterministic choices in the VW growth procedure. This process is guaranteed to produce T provided that the growth procedure is never forced to visit a node corresponding to a leaf of T. But the global safety condition guarantees that any unfinished internal node of T has a weight as least as large as any leaf node of T. o We now give a way of mapping ?i-trees into VW-trees. More specifically, for any ?i-tree T we define VW(T) to be the result of assigning each node m in T the weight PmI(qm), each internal node a visitation count equal to its number of children, and each leaf node a visitation count equal to 1. We now have the following lemmas. Lemma 5.3 If T is grown by a I-boosting run of the multi-branch procedure then VW(T) is locally well-formed. Proof: Note that the children of an internal node m are derived by selecting a branching function h for the node m. Since the run is I-boosting we have ~(St, h)/g(IRhi) 2': II(qt). Therefore ~(St, h) = (I(tit) - 1St (n)) 2': I(tit)(l e(IRhl)). This implies that Ist(Th) ~ e(IRhDI(qt). Multiplying by Pt and transforming the result into weights in the tree VW(T) gives the desired result. 0 The following lemma now suffices for theorem 4.1. Lemma 5.4 If T is grown by a I-boosting run of the multi-branch procedure then VW(T) is globally safe. Proof: First note that the following is an invariant of a I-boosting run of the multi-branch procedure. max Wt < min Wt tEL(VW(T)) - mEN(VW(T)) 306 Y. Mansour and D. MeAl/ester The proof is a simple induction on ,-boosting tree growth using the fact that the procedure always expands a leaf node of maximal weight. We must now show that for every internal node m and every leaf £ we have that Wi ~ e(k -1)wm/(k -1) where k is the number of children of m. Note that if k = 2 then this reduces to Wi ~ Wm which follows from the above invariant. So we can assume without loss of generality that k > 2. Also, since e( k) / k < e( k - 1) / (k - 1), it suffices to show that Wi ~ e(k)wm/k. Let m be an internal node with k > 2 children and let T' be the tree at the time m was selected for expansion. Let Wi be the maximum weight of a leaf in the final tree T. By the definition of the acceptability condition, in the last s/2 iterations we are performing only binary branching. Each binary expansion reduces the index by at least , times the weight of the selected node. Since the sequence of nodes selected in the multi-branch procedure has non-increasing weights, we have that in any iteration the weight of the selected node is at least Wi . Since there are at least s/2 binary expansions after the expansion of m, each of which reduces I by at least ,Wi, we have that s,wd2 ~ I(T') so Wi ~ 2I(T')/(/s). The acceptability condition can be written as 2/(/s) ~ e(k)/(kIT'1) which now yields WI ~ I(T')e(k)/(kIT'I). But we have that I(T')/IT'I ~ Wm which now yields WI ~ e(k)wm/k as desired. 0 References [1] Leo Breiman, Jerome H. Friedman, Richard A. Olshen, and Charles J. Stone. Classification and Regression Trees. Wadsworth International Group, 1984. [2] Tom Dietterich, Michael Kearns and Yishay Mansour. Applying the Weak Learning Framework to understand and improve C4.5. In Proc. of Machine Learning, 96-104, 1996. [3] Yoav Freund. Boosting a weak learning algorithm by majority. Information and Computation, 121(2):256-285, 1995. [4] Yoav Freund and Robert E. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. In Computational Learning Theory: Second European Conference, EuroCOLT '95, pages 23-37. SpringerVerlag, 1995. [5] Yoav Freund and Robert E. Schapire. Experiments with a new boosting algorithm. In Machine Learning: Proceedings of the Thirteenth International Conference, pages 148-156, 1996. [6] Michael Kearns and Yishay Mansour. On the boosting ability of top-down decision tree learning. In Proceedings of the Twenty-Eighth ACM Symposium on the Theory of Computing, pages 459-468,1996. [7] J. Ross Quinlan. C4.5: Programs for Machine Learning. Morgan Kaufmann, 1993. [8] Robert E. Schapire. The strength of weak learnability. Machine Learning, 5(2):197-227, 1990.
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Effects of Spatial and Temporal Contiguity on the Acquisition of Spatial Information Thea B. Ghiselli-Crippa and Paul W. Munro Department of Information Science and Telecommunications University of Pittsburgh Pittsburgh, PA 15260 tbgst@sis.pitt.edu, munro@sis.pitt.edu Abstract Spatial information comes in two forms: direct spatial information (for example, retinal position) and indirect temporal contiguity information, since objects encountered sequentially are in general spatially close. The acquisition of spatial information by a neural network is investigated here. Given a spatial layout of several objects, networks are trained on a prediction task. Networks using temporal sequences with no direct spatial information are found to develop internal representations that show distances correlated with distances in the external layout. The influence of spatial information is analyzed by providing direct spatial information to the system during training that is either consistent with the layout or inconsistent with it. This approach allows examination of the relative contributions of spatial and temporal contiguity. 1 Introduction Spatial information is acquired by a process of exploration that is fundamentally temporal, whether it be on a small scale, such as scanning a picture, or on a larger one, such as physically navigating through a building, a neighborhood, or a city. Continuous scanning of an environment causes locations that are spatially close to have a tendency to occur in temporal proximity to one another. Thus, a temporal associative mechanism (such as a Hebb rule) can be used in conjunction with continuous exploration to capture the spatial structure of the environment [1]. However, the actual process of building a cognitive map need not rely solely on temporal associations, since some spatial information is encoded in the sensory array (position on the retina and proprioceptive feedback). Laboratory studies show different types of interaction between the relative contributions of temporal and spatial contiguities to the formation of an internal representation of space. While Clayton and Habibi's [2] series of recognition priming experiments indicates that priming is controlled only by temporal associations, in the work of McNamara et al. [3] priming in recognition is observed only when space and time are both contiguous. In addition, Curiel and Radvansky's [4] work shows that the effects of spatial and temporal contiguity depend on whether location or identity information is emphasized during learning. Moreover, other experiments ([3]) also show how the effects clearly depend on the task and can be quite different if an explicitly spatial task is used (e.g., additive effects in location judgments). 18 labels labels labels labels T. B. Ghiselli-Crippa and P W. Munro coordinates labels (A coeff.) labels coordinates (B coeff.) Figure 1: Network architectures: temporal-only network (left); spatio-temporal network with spatial units part of the input representation (center); spatio-temporal network with spatial units part of the output representation (right). 2 Network architectures The goal of the work presented in this paper is to study the structure of the internal representations that emerge from the integration of temporal and spatial associations. An encoder-like network architecture is used (see Figure 1), with a set of N input units and a set of N output units representing N nodes on a 2-dimensional graph. A set of H units is used for the hidden layer. To include space in the learning process, additional spatial units are included in the network architecture. These units provide a representation of the spatial information directly available during the learning/scanning process. In the simulations described in this paper, two units are used and are chosen to represent the (x, y) coordinates of the nodes in the graph. The spatial units can be included as part of the input representation or as part of the output representation (see Figure 1, center and right panels): both choices are used in the experiments, to investigate whether the spatial information could better benefit training as an input or as an output [5]. In the second case, the relative contribution of the spatial information can be directly manipulated by introducing weighting factors in the cost function being minimized. A two-term cost function is used, with a cross-entropy term for the N label units and a squared error term for the 2 coordinate units, ri indicates the actual output of unit i and ti its desired output. The relative influence of the spatial information is controlled by the coefficients A and B. 3 Learning tasks The left panel of Figure 2 shows an example of the type of layout used; the effective layout used in the study consists of N = 28 nodes. For each node, a set of neighboring nodes is defined, chosen on the basis of how an observer might scan the layout to learn the node labels and their (spatial) relationships; in Figure 2, the neighborhood relationships are represented by lines connecting neighboring nodes. From any node in the layout, the only allowed transitions are those to a neighbor, thus defining the set of node pairs used to train the network (66 pairs out of C(28, 2) = 378 possible pairs). In addition, the probability of occurrence of a particular transition is computed as a function of the distance to the corresponding neighbor. It is then possible to generate a sequence of visits to the network nodes, aimed at replicating the scanning process of a human observer studying the layout. Spatiotemporal Contiguity Effects on Spatial Information Acquisition 19 knife cup coin eraser eraser button Figure 2: Example of a layout (left) and its permuted version (right). Links represent allowed transitions. A larger layout of 28 units was used in the simulations. The basic learning task is similar to the grammar learning task of Servan-Schreiber et al. [6] and to the neighborhood mapping task described in [1] and is used to associate each of the N nodes on the graph and its (x, y) coordinates with the probability distribution of the transitions to its neighboring nodes. The mapping can be learned directly, by associating each node with the probability distribution of the transitions to all its neighbors: in this case, batch learning is used as the method of choice for learning the mapping. On the other hand, the mapping can be learned indirectly, by associating each node with itself and one of its neighbors, with online learning being the method of choice in this case; the neighbor chosen at each iteration is defined by the sequence of visits generated on the basis of the transition probabilities. Batch learning was chosen because it generally converges more smoothly and more quickly than online learning and gives qualitatively similar results. While the task and network architecture described in [1] allowed only for temporal association learning, in this study both temporal and spatial associations are learned simultaneously, thanks to the presence of the spatial units. However, the temporalonly (T-only) case, which has no spatial units, is included in the simulations performed for this study, to provide a benchmark for the evaluation of the results obtained with the spatio-temporal (S-T) networks. The task described above allows the network to learn neighborhood relationships for which spatial and temporal associations provide consistent information, that is, nodes experienced contiguously in time (as defined by the sequence) are also contiguous in space (being spatial neighbors). To tease apart the relative contributions of space and time, the task is kept the same, but the data employed for training the network is modified: the same layout is used to generate the temporal sequence, but the x , y coordinates of the nodes are randomly permuted (see right panel of Figure 2). If the permuted layout is then scanned following the same sequence of node visits used in the original version, the net effect is that the temporal associations remain the same, but the spatial associations change so that temporally neighboring nodes can now be spatially close or distant: the spatial associations are no longer consistent with the temporal associations. As Figure 4 illustrates, the training pairs (filled circles) all correspond to short distances in the original layout, but can have a distance anywhere in the allowable range in the permuted layout. Since the temporal and spatial distances were consistent in the original layout, the original spatial distance can be used as an indicator of temporal distance and Figure 4 can be interpreted as a plot of temporal distance vs. spatial distance for the permuted layout. The simulations described in the following include three experimental conditions: temporal only (no direct spatial information available); space and time consistent (the spatial coordinates and the temporal sequence are from the same layout); space and time inconsistent (the spatial coordinates and the temporal sequence are from different layouts). 20 T. B. Ghise/li-Crippa and P. W. Munro Hidden unit representations are compared using Euclidean distance (cosine and inner product measures give consistent results); the internal representation distances are also used to compute their correlation with Euclidean distances between nodes in the layout (original and permuted). The correlations increase with the number of hidden units for values of H between 5 and 10 and then gradually taper off for values greater than 10. The results presented in the remainder of the paper all pertain to networks trained with H = 20 and with hidden units using a tanh transfer function; all the results pertaining to S-T networks refer to networks with 2 spatial output units and cost function coefficients A = 0.625 and B = 6.25. 4 Results Figure 3 provides a combined view of the results from all three experiments. The left panel illustrates the evolution of the correlation between internal representation distances and layout (original and permuted) distances. The right panel shows the distributions of the correlations at the end of training (1000 epochs). The first general result is that, when spatial information is available and consistent with the temporal information (original layout), the correlation between hidden unit distances and layout distances is consistently better than the correlation obtained in the case of temporal associations alone. The second general result is that, when spatial information is available but not consistent with the temporal information (permuted layout), the correlation between hidden unit distances and original layout distances (which represent temporal distances) is similar to that obtained in the case of temporal associations alone, except for the initial transient. When the correlation is computed with respect to the permuted layout distances, its value peaks early during training and then decreases rapidly, to reach an asymptotic value well below the other three cases. This behavior is illustrated in the box plots in the right panel of Figure 3, which report the distribution of correlation values at the end of training. 4.1 Temporal-only vs. spatio-temporal As a first step in this study, the effects of adding spatial information to the basic temporal associations used to train the network can be examined. Since the learning task is the same for both the T-only and the S-T networks except for the absence or presence of spatial information during training, the differences observed can be attributed to the additional spatial information available to the S-T networks. The higher correlation between internal representation distances and original layout distances obtained when spatial information is 0 ~ ., ., 0 S and T CO"Isistent 0 i:i T-o" ~ ~ .. Sand T InCOnsistent .. -===s: 8 0 0 ii (corr with T distance) ... • 8 " " 0 0 ........... S and T Ir'ICOOSlStent E:2 '" (corr. Wflh S distance) N --'----' ci 0 0 0 0 0 200 400 600 800 1000 SandT T-only SandT SandT con_atent Inconsistent ineon.stant Ollnber 01 epochs (corr " th T ast ) (corr wth 5 dst ) Figure 3: Evolution of correlation during training (0 - 1000 epochs) (left). Distributions of correlations at the end of training (1000 epochs) (right). Spatiotemporal Contiguity Effects on Spatial Information Acquisition 21 N 0 ., 0 ", '" E 0 ~ ... 0 N 0 0 0 00 02 04 08 1 0 12 Figure 4: Distances in the original layout (x) vs_ distances in the permuted layout (y)_ The 66 training pairs are identified by filled circles_ dHU = 0.6 + 3.4dT + 0.3ds - 2.1(dT)2 + 0.4(d S )2 - 0.4dT ds 2 5 15 05 15 " Figure 5: Similarities (Euclidean distances) between internal representations developed by a S-T network (after 300 epochs)_ Figure 4 projects the data points onto the x, y plane_ available (see Figure 3) is apparent also when the evolution of the internal representations is examined_ As Figure 6 illustrates, the presence of spatial information results in better generalization for the pattern pairs outside the training set While the distances between training pairs are mapped to similar distances in hidden unit space for both the T-only and the S-T networks, the T-only network tends to cluster the non-training pairs into a narrow band of distances in hidden unit space. In the case of the S-T network instead, the hidden unit distances between non-training pairs are spread out over a wider range and tend to reflect the original layout distances. 4.2 Permuted layout As described above, with the permuted layout it is possible to decouple the spatial and temporal contributions and therefore study the effects of each. A comprehensive view of the results at a particular point during training (300 epochs) is presented in Figure 5, where the x, y plane represents temporal distance vs. spatial distance (see also Figure 4) and the z axis represents the similarity between hidden unit representations. The figure also includes a quadratic regression surface fitted to the data points. The coefficients in the equation of the surface provide a quantitative measure of the relative contributions of spatial (ds) and temporal distances (dT ) to the similarity between hidden unit representations (dHU ): (2) In general, after the transient observed in early training (see Figure 3), the largest and most significant coefficients are found for dT and (dT?, indicating a stronger dependence of dHU on temporal distance than on spatial distance. The results illustrated in Figure 5 represent the situation at a particular point during training (300 epochs). Similar plots can be generated for different points during training, to study the evolution of the internal representations. A different view of the evolution process is provided by Figure 7, in which the data points are projected onto the x,Z plane (top panel) and the y,z plane (bottom panel) at four different times during training. In the top panel, 14 22 T. B. Ghiselli-Crippa and P W Munro ::: ::: ~ ~ ~ ~. ~ .~. _ 0 ~ ,.. ~ _ 0 • N ,.. • N ~ ~ , 0 ~ ~ ~ 00 ~ ~, . , ~ ~ ~ ~ ~ ~ ~ , ' ',~-, ',' ~ -... ~ , ~ ~ ~ ~ ~ ~ 00 02 " 06 O. " 12 00 02 .. 06 .. " 12 00 02 .. 06 .. " " 00 02 .. .. .. " 12 "_d "_d "_d " _d ::: ::: ::: ::: ~ ~ ~ roo ~ ~ ~ o : s , e , rIP 0 00 :; ~ ~' ;; ~ _ 0 <P f/Po DO _ 0 ,.~,o 0 • , .' • N .. . • N ~ ~ tP DO 0 ~ : ~ : ~ . , ~ , .. ~ ~, ~ , " i ~ ',' ~ .~ ~ ~ , , ~ ~ .:. ~ .I' ~ ~ ~ g ~ 00 02 O. os .. 10 12 00 02 " 06 .. " 12 00 02 " .. .. 10 12 00 02 " O. o. " 12 "-' "-' "-' "-' Figure 6: Internal representation distances vs. original layout distances: S-T network (top) vs. T-only network (bottom). The training pairs are identified by filled circles. The presence of spatial information results in better generalization for the pairs outside the training set. the internal representation distances are plotted as a function of temporal distance (i.e., the spatial distance from the original layout), while in the bottom panel they are plotted as a function of spatial distance (from the permuted layout). The higher asymptotic correlation between internal representation distances and temporal distances, as opposed to spatial distances (see Figure 3), is apparent also from the examination of the evolutionary plots, which show an asymptotic behavior with respect to temporal distances (see Figure 7, top panel) very similar to the T-only case (see Figure 6, bottom panel). 5 Discussion The first general conclusion that can be drawn from the examination of the results described in the previous section is that, when the spatial information is available and consistent with the temporal information (original layout), the similarity structure of the hidden unit representations is closer to the structure of the original layout than that obtained by using temporal associations alone. The second general conclusion is that, when the spatial information is available but not consistent with the temporal information (permuted layout), the similarity structure of the hidden unit representations seems to correspond to temporal more than spatial proximity. Figures 5 and 7 both indicate that temporal associations take precedence over spatial associations. This result is in agreement with the results described in [1], showing how temporal associations (plus some high-level constraints) significantly contribute to the internal representation of global spatial information. However, spatial information certainly is very beneficial to the (temporal) acquisition of a layout, as proven by the results obtained with the S-T network vs. the T-only network. In terms of the model presented in this paper, the results illustrated in Figures 5 and 7 can be compared with the experimental data reported for recognition priming ([2], [3], [4]), with distance between internal representations corresponding to reaction time. The results of our model indicate that distances in both the spatially far and spatially close condition appear to be consistently shorter for the training pairs (temporally close) than for the nontraining pairs (temporally distant), highlighting a strong temporal effect consistent with the data reported in [2] and [4] (for spatially far pairs) and in [3] (only for the spatially close Spatiotemporal Contiguity Effects on Spatial Information Acquisition ~ ; ~-' ~. ~~. . ~ 0_ Sl ........... .. ri 0 0 ~'--_____ -.J 00 0 2 O. 0 1 01 10 12 In_d (T} ~ L..-____ -.l 00 02 0" 01 01 '0 12 In_d (S) j!I!A " ...... . ,. 0 lfIiiIo '0' ~ ,. ~ ~ • 110 0 \ .. 00 02 O. 01 01 10 1 2 In_d(TI 00 0.2 o. ot 01 10 12 .. _d(S) 00 02 0.4 01 01 10 12 l'I_d (T) 0.0 02 04 0 8 all 1 0 1 2 ... u:I (S) 0 0 02 O. 01 oa 10 12 In_den 23 ~l.-_____ -.J 00 02 O. 06 oa 10 12 !rUi (S) Figure 7: Internal representation distances vs. temporal distances (top) and vs. spatial distances (bottom) for a S-T network (permuted layout). The training pairs are identified by filled circles. The asymptotic behavior with respect to temporal distances (top panel) is similar to the T-only condition. The bottom panel indicates a weak dependence on spatial distances. case). For the training pairs (temporally close), slightly shorter distances are obtained for spatially close pairs vs. spatially far pairs; this result does not provide support for the experimental data reported in either [3] (strong spatial effect) or [2] (no spatial effect). For the non-training pairs (temporally distant), long distances are found throughout, with no strong dependence on spatial distance; this effect is consistent with all the reported experimental data. Further simulations and statistical analyses are necessary for a more conclusive comparison with the experimental data. References [1] Ghiselli-Crippa, TB. & Munro, P.w. (1994). Emergence of global structure from local associations. In J.D. Cowan, G. Tesauro, & J. Alspector (Eds.), Advances in Neural Information Processing Systems 6, pp. 1101-1108. San Francisco, CA: Morgan Kaufmann. [2] Clayton, K.N. & Habibi, A. (1991). The contribution of temporal contiguity to the spatial priming effect. Journal of Experimental Psychology: Learning. Memory. and Cognition 17:263-271. [3] McNamara, TP., Halpin. J.A. & Hardy, J.K. (1992). Spatial and temporal contributions to the structure of spatial memory. Journal of Experimental Psychology: Learning. Memory. and Cognition 18:555-564. [4] Curiel, J.M. & Radvansky, G.A. (1998). Mental organization of maps. Journal of Experimental Psychology: Learning. Memory. and Cognition 24:202-214. [5] Caruana, R. & de Sa, VR. (1997). Promoting poor features to supervisors: Some inputs work better as outputs. In M.e. Mozer, M.I. Jordan, & T Petsche (Eds.), Advances in Neural Information Processing Systems 9, pp. 389-395. Cambridge, MA: MIT Press. [6] Servan-Schreiber, D., Cleeremans, A. & McClelland, J.L. (1989). Learning sequential structure in simple recurrent networks. In D.S. Touretzky (Ed.), Advances in Neural Information Processing Systems 1, pp. 643-652. San Mateo, CA: Morgan Kaufmann. Neural Representation of Multi-Dimensional Stimuli Christian W. Eurich, Stefan D. Wilke and Helmut Schwegler Institut fUr Theoretische Physik Universitat Bremen, Germany (eurich,swilke,schwegler)@physik.uni-bremen.de Abstract The encoding accuracy of a population of stochastically spiking neurons is studied for different distributions of their tuning widths. The situation of identical radially symmetric receptive fields for all neurons, which is usually considered in the literature, turns out to be disadvantageous from an information-theoretic point of view. Both a variability of tuning widths and a fragmentation of the neural population into specialized subpopulations improve the encoding accuracy. 1 Introduction The topic of neuronal tuning properties and their functional significance has focused much attention in the last decades. However, neither empirical findings nor theoretical considerations have yielded a unified picture of optimal neural encoding strategies given a sensory or motor task. More specifically, the question as to whether narrow tuning or broad tuning is advantageous for the representation of a set of stimulus features is still being discussed. Empirically, both situations are encountered: small receptive fields whose diameter is less than one degree can, for example, be found in the human retina [7], and large receptive fields up to 1800 in diameter occur in the visual system of tongue-projecting salamanders [10]. On the theoretical side, arguments have been put forward for small [8] as well as for large [5, 1,9, 3, 13] receptive fields. In the last years, several approaches have been made to calculate the encoding accuracy of a neural population as a function of receptive field size [5, 1,9,3, 13]. It has turned out that for a firing rate coding, large receptive fields are advantageous provided that D 2: 3 stimulus features are encoded [9, 13]. For binary neurons, large receptive fields are advantageous also for D = 2 [5,3]. However, so far only radially symmetric tuning curves have been considered. For neural populations which lack this symmetry, the situation may be very different. Here we study the encoding accuracy of a popUlation of stochastically spiking neurons. A Fisher information analysis performed on different distributions of tunings widths will indeed reveal a much more detailed picture of neural encoding strategies. J J 6 C. W. Eurich. S. D. Wilke and H. Schwegler 2 Model Consider a D-dimensional stimulus space, X. A stimulus is characterized by a position x = (Xl, ... , XD) E X, where the value of feature i, Xi (i = 1, ... , D), is measured relative to the total range of values in the i-th dimension such that it is dimensionless. Information about the stimulus is encoded by a popUlation of N stochastically spiking neurons. They are assumed to have independent spike generation mechanisms such that the joint probability distribution for observing n = (n(l), ... ,n(k), ... ,n(N») spikes within a time interval T, Ps(n; x), can be written in the form N Ps(n;x) = II ps(k) (n(k); x), (1) k=l where Ps(k) (n(k); x) is the single-neuron probability distribution of the number of observed spikes given the stimulus at position x. Note that (1) does not exclude a correlation of the neural firing rates, i.e., the neurons may have common input or even share the same tuning function. The firing rates depend on the stimulus via the local values of the tuning functions, such that Ps(k) (n(k); x) can be written in the form Ps(k) (n(k); x) = S (n(k), j(k) (x), T), where the tuning function of neuron k, j(k) (x), gives its mean firing rate in response to the stimulus at position x. We assume here a form of the tuning function that is not necessarily radially symmetric, f(') (x) = F4> (t (Xi ~~r) )2) =, F¢ ( e( ')2) , (2) where e(k) = (c~k), ... , c};») is the center of the tuning curve of neuron k, O'~k) is its tuning width in the i-th dimension, dk)2 := (Xi c~k»)2/O'ik)2 for i = 1, ... ,D, and ~(k)2 := ~~k)2 + ... + ~~)2. F > 0 denotes the maximal firing rate of the neurons, which requires that maxz~o fj>(z) = 1. We assume that the tuning widths O't), . .. ,O'~) of each neuron k are drawn from a distribution PO' (0'1, ... ,O'D). For a population oftuning functions with centers e(l), ... , e(N), a density 1}(x) is introduced according to 1}(x) := L:~=l 8(x e(k»). The encoding accuracy can be quantified by the Fisher information matrix, J, which is defined as (3) where E[ ... J denotes the expectation value over the probability distribution P(n; x) [2]. The Fisher information yields a lower bound on the expected error of an unbiased estimator that retrieves the stimulus x from the noisy neural activity (Cramer-Rao inequality) [2]. The minimal estimation error for the i-th feature Xi, ti,min, is given by t;,min = (J- 1 )ii which reduces to t;,min = 1/ Jii(X) if J is diagonal. We shall now derive a general expression for the popUlation Fisher information. In the next chapter, several cases and their consequences for neural encoding strategies will be discussed. For model neuron (k), the Fisher information (3) reduces to (k) . (k) (k) _ 1 ( (k)2 ) (k) (k) Jij (X'O'I ""'O'D) (k) (k)Aq.. ~ ,F,T ~i ~j , O'i O'j (4) Neural Representation of Multi-Dimensional Stimuli 117 where the dependence on the tuning widths is indicated by the list of arguments. The function A.p depends on the shape of the tuning function and is given in [13]. The independence assumption (1) implies that the population Fisher information is the sum of h ·b· f h . d··d I ",N J(k)( (k) (k)) U7 t e contn utlOns 0 t e III IVI ua neurons, L.Jk=1 ij x; 0"1 , ... ,0" D . ne now define a population Fisher information which is averaged over the distribution of tuning widths Pt:T(0"1, . .. ,O"D): N (Jij (x)) 17 = L / d0"1 . .. dO"D Pt:T(0"1,· .. , O"D) Ji~k) (x; 0"1, · .. , O"D) . (5) k= 1 Introducing the density of tuning curves, 1J(x), into (5) and assuming a constant distribution, 1J(x) == 1J == const., one obtains the result that the population Fisher information becomes independentofx and that the off-diagonal elements of J vanish [13]. The average population Fisher information then becomes D ( ) / flt:l 0"1) ~ (Jij)t:T = 1J K.p F, r, D \ 0"; 17 Vij, (6) where K.p depends on the geometry of the tuning curves and is defined in [13]. 3 Results In this section, we consider different distributions of tuning widths in (6) and discuss advantageous and disadvantageous strategies for obtaining a high representational accuracy in the neural population. Radially symmetric tuning curves. For radially symmetric tuning curves of width a, the tuning-width distribution reads D Pt:T(O"l, .. . ,O"D) = II O(O"i -a); i=l see Fig. 1 a for a schematic visualization of the arrangement of the tuning widths for the case D = 2. The average population Fisher information (6) for i = j becomes (Jii)t:T = 1JDK.p(F, r, D) aD - 2 , (7) a result already obtained by Zhang and Sejnowski [13]. Equation (7) basically shows that the minimal estimation error increases with a for D = 1, that it does not depend on a for D = 2, and that it decreases as a increases for D 2: 3. We shall discuss the relevance of this case below. Identical tuning curves without radial symmetry. Next we discuss tuning curves which are identical but not radially symmetric; the tuning-width distribution for this case is D Pt:T(0"1, . .. ,O"D) = II O(O"i - ad, i=l where ai denotes the fixed width in dimension i. For i = j, the average population Fisher information (6) reduces to [11,4] flD ( ) 1=1 0"1 (Jii)t:T = 1JDK.p F, r, D -2 . O"i (8) 118 (a) / (c) b b2 _ (b) (d) . . . c. W. Eurich, S. D. Wilke and H. Schwegler Figure 1: Visualization of different distributions of tuning widths for D = 2. (a) Radially symmetric tuning curves. The dot indicates a fixed (j, while the diagonalline symbolizes a variation in (j discussed in [13]. (b) Identical tuning curves which are not radially symmetric. (c) Tuning widths uniformly distributed within a small rectangle. (d) Two sUbpopulations each of which is narrowly tuned in one dimension and broadly tuned in the other direction. Equation (8) contains (7) as a special case. From (8) it becomes immediately clear that the expected minimal square encoding error for the i-th stimulus feature, €~ min = 1/ (Jii(X))u, depends on i, i. e., the population specializes in certain features. The error obtained in dimension i thereby depends on the tuning widths in all dimensions. Which encoding strategy is optimal for a population whose task it is to encode a single feature, say feature i, with high accuracy while not caring about the other dimensions? In order to answer this question, we re-write (8) in terms of receptive field overlap. For the tuning functions f(k) (x) encountered empirically, large values ofthe single-neuron Fisher information (4) are typically restricted to a region around the center of the tuning function, c(k). The fraction p({3) of the Fisher information that falls into a region ED J~(k)2 ~ (3 aroundc(k) is given by f D "",D (k) ( ) d X L....i=l Jii X p({3) := E; dD 2:~ J~~) ( ) X t=l u X X j3 f d~ ~D+l At/>(e, F, T) o 00 f d~ ~D+l At/>(~2, F, T) o (9) where the index (k) was dropped because the tuning curves are assumed to have identical shapes. Equation (9) allows the definition of an effective receptive field, RF~~, inside of which neuron k conveys a major fraction Po of Fisher information, RF~~ := {xl~ ~ {3o} , where (3o is chosen such that p({3o) = Po. The Fisher information a neuron k carries is small unless x E RF~~. This has the consequence that a fixed stimulus x is actually encoded only by a subpopulation of neurons. The point x in stimulus space is covered by 27rD/2({30)D D _ Ncode:= 1] Dr(D/2) }1 (Jj (10) receptive fields. With the help of (10), the average population Fisher information (8) can be re-written as (11) Equation (11) can be interpreted as follows: We assume that the population of neurons encodes stimulus dimension i accurately, while all other dimensions are of secondary importance. The average population Fisher information for dimension i, (Jii ) u, is determined by the tuning width in dimension i, (ji, and by the size of the active subpopulation, Ncode ' There is a tradeoff between these quantities. On the one hand, the encoding error can be decreased by decreasing (ji, which enhances the Fisher information carried by each single Neural Representation of Multi-Dimensional Stimuli 119 neuron. Decreasing ai, on the other hand, will also shrink the active subpopulation via (10). This impairs the encoding accuracy, because the stimulus position is evaluated from the activity of fewer neurons. If (11) is valid due to a sufficient receptive field overlap, Ncode can be increased by increasing the tuning widths, aj, in all other dimensions j i- i. This effect is illustrated in Fig. 2 for D = 2. X2 X2 c=:> II"\.. , \ U x2, s x2,s Figure 2: Encoding strategy for a stimulus characterized by parameters Xl,s and X2,s' Feature Xl is to be encoded accurately. Effective receptive field shapes are indicated for both populations. If neurons are narrowly tuned in X2 (left), the active population (solid) is small (here: Ncode = 3). Broadly tuned receptive fields for X2 (right) yield a much larger population (here: Ncode = 27) thus increasing the encoding accuracy. It shall be noted that although a narrow tuning width ai is advantageous, the limit ai ---t 0 yields a bad representation. For narrowly tuned cells, gaps appear between the receptive fields: The condition 17(X) == const. breaks down, and (6) is no longer valid. A more detailed calculation shows that the encoding error diverges as ai --* 0 [4]. The fact that the encoding error decreases for both narrow tuning and broad tuning - due to (11) - proves the existence of an optimal tuning width, An example is given in Fig. 3a. 0.8 ;to.6 ~ A N~O.4 w v 0.2 3 rTI~--~------~----~------~ 1\ I i 1\ Ii I I I I I ; 2 1\ I , (b) ~~~~;::~-:.~~;: ---------------- --O'----~--~--~-----'-------' o 0.5 1 A 1.5 2 Figure 3: (a) Example for the encoding behavior with narrow tuning curves arranged on a regular lattice of dimension D = 1 (grid spacing ~). Tuning curves are Gaussian, and neural firing is modeled as a Poisson process, Dots indicate the minimal square encoding error averaged over a uniform distribution of stimuli, (E~in)' as a function ofa. The minimum is clearly visible. The dotted line shows the corresponding approximation according to (8). The inset shows Gaussian tuning curves of optimal width, aopt ~ 0.4~. (b) 9D()..) as a function of ).. for different values of D. 120 c. W. Eurich, S. D. Wilke and H. Schwegler Narrow distribution of tuning curves. In order to study the effects of encoding the stimulus with distributed tuning widths instead of identical tuning widths as in the previous cases, we now consider the distribution D Pu(lT1,'" ,lTD) = g :i e [lTi - (O'i - i)] e [(O'i + i) -lTi] , (12) where e denotes the Heaviside step function. Equation (12) describes a uniform distribution in a D-dimensional cuboid of size b1, ... , b D around (0'1, .. . 0' D); cf. Fig. 1 c. A straightforward calculation shows that in this case, the average population Fisher information (6) for i = j becomes ( n~l 0'1 { 1 (bi ) 2 [( bi ) 4] } (Jii)u = f/DKtj) F, T, D) O'~ 1 + 12 O'i + 0 O'i . (13) A comparison with (8) yields the astonishing result that an increase in bi results in an increase in the i-th diagonal element of the average population Fisher information matrix and thus in an improvement in the encoding of the i-th stimulus feature, while the encoding in dimensions j :f. i is not affected. Correspondingly, the total encoding error can be decreased by increasing an arbitrary number of edge lengths of the cube. The encoding by a population with a variability in the tuning curve geometries as described is more precise than that by a uniform population. This is true/or arbitrary D. Zhang and Sejnowski [13] consider the more artificial situation of a correlated variability ofthe tuning widths: tuning curves are always assumed to be radially symmetric. This is indicated by the diagonal line in Fig. 1 a. A distribution of tuning widths restricted to this subset yields an average population Fisher information ex: (O'D-2) and does not improve the encoding for D = 2 or D=3. Fragmentation into D subpopulations. Finally, we study a family of distributions of tuning widths which also yields a lower minimal encoding error than the uniform population. Let the density of tuning curves be given by 1 D Pu(lT1,'" ,lTD) = D L 6(lTi - AO') II 6(lTj - 0'), i=l j¥-i (14) where A > O. For A = 1, the population is uniform as in (7). For A :f. 1, the population is split up into D subpopulations; in subpopulation i, lTi is modified while lTj == 0' for j :f. i. See Fig. Id for an example. The diagonal elements ofthe average population Fisher information are -D-2 {1 + (D - I)A2 } (Jii)u = f/DKtj)(F, T, D) IT DA ' (15) where the term in brackets will be abbreviated as 9D(A). (Jii)u does not depend on i in this case because of the symmetry in the sUbpopulations. Equation (15) and the uniform case (7) differ by 9D(A) which will now be discussed. Figure 3b shows 9D(A) for different values of D. For A = 1, 9D(A) = 1 and (7) is recovered as expected. 9D(A) = 1 also holds for A = 1/ (D - 1) < 1: narrowing one tuning width in each subpopulation will at first decrease the resolution provided D 2: 3; this is due to the fact that Ncode is decreased. For A < 1/(D - 1), however, 9D(A) > 1, and the resolution exceeds (Jii)u in (7) because each neuron in the i-th subpopulation carries a high Fisher information in the i-th dimension. D = 2 is a special case where no impairment of encoding occurs because the effect of a decrease of Ncode is less pronounced. Interestingly, an increase in A also yields an improvement in the encoding accuracy. This is a combined effect resulting from an increase in Ncode on the one hand and the existence of D subpopulations, D - 1 of Neural Representation of Multi-Dimensional Stimuli 121 which maintain their tuning widths in each dimension on the other hand. The discussion of 9D(>") leads to the following encoding strategy. For small >.., (Jii)u increases rapidly, which suggests a fragmentation of the population into D subpopulations each of which encodes one feature with high accuracy, i.e., one tuning width in each subpopulation is small whereas the remaining tuning widths are broad. Like in the case discussed above, the theoretical limit of this method is a breakdown of the approximation of TJ == const. and the validity of (6) due to insufficient receptive field overlap. 4 Discussion and Outlook We have discussed the effects of a variation of the tuning widths on the encoding accuracy obtained by a population of stochastically spiking neurons. The question of an optimal tuning strategy has turned out to be more complicated than previously assumed. More specifically, the case which focused most attention in the literature - radially symmetric receptive fields [5, 1,9, 3, 13] - yields a worse encoding accuracy than most other cases we have studied: uniform populations with tuning curves which are not radially symmetric; distributions of tuning curves around some symmetric or non-symmetric tuning curve; and the fragmentation of the population into D subpopulations each of which is specialized in one stimulus feature. In a next step, the theoretical results will be compared to empirical data on encoding properties of neural popUlations. One aspect is the existence of sensory maps which consist of neural subpopulations with characteristic tuning properties for the features which are represented. For example, receptive fields of auditory neurons in the midbrain of the barn owl have elongated shapes [6]. A second aspect concerns the short-term dynamics of receptive fields. Using single-unit recordings in anaesthetized cats, Worgotter et al. [12] observed changes in receptive field size taking place in 50-lOOms. Our findings suggest that these dynamics alter the resolution obtained for the corresponding stimulus features. The observed effect may therefore realize a mechanism of an adaptable selective signal processing. References [1] Baldi, P. & HeiJigenberg, W. (1988) BioI. Cybern. 59:313-318. [2] Deco, G. & Obradovic, D. (1997) An Information-Theoretic Approach to Neural Computing. New York: Springer. [3] Eurich, C. W. & Schwegler, H. (1997) BioI. Cybern. 76: 357-363. [4] Eurich, C. W. & Wilke, S. D. (2000) NeuraL Compo (in press). [5] Hinton, G. E., McClelland, J. L. & Rumelhart, D. E (1986) In Rumelhart, D. E. & McClelland, J. L. (eds.), ParaLLeL Distributed Processing, Vol. 1, pp. 77-109. Cambridge MA: MIT Press. [6] Knudsen, E. I. & Konishi, M. (1978) Science 200:795-797. [7] Kuffter, S. W. (1953) 1. Neurophysiol. 16:37-68. [8] Lettvin, J. Y., Maturana, H. R., McCulloch, W. S. & Pitts, W. H. (1959) Proc. Inst. Radio Eng. NY 47:1940-1951. [9] Snippe, H. P. & Koenderink, J. J. (1992) BioI. Cybern. 66:543-551. [10] Wiggers, W., Roth, G., Eurich, C. W. & Straub, A. (1995) J. Camp. Physiol. A 176:365-377. [11] Wilke, S. D. & Eurich, C. W. (1999) In Verleysen, M. (ed.), ESANN 99, European Symposium on Artificial Neural Networks, pp. 435-440. Brussels: D-Facto. [12] Worgotter, F., Suder, K., Zhao, Y., Kerscher, N., Eysel, U. T. & Funke, K. (1998) Nature 396:165-168. [13] Zhang, K. & Sejnowski, T. J. (1999) NeuraL Compo 11:75-84.
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Model Selection for Support Vector Machines Olivier Chapelle*,t, Vladimir Vapnik* * AT&T Research Labs, Red Bank, NJ t LIP6, Paris, France { chapelle, vlad} @research.au.com Abstract New functionals for parameter (model) selection of Support Vector Machines are introduced based on the concepts of the span of support vectors and rescaling of the feature space. It is shown that using these functionals, one can both predict the best choice of parameters of the model and the relative quality of performance for any value of parameter. 1 Introduction Support Vector Machines (SVMs) implement the following idea: they map input vectors into a high dimensional feature space, where a maximal margin hyperplane is constructed [6]. It was shown that when training data are separable, the error rate for SVMs can be characterized by (1) where R is the radius ofthe smallest sphere containing the training data and M is the margin (the distance between the hyperplane and the closest training vector in feature space). This functional estimates the VC dimension of hyperplanes separating data with a given margin M. To perform the mapping and to calculate Rand M in the SVM technique. one uses a positive definite kernel K(x, x') which specifies an inner product in feature space. An example of such a kernel is the Radial Basis Function (RBF). K(x, x') = e-llx-x'II2/20'2. This kernel has a free parameter (7 and more generally, most kernels require some parameters to be set. When treating noisy data with SVMs. another parameter. penalizing the training errors. also needs to be set. The problem of choosing the values of these parameters which minimize the expectation of test error is called the model selection problem. It was shown that the parameter of the kernel that minimizes functional (1) provides a good choice for the model: the minimum for this functional coincides with the minimum of the test error [1]. However. the shapes of these curves can be different. In this article we introduce refined functionals that not only specify the best choice of parameters (both the parameter of the kernel and the parameter penalizing training error). but also produce curves which better reflect the actual error rate. Model Selection for Support Vector Machines 231 The paper is organized as follows. Section 2 describes the basics of SVMs, section 3 introduces a new functional based on the concept of the span of support vectors, section 4 considers the idea of rescaling data in feature space and section 5 discusses experiments of model selection with these functionals. 2 Support Vector Learning We introduce some standard notation for SVMs; for a complete description, see [6]. Let (Xi, Yih <i<l be a set of training examples, Xi E jRn which belong to a class labeled by Yi E {-f, f}. The decision function given by a SVM is : (2) where the coefficients a? are obtained by maximizing the following functional: lIt W(a) = Lai - 2' L aiajYiYjK(Xi,Xj) i=l i,j=l (3) under constraints t L aiYi = 0 and 0 ~ ai ~ C i = 1, ... , f. i=l C is a constant which controls the tradeoff between the complexity of the decision function and the number of training examples misclassified. SVM are linear maximal margin classifiers in a high-dimensional feature space where the data are mapped through a non-linear function <p(x) such that <P(Xi) . <p(Xj) = K(Xi,Xj). The points Xi with ai > 0 are called support vectors. We distinguish between those with o < ai < C and those with ai = C. We call them respectively support vectors of the first and second category. 3 Prediction using the span of support vectors The results introduced in this section are based on the leave-one-out cross-validation estimate. This procedure is usually used to estimate the probability of test error of a learning algorithm. 3.1 The leave-one-out procedure The leave-one-out procedure consists of removing from the training data one element, constructing the decision rule on the basis of the remaining training data and then testing the removed element. In this fashion one tests all f elements of the training data (using f different decision rules). Let us denote the number of errors in the leave-one-out procedure by £(Xl' Yl, ... , Xl, Yl) . It is known [6] that the the leave-one-out procedure gives an almost unbiased estimate of the probability of test error: the expectation of test error for the machine trained on f - 1 examples is equal to the expectation of 1£(Xl' Yl, ... , Xl, Yt). We now provide an analysis of the number of errors made by the leave-one-out procedure. For this purpose, we introduce a new concept, called the span of support vectors [7]. 232 O. Chapelle and V. N. Vapnik 3.2 Span of support vectors Since the results presented in this section do not depend on the feature space, we will consider without any loss of generality, linear SVMs, i.e. K (Xi, Xj) = Xi . Xj. Suppose that 0° = (a?, ... , a~) is the solution of the optimization problem (3). For any fixed support vector xp we define the set Ap as constrained linear combinations of the support vectors of the first category (Xi)i:;t:p : . t Ai = 1, 0 ~ a? + Yiypa~Ai ~ c} . t=l , t#p (4) Note that Ai can be less than O. We also define the quantity Sp, which we call the span of the support vector xp as the minimum distance between xp and this set (see figure 1) (5) t... .. AI 2= +inf ·· · '' - - ' ~ .. 2,, t...3 = -inf Figure 1: Three support vectors with al = a2 = a3/2. The set Al is the semi-opened dashed line. It was shown in [7] that the set Ap is not empty and that Sp = d(xp, Ap) ~ Dsv, where D sv is the diameter of the smallest sphere containing the support vectors. Intuitively, the smaller Sp = d(xp, Ap) is, the less likely the leave-one-out procedure is to make an error on the vector xp' Formally, the following theorem holds : Theorem 1 [7 J If in the leave-one-out procedure a support vector xp corresponding to o < a p < C is recognized incorrectly, then the following inequality holds aO > 1 p Sp max(D, 1/.JC)· This theorem implies that in the separable case (C = (0), the number of errors made by the leave-one-out procedure is bounded as follows: £(Xl' Yl, .'" Xl, Yl) ~ 2:p a~ maxp SpD = maxp SpD / M2 , because 2: a~ = 1/ M2 [6]. This is already an improvement compared to functional (I), since Sp ~ Dsv. But depending on the geometry of the support vectors the value of the span Sp can be much less than the diameter D sv of the support vectors and can even be equal to zero. We can go further under the assumption that the set of support vectors does not change during the leave-one-out procedure, which leads us to the following theorem: Model Selection for Support Vector Machines 233 Theorem 2 If the sets of support vectors of first and second categories remain the same during the leave-one-out procedure. then for any support vector xp. the following equality holds: yp[fO(xp) - fP(xp)] = o~S; where fO and fP are the decisionfunction (2) given by the SVM trained respectively on the whole training set and after the point xp has been removed. The proof of the theorem follows the one of Theorem 1 in [7]. The assumption that the set of support vectors does not change during the leave-one-out procedure is obviously not satisfied in most cases. Nevertheless, the proportion of points which violate this assumption is usually small compared to the number of support vectors. In this case, Theorem 2 provides a good approximation of the result of the leave-one procedure, as pointed out by the experiments (see Section 5.1, figure 2). As already noticed in [1], the larger op is, the more "important" in the decision function the support vector xp is. Thus, it is not surprising that removing a point xp causes a change in the decision function proportional to its Lagrange multiplier op. The same kind of result as Theorem 2 has also been derived in [2], where for SVMs without threshold, the following inequality has been derived: yp(f°(xp) - fP(xp)) ~ o~K(xp,xp). The span Sp takes into account the geometry of the support vectors in order to get a precise notion of how "important" is a given point. The previous theorem enables us to compute the number of errors made by the leave-oneout procedure: Corollary 1 Under the assumption of Theorem 2, the test error prediction given by the leave-one-out procedure is (6) Note that points which are not support vectors are correctly classified by the leave-one-out procedure. Therefore t/. defines the number of errors of the leave-one-out procedure on the entire training set. Under the assumption in Theorem 2, the box constraints in the definition of Ap (4) can be removed. Moreover, if we consider only hyperplanes passing through the origin, the constraint E Ai = 1 can also be removed. Therefore, under those assumptions, the computation of the span Sp is an unconstrained minimization of a quadratic form and can be done analytically. For support vectors of the first category, this leads to the closed form S~ = l/(KsMpp, where Ksv is the matrix of dot products between support vectors of the first category. A similar result has also been obtained in [3]. In Section 5, we use the span-rule (6) for model selection in both separable and nonseparable cases. 4 Rescaling As we already mentioned, functional (1) bounds the VC dimension of a linear margin classifier. This bound is tight when the data almost "fills" the surface of the sphere enclosing the training data, but when the data lie on a flat ellipsoid, this bound is poor since the radius of the sphere takes into account only the components with the largest deviations. The idea we present here is to make a rescaling of our data in feature space such that the radius of the sphere stays constant but the margin increases, and then apply this bound to our rescaled data and hyperplane. 234 0. Chapelle and V. N. Vapnik Let us first consider linear SVMs, i.e. without any mapping in a high dimensional space. The rescaling can be achieved by computing the covariance matrix of our data and rescaling according to its eigenvalues. Suppose our data are centered and let ('PI' ... ,'Pn) be the normalized eigenvectors of the covariance matrix of our data. We can then compute the smallest enclosing box containing our data, centered at the origin and whose edges are parallels to ('PI' ... , 'Pn)' This box is an approximation of the smallest enclosing ellipsoid. The length of the edge in the direction 'P k is J-Lk = maxi IXi . 'P k I. The rescaling consists of the following diagonal transformation: D : x --t Dx = LJ-Lk(X' 'Pk) 'Pk' k Let us consider Xi = D-I xi and w = Dw. The decision function is not changed under this transformation since w . Xi = W . xi and the data Xi fill a box of side length 1. Thus, in functional (l), we replace R2 by 1 and 1/ M2 by w2 . Since we rescaled our data in a box, we actually estimated the radius of the enclosing ball using the foo-norm instead of the classical f2-norm. Further theoretical works needs to be done to justify this change of norm. In the non-linear case, note that even if we map our data in a high dimensional feature space, they lie in the linear subspace spanned by these data. Thus, if the number of training data f is not too large, we can work in this subspace of dimension at most f. For this purpose, one can use the tools of kernel PCA [5] : if A is the matrix of normalized eigenvectors of the Gram matrix Kij = K (Xi, Xj) and (>'d the eigenvalues, the dot product Xi . 'P k is replaced by v'XkAik and W· 'Pk becomes v'XkL:i AikYiO'i. Thus, we can still achieve the diagonal transformation A and finally functional (1) becomes L >.~ max Ark (2: AikYiO'i)2 . ~ k i 5 Experiments To check these new methods, we performed two series of experiments. One concerns the choice of (7, the width of the RBF kernel, on a linearly separable database, the postal database. This dataset consists of 7291 handwritten digit of size 16x16 with a test set of 2007 examples. Following [4], we split the training set in 23 subsets of 317 training examples. Our task consists of separating digit 0 to 4 from 5 to 9. Error bars in figures 2a and 3 are standard deviations over the 23 trials. In another experiment, we try to choose the optimal value of C in a noisy database, the breast-cancer database! . The dataset has been split randomly 100 times into a training set containing 200 examples and a test set containing 77 examples. Section 5.1 describes experiments of model selection using the span-rule (6), both in the separable case and in the non-separable one, while Section 5.2 shows VC bounds for model selection in the separable case both with and without rescaling. 5.1 Model selection using the span-rule In this section, we use the prediction of test error derived from the span-rule (6) for model selection. Figure 2a shows the test error and the prediction given by the span for different values of the width (7 of the RBF kernel on the postal database. Figure 2b plots the same functions for different values of C on the breast-cancer database. We can see that the method predicts the correct value of the minimum. Moreover, the prediction is very accurate and the curves are almost identical. I Available from http; I Ihorn. first. gmd. del "'raetsch/da ta/breast-cancer Model Selection for Support Vector Machines 40,-----~---~r=_=_ -""T==es=:'t=er=ro=r ='ll 35 30 25 g20 UJ 15 10 5 ~6 Span prediction i" " , -4 -2 0 2 4 Log sigma (a) choice of (T in the postal database 6 235 o 2 4 6 8 10 12 Loge (b) choice of C in the breast-cancer database Figure 2: Test error and its prediction using the span-rule (6). The computation of the span-rule (6) involves computing the span Sp (5) for every support vector. Note, however, that we are interested in the inequality S; ::; Yp!(xp)/a~, rather than the exact value of the span Sp. Thus, while minimizing Sp = d(xp, Ap), if we find a point x* E Ap such that d(xp, x*)2 ::; Yp! (xp ) / a~, we can stop the minimization because this point will be correctly classified by the leave-one-out procedure. It turned out in the experiments that the time required to compute the span was not prohibitive, since it is was about the same than the training time. There is a noteworthy extension in the application of the span concept. If we denote by e one hyperparameter of the kernel and if the derivative 8K(;~'Xi) is computable, then it is possible to compute analytically 8 ~ aiS~~y;fO(x;) , which is the derivative of an upper bound of the number of errors made by the leave-one-out procedure (see Theorem 2). This provides us a more powerful technique in model selection. Indeed, our initial approach was to choose the value of the width (T of the RBF kernel according to the minimum of the span-rule. In our case, there was only hyperparamter so it was possible to try different values of (T. But, if we have several hyperparameters, for example one (T per component, _~ (Xk- Xj,)2 K(x, x') = e k 2<T~ , it is not possible to do an exhaustive search on all the possible values of of the hyperparameters. Nevertheless, the previous remark enables us to find their optimal value by a classical gradient descent approach. Preliminary results seem to show that using this approach with the previously mentioned kernel improve the test error significantely. 5.2 VC dimension with rescaling In this section, we perform model selection on the postal database using functional (1) and its rescaled version. Figure 3a shows the values of the classical bound R2 / M2 for different values of (T. This bound predicts the correct value for the minimum, but does not reflect the actual test error. This is easily understandable since for large values of (T, the data in input space tend to be mapped in a very flat ellipsoid in feature space, a fact which is not taken into account [4]. Figure 3b shows that by performing a rescaling of our data, we manage to have a much tighter bound and this curve reflects the actual test error, given in figure 2a. 236 18000'---~--~--~--r=~==~=.=~ 16000 14000 12000 E 10000 '6 ~ 8000 6000 4000 2000 4 6 ~L-==~===c~~ ____ ~ __ ~ __ ~ ....., -4 -2 0 2 Log sigma (a) without rescaling 0. Chape/le and V. N. Vapnik 120 VC Dimension with rescali 100 80 E ~ 60 > 40 20 ~ -4 -2 0 2 4 6 Log sigma (b) with rescaling Figure 3: Bound on the VC dimension for different values of ~ on the postal database. The shape of the curve with rescaling is very similar to the test error on figure 2. 6 Conclusion In this paper, we introduced two new techniques of model selection for SVMs. One is based on the span, the other is based on rescaling of the data in feature space. We demonstrated that using these techniques, one can both predict optimal values for the parameters of the model and evaluate relative performances for different values of the parameters. These functionals can also lead to new learning techniques as they establish that generalization ability is not only due to margin. Acknowledgments The authors would like to thank Jason Weston and Patrick Haffner for helpfull discussions and comments. References [1] C. J. C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):121-167, 1998. [2] T. S. Jaakkola and D. Haussler. Probabilistic kernel regression models. In Proceedings of the J 999 Conference on AI and Statistics, 1999. [3] M. Opper and O. Winther. Gaussian process classification and SVM: Mean field results and leave-one-out estimator. In Advances in Large Margin Classifiers. MIT Press, 1999. to appear. [4] B. SchOlkopf, J. Shawe-Taylor, A. 1. Smola, and R. C. Williamson. Kernel-dependent Support Vector error bounds. In Ninth International Conference on Artificial Neural Networks, pp. 304 309 [5] B. SchOlkopf, A. Smola, and K.-R. Muller. Kernel principal component analysis. In Artificial Neural Networks -ICANN'97, pages 583 - 588, Berlin, 1997. Springer Lecture Notes in Computer Science, Vol. 1327. [6] V. Vapnik. Statistical Learning Theory. Wiley, New York, 1998. [7] V. Vapnik and O. Chapelle. Bounds on error expectation for SVM. Neural Computation, 1999. Submitted.
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Memory Capacity of Linear vs. Nonlinear Models of Dendritic Integration Panayiota Poirazi* Biomedical Engineering Department University of Southern California Los Angeles, CA 90089 poirazi@sc/. usc. edu Bartlett W. Mel* Biomedical Engineering Department University of Southern California Los Angeles, CA 90089 mel@lnc.usc.edu Abstract Previous biophysical modeling work showed that nonlinear interactions among nearby synapses located on active dendritic trees can provide a large boost in the memory capacity of a cell (Mel, 1992a, 1992b). The aim of our present work is to quantify this boost by estimating the capacity of (1) a neuron model with passive dendritic integration where inputs are combined linearly across the entire cell followed by a single global threshold, and (2) an active dendrite model in which a threshold is applied separately to the output of each branch, and the branch subtotals are combined linearly. We focus here on the limiting case of binary-valued synaptic weights, and derive expressions which measure model capacity by estimating the number of distinct input-output functions available to both neuron types. We show that (1) the application of a fixed nonlinearity to each dendritic compartment substantially increases the model's flexibility, (2) for a neuron of realistic size, the capacity of the nonlinear cell can exceed that of the same-sized linear cell by more than an order of magnitude, and (3) the largest capacity boost occurs for cells with a relatively large number of dendritic subunits of relatively small size. We validated the analysis by empirically measuring memory capacity with randomized two-class classification problems, where a stochastic delta rule was used to train both linear and nonlinear models. We found that large capacity boosts predicted for the nonlinear dendritic model were readily achieved in practice. -http://lnc.usc.edu 158 P. Poirazi and B. W. Mel 1 Introduction Both physiological evidence and connectionist theory support the notion that in the brain, memories are stored in the pattern of learned synaptic weight values. Experiments in a variety of neuronal preparations however, inQicate that the efficacy of synaptic transmission can undergo substantial fluctuations up or down, or both, during brief trains of synaptic stimuli. Large fluctuations in synaptic efficacy on short time scales seem inconsistent with the conventional connectionist assumption of stable, high-resolution synaptic weight values. Furthermore, a recent experimental study suggests that excitatory synapses in the hippocampus-a region implicated in certain forms of explicit memory-may exist in only a few long-term stable states, where the continuous grading of synaptic strength seen in standard measures of long-term potentiation (LTP) may exist only in the average over a large population of two-state synapses with randomly staggered thresholds for learning (Petersen, Malenka, Nicoli, & Hopfield, 1998). According to conventional connectionist notions, the possibility that individual synapses hold only one or two bits of long-term state information would seem to have serious implications for the storage capacity of neural tissue. Exploration of this question is one of the main themes of this paper. In a related vein, we have found in previous biophysical modeling studies that nonlinear interactions between synapses co-activated on the same branch of an active dendritic tree could provide an alternative form of long-term storage capacity. This capacity, which is largely orthogonal to that tied up in conventional synaptic weights, is contained instead in the spatial permutation of synaptic connections onto the dendritic tree-which could in principle be modified in the course of learning or development (Mel, 1992a, 1992b). In a more abstract setting, we recently showed that a large repository of model flexibility lies in the choice as to which of a large number of possible interaction terms available in high dimension is actually included in a learning machine's discriminant function, and that the excess capacity contained in this "choice flexibility" can be quantified using straightforward counting arguments (Poirazi & Mel, 1999). 2 Two Alternative Models of Dendritic Integration In this paper, we use a similar function-counting approach to address the more biologically relevant case of a neuron with mUltiple quasi-independent dendritic compartments (fig. 1). Our primary objective has been to compare the memory capacity of a cell assuming two different modes of dendritic integration. According to the linear model, the neuron's activation level aL(x) prior to thresholding is given by a weighted sum of of its inputs over the cell as a whole. According to the nonlinear model, the k synaptic inputs to each branch are first combined linearly, a static (e.g. sigmoidal) nonlinearity is applied to each of the m branch subtotals, and the resulting branch outputs are summed to produce the cell's overall activity aN{x): (1) The expressions for aL and aN were written in similar form to emphasize that the models have an identical number of synaptic weights, differing only in the presence or absence of a fixed nonlinear function g applied to the branch subtotals. Though individual synaptic weights in both models are constrained to have a value of 1, any of the d input lines may form multiple connections on the same or different Memory Capacity of Linear vs. Nonlinear Models of Dendritic Integration 159 m 3 • ' . . , I , . Figure 1: A cell is modeled as a set of m identical branches connected to a soma, where each branch contains k synaptic contacts driven by one of d distinct input lines. branches as a means of representing graded synaptic strengths. Similarly, an input line which forms no connection has an implicit weight of O. In light of this restriction to positive (or zero) weight values, both the linear and nonlinear models are split into two opponent channels a+ and a- dedicated to positive vs. negative coefficients, respectively. This leads to a final output for each model: yL(x) = sgn [at(x) - aL(x)] YN(X) = sgn [a;t(x) - aiV(x)] (2) where the sgn operator maps the total activation level into a class label of {-I, I}. In the following, we derive expressions for the number of distinct parameter st.ates available to the linear vs. nonlinear models, a measure which we have found to be a reliable predictor of storage capacity under certain restrictions (Poirazi & Mel, 1999). Based on these expressions, we compute the capacity boost provided by the branch nonlinearity as a function of the number of branches m, synaptic sites per branch k, and input space dimensionality d. Finally, we test the predictions of the analytical model by training both linear and nonlinear models on randomized classification problems using a stochastic delta rule, and empirically measure and compare the storage capacities of the two models. 3 Results 3.1 Counting Parameter States: Linear vs. Nonlinear Model We derived expressions for BLand B N, which estimate the total number of parameter bits available to the linear vs. nonlinear models, respectively: (( k+d-1) 1) B N = 2log2 k m + m BL = 2log2 ( S+d-1) S (3) These expressions estimate the number of non-redundant states in each neuron type, i.e., those assignments of input lines to dendritic sites which yield distinct 160 P Poirazi and B. W Mel input-output functions YL or YN· These formulae are plotted in figure 2A with d = 100, where each curve represents a cell with a fixed number of branches (indicated by m). In each case, the capacity increases steadily as the number of synapses per branch, k, is increased. The logarithmic growth in the capacity of the linear model (evident in an asymptotic analysis of the expression for B L) is shown at the bottom of the graph (circles), from which it may be seen that the boost in capacity provided by the dendritic branch nonlinearity increases steadily with the number of synaptic sites. For a cell with 100 branches containing 100 synaptic sites each, the capacity boost relative to the linear model exceeds a factor of 20. Figure 2B shows that for a given total number of synaptic sites, in this case s = m· k = 10,000, the capacity of the nonlinear cell is maximized for a specific choice of m and k. The peak of each of the three curves (computed for different values of d) occurs for a cell containing 1,250 branches with 8 synapses each. However, the capacity is only moderately sensitive to the branch count: the capacity of a cell with 100 branches of 100 synapses each, for example, lies within a factor of two of the optimal configuration. The linear cell capacities can be found at the far right edge of the plot (m = 10,000), since a nonlinear model with one synapse per branch has a number of trainable states identical to that of a linear model. 3.2 Validating the Analytical Model To test the predictions of the analytical model, we trained both linear and nonlinear cells on randomized two-class classification problems. Training samples were drawn from a 40-dimensional spherical Gaussian distribution and were randomly assigned positive or negative labels-in some runs, training patterns were evenly divided between positive and negative labels, with similar results. Each of the 40 original input dimensions was recoded using a set of 10 I-dimensional binary, nonoverlapping receptive fields with centers spaced along each dimension such that all receptive fields would be activated equally often. This manipulation mapped the original 40-dimensional learning problem into 400 dimensions, thereby increasing the discriminability of the training samples. The relative memory capacity of linear vs. nonlinear cells was then determined empirically by comparing the number of training patterns learnable at a fixed error rate of 2%. The learning rule used for both cell types was similar to the "clusteron" learning rule described in (Mel, 1992a), and involved two mechanisms known to contribute to neural development: (1) random activity-independent synapse formation, and (2) activity-dependent synapse stabilization. In each iteration, a set of 25 synapses was chosen at random, and the "worst" synapse was identified based on the correlation over the training set of (i) the input's pre-synaptic activity, (ii) the post-synaptic activity (Le. the local nonlinear branch response for the nonlinear energy model or a constant of 1 for the linear model), and (iii) a global "delta" signal with a value of a if the cell responded correctly to the input pattern, or ±l if the cell responded incorrectly. The poorest-performing synapse on the branch was then targeted for replacement with a new synapse drawn at random from the d input lines. The probability that the replacement actually occurred was given by a Boltzmann equation based on the difference in the training set error rates before and after the replacement. A "temperature" variable was gradually lowered over the course of the simulation, which was terminated when no further improvement in error rates was seen. Results of the learning runs are shown in fig. 3 where the analytical capacity (measured in bits) was scaled to the numerical capacity (measured in training patterns Memory Capacity of Linear vs. Nonlinear Models of Dendritic Integration 161 A Capacity of Linear vs. Nonlinear Model for Various Geometries x 10' 8.---~----~--~----~---, m=lOOO ~ 7 6 ~ § 5 >. '0 4 !IS 1t3 U 2 d = 100 Nonlinear Model H r m- 000 m Linear Model 2000 4000 6000 8000 10000 ~~l Syn:p:c Sires ~ B Capacity of Linear VS. Nonlinear Model for Different Input Space Dimensions x10' 14 .t" ....... 12 : d= lOO~ j " , , Nonlinear Model s = 10,000 10 j , '(i.) , .... co ....... '"'. Linear ~del l~ o 2000 4000 6000 8000 10000 Number of Branches (m) * Figure 2: Comparison of linear vs. nonlinear model capacity as a function of branch geometry. A. Capacity in bits for linear and several nonlinear cells with different branch counts (for d = 100). For each curve indexed by branch count m, sites per branch k increases from left to right as indicated iconically beneath the x-axis. For all cells, capacity increases with an increasing number of sites, though the capacity of the linear model grows logarithmically, leading to an increasingly large capacity boost for the size-matched nonlinear cells. B. Capacity of a nonlinear model with 10,000 sites for different values of input space dimension d. Branch count m grows along the x-axis. Cells at right edge of plot contain only one synapse per branch, and thus have a number of modifiable parameters (and hence capacity) equivalent to that of the linear model. All three curves show that there exist an optimal geometry which maximizes the capacity of the nonlinear model (in this case 1,250 branches with 8 synapses each). learned at 2% error). Two key features of the theoretical curves (dashed lines) are echoed in the empirical performance curves (solid lines), including the much larger storage capacity of the nonlinear cell model, and the specific cell geometry which maximizes the capacity boost. 4 Discussion We found using both analytical and numerical methods that in the limit of lowresolution synaptic weights, application of a fixed output nonlinearity to each compartment of a dendritic tree leads to a significant boost in capacity relative to a cell whose post-synaptic integration is linear. For example, given a cell with 10,000 synaptic contacts originating from 400 distinct input lines, the analysis predicts a 23-fold increase in capacity for the nonlinear cell, while numerical simulations using a stochastic delta rule actually achieve a I5-fold boost. Given that a linear and a nonlinear model have an identical number of synaptic contacts with uniform synaptic weight values, what accounts for the capacity boost? The principal insight gained in this work is that the attachment of a fixed nonlinearity to each branch in a neuron substantially increases its underlying "model 162 70 I \ 6 1,+ \ - _. Analytical (Bits/14) Numerical (Training Patterns) P. Poirazi and B. W. Mel Figure 3: Comparison of capacity boost predicted by analysis vs. that observed empirically when linear and nonlinear models were trained using the same I, \ >. 50 \. '13 \ Nonlinear Model stochastic delta rule. Dashed lines: analytical curves for linear vs. nonlinear model for a cell with 10,000 sites show capacity for varying cell geometries. Solid lines: empirical performance for same two cells at 2% error criterion, using a subunit nonlinearity g(x) = xlO (similar results were seen using a sigmoidal nonlinearity, though the parameters of the optimal sigmoid depended on the cell geometry). ~ 40 \ 03 ____________________ ~, <..) 30 '" 2 , , , , , , ,. Linear Model For both analytical and numeri, 2 cal curves, peak capacity is seen o x10 o 10 20 30 40 50 60 70 80 90 100 for cell with 1,000 branches (10 Number of Branches (m) synapses per branch) .. Cap~city * exceeds that of same-sIzed lmear ---I...... .:Jkmodel by a factor of 15 at the m ~ peak, and by more than a factor of 7 for cells ranging from about 3 to 60 synapses per branch (horizontal dotted line). flexibility" , i.e. confers upon the cell a much larger choice of distinct input-output relations from which to select during learning. This may be illustrated as follows. For the linear model, branching structure is irrelevant so that Y L depends only on the number of input connections formed from each of the d input lines. All spatial permutations of a set of input connections are thus interchangeable and produce identical cell responses. This massive redundancy confines the capacity of the linear model to grow only logarithmically with an increasing number of synaptic sites (fig. 1A), an unfortunate limitation for a brain in which the formation of large numbers of synaptic contacts between neurons is routine. In contrast, the model with nonlinear subunits contains many fewer redundancies: most spatial permutations of the same set of input connections lead to non-identical values of YN, since an input x swapped from branch bi to branch b2 leads to the elimination of the k - 1 interaction terms involving x on branch bi and the creation of k -1 new interaction terms on branch b2 • Interestingly, the particular form of the branch nonlinearity has virtually no effect on the capacity of the cell as far as the counting arguments are concerned (though it can have a profound effect on the cell's "representational bias"-see below), since the principal effect of the nonlinearity in our capacity calculations is to break the symmetry among the different branches. The issue of representational bias is a critical one, however, and must be considered when attempting to predict absolute or relative performance rates for particular classifiers confronted with specific learning problems. Thus, intrinsic differences in the geometry of linear vs. nonlinear discriminant functions mean that the paramMemory Capacity of Linear vs. Nonlinear Models of Dendritic Integration 163 eters available to the two models may be better or worse suited to solve a given learning problem, even if the two models were equated for total parameter flexibility. While such biases are not taken into account in our analysis, they could nonetheless have a substantial effect on measured error rates-and could thus throw a performance advantage to one machine or the other. One danger is that performance differences measured empirically could be misinterpreted as arising from differences in underlying model capacity, when in fact they arise from differential suitability of the two classifiers for the learning problem at hand. To avoid this difficulty, the random classification problems we used to empirically assess memory capacity were chosen to level the playing field for the linear vs. nonlinear cells, since in a previous study we found that the coefficients on linear vs. nonlinear (quadratic) terms were about equally efficient as featUres for this task. In this way, differences in measured performance on these tasks were primarily attributable to underlying capacity differences, rather than differences in representational bias. This experimental control permitted more meaningful comparisons between our analytical and empirical tests (fig. 3). The problem of representational bias crops up in a second guise, wherein the analytical expressions for capacity in eq. 1 can significantly overestimate the actual performance of the cell. This occurs when a particular ensemble of learning problems fails to utilize all of the entropy available in the cell's parameter space-for example, by requiring the cell to visit only a small subset of its parameter states relatively often. This invalidates the maximum parameter entropy assumption made in the derivation of eq. 1, so that measured performance will tend to fall below predicted values. The actual performance of either model when confronted with an ensemble of learning problems will thus be determined by (1) the number of trainable parameters available to the neuron (as measured by eq. 1), (2) the suitability of the neuron's parameters for solving the assigned learning problems, and (3) the utilization of parameters, which relates to the entropy in the joint probability of the parameter values averaged over the ensemble of learning problems. In our comparisons here of linear and nonlinear cells, we we have calculated (1), and have attempted to control for (2) and (3). In conclusion, our results build upon the results of earlier biophysical simulations, and indicate that in the limit of a large number of low-resolution synaptic weights, nonlinear dendritic processing could nonetheless have a major impact on the storage capacity of neural tissue. References Mel, B. W. (1992a). The clusteron: Toward a simple abstraction for a complex neuron. In Moody, J., Hanson, S., & Lippmann, R. (Eds.), Advances in Neural Information Processing Systems, vol. 4, pp. 35-42: Morgan Kaufmann, San Mateo, CA. Mel, B. W. (1992b). NMDA-based pattern discrimination in a modeled cortical neuron. Neural Comp., 4, 502-516. Petersen, C. C. H., Malenka, R. C., Nicoll, R. A., & Hopfield, J. J. (1998). All-ornone potentiation and CA3-CA1 synapses. Proc. Natl. Acad. Sci. USA, 95, 4732-4737. Poirazi, P., & Mel, B. W. (1999). Choice and value flexibility jointly contribute to the capacity of a subsampled quadratic classifier. Neural Comp., in press.
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State Abstraction in MAXQ Hierarchical Reinforcement Learning Thomas G. Dietterich Department of Computer Science Oregon State University Corvallis, Oregon 97331-3202 tgd@cs.orst.edu Abstract Many researchers have explored methods for hierarchical reinforcement learning (RL) with temporal abstractions, in which abstract actions are defined that can perform many primitive actions before terminating. However, little is known about learning with state abstractions, in which aspects of the state space are ignored. In previous work, we developed the MAXQ method for hierarchical RL. In this paper, we define five conditions under which state abstraction can be combined with the MAXQ value function decomposition. We prove that the MAXQ-Q learning algorithm converges under these conditions and show experimentally that state abstraction is important for the successful application of MAXQ-Q learning. 1 Introduction Most work on hierarchical reinforcement learning has focused on temporal abstraction. For example, in the Options framework [1,2], the programmer defines a set of macro actions ("options") and provides a policy for each. Learning algorithms (such as semi-Markov Q learning) can then treat these temporally abstract actions as if they were primitives and learn a policy for selecting among them. Closely related is the HAM framework, in which the programmer constructs a hierarchy of finitestate controllers [3]. Each controller can include non-deterministic states (where the programmer was not sure what action to perform). The HAMQ learning algorithm can then be applied to learn a policy for making choices in the non-deterministic states. In both of these approaches-and in other studies of hierarchical RL (e.g., [4, 5, 6])-each option or finite state controller must have access to the entire state space. The one exception to this-the Feudal-Q method of Dayan and Hinton [7]introduced state abstractions in an unsafe way, such that the resulting learning problem was only partially observable. Hence, they could not provide any formal results for the convergence or performance of their method. Even a brief consideration of human-level intelligence shows that such methods cannot scale. When deciding how to walk from the bedroom to the kitchen, we do not need to think about the location of our car. Without state abstractions, any RL method that learns value functions must learn a separate value for each state of the State Abstraction in MAXQ Hierarchical Reinforcement Learning 995 world. Some argue that this can be solved by clever value function approximation methods-and there is some merit in this view. In this paper, however, we explore a different approach in which we identify aspects of the MDP that permit state abstractions to be safely incorporated in a hierarchical reinforcement learning method without introducing function approximations. This permits us to obtain the first proof of the convergence of hierarchical RL to an optimal policy in the presence of state abstraction. We introduce these state abstractions within the MAXQ framework [8], but the basic ideas are general. In our previous work with MAXQ, we briefly discussed state abstractions, and we employed them in our experiments. However, we could not prove that our algorithm (MAXQ-Q) converged with state abstractions, and we did not have a usable characterization of the situations in which state abstraction could be safely employed. This paper solves these problems and in addition compares the effectiveness of MAXQ-Q learning with and without state abstractions. The results show that state abstraction is very important, and in most cases essential, to the effective application of MAXQ-Q learning. 2 The MAXQ Framework Let M be a Markov decision problem with states S, actions A, reward function R(s/ls, a) and probability transition function P(s/ls, a). Our results apply in both the finite-horizon undiscounted case and the infinite-horizon discounted case. Let {Mo, .. . ,Mn} be a set of subtasks of M, where each subtask Mi is defined by a termination predicate Ti and a set of actions Ai (which may be other subtasks or primitive actions from A). The "goal" of subtask Mi is to move the environment into a state such that Ti is satisfied. (This can be refined using a local reward function to express preferences among the different states satisfying Ti [8], but we omit this refinement in this paper.) The subtasks of M must form a DAG with a single "root" node-no subtask may invoke itself directly or indirectly. A hierarchical policy is a set of policies 1r = {1ro, ... , 1r n}, one for each subtask. A hierarchical policy is executed using standard procedure-call-and-return semantics, starting with the root task Mo and unfolding recursively until primitive actions are executed. When the policy for Mi is invoked in state s, let P(SI, Nls, i) be the probability that it terminates in state Sl after executing N primitive actions. A hierarchical policy is recursively optimal if each policy 1ri is optimal given the policies of its descendants in the DAG. Let V(i, s) be the value function for subtask i in state s (Le., the value of following some policy starting in s until we reach a state Sl satisfying Ti (S/) ) • Similarly, let Q(i, s,j) be the Q value for subtask i of executing child action j in state sand then executing the current policy until termination. The MAXQ value function decomposition is based on the observation that each subtask Mi can be viewed as a Semi-Markov Decision problem in which the reward for performing action j in state s is equal to V(j, s), the value function for subtask j in state s. To see this, consider the sequence of rewards rt that will be received when we execute child action j and then continue with subsequent actions according to hierarchical policy 1r: Q(i, s,j) = E{rt + ,rt+l + ,2rt+2 + .. ' Ist = S,1r} The macro action j will execute for some number of steps N and then return. Hence, we can partition this sum into two terms: 996 T. G. Dietterich The first term is the discounted sum ofrewards until subtask j terminates-V(j, s). The second term is the cost of finishing subtask i after j is executed (discounted to the time when j is initiated). We call this second term the completion function, and denote it C(i,s,j). We can then write the Bellman equation as Q(i,s,j) L P(s',Nls,j)· [V(j,s) +,N m.,?-xQ(i,s',j')] s',N J V(j, s) + C(i, s,j) To terminate this recursion, define V (a, s) for a primitive action a to be the expected reward of performing action a in state s. The MAXQ-Q learning algorithm is a simple variation of Q learning in which at subtask M i , state s, we choose a child action j and invoke its (current) policy. When it returns, we observe the resulting state s' and the number of elapsed time steps N and update C(i, s,j) according to C(i, s, j) := (1 - Ut)C(i, s, j) + Ut .,N[max V(a', s') + C(i, s', a')]. a' To prove convergence, we require that the exploration policy executed during learning be an ordered GLIE policy. An ordered policy is a policy that breaks Q-value ties among actions by preferring the action that comes first in some fixed ordering. A GLIE policy [9] is a policy that (a) executes each action infinitely often in every state that is visited infinitely often and (b) converges with probability 1 to a greedy policy. The ordering condition is required to ensure that the recursively optimal policy is unique. Without this condition, there are potentially many different recursively optimal policies with different values, depending on how ties are broken within subtasks, subsubtasks, and so on. Theorem 1 Let M = (S, A, P, R) be either an episodic MDP for which all deterministic policies are proper or a discounted infinite horizon MDP with discount factor,. Let H be a DAG defined over subtasks {Mo, ... ,Mk}. Let Ut(i) > 0 be a sequence of constants for each subtask Mi such that T T lim L Ut(i) = 00 T-too and lim '" u;(i) < 00 T-too~ (1) t=l t=l Let 7rx (i, s) be an ordered GLIE policy at each subtask Mi and state s and assume that IVt (i, s) I and ICt (i, s, a) I are bounded for all t, i, s, and a. Then with probability 1, algorithm MAXQ-Q converges to the unique recursively optimal policy for M consistent with Hand 7r x . Proof: (sketch) The proof is based on Proposition 4.5 from Bertsekas and Tsitsiklis [10] and follows the standard stochastic approximation argument due to [11] generalized to the case of non-stationary noise. There are two key points in the proof. Define Pt(s',Nls,j) to be the probability transition function that describes the behavior of executing the current policy for subtask j at time t. By an inductive argument, we show that this probability transition function converges (w.p. 1) to the probability transition function of the recursively optimal policy for j. Second, we show how to convert the usual weighted max norm contraction for Q into a weighted max norm contraction for C. This is straightforward, and completes the proof. What is notable about MAXQ-Q is that it can learn the value functions of all subtasks simultaneously-it does not need to wait for the value function for subtask j to converge before beginning to learn the value function for its parent task i. This gives a completely online learning algorithm with wide applicability. State Abstraction in MAXQ Hierarchical Reinforcement Learning 4 3 2 1 o R G 0 y B o 1 234 Figure 1: Left: The Taxi Domain (taxi at row 3 column 0) . Right: Task Graph. 3 Conditions for Safe State Abstraction 997 To motivate state abstraction, consider the simple Taxi Task shown in Figure 1. There are four special locations in this world, marked as R(ed), B(lue), G(reen), and Y(ellow). In each episode, the taxi starts in a randomly-chosen square. There is a passenger at one of the four locations (chosen randomly), and that passenger wishes to be transported to one of the four locations (also chosen randomly). The taxi must go to the passenger's location (the "source"), pick up the passenger, go to the destination location (the "destination"), and put down the passenger there. The episode ends when the passenger is deposited at the destination location. There are six primitive actions in this domain: (a) four navigation actions that move the taxi one square North, South, East, or West, (b) a Pickup action, and (c) a Putdown action. Each action is deterministic. There is a reward of -1 for each action and an additional reward of +20 for successfully delivering the passenger. There is a reward of -10 if the taxi attempts to execute the Putdown or Pickup actions illegally. If a navigation action would cause the taxi to hit a wall, the action is a no-op, and there is only the usual reward of -1. This task has a hierarchical structure (see Fig. 1) in which there are two main sub-tasks: Get the passenger (Get) and Deliver the passenger (Put). Each of these subtasks in turn involves the subtask of navigating to one of the four locations (Navigate(t); where t is bound to the desired target location) and then performing a Pickup or Putdown action. This task illustrates the need to support both temporal abstraction and state abstraction. The temporal abstraction is obvious-for example, Get is a temporally extended action that can take different numbers of steps to complete depending on the distance to the target. The top level policy (get passenger; deliver passenger) can be expressed very simply with these abstractions. The need for state abstraction is perhaps less obvious. Consider the Get subtask. While this subtask is being solved, the destination of the passenger is completely irrelevant- it cannot affect any of the nagivation or pickup decisions. Perhaps more importantly, when navigating to a target location (either the source or destination location of the passenger), only the taxi's location and identity ofthe target location are important. The fact that in some cases the taxi is carrying the passenger and in other cases it is not is irrelevant. We now introduce the five conditions for state abstraction. We will assume that the state s of the MDP is represented as a vector of state variables. A state abstraction can be defined for each combination of subtask Mi and child action j by identifying a subset X of the state variables that are relevant and defining the value function and the policy using only these relevant variables. Such value functions and policies 998 T. G. Dietterich are said to be abstract. The first two conditions involve eliminating irrelevant variables within a subtask of the MAXQ decomposition. Condition 1: Subtask Irrelevance. Let Mi be a subtask of MDP M. A set of state variables Y is irrelevant to sub task i if the state variables of M can be partitioned into two sets X and Y such that for any stationary abstract hierarchical policy 7r executed by the descendants of M i , the following two properties hold: (a) the state transition probability distribution P7r(5',NI5,j) for each child action j of Mi can be factored into the product of two distributions: P7r(x',y',Nlx,y,j) = P7r(x',Nlx,j)' P7r(y'lx,y,j), (2) where x and x' give values for the variables in X, and y and y' give values for the variables in Y; and (b) for any pair of states 51 = (x, yr) and 52 = (x, Y2) and any child action j, V 7r(j, 51) = V7r(j, 52)' In the Taxi problem, the source and destination of the passenger are irrelevant to the Navigate(t) subtask-only the target t and the current taxi position are relevant. The advantages of this form of abstraction are similar to those obtained by Boutilier, Dearden and Goldszmidt [12] in which belief network models of actions are exploited to simplify value iteration in stochastic planning. Condition 2: Leaf Irrelevance. A set of state variables Y is irrelevant for a primitive action a if for any pair of states 51 and 52 that differ only in their values for the variables in Y, L P(5~151' a)R(5~151' a) = L P(5~152' a)R(s~152' a). s' 1 s' 2 This condition is satisfied by the primitive actions North, South, East, and West in the taxi task, where all state variables are irrelevant because R is constant. The next two conditions involve "funnel" actions- macro actions that move the environment from some large number of possible states to a small number of resulting states. The completion function of such subtasks can be represented using a number of values proportional to the number of resulting states. Condition 3: Result Distribution Irrelevance (Undiscounted case.) A set of state variables }j is irrelevant for the result distribution of action j if, for all abstract policies 7r executed by M j and its descendants in the MAXQ hierarchy, the following holds: for all pairs of states 51 and 52 that differ only in their values for the state variables in }j, V 5' P7r(5'151,j) = P7r(5'152,j). Consider, for example, the Get subroutine under an optimal policy for the taxi task. Regardless of the taxi's position in state 5, the taxi will be at the passenger's starting location when Get finishes executing (Le., because the taxi will have just completed picking up the passenger). Hence, the taxi's initial position is irrelevant to its resulting position. (Note that this is only true in the undiscounted settingwith discounting, the result distributions are not the same because the number of steps N required for Get to finish depends very much on the starting location of the taxi. Hence this form of state abstraction is rarely useful for cumulative discounted reward.) Condition 4: Termination. Let Mj be a child task of Mi with the property that whenever Mj terminates, it causes Mi to terminate too. Then the completion State Abstraction in MAXQ Hierarchical Reinforcement Learning 999 cost C ( i, s, j) = 0 and does not need to be represented. This is a particular kind of funnel action- it funnels all states into terminal states for Mi' For example, in the Taxi task, in all states where the taxi is holding the passenger, the Put subroutine will succeed and result in a terminal state for Root. This is because the termination predicate for Put (i.e., that the passenger is at his or her destination location) implies the termination condition for Root (which is the same). This means that C(Root, s, Put) is uniformly zero, for all states s where Put is not terminated. Condition 5: Shielding. Consider subtask Mi and let s be a state such that for all paths from the root of the DAG down to M i , there exists a subtask that is terminated. Then no C values need to be represented for subtask Mi in state s, because it can never be executed in s. In the Taxi task, a simple example of this arises in the Put task, which is terminated in all states where the passenger is not in the taxi. This means that we do not need to represent C(Root, s, Put) in these states. The result is that, when combined with the Termination condition above, we do not need to explicitly represent the completion function for Put at all! By applying these abstraction conditions to the Taxi task, the value function can be represented using 632 values, which is much less than the 3,000 values required by flat Q learning. Without state abstractions, MAXQ requires 14,000 values! Theorem 2 (Convergence with State Abstraction) Let H be a MAXQ task graph that incorporates the five kinds of state abstractions defined above. Let 7r x be an ordered GLIE exploration policy that is abstract. Then under the same conditions as Theorem 1, MAXQ-Q converges with probability 1 to the unique recursively optimal policy 7r; defined by 7r x and H . Proof: (sketch) Consider a subtask Mi with relevant variables X and two arbitrary states (x, Yl) and (x, Y2). We first show that under the five abstraction conditions, the value function of 7r; can be represented using C(i,x,j) (Le., ignoring the Y values). To learn the values of C(i,x,j) = L:xl,NP(xl,Nlx,j)V(i,x'), a Q-learning algorithm needs samples of x' and N drawn according to P(x' , Nlx,j). The second part of the proof involves showing that regardless of whether we execute j in state (x, Yl) or in (x, Y2), the resulting x' and N will have the same distribution, and hence, give the correct expectations. Analogous arguments apply for leaf irrelevance and V (a, x). The termination and shielding cases are easy. 4 Experimental Results We implemented MAXQ-Q for a noisy version of the Taxi domain and for Kaelbling's HDG navigation task [5] using Boltzmann exploration. Figure 2 shows the performance of flat Q and MAXQ-Q with and without state abstractions on these tasks. Learning rates and Boltzmann cooling rates were separately tuned to optimize the performance of each method. The results show that without state abstractions, MAXQ-Q learning is slower to converge than flat Q learning, but that with state abstraction, it is much faster. 5 Conclusion This paper has shown that by understanding the reasons that state variables are irrelevant, we can obtain a simple proof of the convergence of MAXQ-Q learning 1000 200 MAXQ+Abscradion 0 1 ·200 .~ i -e j -600 -BOO · 1000 0 20000 40000 l ,\ FIalQ 1-\ f LX.\' ~ ~fo~ \ 60000 80000 100000 Primidve Actions '! ~ .~ ~ g ::E 120000 140000 160000 · 20 ..0 -60 -SO -100 -120 · 140 T. G. Dietterich 200000 400000 600000 800000 le+06 1.2e+06 l .~ Primitive Actioru Figure 2: Comparison of MAXQ-Q with and without state abstraction to flat Q learning on a noisy taxi domain (left) and Kaelbling's HDG task (right). The horizontal axis gives the number of primitive actions executed by each method. The vertical axis plots the average of 100 separate runs. under state abstraction. This is much more fruitful than previous efforts based only on weak notions of state aggregation [10], and it suggests that future research should focus on identifying other conditions that permit safe state abstraction. References [1) D. Precup and R. S. Sutton, "Multi-time models for temporally abstract planning," in NIPS10, The MIT Press, 1998. [2) R. S. Sutton, D. Precup, and S. Singh, "Between MDPs and Semi-MDPs: Learning, planning, and representing knowledge at multiple temporal scales," tech. rep., Univ. Mass., Dept. Compo Inf. Sci., Amherst, MA, 1998. [3] R. Parr and S. Russell, "Reinforcement learning with hierarchies of machines," in NIPS-10, The MIT Press, 1998. [4) S. P. Singh, "Transfer of learning by composing solutions of elemental sequential tasks," Machine Learning, vol. 8, p. 323, 1992. [5) L. P. Kaelbling, "Hierarchical reinforcement learning: Preliminary results," in Proceedings ICML-l0, pp. 167-173, Morgan Kaufmann, 1993. [6) M. Hauskrecht, N. Meuleau, C. Boutilier, L. Kaelbling, and T .. Dean, "Hierarchical solution of Markov decision processes using macro-actions," tech. rep., Brown Univ., Dept. Compo Sci., Providence, RI, 1998. [7) P. Dayan and G. Hinton, "Feudal reinforcement learning," in NIPS-5, pp. 271- 278, San Francisco, CA: Morgan Kaufmann, 1993. [8) T. G. Dietterich, "The MAXQ method for hierarchical reinforcement learning," in ICML-15, Morgan Kaufmann, 1998. [9) S. Singh, T. Jaakkola, M. L. Littman, and C. Szpesvari, "Convergence results for single-step on-policy reinforcement-learning algorithms," tech. rep., Univ. Col., Dept. Compo Sci., Boulder, CO, 1998. [10) D. P. Bertsekas and J. N. Tsitsiklis, Neu.ro-Dynamic Programming. Belmont, MA: Athena Scientific, 1996. [11) T. Jaakkola, M. 1. Jordan, and S. P. Singh, "On the convergence of stochastic iterative dynamic programming algorithms," Neur. Comp., vol. 6, no. 6, pp. 1185- 1201, 1994. [12) C. Boutilier, R. Dearden, and M. Goldszmidt, "Exploiting structure in policy construction," in Proceedings IJCAI-95, pp. 1104- 1111, 1995.
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Modeling High-Dimensional Discrete Data with Multi-Layer Neural Networks Yoshua Bengio Dept.IRO Universite de Montreal Montreal, Qc, Canada, H3C 317 bengioy@iro.umontreal.ca Abstract Samy Bengio * IDIAP CP 592, rue du Simplon 4, 1920 Martigny, Switzerland bengio@idiap.ch The curse of dimensionality is severe when modeling high-dimensional discrete data: the number of possible combinations of the variables explodes exponentially. In this paper we propose a new architecture for modeling high-dimensional data that requires resources (parameters and computations) that grow only at most as the square of the number of variables, using a multi-layer neural network to represent the joint distribution of the variables as the product of conditional distributions. The neural network can be interpreted as a graphical model without hidden random variables, but in which the conditional distributions are tied through the hidden units. The connectivity of the neural network can be pruned by using dependency tests between the variables. Experiments on modeling the distribution of several discrete data sets show statistically significant improvements over other methods such as naive Bayes and comparable Bayesian networks, and show that significant improvements can be obtained by pruning the network. 1 Introduction The curse of dimensionality hits particularly hard on models of high-dimensional discrete data because there are many more possible combinations of the values of the variables than can possibly be observed in any data set, even the large data sets now common in datamining applications. In this paper we are dealing in particular with multivariate discrete data, where one tries to build a model of the distribution of the data. This can be used for example to detect anomalous cases in data-mining applications, or it can be used to model the class-conditional distribution of some observed variables in order to build a classifier. A simple multinomial maximum likelihood model would give zero probability to all of the combinations not encountered in the training set, i.e., it would most likely give zero probability to most out-of-sample test cases. Smoothing the model by assigning the same non-zero probability for all the unobserved cases would not be satisfactory either because it would not provide much generalization from the training set. This could be obtained by using a multivariate multinomial model whose parameters B are estimated by the maximum a-posteriori (MAP) principle, i.e., those that have the greatest probability, given the training data D, and using a diffuse prior PCB) (e.g. Dirichlet) on the parameters. A graphical model or Bayesian network [6, 5) represents the joint distribution of random variables Zl ... Zn with n P(ZI ... Zn) = II P(ZiIParentsi) i=l °Part of this work was done while S.B. was at CIRANO, Montreal, Qc. Canada. Modeling High-Dimensional Discrete Data with Neural Networks 401 where Parentsi is the set of random variables which are called the parents of variable i in the graphical model because they directly condition Zi, and an arrow is drawn, in the graphical model, to Zi, from each of its parents. A fully connected "left-to-right" graphical model is illustrated in Figure 1 (left), which corresponds to the model n P(ZI . .. Zn) = II P(ZiIZl ... Zi-r) . (1) i= l Figure 1: Left: a fully connected "left-to-right" graphical model. Right: the architecture of a neural network that simulates a ful1y connected "left-to-right" graphical model. The observed values Zi = Zi are encoded in the corresponding input unit group. hi is a group of hidden units. gi is a group of output units, which depend on Zl ... Zi-l , representing the parameters of a distribution over Zi. These conditional probabilities P(ZiIZl . . . Zi-r) are multiplied to obtain the joint distribution. Note that this representation depends on the ordering of the variables (in that all previous variables in this order are taken as parents). We call each combination of the values of Parentsi a context. In the "exact" model (with the full table of all possible contexts) all the orders are equivalent, but if approximations are used, different predictions could be made by different models assuming different orders. In graphical models, the curse of dimensionality shows up in the representation of conditional distributions P(ZiIParentsi) where Zi has many parents. If Zj E Parentsi can take nj values, there are TIj nj different contexts which can occur in which one would like to estimate the distribution of Zi. This serious problem has been addressed in the past by two types of approaches, which are sometimes combined: 1. Not modeling all the dependencies between all the variables: this is the approach mainly taken with most graphical models or Bayes networks [6, 5]. The set of independencies can be assumed using a-priori or human expert knowledge or can be learned from data. See also [2] in which the set Parentsi is restricted to at most one element, which is chosen to maximize the correlation with Zi. 2. Approximating the mathematicalform of the joint distribution with a form that takes only into account dependencies of lower order, or only takes into account some of the possible dependencies, e.g., with the Rademacher-Walsh expansion or multi-binomial [1,3], which is a low-order polynomial approximation of a full joint binomial distribution (and is used in the experiments reported in this paper). The approach we are putting forward in this paper is mostly of the second category, although we are using simple non-parametric statistics of the dependency between pairs of variables to further reduce the number of required parameters. In the multi-binomial model [3], the joint distribution of a set of binary variables is approximated by a polynomial. Whereas the "exact" representation of P( Zl = Zl , ... Zn = zn) as a function of Z l . . . Zn is a polynomial of degree n, it can be approximated with a lower 402 Y. Bengio and S. Bengio degree polynomial, and this approximation can be easily computed using the RademacherWalsh expansion [1] (or other similar expansions, such as the Bahadur-Lazarsfeld expansion [1]). Therefore, instead of having 2n parameters, the approximated model for P(Zl , . . . Zn) only requires O(nk) parameters. Typically, order k = 2 is used. The model proposed here also requires O(n2 ) parameters, but it allows to model dependencies between tuples of variables, with more than 2 variables at a time. In previous related work by Frey [4], a fully-connected graphical model is used (see Figure 1, left) but each of the conditional distributions is represented by a logistic, which take into account only first-order dependency between the variables: 1 P(Zi = llZl ... Zi-d = ( L Z )' 1 + exp -Wo j<i Wj j In this paper, we basically extend Frey's idea to using a neural network with a hidden layer, with a particular architecture, allowing multinomial or continuous variables, and we propose to prune down the network weights. Frey has named his model a Logistic Autoregressive Bayesian Network or LARC. He argues that the prior variances on the logistic weights (which correspond to inverse weight decays) should be chosen inversely proportional to the number of conditioning variables (i.e. the number of inputs to the particular output neuron). The model was tested on a task of learning to classify digits from 8x8 binary pixel images. Models with different orderings of the variables were compared and did not yield significant differences in performance. When averaging the predictive probabilities from 10 different models obtained by considering 10 different random orderings, Frey obtained small improvements in likelihood but not in classification. The model performed better or equivalently to other models tested: CART, naive Bayes, K-nearest neighbors, and various Bayesian models with hidden variables (Helmholtz machines). These results are impressive, taking into account the simplicity of the LARC model. 2 Proposed Architecture The proposed architecture is a "neural network" implementation of a graphical model where all the variables are observed in the training set, with the hidden units playing a significant role to share parameters across different conditional distributions. Figure 1 (right) illustrates the model in the simpler case of a fully connected (Ieft-to-right) graphical model (Figure 1, left). The neural network represents the parametrized function jo(zt, . .. , zn) = log(?O(Zl = Zl,· · ., Zn = zn)) (2) approximating the joint distribution of the variables, with parameters 0 being the weights of the neural network. The architecture has three layers, with each layer organized in groups associated to each of the variables. The above log-probability is computed as the sum of conditional log-probabilities n jO(Zl , . .. , zn) = L 109(P(Zi = zilgi(zl, .. . , Zi-l))) i=l where gi(Zt, ... , zi-d is the vector-valued output of the i-th group of output units, and it gives the value of the parameters of the distribution of Zi when Zl = Zl , Z2 = Z2, .. . , Zi-l = Zi-l' For example, in the ordinary discrete case, gi may be the vector of probabilities associated with each of the possible values of the multinomial random variable Zi. In this case, we have P(Zi = i'lgi) = gi,i' In this example, a softmax output for the i-th group may be used to force these parameters to be positive and sum to 1, i.e., gi,i' = g' Lil e i ,i' Modeling High-Dimensional Discrete Data with Neural Networks 403 where g~ i' are linear combinations of the hidden units outputs, with i' ranging over the number of elements of the parameter vector associated with the distribution of Zi (for a fixed value of Zl ... Zi-l). To guarantee that the functions gi(Zl, ... , Zi-l) only depend on Zl ... Zi-l and not on any of Zi ... Zn, the connectivity struture of the hidden units must be constrained as follows: mj g~,i' = bi,i' + 2: 2: Wi,i' ,j,j' hj,j' j~i j'=1 where the b's are biases and the w's are weights of the output layer, and the hj,j' is the output of the j'-th unit (out of mj such units) in the j-th group of hidden layer nodes. It may be computed as follows: nk hj,j' = tanh(cj,j' + 2: 2: Vj ,j' ,k,k' Zk ,k') k<j k'=l where the c's are biases and the v's are the weights of the hidden layer, and Zk,k' is k'-th element of the vectorial input representation of the value Zk = Zk. For example, in the binary case (Zi = 0 or 1) we have used only one input node, i.e., Zi binomial -t Zi,O = Zi and in the multinomial case we use the one-hot encoding, Zi E {O, 1, ... ni - I} -t Zi ,i' = 8Zi ,i' where 8i ,i' = 1 if i = i' and 0 otherwise. The input layer has n - 1 groups because the value Zn = Zn is not used as an input. The hidden layer also has n 1 groups corresponding to the variables j = 2 to n (since P(Z.) is represented unconditionally in the first output group, its corresponding group does not need any hidden units or inputs, but just has biases). 2.1 Discussion The number of free parameters of the model is O(n2 H) where H = maXi mj is the maximum number of hidden units per hidden group (i.e., associated with one of the variables). This is basically quadratic in the number of variables, like the multi-binomial approximation that uses a polynomial expansion of the joint distribution. However, as H is increased, representation theorems for neural networks suggest that we should be able to approximate with arbitrary precision the true joint distribution. Of course the true limiting factor is the amount of data, and H should be tuned according to the amount of data. In our experiments we have used cross-validation to choose a value of mj = H for all the hidden groups. In this sense, this neural network representation of P(ZI ... Zn) is to the polynomial expansions (such as the multi-binomial) what ordinary multilayer neural networks for function approximation are to polynomial function approximators. It allows to capture high-order dependencies, but not all of them. It is the number of hidden units that controls "how many" such dependencies will be captured, and it is the data that "chooses" which of the actual dependencies are most useful in maximizing the likelihood. Unlike Bayesian networks with hidden random variables, learning with the proposed architecture is very simple, even when there are no conditional independencies. To optimize the parameters we have simply used gradient-based optimization methods, either using conjugate or stochastic (on-line) gradient, to maximize the total log-likelihood which is the sum of values of f (eq. 2) for the training examples. A prior on the parameters can be incorporated in the cost function and the MAP estimator can be obtained as easily, by maximizing the total log-likelihood plus the log-prior on the parameters. In our experiments we have used a "weight decay" penalty inspired by the analysis of Frey [4], with a penalty proportional to the number of weights incoming into a neuron. 404 Y. Bengio and S. Bengio However, it is not so clear how the distribution could be generally marginalized, except by summing over possibly many combinations of the values of variables to be integrated. Another related question is whether one could deal with missing values: if the total number of values that the missing variables can take is reasonably small, then one can sum over these values in order to obtain a marginal probability and maximize this probability. If some variables have more systematically missing values, they can be put at the end of the variable ordering, and in this case it is very easy to compute the marginal distribution (by taking only the product of the output probabilities up to the missing variables). Similarly, one can easily compute the predictive distribution of the last variable given the first n - 1 variables. The framework can be easily extended to hybrid models involving both continuous and discrete variables. In the case of continuous variables, one has to choose a parametric form for the distribution of the continuous variable when all its parents (i.e., the conditioning context) are fixed. For example one could use a normal, log-normal, or mixture of normals. Instead of having softmax outputs, the i-th output group would compute the parameters of this continuous distribution (e.g., mean and log-variance). Another type of extension allows to build a conditional distribution, e.g., to model P(ZI ... ZnlXl ... Xm). One just adds extra input units to represent the values of the conditioning variables Xl ... X m . Finally, an architectural extension that we have implemented is to allow direct input-tooutput connections (still following the rules of ordering which allow gi to depend only on Zl ... Zi-l). Therefore in the case where the number of hidden units is 0 (H = 0) we obtain the LARC model proposed by Frey [4]. 2.2 Choice of topology Another type of extension of this model which we have found very useful in our experiments is to allow the user to choose a topology that is not fully connected (Ieft-to-right). In our experiments we have used non-parametric tests to heuristically eliminate some of the connections in the network, but one could also use expert or prior knowledge, just as with regular graphical models, in order to cut down on the number of free parameters. In our experiments we have used for a pairwise test of statistical dependency the Kolmogorov-Smirnov statistic (which works both for continuous and discrete variables). The statistic for variables X and Y is s = Jl sup IP(X :::; Xi, Y :::; Yi) - P(X :::; Xi)P(Y :::; Yi) I i where l is the number of examples and P is the empirical distribution (obtained by counting over the training data). We have ranked the pairs according to their value of the statistic s, and we have chosen those pairs for which the value of statistic is above a threshold value s*, which was chosen by cross-validation. When the pairs {(Zi' Zj)} are chosen to be part of the model, and assuming without loss of generality that i < j for those pairs, then the only connections that are kept in the network (in addition to those from the k-th hidden group to the k-th output group) are those from hidden group i to output group j, and from input group i to hidden group j, for every such (Zi' Zj) pair. 3 Experiments In the experiments we have compared the following models: • Naive Bayes: the likelihood is obtained as a product of multinomials (one per variable). Each multinomial is smoothed with a Dirichlet prior. • Multi-Binomial (using Rademacher-Walsh expansion of order 2) [3]. Since this only handles the case of binary data, it was only applied to the DNA data set. • A simple graphical model with the same pairs of variables and variable ordering as selected for the neural network, but in which each of the conditional distribution is modeled Modeling High-Dimensional Discrete Data with Neural Networks 405 by a separate multinomial for each of the conditioning context. This works only if the number of conditioning variables is small so in the Mushroom, Audiology, and Soybean experiments we had to reduce the number of conditioning variables (following the order given by the above tests). The multinomials are also smoothed with a Dirichlet prior. • Neural network: the architecture described above, with or without hidden units (i.e., LARC), with or without pruning. 5-fold cross-validation was used to select the number of hidden units per hidden group and the weight decay for the neural network and LARC. Cross-validation was also used to choose the amount of pruning in the neural network and LARC, and the amount of smoothing in the Dirichlet priors for the muItinomials of the naive Bayes model and the simple graphical model. 3.1 Results All four data sets were obtained on the web from the VCI Machine Learning and STATLOG databases. Most of these are meant to be for classification tasks but we have instead ignored the classification and used the data to learn a probabilistic model of all the input features. • DNA (from STATLOG): there are 180 binary features. 2000 cases were used for training and cross-validation, and 1186 for testing. • Mushroom (from VCI): there are 22 discrete features (taking each between 2 and 12 values). 4062 cases were used for training and cross-validation, and 4062 for testing. • Audiology (from VCI): there are 69 discrete features (taking each between 2 and 7 values). 113 cases are used for training and 113 for testing (the original train-test partition was 200 + 26 and we concatenated and re-split the data to obtain more significant test figures). • Soybean (from VCI): there are 35 discrete features (taking each between 2 and 8 values). 307 cases are used for training and 376 for testing. Table 1 clearly shows that the proposed model yields promising results since the pruned neural network was superior to all the other models in all 4 cases, and the pairwise differences with the other models are statistically significant in all 4 cases (except Audiology, where the difference with the network without hidden units, LARC, is not significant). 4 Conclusion In this paper we have proposed a new application of multi-layer neural networks to the modelization of high-dimensional distributions, in particular for discrete data (but the model could also be applied to continuous or mixed discrete / continuous data). Like the polynomial expansions [3] that have been previously proposed for handling such high-dimensional distributions, the model approximates the joint distribution with a reasonable (O( n 2 )) number of free parameters but unlike these it allows to capture high-order dependencies even when the number of parameters is small. The model can also be seen as an extension of the previously proposed auto-regressive logistic Bayesian network [4], using hidden units to capture some high-order dependencies. Experimental results on four data sets with many discrete variables are very encouraging. The comparisons were made with a naive Bayes model, with a multi-binomial expansion, with the LARC model and with a simple graphical model, showing that a neural network did significantly better in terms of out-of-sample log-likelihood in all cases. The approach to pruning the neural network used in the experiments, based on pairwise statistical dependency tests, is highly heuristic and better results might be obtained using approaches that take into account the higher order dependencies when selecting the conditioning variables. Methods based on pruning the fully connected network (e.g., with a "weight elimination" penalty) should also be tried. Also, we have not tried to optimize 406 Y Bengio and S. Bengio DNA Mushroom mean (stdev) p-value mean (stdev) p-value naive Bayes 100.4 (.18) <le-9 47.00 (.29) <le-9 multi-Binomial order 2 117.8(.01) <le-9 ordinary graph. model 108.1 (.06) <le-9 44.68 (.26) <le-9 LARC 83.2 (.24) 7e-5 42.51 (.16) <le-9 prunedLARC 91.2(.15) <le-9 43.87 (.13) < le-9 full-conn. neural net. 120.0 (.02) <le-9 33.58 (.01) <le-9 pruned neural network 82.9 (.21) 31.25 (.04) Audiology Soybean mean (stdev) p-value mean (stdev) p-value naive Bayes 36.40 (2.9) <le-9 34.74 (1.0) <le-9 multi-Binomial order 2 ordinary graph. model 16.56 (.48) 6.8e-4 43.65 (.07) <le-9 LARC 17.69 (.65) <le-9 16.95 (.35) 5.5e-4 prunedLARC 16.69 (.41) 0.20 19.06 (.43) <le-9 full-conn. neural net. 17.39 (.58) <le-9 21.65 (.43) <le-9 pruned neural network 16.37 (.45) 16.55 (.27) Table 1: Average out-of-sample negative log-likelihood obtained with the various models on four data sets (standard deviations of the average in parenthesis and p-value to test the null hypotheses that a model has same true generalization error as the pruned neural network). The pruned neural network was better than all the other models in in all cases, and the pair-wise difference is always statistically significant (except with respect to the pruned LARC on Audiology). the order of the variables, or combine different networks obtained with different orders, like [4]. References [1] RR Bahadur. A representation of the joint distribution of responses to n dichotomous items. In ed. H. Solomon, editor, Studies in Item Analysis and Predictdion, pages 158-168. Stanford University Press, California, 1961. [2] c.K. Chow. A recognition method using neighbor dependence. IRE Trans. Elec. Comp., EC-l1 :683-690, October 1962. [3] RO. Duda and P.E. Hart. Pattern Classification and Scene Analysis. Wiley, New York, 1973. [4] B. Frey. Graphical models for machine learning and digital communication. MIT Press, 1998. [5] Steffen L. Lauritzen. The EM algorithm for graphical association models with missing data. Computational Statistics and Data Analysis, 19:191-201,1995. [6] Judea Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, 1988.
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A generative model for attractor dynamics Richard S. Zemel Department of Psychology University of Arizona Tucson, AZ 85721 zemel@u.arizona.edu Michael C. Mozer Department of Computer Science University of Colorado Boulder, CO 80309-0430 mozer@colorado.edu Abstract Attractor networks, which map an input space to a discrete output space, are useful for pattern completion. However, designing a net to have a given set of attractors is notoriously tricky; training procedures are CPU intensive and often produce spurious afuactors and ill-conditioned attractor basins. These difficulties occur because each connection in the network participates in the encoding of multiple attractors. We describe an alternative formulation of attractor networks in which the encoding of knowledge is local, not distributed. Although localist attractor networks have similar dynamics to their distributed counterparts, they are much easier to work with and interpret. We propose a statistical formulation of localist attract or net dynamics, which yields a convergence proof and a mathematical interpretation of model parameters. Attractor networks map an input space, usually continuous, to a sparse output space composed of a discrete set of alternatives. Attractor networks have a long history in neural network research. Attractor networks are often used for pattern completion, which involves filling in missing, noisy, or incorrect features in an input pattern. The initial state of the attractor net is typically determined by the input pattern. Over time, the state is drawn to one of a predefined set of states-the attractors. Attractor net dynamics can be described by a state trajectory (Figure 1a). An attractor net is generally implemented by a set of visible units whose activity represents the instantaneous state, and optionally, a set of hidden units that assist in the computation. Attractor dynamics arise from interactions among the units. In most formulations of afuactor nets,2,3 the dynamics can be characterized by gradient descent in an energy landscape, allowing one to partition the output space into attractor basins. Instead of homogeneous attractor basins, it is often desirable to sculpt basins that depend on the recent history of the network and the arrangement of attractors in the space. In psychological models of human cognition, for example, priming is fundamental: after the model visits an attractor, it should be faster to fall into the same attractor in the near future, i.e., the attractor basin should be broadened. 1 ,6 Another property of attractor nets is key to explaining behavioral data in psychological and neurobiological models: the gang effect, in which the strength of an attractor is influenced by other attractors in its neighborhood. Figure 1b illustrates the gang effect: the proximity of the two rightmost afuactors creates a deeper attractor basin, so that if the input starts at the origin it will get pulled to the right. A Generative Model for Attractor Dynamics 81 k , " , " ,/ "------ .. Figure 1: (a) A two-dimensional space can be carved into three regions (dashed lines) by an attractor net. The dynamics of the net cause an input pattern (the X) to be mapped to one of the attractors (the O's). The solid line shows the temporal trajectory of the network state. (b) the actual energy landscape for a localist attractor net as a function of y, when the input is fixed at the origin and there are three attractors, W = ((-1,0), (1,0), (1, -A)), with a uniform prior. The shapes of attractor basins are influenced by the proximity of attractors to one another (the gang effect). The origin of the space (depicted by a point) is equidistant from the attractor on the left and the attractor on the upper right, yet the origiri clearly lies in the basin of the right attractors. 1bis effect is an emergent property of the distribution of attractors, and is the basis for interesting dynamics; it produces the mutually reinforcing or inhibitory influence of similar items in domains such as semantics,9 memory,lO,12 and olfaction.4 Training an attract or net is notoriously tricky. Training procedures are CPU intensive and often produce spurious attractors and ill-conditioned attractor basins. 5,11 Indeed, we are aware of no existing procedure that can robustly translate an arbitrary specification of an attractor landscape into a set of weights. These difficulties are due to the fact that each connection participates in the specification of multiple attractors; thus, knowledge in the net is distributed over connections. We describe an alternative attractor network model in which knowledge is localized, hence the name localist attractor network. The model has many virtues, including: a trivial procedure for wiring up the architecture given an attractor landscape; eliminating spurious attractors; achieving gang effects; providing a clear mathematical interpretation of the model parameters, which clarifies how the parameters control the qualitative behavior of the model (e.g., the magnitude of gang effects); and proofs of convergence and stability. A Iocalist attractor net consists of a set of n state units and m attractor units. Parameters associated with an attractor unit i encode the location of the attractor, denoted Wi, and its "pull" or strength, denoted 7ri, which influence the shape of the attractor basin. Its activity at time t, qi(t), reflects the normalized distance from the attractor center to the current state, y(t), weighted by the attractor strength: g(y, w, 0") 7rjg(y(t), Wi, O"(t)) L:j 7rjg(y(t), Wj, O"(t)) exp( -\y - w\2/20"2) Thus, the attractors form a layer of normalized radial-basis-function units. (1) (2) The input to the net, &, serves as the initial value of the state, and thereafter the state is pulled toward attractors in proportion to their activity. A straightforward 82 R. S. Zemel and M. C. Mozer expression of this behavior is: (3) where a(l) = Ion the first update and a(t) = 0 fort> 1. More generally, however, one might want to gradually reduce a over time, allowing for a persistent effect of the external input on the asymptotic state. The variables O"(t) and a(t) are not free parameters of the model, but can be derived from the formalism we present below. The localist attractor net is motivated by a generative model of the input based on the attractor distribution, and the network dynamics corresponds to a search for a maximum likelihood interpretation of the observation. In the following section, we derive this result, and then present simulation studies of the architecture. 1 A MAXIMUM LIKELIHOOD FORMULATION The starting point for the statistical formulation of a localist attractor network is a mixture of Gaussians model. A standard mixture of Gaussians consists of m Gaussian density functions in n dimensions. Each Gaussian is parameterized by a mean, a covariance matrix, and a mixture coefficient. The mixture model is generative, i.e., it is considered to have produced a set of observations. Each observation is generated by selecting a Gaussian based on the mixture coefficients and then stochastically selecting a point from the corresponding density function. The model parameters are adjusted to maximize the likelihood of a set of observations. The Expectation-Maximization (EM) algorithm provides an efficient procedure for estimating the parameters.The Expectation step calculates the posterior probability qi of each Gaussian for each observation, and the Maximization step calculates the new parameters based on the previous values and the set of qi. The mixture of Gaussians model can provide an interpretation for a localist attractor network, in an unorthodox way. Each Gaussian corresponds to an attractor, and an observation corresponds to the state. Now, however, instead of fixing the observation and adjusting the Gaussians, we fix the Gaussians and adjust the observation. If there is a single observation, and a = 0 and all Gaussians have uniform spread 0", then Equation 1 corresponds to the Expectation step, and Equation 3 to the Maximization step in this unusual mixture model. Unfortunately, this simple characterization of the localist attractor network does not produce the desired behavior. Many situations produce partial solutions, in which the observation does not end up at an attractor. For example, if two unidimensional Gaussians overlap significantly, the most likely value for the observation is midway between them rather than at the mean of either Gaussian. We therefore extend this mixture-of-Gaussians formulation to better characterize the localist attractor network. As in the simple model, each of the m attractors is a Gaussian generator, the mean of which is a location in the n-dimensional state space. The input to the net, e, is considered to have been generated by a stochastic selection of one of the attractors, followed by the addition of zero-mean Gaussian noise with variance specified by the attractor. Given a particular observation e, the an attractor's posterior probability is the normalized Gaussian probability of e, weighted by its mixing proportion. This posterior distribution for the attractors corresponds to a distribution in state space that is a weighted sum of Gaussians. We then consider the attractor network as encoding this distribution over states implied by the attractor posterior probabilities. At anyone time, however, the attractor network can only represent a single position in state space, rather than A Generative Model for Attractor Dynamics 83 the entire distribution over states. This restriction is appropriate when the state is an n-dimensional point represented by the pattern of activity over n state units. To accommodate this restriction, we change the standard mixture of Gaussians generative model by interjecting an intermediate level between the attractors and the observation. The first generative level consists of the discrete attractors, the second is the state space, and the third is the observation. Each observation is considered to have been generated by moving down this hierarchy: 1. select an attractor x = i from the set of attractors 2. select a state (i.e., a pattern of activity across the state units) based on the preferred location of that attractor: y = Wi + Ny 3. select an observation z = yG + Nz The observation z produced by a particular state y depends on the generative weight matrix G. In the networks we consider here, the observation and state spaces are identical, so G is the identity matrix, but the formulation allows for z to lie in some other space. Ny and Nz describe the zero-mean, spherical Gaussian noise introduced at the two levels, with deviations (1 y and (1 z, respectively. In comparison with the 2-level Gaussian mixture model described above, this 3level model is more complicated but more standard: the observation & is preserved as stable data, and rather than the model manipulating the data here it can be viewed as iteratively manipulating an internal representation that fits the observation and attractor structure. The attractor dynamics correspond to an iterative search through state space to find the most likely single state that: (a) was generated by the mixture of Gaussian attractors, and (b) in tum generated the observation. Under this model, one could fit an observation & by finding the posterior distribution over the hidden states (X and Y) given the observation: (X = i Y = ylZ = &) = p(&ly, i)p(y, i) = p(&IY)1riP(yli) (4) p, p(&) Jy p(&IY)L:i1riP(Yli)dy where the conditional distributions are Gaussian: p(Y = ylX = i) = 9(ylwi, (1y) and p(&IY = y) = 9(&ly, (1z). Evaluating the distribution in Equation 4 is tractable, because the partition function is a sum of a set of Gaussian integrals. Due to the restriction that the network cannot represent the entire distribution, we do not directly evaluate this distribution but instead adopt a mean-field approach, in which we approximate the posterior by another distribution Q(X, YI&). Based on this approximation, the network dynamics can be seen as minimizing an objective function that describes an upper bound on the negative log probability of the observation given the model and mean-field parameters. In this approach, one can choose any form of Q to estimate the posterior distribution, but a better estimate allows the network to approach a maximum likelihood solution.13 We select a simple posterior: Q(X, Y) = qid(Y = y), where qi = Q(X = i) is the responsibility assigned to attractor i, and y is the estimate of the state that accounts for the observation. The delta function over Y is motivated by the restriction that the explanation of an input consists of a single state. Given this posterior distribution, the objective for the network is to minimize the free energy F, described here for a particular input example &: ~ J (. Q(X = i, Y = y) F(q,yl&) = ~ QX=t,Y=y)lnp(&,X=i,Y=y)dy , L qi In qi. -lnp(&IY) - L qi lnp(yli) . 1r, . , , 84 R. S. Zemel and M C. Mozer where 7r; is the prior probability (mixture coefficient) associated with attractor i. These priors are parameters of the generative model, as are (fy, u z , and w. F(q, yl&) = 2: qdn qi. + 2 \ 1& - Yl2 + 2 \ 2: qi!y - wd2 + n In((fy(fz) (5) i 7r, U z (f Y i Given an observation, a good set of mean-field parameters can be determined by alternating between updating the generative parameters and the mean-field parameters. The update procedure is guaranteed to converge to a minimum of F, as long as the updates are done asynchronously and each update minimizes F with respect to a parameter.s The update equations for the mean-field parameters are: 2&+ 2E . . y = (f y (f z i q, w, (6) (f2 + (f2 y z qi 7riP(Y Ii) (7) E j 7rjp(yli) In our simulations, we hold most of the parameters of the generative model constant, such as the priors 7r, the weights w, and the generative noise in the observation, (f z. The only aspect that changes is the generative noise in the state, (f Y' which is a single parameter shared by all attractors: 2 1 ~ I' 12 (fy = d L..Jqi Y -Wi i (8) The updates of Equations 6-8 can be in any order. We typically initialize the state y to & at time 0, and then cyclically update the qi, (fy, then y. This generative model avoids the problem of spurious attractors described above for the standard Gaussian mixture model. Intuition into how the model avoids spurious attractors can be gained by inspecting the update equations. These equations effectively tie together two processes: moving y closer to some Wi than the others, and increasing the corresponding responsibility qi. As these two processes evolve together, they act to descrease the noise (fy, which accentuates the pull of the attractor. Thus stable points that do not correspond to the attractors are rare. 2 SIMULATION STUDIES To create an attractor net, we specify the parameters (7ri' w;) associated with the attractors based on the desired structure of the energy landscape (e.g., Figure Ib). The only remaining free parameter, (f z, plays an important role in determining how responsive the system is to the external input. We have conducted several simulation studies to explore properties of localist attractor networks. Systematic investigations with a 200-dimensional state space and 200 attractors, randomly placed at corners of the 200-D hypercube, have demonstrated that spurious responses are exceedingly rare unless more than 85% of an input's features are distorted (Figure 2), and that manipulating parameters such as noise and prior probabilities has the predicted effects. We have also conducted studies of localist attractor networks in the domain of visual images of faces. These simulations have shown that gang effects arise when there is structure among the attractors. For example, when the attractor set consists of a single view of several different faces, and multiple views of one face, then an input that is a morphed face-a linear combination of one of the single-view faces and one view of the gang face-will end up in the gang attractor even when the initial weighting assigned to the gang face was less than 40%. A Generative Modelfor Attractor Dynamics 100 ~~~==~~~~--------~~loo % Missing features 85 Figure 2: The input must be severely corrupted before the net makes spurious (final state not at an attractor) or adulterous (final state at a neighbor of the generating attractor) responses. (a) The percentage of spurious responses increases as (Tz is increased. (b) The percentage of adulterous responses increases as (T z is decreased. To test the architecture on a larger, structured problem, we modeled the domain of three-letter English words. The idea is to use the attractor network as a content addressable memory which might, for example, be queried to retrieve a word with P in the third position and any letter but A in the second position, a word such as HIP. The attractors consist of the 423 three-letter English words, from ACE to ZOO. The state space of the attractor network has one dimension for each of the 26 letters of the English alphabet in each of the 3 positions, for a total of 78 dimensions. We can refer to a given dimension by the letter and position it encodes, e.g., P3 denotes the dimension corresponding to the letter P in the third position of the word. The attractors are at the comers of a [-1, +1]18 hypercube. The attractor for a word such as HIP is located at the state having value -Ion all dimensions except for HI, h, and P3 which have value +1. The external input specifies a state that constrains the solution. For example, one might specify lip in the third position" by setting the external input to +1 on dimension P 3 and to -Ion dimensions ll'3, for all letters ll' other than P. One might specify the absence of a constraint in a particular letter position, p, by setting the external input to a on dimensions ll'p, for all letters ll'. The network's task is to settle on a state corresponding to one of the words, given soft constraints on the letters. The interactive-activation model of word perception 7 performs a similar computation, and our implementation exhibits the key qualitative properties of their model. If the external input specifies a word, of course the attractor net will select that word. Interesting queries are those in which the external input underconstrains or overconstrains the solution. We illustrate with one example of the network's behavior, in which the external input specifies D1, E2, and G3 • Because DEG is a nonword, no attractor exists for that state. The closest attractors share two letters with DEG, e.g., PEG, BEG, DEN, and DOG. Figure 3 shows the effect of gangs on the selection of a response, BEG. 3 CONCLUSION Localist attractor networks offer an attractive alternative to standard attractor networks, in that their dynamics are easy to control and adapt. We described a statistical formulation of a type of localist attractor, and showed that it provides a Lyapunov function for the system as well as a mathematical interpretation for the network parameters. The dynamics of this system are derived not from intuitive arguments but from this formal mathematical model. Simulation studies show that the architecture achieves gang effects, and spurious attractors are rare. This approach is inefficient if the attractors have compositional structure, but for many applications of pattern recognition or associative memory, the number of items 86 R. S. Zemel and M C. Mozer Iteration I ~·;s~., ~·; ~: f.· f.j.~·.;:: BEG ;';;,"',' :.':(; l;,;)f ,:~;.:,,: >.~~~-t".r::';"'~ '.~r.': DEN DEW F,.; :.:':. DIG ,,':i_ ~:, ~ ~~::, DOG '.~':";:, ~y'.)' :;',~ :_'>":' .• ':.';.:; DUG ~ ~ .... ' j:. ~.:;?:( ;: _""- '.~ ~. f:.:: ~+t: ,' :: "; .§.. •• :; " ; '~: 3: '~~ ', ~,': :? ' :. . ~:. :.: :.< .. 'a, ~ ': L ?<~ ; ,~.: , ;.:~: ! ~ ;. ':.' ~;.;:: :. ; .. ( ~ .. '.':~ ",j' I n .. h' .~:)~.: ,'<};~ '';'Uoli <::~'i . ;;,.' .. ;.U'> u:':,: LEG .F· ,:;'.:: ?'>:'?J 1:/-:: 1:S~ ':·:: .... ~ , ~ '<f" ' /:.\~~ ~.!.;, PEa -'::'.? ":r.:r :.~.!. ,.y Iteration 2 LEG PEG Iteration 3 lEG lEG PEG Iteration 4 lEG Iteration 5 lEG Figure 3: Simulation of the 3-letter word attractor network, queried with DEC. Each frame shows the relative activity of attractor units at various points in processing. Activity in each frame is normalized such that the most active unit is printed in black ink; the lighter the ink color, the less active the unit. Only attractor units sharing at least one letter with DEC are shown. The selection, BEG, is a product of a gang effect. The gangs in this example are formed by words sharing two letters. The most common word beginnings are PE- (7 instances) and DI- (6); the most common word endings are -AC (10) and -ET (10); the most common first-last pairings are B-G (5) and D-G (3). One of these gangs supports B1, two support E2, and three support G3, hence BEG is selected. being stored is small. The approach is especially useful in cases where attractor locations are known, and the key focus of the network is the mutual influenre of the attractors, as in many cognitive modelling studies. References [1) Becker, s., Moscovitch, M, Belumann,M, & Joordens, S. (1997). Long-term semantic priming: A computational account and empirical evidence. TOIlTtUIl of Experimental Psychology: Learning, Memory, & Cognition, 23(5), 10591082. [2) Golden, R (1988). Probabilistic characterization ofneura1modelcomputations. In D. Z. Anderson (Ed.), Neural Infrmnation Processing Systems (pp. 310-316). American Institute of PhysiCS. [3) Hopfield, J. J. (1982). Neural networks and physical systems with emergent collective computational abilities. Proceedings of the NationlU Aaulemy of Sciences, 79,2554-2558. [4) Kay, LM., Lancaster. L.R, & Freeman W.J. (1996). Reafference and attractors in the olfactory system during odor recognition. Int J Neural Systems, 7(4),489-95. [5) Mathis, D. (1997). A computational theory of consciousness in cognition. Unpublished Doctoral Dissertation. Boulder. CO: Department of Computer Science, University of Colorado. [6) Mathis, D., & Mozer, M. C. (1996). Conscious and unconscious perception: A computational theory. In G. Cottrell (Ed.), Proceedings of the Eighteenth Annual Conference of the Cognitive Science Society (pp. 324-328). Erlbaum. [7) McC1e11and, J. L. & Rumelhart, D. E. (1981). An interactive activation model of context effects in letter perception: Part L An account of basic findings. Psychological Reuiew, 88,375-407. [8) Neal. R M & Hinton, G. E. (1998). A view of the EM algorithm that justifies incremental, sparse, and other variants. In M I. Jordan (Ed.), Learning in Graphical Models. Kluwer Academic Press. [9) McRae, K., de Sa, V. R, & SeidenbeIg, M S. (1997) On the nature and scope of featural representations of word meaning. TournaI of Experimental Psychology; General, 126(2),99-130. [10) Redish, A. D. & Touretzky, D. S. (1998). The role of the hippocampus in solving the Morris water maze. Neural Computation, 10(1), 73-111. [11) Rodrigues, N. c., & Fontanari, J. F. (1997). Multiva1ley structure of attractor neural networks. Journal of Physics A (Mathematical and General), 30, 7945-7951. [12) Samsonovich, A. & McNaughton, B. L. (1997) Path integration and cognitive mapping in a continuous attractor neural network model TournaI ofNeumscience, 17(15),5900-5920. [13) Saul, L.K., Jaakkola, T., & Jordan, ML (1996). Mean field theory for sigmoid belief networks. Tounud of AI Resetm:h, 4, 61-76. PART II NEUROSCIENCE
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An Environment Model for N onstationary Reinforcement Learning Samuel P. M. Choi Dit-Yan Yeung Nevin L. Zhang pmchoi~cs.ust.hk dyyeung~cs.ust.hk lzhang~cs.ust.hk Department of Computer Science, Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong Abstract Reinforcement learning in nonstationary environments is generally regarded as an important and yet difficult problem. This paper partially addresses the problem by formalizing a subclass of nonstationary environments. The environment model, called hidden-mode Markov decision process (HM-MDP), assumes that environmental changes are always confined to a small number of hidden modes. A mode basically indexes a Markov decision process (MDP) and evolves with time according to a Markov chain. While HM-MDP is a special case of partially observable Markov decision processes (POMDP), modeling an HM-MDP environment via the more general POMDP model unnecessarily increases the problem complexity. A variant of the Baum-Welch algorithm is developed for model learning requiring less data and time. 1 Introduction Reinforcement Learning (RL) [7] is a learning paradigm based upon the framework of Markov decision process (MDP). Traditional RL research assumes that environment dynamics (i.e., MDP parameters) are always fixed (Le., stationary). This assumption, however, is not realistic in many real-world applications. In elevator control [3], for instance, the passenger arrival and departure rates can vary significantly over one day, and should not be modeled by a fixed MDP. Nonetheless, RL in nonstationary environments is regarded as a difficult problem. In fact, it is an impossible task if there is no regularity in the ways environment dynamics change. Hence, some degree of regularity must be assumed. Typically, nonstationary environments are presummed to change slowly enough such that online RL algorithms can be employed to keep track the changes. The online approach is memory less in the sense that even if the environment ever revert to the previously learned dynamics, learning must still need to be started all over again. 988 S. P. M Choi, D.-y' Yeung and N. L. Zhang 1.1 Our Proposed Model This paper proposes a formal model [1] for the nonstationary environments that repeats their dynamics in certain ways. Our model is inspired by the observations from the real-world nonstationary tasks with the following properties: Property 1. Environmental changes are confined to a small number of modes, which are stationary environments with distinct dynamics. The environment is in exactly one of these modes at any given time. This concept of modes seems to be applicable to many real-world tasks. In an elevator control problem, for example, the system might operate in a morning-rush-hour mode, an evening-rush-hour mode and a non-rush-hour mode. One can also imagine similar modes for other control tasks, such as traffic control and dynamic channel allocation [6]. Property 2. Unlike states, modes cannot be directly observed; the current mode can only be estimated according to the past state transitions. It is analogous to the elevator control example in that the passenger arrival rate and pattern can only be inferred through the occurrence of pick-up and drop-off requests. Property 3. Mode transitions are stochastic events and are independent of the control system's responses. In the elevator control problem, for instance, the events that change the current mode of the environment could be an emergency meeting in the administrative office, or a tea break for the staff on the 10th floor. Obviously, the elevator's response has no control over the occurrence of these events. Property 4. Mode transitions are relatively infrequent. In other words, a mode is more likely to retain for some time before switching to another one. If we consider the emergency meeting example, employees on different floors take time to arrive at the administrative office, and thus would generate a similar traffic pattern (drop-off requests on the same floor) for some period of time. Property 5. The number of states is often substantially larger than the number of modes. This is a common property for many real-world applications. In the elevator example, the state space comprises all possible combinations of elevator positions, pick-up and drop-off requests, and certainly would be huge. On the other hand, the mode space could be small. For instance, an elevator control system can simply have the three modes as described above to approximate the reality. Based on these properties, an environment model is proposed by introducing a mode variable to capture environmental changes. Each mode specifies an MDP and hence completely determines the current state transition function and reward function (property 1). A mode, however, is not directly observable (property 2), and evolves with time according to a Markov process (property 3). The model is therefore called hidden-mode model. Note that our model does not impose any constraint to satisfy properties 4 and 5. In other words, the hidden-mode model can work for environments without these two properties. Nevertheless, as will be shown later, these properties can improve learning in practice. 1.2 Related Work Our hidden-mode model is related to a non stationary model proposed by Dayan and Sejnowski [4]. Although our model is more restrictive in terms of representational power, it involves much fewer parameters and is thus easier to learn. Besides, other than the number of possible modes, we do not assume any other knowledge about An Environment Model for Nonstationary Reinforcement Learning 989 the way environment dynamics change. Dayan and Sejnowski, on the other hand, assume that one knows precisely how the environment dynamics change. The hidden-mode model can also be viewed as a special case of the hidden-state model, or partially observable Markov decision process (POMDP). As will be shown later, a hidden-mode model can always be represented by a hidden-state model through state augmentation. Nevertheless, modeling a hidden-mode environment via a hidden-state model will unnecessarily increase the problem complexity. In this paper, the conversion from the former to the latter is also briefly discussed. 1.3 Our Focus There are two approaches for RL. Model-based RL first acquires an environment model and then, from which, an optimal policy is derived. Model-free RL, on the contrary, learns an optimal policy directly through its interaction with the environment. This paper is concerned with the first part of the model-based approach, i.e., how a hidden-mode model can be learned from experience. We will address the policy learning problem in a separate paper. 2 Hidden-Mode Markov Decision Processes This section presents our hidden-mode model. Basically, a hidden-mode model is defined as a finite set of MDPs that share the same state space and action space, with possibly different transition functions and reward functions. The MDPs correspond to different modes in which a system operates. States are completely observable and their transitions are governed by an MDP. In contrast, modes are not directly observable and their transitions are controlled by a Markov chain. We refer to such a process as a hidden-mode Markov decision process (HM-MDP). An example of HM-MDP is shown in Figure l(a). (a) A 3-mode, 4-state, I-action HM-MDP Time • Mode Action ... ... StaIC (b) The evolution of an HM-MDP. The arcs indicate dependencies between the variables Figure 1: An HM -MDP Formally, an HM-MDP is an 8-tuple (Q,S,A,X,Y,R,rr,'l'), where Q, S and A represent the sets of modes, states and actions respectively; the mode transition function X maps mode m to n with a fixed probability Xmn; the state transition function Y defines transition probability, Ym(8, a, s'), from state 8 to 8' given mode m and action a; the stochastic reward function R returns rewards with mean value r m (8, a); II and 'l1 denote the prior probabilities of the modes and of the states respectively. The evolution of modes and states over time is depicted in Figure 1 (b). 990 S. P. M. Choi, D.-y' Yeung and N. L. Zhang HM-MDP is a subclass of POMDP. In other words, the former can be reformulated as a special case of the latter. Specifically, one may take an ordered pair of any mode and observable state in the HM-MDP as a hidden state in the POMDP, and any observable state of the former as an observation of the latter. Suppose the observable states sand s' are in modes m and n respectively. These two HMMDP states together with their corresponding modes form two hidden states (m, s) and (n, s') for its POMDP counterpart. The transition probability from (m, s) to (n, s') is then simply the mode transition probability Xmn multiplied by the state transition probability Ym(s, a, s'). For an M-mode, N-state, K-action HM-MDP, the equivalent POMDP thus has N observations and M N hidden states. Since most state transition probabilities are collapsed into mode transition probabilities through parameter sharing, the number of parameters in an HM-MDP (N2 M K + M2) is much less than that of its corresponding POMDP (M2 N 2 K). 3 Learning a Hidden-Mode Model There are now two ways to learn a hidden-mode model. One may learn either an HM-MDP, or an equivalent POMDP instead. POMDP models can be learned via a variant of the Baum-Welch algorithm [2]. This POMDP Baum-Welch algorithm requires 8(M2 N 2T) time and 8(M2 N 2 K) storage for learning an M-mode, Nstate, K-action HM-MDP, given T data items. A similar idea can be applied to the learning of an HM-MDP. Intuitively, one can estimate the model parameters based on the expected counts of the mode transitions, computed by a set of auxiliary variables. The major difference from the original algorithm is that consecutive state transitions, rather than the observations, are considered. Additional effort is thus needed for handling the boundary cases. This HM-MDP Baum-Welch algorithm is described in Figure 2. 4 Empirical Studies This section empirically examines the POMDP Baum-Welch1 and HM-MDP BaumWelch algorithms. Experiments based on various randomly generated models and some real-world environments were conducted. The results are quite consistent. For illustration, a simple traffic control problem is presented. In this problem, one direction of a two-way traffic is blocked, and cars from two different directions (left and right) are forced to share the remaining road. To coordinate the traffic, two traffic lights equipped with sensors are set. The system then has two possible actions: either to signal cars from the left or cars from the right to pass. For simpliCity, we assume discrete time steps and uniform speed of the cars. The system has 8 possible states; they correspond to the combinations of whether there are cars waiting on the left and the right directions, and the stop signal position in the previous time step. There are 3 traffic modes. The first one has cars waiting on the left and the right directions with probabilities 0.3 and 0.1 respectively. In the second mode, these probabilities are reversed. For the last one, both probabilities are 0.3. In addition, the mode transition probability is 0.1. A cost of -1.0 results if lChrisman's algorithm also attempts to learn a minimal possible number of states. Our paper concerns only with learning the model parameters. An Environment Model for Nonstationary Reinforcement Learning Given a collection of data and an initial model parameter vector 0. repeat 0=0 Compute forward variables (Xt. (Xl (i) = 1/;$1 (X2(i) = 1I"i 1/;$1 Yi(SI, al,S2) (Xt+l(j) = L:iEQ (Xt(i) Xii Yi(St,at,St+l) Compute backward variables (3t . (3T(i) = 1 (3t(i) = LiEQXii Yi(St,at,St+I) (3t+I(j) (31(i) = L:iEQ 1I"j Yi(sl ,al,s2) (32(j) Compute the new model parameter 0. _ .. _ L;-2 {. (i,i) Xl] ~T . L....t=1 "'Yt (I) 1Ti = "Yl (i) until maxi I Oi OJ I < to "Ii E Q "Ii E Q "Ii E Q "Ii E Q "Ii E Q "Ii E Q "I i,j E Q "Ii E Q 8(a, b) = {01 a = b af.b Figure 2: HM-MDP Baum-Welch Algorithm a car waits on either side. 991 The experiments were run with the same initial model for data sets of various sizes. The algorithms iterated until the maximum change of the model parameters was less than a threshold of 0.0001. The experiment was repeated for 20 times with different random seeds in order to compute the median. Then the learned models were compared in their POMDP forms using the Kullback-Leibler (KL) distance [5], and the total CPU running time on a SUN Ultra I workstation was measured. Figure 3 (a) and (b) report the results. Generally speaking, both algorithms learn a more accurate environment model as the data size increases (Figure 3 (a)). This result is expected as both algorithms are statistically-based, and hence their performance relies largely on the data size. When the training data size is very small, both algorithms perform poorly. However, as the data size increases, HM-MDP Baum-Welch improves substantially faster than POMDP Baum-Welch. It is because an HM-MDP in general consists of fewer free 992 " o ~~~~~~~~~~~~~~ o !SOO 1000 1501) 2000 2&00 )000 :1500 oKIOO .&500 5000 Wndow9tD (a) Error in transition function s. P M Choi. D.-y' Yeung and N. L. Zhang '0000 ............ ...... -.----.".~ .. /---.-.-.. -.... -----.-.. ~ ... --.... --... 10500L--' OOOJ...--'..."500-2000~~ .... ,,---:"_':::"--=_.,.,.......-:-""':':-:--, .... ...,,...,---:-!,OOO WIndowSiz. (b) Required learning time Figure 3: Empirical results on model learning parameters than its POMDP counterpart. HM-MDP Baum-Welch also runs much faster than POMDP Baum-Welch (Figure 3 (b)). It holds in general for the same reason discussed above. Note that computational time is not necessarily monotonically increasing with the data size. It is because the total computation depends not only on the data size, but also on the number of iterations executed. From our experiments, we noticed that the number of iterations tends to decrease as the data size increases. Larger models have also been tested. While HM-MDP Baum-Welch is able to learn models with several hundred states and a few modes, POMDP Baum-Welch was unable to complete the learning in a reasonable time. Additional experimental results can be found in [1]. 5 Discussions and Future Work The usefulness of a model depends on the validity of the assumptions made. We now discuss the assumptions of HM-MDP, and shed some light on its applicability to real-world nonstationary tasks. Some possible extensions are also discussed. Modeling a nonstationary environment as a number of distinct MDPs. MDP is a flexible framework that has been widely adopted in various applications. Modeling nonstationary environments by distinct MDPs is a natural extension to those tasks. Comparing to POMDP, our model is more comprehensive: each MDP naturally describes a mode of the environment. Moreover, this formulation facilitates the incorporation of prior knowledge into the model initialization step. States are directly observable while modes are not. While completely observable states are helpful to infer the current mode, it is also possible to extend the model to allow partially observable states. In this case, the extended model would be equivalent in representational power to a POMDP. This could be proved easily by showing the reformulation of the two models in both directions. An Environment Model for Nonstationary Reinforcement Learning 993 Mode changes are independent of the agent's responses. This property may not always hold for all real-world tasks. In some applications, the agent's actions might affect the state as well as the environment mode. In that case, an MDP should be used to govern the mode transition process. Mode transitions are relatively infrequent. This is a property that generally holds in many applications. Our model, however, is not limited by this condition. We have tried to apply our model-learning algorithms to problems in which this property does not hold. We find that our model still outperforms POMDP, although the required data size is typically larger for both models. Number of states is substantially larger than the number of modes. This is the key property that significantly reduces the number of parameters in HM-MDP compared to that in POMDP. In practice, introduction of a few modes is sufficient for boosting the system performance. More modes might only help little. Thus a trade-off between performance and response time must be decided. There are additional issues that need to be addressed. First, an efficient algorithm for policy learning is required. Although in principle it can be achieved indirectly via any POMDP algorithm, a more efficient algorithm based on the model-based approach is possible. We will address this issue in a separate paper. Next, the number of modes is currently assumed to be known. We are now investigating how to remove this limitation. Finally, the exploration-exploitation issue is currently ignored. In our future work, we will address this important issue and apply our model to real-world nonstationary tasks. References [1] S. P. M. Choi, D. Y. Yeung, and N. L. Zhang. Hidden-mode Markov decision processes. In IJCAI 99 Workshop on Neural, Symbolic, and Reinforcement Methods for Sequence Learnin9, 1999. [2] L. Chrisman. Reinforcement learning with perceptual aliasing: The perceptual distinctions approach. In AAAI-92, 1992. [3] R. H. Crites and A. G. Barto. Improving elevator performance using reinforcement learning. In D. Touretzky, M. Mozer, and M. Hasselmo, editors, Advances in Neural Information Processing Systems 8, 1996. [4] P. Dayan and T. J. Sejnowski. Exploration bonuses and dual control. Machine Learning, 25(1):5- 22, Oct. 1996. [5J S. Kullback. Information Theory and Statistics. Wiley, New York, NY, USA, 1959. [6] S. Singh and D. P. Bertsekas. Reinforcement learning for dynamic channel allocation in cellular telephone systems. In Advances in Neural Information Processing Systems 9, 1997. [7] R. S. Sutton and A. G. Barto. Reinforcement Learning: An Introduction. The MIT Press, 1998.
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Learning Factored Representations for Partially Observable Markov Decision Processes Department of Computer Science University of Toronto Toronto M5S 2Z9 Canada Brian Sallans Gatsby Computational Neuroscience Unit* University College London London WCIN 3AR U.K. sallans@cs.toronto.edu Abstract The problem of reinforcement learning in a non-Markov environment is explored using a dynamic Bayesian network, where conditional independence assumptions between random variables are compactly represented by network parameters. The parameters are learned on-line, and approximations are used to perform inference and to compute the optimal value function. The relative effects of inference and value function approximations on the quality of the final policy are investigated, by learning to solve a moderately difficult driving task. The two value function approximations, linear and quadratic, were found to perform similarly, but the quadratic model was more sensitive to initialization. Both performed below the level of human performance on the task. The dynamic Bayesian network performed comparably to a model using a localist hidden state representation, while requiring exponentially fewer parameters. 1 Introduction Reinforcement learning (RL) addresses the problem of learning to act so as to maximize a reward signal provided by the environment. Online RL algorithms try to find a policy which maximizes the expected time-discounted reward. They do this through experience by performing sample backups to learn a value function over states or state-action pairs. If the decision problem is Markov in the observable states, then the optimal value function over state-action pairs yields all of the information required to find the optimal policy for the decision problem. When complete know ledge of the environment is not available, states which are different may look the same; this uncertainty is called perceptual aliasing [1], and causes decision problems to have dynamics which are non-Markov in the perceived state . • Correspondence address Learning Factored Representations for POMDPs 1051 1.1 Partially observable Markov decision processes Many interesting decision problems are not Markov in the inputs. A partially observable Markov decision process (POMDP) is a formalism in which it is assumed that a process is Markov, but with respect to some unobserved (i.e. "hidden") random variable. The state of the variable at time t, denoted st, is dependent only on the state at the previous time step and on the action performed. The currently-observed evidence is assumed to be independent of previous states and observations given the current state. The state of the hidden variable is not known with certainty, so a belief state is maintained instead. At each time step, the beliefs are updated by using Bayes' theorem to combine the belief state at the previous time step (passed through a model of the system dynamics) with newly observed evidence. In the case of discrete time and finite discrete state and actions, a POMDP is typically represented by conditional probability tables (CPTs) specifying emission probabilities for each state, and transition probabilities and expected rewards for states and actions. This corresponds to a hidden Markov model (HMM) with a distinct transition matrix for each action. The hidden state is represented by a single random variable that can take on one of K values. Exact belief updates can be computed using Bayes' rule. The value function is not over the discrete state, but over the real-valued belief state. It has been shown that the value function is piecewise linear and convex [2]. In the worst case, the number of linear pieces grows exponentially with the problem horizon, making exact computation of the optimal value function intractable. Notice that the localist representation, in which the state is encoded in a single random variable, is exponentially inefficient: Encoding n bits of information about the state of the process requires 2n possible hidden states. This does not bode well for the abilities of models which use this representation to scale up to problems with high-dimensional inputs and complex non-Markov structure. 1.2 Factored representations A Bayesian network can compactly represent the state of the system in a set of random variables [3]. A two time-slice dynamic Bayesian network (DBN) represents the system at two time steps [4]. The conditional dependencies between random variables from time t to time t + 1, and within time step t, are represented by edges in a directed acyclic graph. The conditional probabilities can be stored explicitly, or parameterized by weights on edges in the graph. If the network is densely-connected then inference is intractable [5]. Approximate inference methods include Markov chain Monte Carlo [6], variational methods [7], and belief state simplification [8]. In applying a DBN to a large problem there are three distinct issues to disentangle: How well does a parameterized DBN capture the underlying POMDP; how much is the DBN hurt by approximate inference; and how good must the approximation of the value function be to achieve reasonable performance? We try to tease these issues apart by looking at the performance of a DBN on a problem with a moderately large state-space and non-Markov structure. 2 The algorithm We use a fully-connected dynamic sigmoid belief network (DSBN) [9], with K units at each time slice (see figure 1). The random variables Si are binary, and conditional proba1052 B. Sa/lans Figure 1: Architecture of the dynamic sigmoid belief network. Circles indicate random variables, where a filled circle is observed and an empty circle is unobserved. Squares are action nodes, and diamonds are rewards. bilities relating variables at adjacent time-steps are encoded in action-specific weights: P(s~+1 = II{sD~=l,at) = a (twi~st) (1) k=l where wi~ is the weight from the ith unit at time step t to the kth unit at time step t + 1, assuming action at is taken at time t. The nonlinearity is the usual sigmoid function: a(x) = 1/1 +exp{ -x}. Note that a bias can be incorporated into the weights by clamping one of the binary units to 1. The observed variables are assumed to be discrete; the conditional distribution of an output given the hidden state is multinomial and parameterized by output weights. The probability of observing an output with value t is given by: K exp {L:~1 Uklst} P(ot = ll{sDk 1) = { } = 101 K t L:m=l exp L:k=l UkmSk (2) where ot E 0 and Ukl denotes the output weight from hidden unit k to output value t. 2.1 Approximate inference Inference in the fully-connected Bayesian network is intractable. Instead we use a variational method with a fully-factored approximating distribution: K P(stlst-1,at- 1,ot) ~ Pst ~ II,u~t(I-,uk)l-S~ (3) k==l where the ,uk are variational parameters to be optimized. This is the standard mean-field approximation for a sigmoid belief network [10]. The parameters,u are optimized by iterating the mean-field equations, and converge in a few iterations. The values of the variational parameters at time t are held fixed while computing the values for step t + 1. This is analogous to running only the forward portion of the HMM forward-backward algorithm [11]. The parameters of the DSBN are optimized online using stochastic gradient ascent in the log-likelihood: U f--(4) Learning Factored Representations for POMDPs 1053 where Wand U are the transition and emission matrices respectively, aw and au are learning rates, the vector J-L contains the fully-factored approximate belief state, and 1/ is a vector of zeros with a one in the otth place. The notation [·]k denotes the kth element of a vector (or kth column of a matrix). 2.2 Approximating the value function Computing the optimal value function is also intractable. If a factored state-space representation is appropriate, it is natural (if extreme) to assume that the state-action value function can be decomposed in the same way: K Q(Pst, at) ~ L Qk (J-Lt, at) t::. Q F(J-L, at) (5) k=l This simplifying assumption is still not enough to make finding the optimal value function tractable. Even if the states were completely independent, each Q k would still be piecewise-linear and convex, with the number of pieces scaling exponentially with the horizon. We test two approximate value functions, a linear approximation: and a quadratic approximation: K L qk,a' J-Lk = [Q]at T . J-L k=l K L ¢k,at J-Lk + qk,at J-Lk + bat k=l [~]at T . (J-L 0 J-L) + [Q]at T . J-L + [blat (6) (7) Where ~, Q and b are parameters of the approximations. The notation [·]i denotes the ith column of a matrix, [.]T denotes matrix transpose and 0 denotes element-wise vector multiplication. We update each term of the factored approximation with a modified Q-Iearning rule [12], which corresponds to a delta-rule where the target for input J-L is rt + 'Y maxa Q F (J-LH I , a): ¢k,at t¢k,at + a J-Lk EB qk,at tqk,at + a J-Lk EB bat tbat + a EB (8) Here a is a learning rate, 'Y is the temporal discount factor, and EB is the Bellman residual: (9) 3 Experimental results The "New York Driving" task [13] involves navigating through slower and faster one-way traffic on a multi-lane highway. The speed of the agent is fixed, and it must change lanes to avoid slower cars and move out of the way of faster cars. If the agent remains in front of a faster car, the driver of the fast car will honk its horn, resulting in a reward of -1.0. Instead of colliding with a slower car, the agent can squeeze past in the same lane, resulting in a reward of -10.0. A time step with no horns or lane-squeezes constitutes clear progress, and is rewarded with +0.1. See [13] for a detailed description of this task. 1054 B. Sal/ans Table 1: Sensory input for the New York driving task Dimension I Size I Values Hear horn 2 yes,no Gaze object 3 truck, shoulder, road Gaze speed 2 looming, receding Gaze distance 3 far, near, nose Gaze refined distance 2 far-half, near-half Gaze colour 6 red, blue, yellow, white, gray, tan A modified version of the New York Driving task was used to test our algorithm. The task was essentially the same as described in [13], except that the "gaze side" and "gaze direction" inputs were removed. See table 1 for a list of the modified sensory inputs. The performance of a number of algorithms and approximations were measured on the task: a random policy; Q-Iearning on the sensory inputs; a model with a localist representation (i.e. the hidden state consisted of a single multinomial random variable) with linear and quadratic approximate value functions; the DSBN with mean-field inference and linear and quadratic approximations; and a human driver. The localist representation used the linear Q-Iearning approximation of [14], and the corresponding quadratic approximation. The quadratic approximations were trained both from random initialization, and from initialization with the corresponding learned linear models (and random quadratic portion). The non-human algorithms were each trained for 100000 iterations, and in each case a constant learning rate of 0.01 and temporal decay rate of 0.9 were used. The human driver (the author) was trained for 1000 iterations using a simple character-based graphical display, with each iteration lasting 0.5 seconds. Stochastic policies were used for all RL algorithms, with actions being chosen from a Boltzmann distribution with temperature decreasing over time: (10) The DSBN had 4 hidden units per time slice, and the localist model used a multinomial with 16 states. The Q-Iearner had a table representation with 2160 entries. After training, each non-human algorithm was tested for 20 trials of 5000 time steps each. The human was tested for 2000 time steps, and the results were renormalized for comparison with the other methods. The results are shown in figure 2. All results were negative, so lower numbers indicate better performance in the graph. The error bars show one standard deviation across the 20 trials. There was little performance difference between the localist representation and the DSBN but, as expected, the DSBN was exponentially more efficient in its hidden-state representation. The linear and quadratic approximations performed comparably, but well below human performance. However, the DSBN with quadratic approximation was more sensitive to initialization. When initialized with random parameter settings, it failed to find a good policy. However, it did converge to a reasonable policy when the linear portion of the quadratic model was initialized with a previously learned linear model. The hidden units in the DSBN encode useful features of the input, such as whether a car was at the "near" or "nose" position. They also encode some history, such as current gaze direction. This has advantages over a simple stochastic policy learned via Q-Iearning: If the Q-Iearner knows that there is an oncoming car, it can randomly select to look left or right. The DSBN systematically looks to the left, and then to the right, wasting fewer actions. Learning Factored Representations for POMDPs 4000 3500 3000 "1:l 2500 ~ ~2000 ~ I 1500 1000 4 Discussion 1055 Figure 2: Results on the New York Driving task for nine algorithms: R=random; Q=Q-Ieaming; LC=linear multinomial; QCR=quadratic multinomial, random init.; QCL=quadratic multinomial, linear init; LD=linear DSBN; QDR=quadratic DSBN, random init.; QDL=quadratic DSBN, linear init.; H=human The DSBN performed better than a standard Q-learner, and comparably to a model with a localist representation, despite using approximate inference and exponentially fewer parameters. This is encouraging, since an efficient encoding of the state is a prerequisite for tackling larger decision problems. Less encouraging was the value-function approximation: When compared to human performance, it is clear that all methods are far from optimal, although again the factored approximation of the DSBN did not hurt performance relative to the localist multinomial representation. The sensitivity to initialization of the quadratic approximation is worrisome, but the success of initializing from a simpler model suggests that staged learning may be appropriate, where simple models are learned and used to initialize more complex models. These findings echo those of [14] in the context of learning a non-factored approximate value function. There are a number of related works, both in the fields of reinforcement learning and Bayesian networks. We use the sigmoid belief network mean-field approximation given in [10], and discussed in the context of time-series models (the "fully factored" approximation) in [15]. Approximate inference in dynamic Bayesian networks has been discussed in [15] and [8]. The additive factored value function was used in the context of factored MDPs (with no hidden state) in [16], and the linear Q-learning approximation was given in [14]. Approximate inference was combined with more sophisticated value function approximation in [17]. To our knowledge, this is the first attempt to explore the practicality of combining all of these techniques in order to solve a single problem. There are several possible extensions. As described above, the representation learned by the DSBN is not tuned to the task at hand. The reinforcement information could be used to guide the learning of the DSBN parameters[18, 13]. Also, if this were done, then the reinforcement signals would provide additional evidence as to what state the POMDP is in, and could be used to aid inference. More sophisticated function approximation could be used [17]. Finally, although this method appears to work in practice, there is no guarantee that the reinforcement learning will converge. We view this work as an encouraging first step, with much further study required. 5 Conclusions We have shown that a dynamic Bayesian network can be used to construct a compact representation useful for solving a decision problem with hidden state. The parameters of the DBN can be learned from experience. Learning occurs despite the use of simple value1056 B. Sallans function approximations and mean-field inference. Approximations of the value function result in good performance, but are clearly far from optimal. The fully-factored assumptions made for the belief state and the value function do not appear to impact performance, as compared to the non-factored model. The algorithm as presented runs entirely on-line by performing "forward" inference only. There is much room for future work, including improving the utility of the factored representation learned, and the quality of approximate inference and the value function approximation. Acknowledgments We thank Geoffrey Hinton, Zoubin Ghahramani and Andy Brown for helpful discussions, the anonymous referees for valuable comments and criticism, and particularly Peter Dayan for helpful discussions and comments on an early draft of this paper. This research was funded by NSERC Canada and the Gatsby Charitable Foundation. References [1] S.D. Whitehead and D.H. Ballard. Learning to perceive and act by trial and error. Machine Learning, 7, 1991. [2] EJ. Sondik. The optimal control of partially observable Markov processes over the infinite horizon: Discounted costs. Operations Research, 26:282-304, 1973. [3] J. Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo, CA, 1988. [4] T. Dean and K. Kanazawa. A model for reasoning about persistence and causation. Computationallntelligence, 5, 1989. [5] Gregory F. Cooper. The computational complexity of probabilistic inference using Bayesian belief networks. Anijiciallntelligence, 42:393-405, 1990. [6] R. M. Neal. Probabilistic inference using Markov chain Monte Carlo methods. Technical Report CRG-TR-93-1, Department of Computer Science, University of Toronto, 1993. [7] M.I. Jordan, Z. Ghahramani, T.S. Jaakkola, and L.K. Saul. An introduction to variational methods for graphical models. Machine Learning, 1999. in press. [8] X. Boyen and D. Koller. Tractable inference for complex stochastic processes. In Proc. UA1'98, 1998. [9] R. M. Neal. Connectionist learning of belief networks. Artijiciallntelligence, 56:71-113, 1992. [10] L. K. Saul, T. Jaakkola, and M. I. Jordan. Mean field theory for sigmoid belief networks. Journal of Artijiciallntelligence Research, 4:61-76, 1996. [11] Lawrence R. Rabiner and Biing-Hwang Juang. An introduction to hidden Markov models. IEEE ASSAP Magazine, 3:4-16, January 1986. [12] CJ.C.H. Watkins and P. Dayan. Q-Iearning. Machine Learning, 8:279-292, 1992. [13] A.K. McCallum. Reinforcement learning with selective perception and hidden state. Dept. of Computer Science, Universiy of Rochester, Rochester NY, 1995. Ph.D. thesis. [14] M.L. Littman, A.R. Cassandra, and L.P. Kaelbling. Learning policies for partially observable environments: Scaling up. In Proc. International Conference on Machine Learning, 1995. [15] Z. Ghahramani and M. I. Jordan. Factorial hidden Markov models. Machine Learning, 1997. [16] D. Koller and R. Parr. Computing factored value functions for policies in structured MDPs. In Proc. lJCA/'99, 1999. [17] A. Rodriguez, R. Parr, and D. Koller. Reinforcement learning using approximate belief states. In S. A. Solla, T. K. Leen, and K.-R. Mtiller, editors, Advances in Neural Information Processing Systems, volume 12. The MIT Press, Cambridge, 2000. [18] L. Chrisman. Reinforcement learning with perceptual aliasing: The perceptual distinctions approach. In Tenth National Conference on AI, 1992.
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Predictive Approaches For Choosing Hyperparameters in Gaussian Processes S. Sathiya Keerthi Mechanical and Production Engg. National University of Singapore S. Sundararajan Computer Science and Automation Indian Institute of Science Bangalore 560 012, India sundar@csa.iisc. ernet. in 10 Kentridge Crescent, Singapore 119260 mpessk@guppy. mpe. nus. edu. sg Abstract Gaussian Processes are powerful regression models specified by parametrized mean and covariance functions. Standard approaches to estimate these parameters (known by the name Hyperparameters) are Maximum Likelihood (ML) and Maximum APosterior (MAP) approaches. In this paper, we propose and investigate predictive approaches, namely, maximization of Geisser's Surrogate Predictive Probability (GPP) and minimization of mean square error with respect to GPP (referred to as Geisser's Predictive mean square Error (GPE)) to estimate the hyperparameters. We also derive results for the standard Cross-Validation (CV) error and make a comparison. These approaches are tested on a number of problems and experimental results show that these approaches are strongly competitive to existing approaches. 1 Introduction Gaussian Processes (GPs) are powerful regression models that have gained popularity recently, though they have appeared in different forms in the literature for years. They can be used for classification also; see MacKay (1997), Rasmussen (1996) and Williams and Rasmussen (1996). Here, we restrict ourselves to regression problems. Neal (1996) showed that a large class of neural network models converge to a Gaussian Process prior over functions in the limit of an infinite number of hidden units. Although GPs can be created using infinite networks, often GPs are specified directly using parametric forms for the mean and covariance functions (Williams and Rasmussen (1996)). We assume that the process is zero mean. Let ZN = {XN,yN} whereXN = {xCi): i = 1, ... ,N}andYN = {y(i): i = 1, ... ,N}. Here,y(i) represents the output corresponding to the input vector xCi). Then, the Gaussian prior over the functions is given by (1) where eN is the covariance matrix with (i,j)th element [CN]i,j C(x(i),x(j);8) and C(.; 8) denotes the parametrized covariance function. Now, assuming that the 632 S. Sundararajan and S. S. Keerthi observed output tN is modeled as tN = YN + eN and eN is zero mean multivariate Gaussian with covariance matrix 0'2IN and is independent of YN, we get (t IX 9) = exp(-t~Ci\/tN) p N N, (27r)~ICNli (2) where eN = eN + 0'2IN. Therefore, [eN kj = [eN kj + 0'2 bi,j, where bi,j = 1 when i = j and zero otherwise. Note that 9 = (9,0'2) is the new set of hyperparameters. Then, the predictive distribution ofthe output yeN + 1) for a test case x(N + 1) is also Gaussian with mean and variance (3) and O';(N+1) = bN+1 k~+1 C;/kN +1 (4) where bN+1 C(x(N + 1), x(N + 1); 9) and kN+l is an N x 1 vector with ith element given by C(x(N + 1),x(i); 9). Now, we need to specify the covariance function C(.; 9). Williams and Rasmussen (1996) found the following covariance function to work well in practice. M M C(x(i), x(j); 9) = ao + al L xp(i)xp(j) + voexp( - ~ L Wp (xp(i) - Xp(j))2) (5) p=1 p=1 where xp(i) is the pth component of ith input vector xCi). The wp are the Automatic Relevance Determination (ARD) parameters. Note that C(x(i), x(j); 9) = C(x(i), x(j); 9) + 0'2bi ,j' Also, all the parameters are positive and it is convenient to use logarithmic scale. Hence, 9 is given by log (ao, aI, vo, WI, ... ,W M, 0'2). Then, the question is: how do we handle 9? More sophisticated techniques like Hybrid Monte Carlo (HMC) methods (Rasmussen (1996) and Neal (1997)) are available which can numerically integrate over the hyperparameters to make predictions. Alternately, we can estimate 9 from the training data. We restrict to the latter approach here. In the classical approach, 9 is assumed to be deterministic but unknown and the estimate is found by maximizing the likelihood (2). That is, 9ML = argijaz p(tNIXN' 9). In the Bayesian approach, 9 is assumed to be random and a prior p( 9) is specified. Then, the MAP estimate 9MP is obtained as 9MP = argijaz p(tNIXN,9)p(9) with the motivation that the the predictive distribution p(y(N + 1)lx(N + 1), ZN) can be approximated as p(y(N + 1)lx(N + 1),ZN,9MP)' With this background, in this paper we propose and investigate different predictive approaches to estimate the hyperparameters from the training data. 2 Predictive approaches for choosing hyperparameters Geisser (1975) proposed Predictive Sample Reuse (PSR) methodology that can be applied for both model selection and parameter estimation problems. The basic idea is to define a partition scheme peN, n, r) such that pJJ~n = (ZX; -n; Z~) is ith partition belonging to a set r of partitions with Z}V -n' Z~ representing the N - n retained and n omitted data sets respectively. Then, the unknown 9 is estimated (or a model M j is chosen among a set of models indexed by j = 1, ... , J) by means of optimizing a predictive measure that measures the predictive performance on the omitted observations X~ by using the retained observations ZX;_n averaged over the partitions (i E r). In the special case of n = 1, we have the leave one out strategy. Note that this approach was independently presented in the Predictive Approaches for Choosing Hyperparameters in Gaussian Processes 633 name of cross-validation (CV) by Stone (1974). The well known examples are the standard CV error and negative of average predictive likelihood. Geisser and Eddy (1979) proposed to maximize n~l p(t(i)lx(i), Z<;}, Mj ) (known as Geisser's surrogate Predictive Probability (GPP)) by synthesizing Bayesian and PSR methodology in the context of (parametrized) model selection. Here, we propose to maximize n~l p(t(i)lx(i), Z<;}, 0) to estimate 0, where Z<;} is obtained from ZN by removing the ith sample. Note that p(t(i)lx(i), Zr;) ,0) is nothing but the predictive distribution p(y(i)lx(i), Zr;), 0) evaluated at y(i) = t(i). Also, we introduce the notion of Geisser's Predictive mean square Error (GPE) defined as ~ 2:~1 E((y(i) - t(i))2) (where the expectation operation is defined with respect to p(y(i)lx(i), Zr;), 0)) and propose to estimate 0 by minimizing GPE. 2.1 Expressions for GPP and its gradient The objective function corresponding to GPP is given by 1 N . G(O) - N L log(P(t(i)lx(i), Z~, 0) i=l (6) From (3) and (4) we get G(O) = ~ ;... (t(i) - y(i))2 N ~ 20'2 . i=l 11(~) 1 N 1 + 2N L log O';(i) + '2 log 27l' i=l (7) where y(i) = [c~i)JT[C~)J-lt~ and O';(i) = Cii [c~i)JT[C~)-lC~i). Here, C~ is an N - 1 x N - 1 matrix obtained from C N by removing the ith column and ith row. Similarly, t<;} and c~i) are obtained from tN and Ci (Le., ith column of CN) respectively by removing the ith element. Then, G(O) and its gradient can be computed efficiently using the following result. Theorem 1 The objective junction G (0) under the Gaussian Process model is given by 1 N q'fv(i) 1 N _ 1 G(O) = 2N tt Cii 2N ~ logcii + "210g27l' (8) where Cii denotes the ith diagonal entry of C-r/ and qN (i) denotes the ith element of qN = C~;ItN' Its gradient is given by 8G(O) = _1 t (1 + q~(i)) (Si,i) + ~ t qN(i)(r~(i)) (9) 80J· 2N . Cii Cii N . 1 Cii ~=l a= h -T8CNC-18CNC-1t d C-1t H were Bj,i Ci 80; Ci, rj N 80; N N an qN N N· ere, Ci denotes the ith column of the matrix c~;I . Thus, using (8) and (9) we can compute the GPP and its gradient. We will give meaningful interpretation to the different terms shortly. 2.2 Expressions for CV function and its gradient We define the CV function as H(O) N ~ L (t(i) - y(i))2 i=l (10) 634 S. Sundararajan and S. S. Keerthi where y(i) is the mean of the conditional predictive distribution as given above. Now, using the following result we can compute R((}) efficiently. Theorem 2 The CV function R ((}) under the Gaussian model is given by R((}) = ~ ~ (q~(i))2 N ~ C;. i=l ' and its gradient is given by where Sj,i,rj,qN(i) and Cii are as defined in theorem 1. 2.3 Expressions for GPE and its gradient The G PE function is defined as (11) N GE((}) = ~ L / (t(i) - y(i))2 p(y(i)lx(i), Z~, (}) dy(i) (13) i=l which can be readily simplified to N 1 N GE((}) = ~ L (t(i) - y(i))2 + N L a~(i) i=l i=l (14) On comparing (14) with (10), we see that while CV error minimizes the deviation from the predictive mean, GPE takes predictive variance also into account. Now, the gradient can be written as (15) 8C- 1 8C- 1 where we have used the results a~(i) = C!i' ~Oiji = e[ 8(}~ ei and 8/; = -Cj\/ 88~N CNI . Here ei denotes the ith column vector of the identity matrix IN. J 2.4 Interpretations More insight can be obtained from reparametrizing the covariance function as follows. M 1M C(x(i), x(j); (}) = a2 (ao+ih L xp(i)xp(j)+voexp( - 2 L wp(xp(i)-xPU))2)+Oi,j) p=I p=l (16) where ao = a2 £la, al = a2 aI, Va = a2 Va. Let us define P(x(i), xU); (}) = ~C(x(i), xU); (}). Then PNI = a2 CNI . Therefore, Ci,j = ~ where Ci,j, Pi,j denote the (i, j)th element of the matrices CNI and PNI respectively. From theorem 2 (see (10) and (11)) we have t(i) - y(i) = q~i~i) = c~~iN. Then, we can rewrite (8) as _ _ 1_ N iJ'jy (i) 1 N 1 G ((}) = " - 2N " lOgPii + -2log2rra2 (17) 2Na2 ~ p.. ~ i=l n i=l Predictive Approaches for Choosing Hyperparameters in Gaussian Processes 635 Here, iiN = Pj\hN and, Pi, Pii denote, respectively, the ith column and ith diagonal entry of the matrix Pj\/. Now, by setting the derivative of (17) with respect to a2 to zero, we can infer the noise level as (18) Similarly, the CV error (10) can be rewritten as H(9) = ~ t ii~~.i) i=l Pu (19) Note that H(9) is dependent only on the ratio of the hyperparameters (Le., on ao, aI, vo) apart from the ARD parameters. Therefore, we cannot infer the noise level uniquely. However, we can estimate the ARD parameters and the ratios ao, aI, vo. Once we have estimated these parameters, then we can use (18) to estimate the noise level. Next, we note that the noise level preferred by the GPE criterion is zero. To see this, first let us rewrite (14) under reparametrization as GE (9) = ~ t q~;i) + a2 t ~ N i=l Pii N i=l Pii (20) Since iiN(i) and Pii are independent of a2 , it follows that the GPE prefers zero as the noise level, which is not true. Therefore, this approach can be applied when, either the noise level is known or a good estimate of it is available. 3 Simulation results We carried out simulation on four data sets. We considered MacKay's robot arm problem and its modified version introduced by Neal (1996). We used the same data set as MacKay (2-inputs and 2-outputs), with 200 examples in the training set and 200 in the test set. This data set is referred to as 'data set l' in Table 1. Next, to evaluate the ability of the predictive approaches in estimating the ARD parameters, we carried out simulation on the robot arm data with 6 inputs (Neal's version), denoted as 'data set 2' in Table 1. This data set was generated by adding four further inputs, two of which were copies of the two inputs corrupted by additive zero mean Gaussian noise of standard deviation 0.02 and two further irrelevant Gaussian noise inputs with zero mean and unit variance (Williams and Rasmussen (1996)). The performance measures chosen were average of Test Set Error (normalized by true noise level of 0.0025) and average of negative logarithm of predictive probability (NLPP) (computed from Gaussian density function with (3) and (4)). Friedman's [1 J data sets 1 and 2 were based on the problem of predicting impedance and phase respectively from four parameters of an electrical circuit. Training sets of three different sizes (50, 100, 200) and with a signal-to-noise ratio of about 3:1 were replicated 100 times and for each training set (at each sample f (y(x) -1i(x))2dx size N), scaled integral squared error (ISE = D varD y(x) ) and NLPP were computed using 5000 data points randomly generated from a uniform distribution over D (Friedman (1991)). In the case of GPE (denoted as GE in the tables), we used the noise level estimate generated from Gaussian distribution with mean N LT (true noise level) and standard deviation 0.03 N LT. In the case of CV, we estimated the hyperparameters in the reparametrized form and estimated the noise level using (18). In the case of MAP (denoted as MP in the tables), we used the same prior 636 S. Sundararajan and S. S. Keerthi Table 1: Results on robot arm data sets. Average of normalized test set error (TSE) and negative logarithm of predictive probability (NLPP) for various methods. Data Set: 1 Data Set: 2 TSE NLPP TSE NLPP ML 1.126 -1.512 1.131 -1.512 MP 1.131 -1.511 1.181 -1.489 Gp 1.115 -1.524 1.116 -1.516 CV 1.112 -1.518 1.146 -1.514 GE 1.111 -1.524 1.112 -1.524 Table 2: Results on Friedman's data sets. Average of scaled integral squared error and negative logarithm of predictive probability (given in brackets) for different training sample sizes and various methods. DataSet: 1 Data Set: 2 N = 50 N = 100 N = 200 N = 50 N = 100 N = 200 ML 0.43 7.24 0.19 6.71 0.10 6.49 0.26 1.05 0.16 0.82) 0.11 0.68) MP 0.42 7.18 0.22 6.78 0.12 6.56 0.25 1.01 0.16 0.82) 0.11 0.69) Gp 0.47 7.29 0.20 6.65 0.10 6.44 0.33 1.25 0.20 0.86) 0.12 0.70) cV 0.55 7.27 0.22 6.67 0.10 6.44 0.42 1.36 0.21 0.91) 0.13 0.70) GE 0.35 7.10 0.15 6.60 0.08 6.37 0.28 1.20 0.18 0.85) 0.12 0.63) given in Rasmussen (1996). The GPP approach is denoted as Gp in the tables. For all these methods, conjugate gradient (CG) algorithm (Rasmussen (1996)) was used to optimize the hyperparameters. The termination criterion (relative function error) with a tolerance of 10-7 was used, but with a constraint on the maximum number of CG iterations set to 100. In the case of robot arm data sets, the algorithm was run with ten different initial conditions and the best solution (chosen from respective best objective function value) is reported. The optimization was carried out separately for the two outputs and the results reported are the average TSE, NLPP. In the case of Friedman's data sets, the optimization algorithm was run with three different initial conditions and the best solution was picked up. When N = 200, the optimization algorithm was run with only one initial condition. For all the data sets, both the inputs and outputs were normalized to zero mean and unit variance. From Table 1, we see that the performances (both TSE and NLPP) of the predictive approaches are better than ML and MAP approaches for both the data sets. In the case of data set 2, we observed that like ML and MAP methods, all the predictive approaches rightly identified the irrelevant inputs. The performance of GPE approach is the best on the robot arm data and demonstrates the usefulness of this approach when a good noise level estimate is available. In the case of Friedman's data set 1 (see Table 2), the important observation is that the performances (both ISE and NLPP) of GPP, CV approaches are relatively poor at low sample size (N = 50) and improve very well as N increases. Note that the performances of the predictive approaches are better compared to the ML and MAP methods starting from N = 100 onwards (see NLPP). Again, GPE gives the best performance and the performance at low sample size (N = 50) is also quite good. In the case of Friedman's data set 2, the ML and MAP approaches perform better compared to the predictive approaches except GPE. The performances of GPP and CV improve Predictive Approaches for Choosing Hyperparameters in Gaussian Processes 637 as N increases and are very close to the ML and MAP methods when N 200. Next, it is clear that the MAP method gives the best performance at low sample size. This behavior, we believe, is because the prior plays an important role and hence is very useful. Also, note that unlike data set 1, the performance of GP E is inferior to ML and MAP approaches at low sample sizes and improves over these approaches (see NLPP) as N increases. This suggests that the knowledge of the noise level alone is not the only issue. The basic issue we think is that the predictive approaches estimate the predictive performance of a given model from the training samples. Clearly, the quality of the estimate will become better as N increases. Also, knowing the noise level improves the quality of the estimate. 4 Discussion Simulation results indicate that the size N required to get good estimates of predictive performance will be dependent on the problem. When N is sufficiently large, we find that the predictive approaches perform better than ML and MAP approaches. The sufficient number of samples can be as low as 100 as evident from our results on Friedman's data set 1. Also, MAP approach is the best, when N is very low. As one would expect, the performances of ML and MAP approaches are nearly same as N increases. The comparison with the existing approaches indicate that the predictive approaches developed here are strongly competitive. The overall cost for computing the function and the gradient (for all three predictive approaches) is O(M N3). The cost for making prediction is same as the one required for ML and MAP methods. The proofs of the results and detailed simulation results will be presented in another paper (Sundararajan and Keerthi, 1999). References Friedman, J .H., (1991) Multivariate Adaptive Regression Splines, Ann. of Stat., 19, 1-141. Geisser, S., (1975) The Predictive Sample Reuse Method with Applications, Journal of the American Statistical Association, 70, 320-328. Geisser, S., and Eddy, W.F., (1979) A Predictive Approach to Model Selection, Journal of the American Statistical Association, 74, 153-160. MacKay, D.J.C. (1997) Gaussian Processes - A replacement for neural networks ?, Available in Postscript via URL http://www.wol.ra.phy.cam.ac.uk/mackayj. Neal, R.M. (1996) Bayesian Learning for Neural Networks, New York: Springer-Verlag. Neal, R.M. (1997) Monte Carlo Implementation of Gaussian Process Models for Bayesian Regression and Classification. Tech. Rep. No. 9702, Dept. of Statistics, University of Toronto. Rasmussen, C. (1996) Evaluation of Gaussian Processes and other Methods for Non-Linear Regression, Ph.D. Thesis, Dept. of Computer Science, University of Toronto. Stone, M. (1974) Cross-Validatory Choice and Assessment of Statistical Predictions (with discussion), Journal of Royal Statistical Society, ser.B, 36, 111-147. Sundararajan, S., and Keerthi, S.S. (1999) Predictive Approaches for Choosing Hyperparameters in Gaussian Processes, submitted to Neural Computation, available at: http://guppy.mpe.nus.edu.sgFmpessk/gp/gp.html. Williams, C.K.I., and Rasmussen, C.E. (1996) Gaussian Processes for Regression. In Advances in Neural Information Processing Systems 8, ed. by D.S.Touretzky, M.C.Mozer, and M.E.Hasselmo. MIT Press.
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Acquisition in Autoshaping Sham Kakade Peter Dayan Gatsby Computational Neuroscience Unit 17 Queen Square, London, England, WC1N 3AR. sharn@gatsby.ucl.ac.uk dayan@gatsby.ucl.ac.uk Abstract Quantitative data on the speed with which animals acquire behavioral responses during classical conditioning experiments should provide strong constraints on models of learning. However, most models have simply ignored these data; the few that have attempted to address them have failed by at least an order of magnitude. We discuss key data on the speed of acquisition, and show how to account for them using a statistically sound model of learning, in which differential reliabilities of stimuli playa crucial role. 1 Introduction Conditioning experiments probe the ways that animals make predictions about rewards and punishments and how those predictions are used to their advantage. Substantial quantitative data are available as to how pigeons and rats acquire conditioned responses during autoshaping, which is one of the simplest paradigms of classical conditioning.4 These data are revealing about the statistical, and ultimately also the neural, substrate underlying the ways that animals learn about the causal texture of their environments. In autoshaping experiments on pigeons, the birds acquire a peck response to a lighted key associated (irrespective of their actions) with the delivery of food. One attractive feature of autoshaping is that there is no need for separate 'probe trials' to assess the degree of association formed between the light and the food by the animal- rather, the rate of key pecking during the light (and before the food) can be used as a direct measure of this association. In particular, acquisition speeds are often measured by the number of trials until a certain behavioral criterion is met, such as pecking during the light on three out of four successive trials.4,8,10 As stressed persuasively by Gallistel & Gibbon4 (GG; forthcoming), the critical feature of autoshaping is that there is substantial experimental evidence on how acquisition speed depends on the three critical variables shown in figure 1A. The first is I, the inter-trial interval; the second is T, the time during the trial for which the light is presented; the third is the training schedule, liS, which is the fractional number of deliveries per light some birds were only partially reinforced. Figure 1 makes three key points. First, figure 1B shows that the median number of trials to the acquisition criterion depends on the ratio of I IT, and not on I and T separately - experiments reported for the same I IT are actually performed with I and T differing by more than an order of magnitude.4,8 Second, figure 1B shows convincingly that the number of reinforcements is approximately inversely proportional to I IT the relatively shorter presentation of light, the faster the leamAcquisition in Autoshaping 25 A B 500 . C 'OOOOr---------, vrnotoo E .. . . \~ ' '''' ~ 1': \: \ , ': ;: :::::~::;i~: lime 2 Jr 10 20 50 10 , - -2 · · :~ . 5 • • '0 Figure 1: Autoshaping. A) Experimental paradigm. Top: the light is presented for T seconds every C seconds and is always followed by the delivery of food (filled circle). Bottom: the food is delivered with probability liS = 1/2 per trial. In some cases I is stochastic, with the appropriate mean. B) Log-log plot4 of the number of reinforcements to a given acquisition criterion versus the I IT ratio for S = l. The data are median acquisition times from 12 different laboratories. C) Log-log acquisition curves for various I IT ratios and S values. The main graph shows trials versus S; the inset shows reinforcements versus S. (1999). ing. Third, figure lC shows that partial reinforcement has almost no effect when measured as a function of the number of reinforcements (rather than the number of trials),4, 10 since although it takes S times as many trials to acquire, there are reinforcements on only liS trials. Changing S does not change the effective I IT when measured as a function of reinforcements, so this result might actually be expected on the basis of figure IB, and we only consider S = 1 in this paper. Altogether, the data show that: (1) where n is the number of rewards to the acquisition criterion. Remarkably, these effects seem to hold for over an order of magnitude in both I IT and S. These quantitative data should be a most seductive target for statistically sound models of learning. However, few models have even attempted to capture the strong constraints they provide, and those that have attempted, all fail in critical aspects. The best of them, rate estimation theory4 (RET), is closely related to the Rescorla-Wagnerl3 (RW) model, and actually captures the proportionality in equation 1. However, as shown below, RET grossly overestimates the observed speed of acquisition (underestimating the proportionality constant). Further, RET is designed to account for the time at which a particular, standard, acquisition criterion is met. Figure 2A shows that this is revealing only about the very early stages of learning RET is silent about the remainder of the learning curve. We look at additional quantitative data on learning, which collectively suggest that stimuli compete to predict the delivery of reward. Dayan & Long3 (DL) discussed various statistically inspired competitive models of classical conditioning, concluding with one in which stimuli are differently reliable as predictors of reward. However, DL ignored the data shown in figures 1 and 2, basing their analYSis on conditioning paradigms in which I IT was not a factor. Figures 1 and 2 demand a more sophisticated statistical model building such a model is the focus of this paper. 2 Rate Estimation Theory Gallistel & Gibbon4 (GG; forthcoming) are amongst the strongest proponents of the quantitative relationships in figure 1. To account for them, GG suggest that animals are estimating the rates of rewards one, >'1, for the rate associated with the light and another, >'b, for the rate associated with the background context. The context is the ever-present environment which can itself gain associative value. The overall 26 A 1.5 i ' rite ' S. Kakade and P. Dayan cl40r---~~-~-----' B :~120 '::; 8"100 '" $I 80 <Il ~60 E e 40 ~ 20 100 200 300 400 '~ 0,L.........J~0 ---4 ~8L.J10164~12":-8 -!"256"':-J.!12~00,---J reinforcements prior context reinforcements Figure 2: Additional Autoshaping Data. A) Acquisition of keypecking. The figure shows response rate versus reinforcements.6 The acquisition criterion is satisfied at a relatively early time when the response curve crosses the acquisition criterion line. B) The effects of prior context reinforcements on subsequent acquisition speed. The data are taken from two experiments,I,2 with I IT = 6. predicted reward rate while the light is on is.A1 + .Ab, and the rate without the light is just .Ab· The additive form of the model makes it similar to Rescorla-Wagner's13 (RW) standard delta-rule model, for which the net prediction of the expected reward in a trial is the sum of the associative values of each active predictor (in this case, the context and light). If the rewards are modeled as being just present or absent, the expected value for a reward is just its probability of occurrence. Instead, RET uses rates, which are just probabilities per unit time. GG4 formulated their model from a frequentist viewpoint. However, it is easier to discuss a closely related Bayesian model which suffers from the same underlying problem. Instead of using RW's delta-rule for learning the rates, GG assume that reinforcements come from a constant rate Poisson process, and make sound statistical inferences about the rates given the data on the rewards. Using an improper flat prior over the rates, we can write the joint distribution as: P(.AI.Ab I data) ex P(n I .A1.Abtltb) ex (.AI + .Ab)ne-(AI+Ab)tle-Abtb (2) since all n rewards occur with the light, at rate .AI + .Ab. Here, tl = nT is the total time the light is on, and tb = nI is the total time the light is off. GG take the further important step of relating the inferred rates .AI and .Ab to the decision of the animals to start responding (ie to satisfy the acquisition criterion). GG suggest that acquisition should occur when the animals have strong evidence that the fractional increase in the reward rate, whilst the light is on, is greater than some threshold. More formally, acquisition should occur when: P «.AI + .Ab) l.Ab > J3 I n) = 1 Q (3) where Q is the uncertainty threshold and J3 is slightly greater than one, reflecting the fractional increase. The n that first satisfies equation 3 can be found by integrating the joint probability in equation 2. It turns out that n ex tlltb, which has the approximate, linear dependence on the ratio I IT (as in figure IB), since tt/tb = nT InI = T I I. It also has no dependence on partial reinforcement, as observed in figure 1 C. However, even with a very low uncertainty, Q = 0.001, and a reasonable fractional increase, J3 = 1.5, this model predicts that learning should be more than ten times as fast as observed, since we get n ~ 20 * T I I as opposed to the 300 * T I I observed. Equation 1 can only be satisfied by setting Q between 10-20 and 10-50 (depending on the precise values of I IT and J3)! This spells problems for GG as a normative, ideal detector model of learning it cannot, for instance, be repaired with any reasonable prior for the rates, as Q drops drastically with n. In other circumstances, Acquisition in Autoshaping 27 though, Gallistel, Mark & KingS (forthcoming) have shown that animals can be ideal detectors of changes in rates. One hint of the flaw with GG is that simple manipulations to the context before starting auto shaping (in particular extinction) can produce very rapid learning.2 More generally, the data show that acquisition speed is strongly- controlled by prior rewards being given only in the context (without the light present).2 Figure 2B shows a parametric study of subsequent acquisition speeds during autoshaping as a function of the number of rewards given only with the context. This effect cannot simply be modeled by assuming a different prior distribution for the rates (which does not fix the problem of the speed of acquisition in any case), since the rate at which these prior context rewards were given has little effect on subsequent acquisition speed for a given number of prior reinforcements.9 Note that the data in figure 2B (ie equation 1) suggest that there were about thirty prior rewards in the context this is consistent with the experimental procedures used,8--10 although prior experience was not a carefully controlled factor. 3 The Competitive Model Five sets of constraints govern our new model. First, since animals can be ideal detectors of rates in some circumstances,s we only consider accounts under which their acquisition of responding has a rational statistical basis. Second, the number of reinforcements to acquisition must be n ~ 300 * T / I, as in equation 1. This requires that the constant of proportionality should come from rational, not absurd, uncertainties. Third, pecking rates after the acquisition criterion is satisfied should also follow the form of figure 2A (in the end, we are preventing from a normative account of this by a dearth of data). Fourth, the overallieaming speed should be strongly affected by the number of prior context rewards (figure 2B), but not by the rate at which they were presented. That is, the context, as an established predictor, regardless of the rate it predicts, should be able to substantially block learning to a less established predictor. Finally, the asymptotic accuracy of rate estimates. should satisfy the substantial experimental data on the intrinsic uncertainty in the predictions in the form of a quantitative account called scalar expectancy theory7 (SET). In our model, as in DL, an independent prediction of the rate of reward delivery is made on the basis of each stimulus that is present (wc, for the context; WI for the light). These separate predictions are combined based on estimated reliabilities of the predictions. Here, we present a heuristic version of a more rigorously specified model.12 3.1 Rate Predictions SET7 was originally developed to capture the nature of uncertainty in the way that animals estimate time intervals. Its most important result is that the standard deviation of an estimate is consistently proportional to the mean, even after an asymptotic number of presentations of the interval. Since the estimated time to a reward is just the inverse rate, asymptotic rate estimates might also be expected to have constant coefficients of variation. Therefore, we constrain the standard deviations of rate estimates not to drop below a multiple of their means. Evidence suggests that this multiple is about 0.2.7 RET clearly does not satisfy this constraint as the joint distribution (equation 2) becomes arbitrarily accurate over time. Inspired by Sutton,14 we consider Kalman filter models for independent logpredictions, logwc(m) and logwl(m), on trial m. The output models for the filters 28 s. Kakade and P. Dayan specify the relationship between the predicted and observed rates. We use a simple log-normal, CN, approximation (to an underlying truly Poisson model): P(oc(m) I wc(m» ,... CN(wc(m) , v;) P(ol(m) I wl(m)) ,... CN(wl(m), vt) (4) where o.(m) is the observed average reward whilst predictor * is present, so if a reward occurs with the light in trial m, then ol(m) = l/T and oc(m) = l/C (where C = T + J). The values of v; can be determined, from the Poisson model, to be V 2 - v2 -1 c I • The other part of the Kalman filter is a model of change in the world for the w's: logwc(m) = logwc(m - 1) + €c(m) logwl(m) = log WI (m - 1) + €l(m) €c(m) ,... N(O, (1](1] + 1»-1) (5) €l(m) ,... N(O, (1](1] + 1»-1) (6) We use log(rates) so that there is no inherent scale to change in the world. Here, 1] is a constant chosen to satisfy the SET constraint, imposed as u. = w./..,fii at asymptote. Notice that 1] acts as the effective number of rewards remembered, which will be less than 30, to get the observed coefficient of variation above 0.2. After observing the data from m trials, the posterior distributions for the predictions will become approximately: P(wc(m) I data) '" N(1/C,u;(m» P(wl(m) I data) ,... N(1/T, ut(m» (7) and, in about m = 1] trials, uc(m) -+ (1/C)/..,fii and ul(m) -+ (l/T)/..,fii. This captures the fastest acquisition in figure 2, and also extinction. 3.2 Cooperative Mixture of Experts The two predictions (equation 7) are combined using the factorial experts model of Jacobs et a[11 that was also used by DL. For this, during the presentation of the light (and the context, of course), we consider that, independently, the relationships between the actual reward rate rem) and the outputs wl(m) and wc(m) of 'experts' associated with each stimulus are: P(wl(m)lr(m» '" N(r(m), pJm) , P(wc(m)lr(m»,... N(r(m), p)m) (8) where PI(m)-1 and pc(m)-1 are inverse variances, or reliabilities for the stimuli. These reliabilities reflect the belief as to how close wl(m) and wc(m) are to rem). The estimates are combined, giving P(r(m) I wl(m),wc(m» '" N(T(m) , (Pl(m) + pc(m»-I) rem) = 7f1(m)wl(m) + (1- 7f1(m))wc(m) 7f1(m) = Pl(m)/(Pl(m) + pc(m» The prediction of the reward rate without the light r c (m) is determined just by the context value wc(m). In this formulation, the context can block the light's prediction if it is more reliable (Pc» PI), since 7f1 ~ 0, making the mean rem) ~ wc(m), and this blocking occurs regardless of the context's rate,wc(m). If PI slowly increases, then rem) -+ WI slowly as 7f1 (m) -+ 1. We expect this to model the post-acquisition part of the learning shown in figure 2A. A fully normative model of acquisition would come from a statistically correct account of how the reliabilities should change over time, which, in turn, would come from a statistical model of the expectations the animal has of how predictabilities change in the world. Unfortunately, the slow phase of learning in figure 2A, which should provide the most useful data on these expectations, is almost ubiquitously Acquisition in Autoshaping 29 A 1.5 B lIT ...__-==---1 C 500. .: I .. .... 0.6 0.3 • . ; I '. . . , acquiS110n Criterion 10 100 200 300 400 100 200 300 400 5 10 20 50 reinforcements reinforcements 1fT Figure 3: Satisfaction of the Constraints. A) The fit to the behavioral response curve (figure 2B), using equation 9 and 7r0 = 0.004. B) Possible acquisition curves showing r{m) versus m. The +--7 on the criterion line denotes the range of 15 to 120 reinforcements that are indicated by figure 2B. The -curve is the same as in Fig 3A. The parameters displayed are values for 7r0 in multiples of 7r0 for the center curve. C) A theoretical fit to the data using equation 11. Here,o: = 5% and 7ro..jPo = 0.004. ignored in experiments. We therefore make two assumptions about this, which are chosen to fit the acquisition data, but whose normative underpinnings are unclear. The first assumption, chosen to obtain the slow learning curve, is that: 1ft (m) = tanh 1fom (9) Assuming that the strength of the behavioral response is approximately proportional to r(m) - rc(m), which we will estimate by 1fl(m)(i~h(m) - wc(m)), figure 3A compares the rate of key pecking in the model with the data from figure 2A. Figure 3B shows the effect on the behavioral response of varying 1fo. Within just a half an order magnitude of variation of 1fo, the acquisition speeds (judged at the criterion line shown) due to between 1200 and 0 prior context rewards (figure 2B) can be obtained. Note the slightly counter-intuitive explanation the actual reward rate associated with the light is established very quickly slow learning comes from slow changes in the importance paid to these rates. We make a second assumption that the coefficient of variation of the context's prediction, from equation 8, does not change Significantly for the early trials before the acquisition criterion is met (it could change thereafter). This gives: pc(m) ~ po/wc(m)2 for early m (10) It is plausible that the context is not becoming a relatively worse 'expert' for early m, since no other predictor has yet proven more reliable. Following GG's suggestion, we model acquisition as occurring on trial m if P(r(m) > rc(m)ldata) ~ 1 0:, ie if the animal has sound reasons to expect a higher reward rate with the light. Integrating over the Kalman filter distributions in equation 7 gives the distribution of r(m) - rc(m) for early mas P(r(m) - rc(m) I data) '" N«(tanh 1fom)(1/T - l/C), (pOC2)-1) where O".(m) has dropped out due to 1ft(m) being small at early m. Finding the number of rewards, n, that satisfies the acquisition criterion gives: 0: T n ~ (11) 1foVPO I where the factor of 0: depends on the uncertainty, 0:, used. Figure 3C shows the theoretical fit to the data. 4 Discussion Although a noble attempt, RET fails to satisfy the strong body of constraints under which any acquisition model must labor. Under RET, the acquisition of responding cannot have a rational statistical basis, as the animal's modeled uncertainty in 30 S. Kakade and P. Dayan the association between light and reward at the time of acquisition is below 10-20. Further, RET ignores constraints set forth by the data establishing SET and also data on prior context manipulations. These latter data show that the context, regardless of the rate it predicts, will substantially block learning to a less established predictor. Additive models, such as RET, are unable to capture this effect. We have suggested a model in which each stimulus is like an 'expert' that learns independently about the world. Expert predictions can adapt quickly to changes in contingencies, as they are based on a Kalman filter model, with variances chosen to satisfy the constraint suggested by SET, and they can be combined based on their reliabilities. We have demonstrated the model's close fit to substantial experimental data. In particular, the new model captures the I IT dependence of the number of rewards to acquisition, with a constant of proportionality that reflects rational statistical beliefs. The slow learning that occurs in some circumstances, is due to a slow change in the reliabilities of predictors, not due to the rates being unable to adapt quickly. Although we have not shown it here, the model is also able to account for quantitative data as to the speed of extinction of the association between the light and the reward. The model leaves many directions for future study. In particular, we have not specified a sound statistical basis for the changes in reliabilities given in equations 9 and 10. Such a basis is key to understanding the slow phase of learning. Second, we have not addressed data from more sophisticated conditioning paradigms. For instance, overshadowing, in which multiple conditioned stimuli are similarly predictive of the reward, should be able to be incorporated into the model in a natural way. Acknowledgements We are most grateful to Randy Gallistel and John Gibbon for freely sharing, prior to publication, their many ideas about timing and conditioning. We thank Sam Roweis for comments on an earlier version of the manuscript. Funding is from a NSF Graduate Research Fellowship (SK) and the Gatsby Charitable Foundation. References [1] Balsam, PD, & Gibbon, J (1988). Journal of Experimental Psychology: Animal Behavior Processes, 14: 401-412. [2] Balsam, PD, & Schwartz, AL (1981). Journal of Experimental Psychology: Animal Behavior Processes, 7: 382-393. [3] Dayan, P, & Long, T, (1997) Neural Information Processing Systems, 10:117-124. [4] Gallistel, CR, & Gibbon, J (1999). Time, Rate, and Conditioning. Forthcoming. [5] Gallistel, CR, Mark, TS & King, A (1999). Is the Rat an Ideal Detector of Changes in Rates of Reward? Forthcoming. [6] Gamzu, ER, & Williams, DR (1973). Journal of the Experimental Analysis of Behavior, 19:225-232. [7] Gibbon, J (1977). Psychological Review 84:279-325. [8] Gibbon, J, Baldock, MD, Locurto, C, Gold, L & Terrace, HS (1977). Journal of Experimental Psychology: Animal Behavior Processes, 3: 264-284. [9] Gibbon, J & Balsam, P (1981). In CM Locurto, HS Terrace, & J Gibbon, editors, Autoshaping and Conditioning Theory. 219-253. New York, NY: Academic Press. [10] Gibbon, J, Farrell, L, Locurto, CM, Duncan, JH & Terrace, HS (1980). Animal Learning and Behavior, 8:45-59. [11] Jacobs, RA, Jordan, MI, & Barto, AG (1991). Cognitive Science 15:219-250. [12] Kakade, S & Dayan, P (2000). In preparation. [13] Rescorla, RA & Wagner, AR (1972). In AH Black & WF Prokasy, editors, Classical Conditioning II: Current Research and Theory, 64-69. New York, NY: Appleton-Century-Crofts. [14] Sutton, R (1992). In Proceedings of the 7th Yale Workshop on Adaptive and Learning Systems.
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Transductive Inference for Estimating Values of Functions Olivier Chapelle*, Vladimir Vapnik*,t, Jason Westontt.t,* * AT&T Research Laboratories, Red Bank, USA. t Royal Holloway, University of London, Egham, Surrey, UK. tt Barnhill BioInformatics.com, Savannah, Georgia, USA. { chapelle, vlad, weston} @research.att.com Abstract We introduce an algorithm for estimating the values of a function at a set of test points Xe+!, ... , xl+m given a set of training points (XI,YI), ... ,(xe,Ye) without estimating (as an intermediate step) the regression function. We demonstrate that this direct (transducti ve) way for estimating values of the regression (or classification in pattern recognition) can be more accurate than the traditionalone based on two steps, first estimating the function and then calculating the values of this function at the points of interest. 1 Introduction Following [6] we consider a general scheme of transductive inference. Suppose there exists a function y* = fo(x) from which we observe the measurements corrupted with noise ((Xl, YI)," . (xe, Ye)), Yi = Y; + ~i' (1) Find an algorithm A that using both the given set of training data (1) and the given set of test data (Xl+!,' .. , XHm) selects from a set of functions {x t--+ f (x)} a function Y = f(x) = fA(xlxl,YI, ... ,Xl,Yl,XHI"",XHm) and minimizes at the points of interest the functional R(A) = E (~ (y; - fA(Xilxl,Yl, ... ,Xl,Ye,Xl+l, . .. ,Xl+m))2) i=l+l (2) (3) (4) where expectation is taken over X and~. For the training data we are given the vector X and the value Y, for the test data we are only given x. Usually, the problem of estimating values of a function at points of interest is sol ved in two steps: first in a given set of functions f (x, a), a E A one estimates the regression, i.e the function which minimizes the functional R(a) = J ((y - f(x, a))2dF(x, Y), (5) 422 0. Chapelle, V. N. Vapnik and J. Weston (the inductive step) and then using the estimated function Y = f(x,al) we calculate the values at points of interest yi = f(x;, ae) (6) (the deductive step). Note, however, that the estimation of a function is equivalent to estimating its values in the continuum points of the domain of the function. Therefore, by solving the regression problem using a restricted amount of information, we are looking for a more general solution than is required. In [6] it is shown that using a direct estimation method one can obtain better bounds than through the two step procedure. In this article we develop the idea introduced in [5] for estimating the values of a function only at the given points. The material is organized as follows. In Section 1 we consider the classical (inductive) Ridge Regression procedure, and the leave-one--out technique which is used to measure the quality of its solutions. Section 2 introduces the transductive method of inference for estimation of the values of a function based on this leave-one- out technique. In Section 3 experiments which demonstrate the improvement given by transductive inference compared to inductive inference (in both regression and pattern recognition) are presented. Finally, Section 4 summarizes the results. 2 Ridge Regression and the Leave-One-Out procedure In order to describe our transductive method, let us first discuss the classical twostep (inductive plus deductive) procedure of Ridge Regression. Consider the set of functions linear in their parameters n f(x, a) = L aicPi(x). (7) i=1 To minimize the expected loss (5), where F(x, y) is unknown, we minimize the following empirical functional (the so-called Ridge Regression functional [1]) l 1 ~ 2 2 Remp(a) = e L)Yi - f(Xi, a)) + 1'110.11 i=1 (8) where l' is a fixed positive constant, called the regularization parameter. The minimum is given by the vector of coefficients ae = a(xl, Yl, ... , Xl, Yl) = (KT K + 1'1)-1 KTy (9) where y = (Y1, ... ,Ylf, and K is a matrix with elements: Kij=cPj(Xi), i=I, ... ,£, j=I, ... ,n. (10) (11) The problem is to choose the value l' which provides small expected loss for training on a sample Sl = {(Xl,Yl), .. . ,(Xl,Yl)}. For this purpose, we would like to choose l' such that f"f minimizing (8) also minimizes R = J (Y* - f"f(x* ISl))2dF(x*, y*)dF(Se). (12) Transductive Inference for Estimating Values of Functions 423 Since F(x, y) is unknown one cannot estimate this minimum directly. To solve this problem we instead use the leave-one-out procedure, which is an almost unbiased estimator of (12). The leave-one-out error of an algorithm on the training sample Sf. is (13) The leave-one-out procedure consists of removing from the training data one element (say (Xi, Yi)), constructing the regression function only on the basis of the remaining training data and then testing the removed element. In this fashion one tests all f elements of the training data using f different decision rules. The minimum over, of (13) we consider as the minimum over, of (12) since the expectation of (13) coincides with (12) [2]. For Ridge Regression, one can derive a dosed form expression for the leave- one- out error. Denoting the error incurred by the leave-one-out procedure is [6] 1 f. (Y'_kTA-1KTy)2 T. -_ ~ ~ 'Y loo(r) f L 1 _ kT A-1k. ~=1 ~ 'Y ~ where kt = (i>I(xd··· ,l/>n(Xt)f· Let, = ,0 be the minimum of (15). Then the vector yO = K*(KT K +,0 I)-I KTy where ( I/>(XHI) K*. 1/>1 (XHm) is the Ridge Regression estimate of the unknown values (Ye+l' ... ,Ye+m)' 3 Leave-One-Out Error for Transductive Inference (14) (15) (16) (17) (18) In transductive inference, our goal is to find an algorithm A which minimizes the functional (4) using both the training data (1) and the test data (2). We suggest the following method: predict (Ye+l' ... 'Ye+m) by finding those values which minimize the leave-one-out error of Ridge Regression training on the joint set (Xl, yd,·· . ,(Xl, Yl), (Xl+l, ye+l),·· ., (XHm, Ye+m)' (19) This is achieved in the following way. Suppose we treat the unknown values (Ye+l" .. ,Ye+m) as variables and for some fixed value of these variables we minimize the following empirical functional Remp(aly;, .. ·, y~) = f: m ~(Yi - f(xi,a))2 + . L (y; - f(xi, a))2 +,llaI1 2 . ( f. Hm ) ~=l ~=l+1 (20) This functional differs only in the second term from the functional (8) and corresponds to performing Ridge Regression with the extra pairs (21) 424 O. Chapel/e, V. N. Vapnik and J. Weston Suppose that vector Y" = (Yi, ... , y:n) is taken from some set Y" E Y such that the pairs (21) can be considered as a sample drawn from the same distribution as the pairs (Xl, yi), ... , (Xl, yi)· In this case the leave-one-out error of minimizing (20) over the set (19) approximates the functional (4). We can measure this leaveone-out error using the same technique as in Ridge Regression. Using the closed form (15) one obtains 1 l+m (Y:' _ kT A-I kTY) 2 7loo(rly~, .. ·,y~) = -f-- L ~ t~T ~-1~ + m i=l 1 - ki A-y ki where we denote x = (Xl, ... , Xl+m), Y = (YI, ... , Yl, Yi+1" .. , Yi+m)T, and Kij=<pj(Xi), i=I, ... ,i+m, j=I, ... ,n. kt = (<PI(Xt} ... ,<Pn(xt)f· (22) (23) (24) (25) Now let us rewrite the expression (22) in an equivalent form to separate the terms with Y from the terms with x. Introducing (26) and the matrix M with elements l+m M .. _ " CikCkj tJ ~ 2 k=l Cu (27) we obtain the equivalent expression of (22) (28) In order for the Y" which minimize the leave-one-out procedure to be valid it is required that the pairs (21) are drawn from the same distribution as the pairs (Xl, yi), ... , (Xl, yi)· To satisfy this constraint we choose vectors Y" from the set Y = {Y" : IIY" - y011 -s: R} where the vector yO is the solution obtained from classical Ridge Regression. To minimize (28) under constraint (29) we use the functional where 'Y" is a constant depending on R. (29) (30) Now, to find the values at the given points of interest (2) all that remains is to find the minimum of (30) in Y". Note that the matrix M is obtained using only the vectors X. Therefore, to find the minimum of this functional we rewrite Equation (30) as where (32) Transductive Inference for Estimating Values of Functions 425 and Mo is a e x e matrix, Ml is a e x m matrix and M2 is a m x m matrix. Taking the derivative of (31) in y* we obtain the condition for the solution 2M1Y + 2M2Y* - 2),*Yo + 2),*Y* = 0 (33) which gives the predictions Y* = ()'* 1+ M 2)-1 (-MIY + ),*yO) . (34) In this algorithm (which we will call Transductive Regression) we have two parameters to control: )' and )'*. The choice of )' can be found using the leave-one-out estimator (15) for Ridge Regression. This leaves )'* as the only free parameter. 4 Experiments To compare our one- step transductive approach with the classical two- step approach we performed a series of experiments on regression problems. We also describe experiments applying our technique to the problem of pattern recognition. 4.1 Regression We conducted computer simulations for the regression problem using two datasets from the DELVE repository: boston and kin-32th. The boston dataset is a well- known problem where one is required to estimate house prices according to various statistics based on 13 locational, economic and structural features from data collected by the U.S Census Service in the Boston Massachusetts area. The kin-32th dataset is a realistic simulation of the forward dynamics of an 8 link all-revolute robot arm. The task is to predict the distance of the end-effector from a target, given 32 inputs which contain information on the joint positions, twist angles and so forth. Both problems are nonlinear and contain noisy data. Our objective is to compare our transductive inference method directly with the inductive method of Ridge Regression. To do this we chose the set of basis functions ¢i(X) = exp (-llx - xiI12/2(2), i = 1, ... , e, and found the values of )' and a for Ridge Regression which minimized the leave-one-out bound (15). We then used the same values of these parameters in our transductive approach, and using the basis functions ¢i(X) = exp (-llx - XiW /2(72) , i = 1, . . . , e + m, we then chose a fixed value of)'* . For the boston dataset we followed the same experimental setup as in [4], that is, we partitioned the training set of 506 observations randomly 100 times into a training set of 481 observations and a testing set of 25 observations. We chose the values of)' and a by taking the minimum average leave- one-out error over five more random splits of the data stepping over the parameter space. The minimum was found at )' = 0.005 and log a = 0.7. For our transductive method, we also chose the parameter),* = 10. In Figure la we plot mean squared error (MSE) on the test set averaged over the 100 runs against log a for Ridge Regression and Transductive Regression. Transductive Regression outperforms Ridge Regression, especially at the minimum. To observe the influence of the number of test points m on the generalization ability of our transductive method, we ran further experiments, setting )'* = e/2m for 426 9.2 , 9 , 8.8 8.6 8.4 8.2 8 7.8 7.6 0.4 015 , 0.14 , , , 0.13 g ~0 . 12 , Q) I0.11 0.1 0.09 1 , 1Tranaductive Regression 1 - _ . Ridge Regress.,n 0.6 0.8 Log sigma (a) 1Transductive Regression 1 :' - - - RIdge RegreSSion / , , 1.5 " , - --- , , , 2 2.5 Log sigma (c) , , , , , , , , , , , 3 " 1.2 0. Chapel/e, V. N. Vapnik and J. Weston 8r---~----~----~---------, 7.95 7.9 g7.85 UJ W ~ 7.8 7.75 7.7 7.65;L--~5:--=~10===-15==:"2=0-----:!25 Test Set Size (b) 0.135 O.13,r---~--=--- - - - - - - - - - - - - - - - - - - - - - - - - - - - - 0.125 ~ 0.12 t: ~0 . 115 ;l I- 0.11 0.105 0.1 0.095 0.09''-------:'':50-----.,.1 0~ 0----1~50,------.,.200 ~---:2=-=-' 50 Test Set Size (d) Figure 1: A comparison of Transductive Regression to Ridge Regression on the boston dataset: (a) error rates for varying (J', (b) varying the test set size, m, and on the kin-32fh dataset: (c) error rates for varying (J', (d) varying the test set size. different values of m. In Figure 1b we plot m against MSE on the testing set, at log (J' = 0.7. The results indicate that increasing the test set size gives improved performance in Transductive Regression. For Ridge Regression, of course, the size of the testing set has no influence on the generalization ability. We then performed similar experiments on the kin-32fh dataset. This time, as we were interested in large testing sets giving improved performance for Transductive Regression we chose 100 splits where we took a subset of only 64 observations for training and 256 for testing. Again the leave-one-out estimator was used to find the values, = 0.1 and log (J' = 2 for Ridge Regression, and for Transductive Regression we also chose the parameter ,* = 0.1. We plotted MSE on the testing set against log (J' (Figure 1c) and the size of the test set m for log (J' = 2.75 (also, ,* = 50/m) (Figure 1d) for the two algorithms. For large test set sizes our method outperforms Ridge Regression. 4.2 Pattern Recognition This technique can also be applied for pattern recognition problems by solving them based on minimizing functional (8) with y = ±1. Such a technique is known as a Linear Discriminant (LD) technique. Transductive Inference for Estimating Values of Functions 427 AB ABR SVM TLD Postal 5.5 4.7 Banana 12.3 10.9 11.5 1l.4 Diabetes 26.5 23.8 23.5 23.3 Titanic 22.6 22.6 22.4 22.4 Breast Cancer 30.4 26.5 26.0 25.7 Heart 20.3 16.6 16.0 15.7 Thyroid 4.4 4.6 4.8 4.0 Table 1: Comparison of percentage test error of AdaBoost (A B) , Regularized AdaBoost (ABR), Support Vector Machines (SVM) and Tmnsductive Linear Discrimination (TLD) on seven datasets. Table 1 describes results of experiments on classification in the following problems: 2 class digit recognition (0 - 4 versus 5 - 9) splitting the training set into 23 runs of 317 observations and considering a testing set of 2000 observations, and six problems from the UCI database. We followed the same experimental setup as in [3]: the performance of a classifier is measured by its average error over one hundred partitions of the datasets into training and testing sets. Free parameter(s) are chosen via validation on the first five training datasets. The performance of the transductive LD technique was compared to Support Vector Machines, AdaBoost and Regularized AdaBoost [3]. It is interesting to note that in spite of the fact that LD technique is one of the simplest pattern recognition techniques, transductive inference based upon this method performs well compared to state of the art methods of pattern recognition. 5 Summary In this article we performed transductive inference in the problem of estimating values of functions at the points of interest. We demonstrate that estimating the unknown values via a one- step (transductive) procedure can be more accurate than the traditional two-step (inductive plus deductive) one. References [1] A. Hoerl and R. W. Kennard. Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12(1):55-67, 1970. [2] A. Luntz and V. Brailovsky. On the estimation of characters obtained in statistical procedure of recognition,. Technicheskaya Kibernetica, 1969. [In Russian]. [3] G. Witsch, T. Onoda, and K.-R. Muller. Soft margins for adaboost. Technical report, Royal Holloway, University of London, 1998. TR-98-2l. [4] C. Saunders, A. Gammermann, and V. Vovk. Ridge regression learning algorithm in dual variables. In Proccedings of the 15th International Conference on Machine Learning, pages 515-52l. Morgan Kaufmann, 1998. [5] V. Vapnik. Estimating of values of regression at the point of interest. In Method of Pattern Recognition. Sovetskoe Radio, 1977. [In Russian]. [6] V. Vapnik. Estimation of Dependences Based on Empirical Data. SpringerVerlag, New York, 1982.
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Effective Learning Requires Neuronal Remodeling of Hebbian Synapses Gal Chechik Isaac Meilijson Eytan Ruppin School of Mathematical Sciences Tel-Aviv University Tel Aviv, Israel ggal@math.tau.ac.il isaco@math.tau.ac.il ruppin@math.tau.ac.il Abstract This paper revisits the classical neuroscience paradigm of Hebbian learning. We find that a necessary requirement for effective associative memory learning is that the efficacies of the incoming synapses should be uncorrelated. This requirement is difficult to achieve in a robust manner by Hebbian synaptic learning, since it depends on network level information. Effective learning can yet be obtained by a neuronal process that maintains a zero sum of the incoming synaptic efficacies. This normalization drastically improves the memory capacity of associative networks, from an essentially bounded capacity to one that linearly scales with the network's size. It also enables the effective storage of patterns with heterogeneous coding levels in a single network. Such neuronal normalization can be successfully carried out by activity-dependent homeostasis of the neuron's synaptic efficacies, which was recently observed in cortical tissue. Thus, our findings strongly suggest that effective associative learning with Hebbian synapses alone is biologically implausible and that Hebbian synapses must be continuously remodeled by neuronally-driven regulatory processes in the brain. 1 Introduction Synapse-specific changes in synaptic efficacies, carried out by long-term potentiation (LTP) and depression (LTD) are thought to underlie cortical self-organization and learning in the brain. In accordance with the Hebbian paradigm, LTP and LTD modify synaptic efficacies as a function of the firing of pre and post synaptic neurons. This paper revisits the Hebbian paradigm showing that synaptic learning alone cannot provide effective associative learning in a biologically plausible manner, and must be complemented with neuronally-driven synaptic remodeling. The importance of neuronally driven normalization processes has already been demonstrated in the context of self-organization of cortical maps [1, 2] and in continuous unsupervised learning as in principal-component-analysis networks [3]. In these scenarios normalization is necessary to prevent the excessive growth of synap
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The Relevance Vector Machine Michael E. Tipping Microsoft Research St George House, 1 Guildhall Street Cambridge CB2 3NH, U.K. mtipping~microsoft.com Abstract The support vector machine (SVM) is a state-of-the-art technique for regression and classification, combining excellent generalisation properties with a sparse kernel representation. However, it does suffer from a number of disadvantages, notably the absence of probabilistic outputs, the requirement to estimate a trade-off parameter and the need to utilise 'Mercer' kernel functions. In this paper we introduce the Relevance Vector Machine (RVM), a Bayesian treatment of a generalised linear model of identical functional form to the SVM. The RVM suffers from none of the above disadvantages, and examples demonstrate that for comparable generalisation performance, the RVM requires dramatically fewer kernel functions. 1 Introd uction In supervised learning we are given a set of examples of input vectors {Xn}~=l along with corresponding targets {tn}~=l' the latter of which might be real values (in regression) or class labels (classification). From this 'training' set we wish to learn a model of the dependency of the targets on the inputs with the objective of making accurate predictions of t for previously unseen values of x. In real-world data, the presence of noise (in regression) and class overlap (in classification) implies that the principal modelling challenge is to avoid 'over-fitting' of the training set. A very successful approach to supervised learning is the support vector machine (SVM) [8]. It makes predictions based on a function of the form N y(x) = 2:: wnK(x, x n) + Wo, (1) n=l where {wn} are the model 'weights' and K(·,·) is a kernel function. The key feature of the SVM is that, in the classification case, its target function attempts to minimise the number of errors made on the training set while simultaneously maximising the 'margin' between the two classes (in the feature space implicitly defined by the kernel). This is an effective 'prior' for avoiding over-fitting, which leads to good generalisation, and which furthermore results in a sparse model dependent only on a subset of kernel functions: those associated with training examples Xn that lie either on the margin or on the 'wrong' side of it. State-of-the-art results have been reported on many tasks where SVMs have been applied. The Relevance Vector Machine 653 However, the support vector methodology does exhibit significant disadvantages: • Predictions are not probabilistic. In regression the SVM outputs a point estimate, and in classification, a 'hard' binary decision. Ideally, we desire to estimate the conditional distribution p(tlx) in order to capture uncertainty in our prediction. In regression this may take the form of 'error-bars', but it is particularly crucial in classification where posterior probabilities of class membership are necessary to adapt to varying class priors and asymmetric misclassification costs. • Although relatively sparse, SVMs make liberal use of kernel functions, the requisite number of which grows steeply with the size of the training set. • It is necessary to estimate the error/margin trade-off parameter 'e' (and in regression, the insensitivity parameter If' too). This generally entails a cross-validation procedure, which is wasteful both of data and computation. • The kernel function K(·,·) must satisfy Mercer's condition. In this paper, we introduce the 'relevance vector machine' (RVM), a probabilistic sparse kernel model identical in functional form to the SVM. Here we adopt a Bayesian approach to learning, where we introduce a prior over the weights governed by a set of hyperparameters, one associated with each weight, whose most probable values are iteratively estimated from the data. Sparsity is achieved because in practice we find that the posterior distributions of many of the weights are sharply peaked around zero. Furthermore, unlike the support vector classifier, the nonzero weights in the RVM are not associated with examples close to the decision boundary, but rather appear to represent 'prototypical' examples of classes. We term these examples 'relevance' vectors, in deference to the principle of automatic relevance determination (ARD) which motivates the presented approach [4, 6J. The most compelling feature of the RVM is that, while capable of generalisation performance comparable to an equivalent SVM, it typically utilises dramatically fewer kernel functions. Furthermore, the RVM suffers from none of the other limitations of the SVM outlined above. In the next section, we introduce the Bayesian model, initially for regression, and define the procedure for obtaining hyperparameter values, and thus weights. In Section 3, we give brief examples of application of the RVM in the regression case, before developing the theory for the classification case in Section 4. Examples of RVM classification are then given in Section 5, concluding with a discussion. 2 Relevance Vector Regression Given a dataset of input-target pairs {xn, tn}~=l' we follow the standard formulation and assume p(tlx) is Gaussian N(tIY(x), a 2 ). The mean ofthis distribution for a given x is modelled by y(x) as defined in (1) for the SVM. The likelihood of the dataset can then be written as p(tlw, a2 ) = (27ra 2 )-N/2 exp { - 2:2 lit - ~w)1I2 } , (2) where t = (tl ... tN), W = (wo .. . WN) and ~ is the N x (N + 1) 'design' matrix with ~nm = K(xn , Xm- l) and ~nl = 1. Maximum-likelihood estimation of wand a 2 from (2) will generally lead to severe overfitting, so we encode a preference for smoother functions by defining an ARD Gaussian prior [4, 6J over the weights: N p(wla) = II N(wiIO,ai 1 ), (3) i=O 654 ME. Tipping with 0 a vector of N + 1 hyperparameters. This introduction of an individual hyperparameter for every weight is the key feature of the model, and is ultimately responsible for its sparsity properties. The posterior over the weights is then obtained from Bayes' rule: p(wlt, 0,0'2) = (21r)-(N+l)/21:E1- 1/ 2 exp { -~(w - J.lY:E-1(w - JL)}, (4) with :E = (q,TBq, + A)-I, JL = :Eq, TBt, (5) (6) where we have defined A = diag(ao,al, ... ,aN) and B = 0'-2IN. Note that 0'2 is also treated as a hyperparameter, which may be estimated from the data. By integrating out the weights, we obtain the marginal likelihood, or evidence [2], for the hyperparameters: p(tIO,0'2) = (21r)-N/2IB-1 + q,A -1q,TI- 1/ 2 exp { -~e(B-l + q,A -lq,T)-lt} . (7) For ideal Bayesian inference, we should define hyperpriors over 0 and 0'2, and integrate out the hyperparameters too. However, such marginalisation cannot be performed in closed-form here, so we adopt a pragmatic procedure, based on that of MacKay [2], and optimise the marginal likelihood (7) with respect to 0 and 0'2, which is essentially the type II maximum likelihood method [1]. This is equivalent to finding the maximum of p(o, 0'2It), assuming a uniform (and thus improper) hyperprior. We then make predictions, based on (4), using these maximising values. 2.1 Optimising the hyperparameters Values of 0 and 0'2 which maximise (7) cannot be obtained in closed form, and we consider two alternative formulae for iterative re-estimation of o . First, by considering the weights as 'hidden' variables, an EM approach gives: new 1 1 a i = -( 2) 2' Wi p(wlt,Q,u2) Eii + J-Li (8) Second, direct differentiation of (7) and rearranging gives: new 'Yi (9) ai = 2' J-Li where we have defined the quantities 'Yi = 1 - aiEii, which can be interpreted as a measure of how 'well-determined' each parameter Wi is by the data [2]. Generally, this latter update was observed to exhibit faster convergence. For the noise variance, both methods lead to the same re-estimate: (10) In practice, during re-estimation, we find that many of the ai approach infinity, and from (4), p(wilt,0,0'2) becomes infinitely peaked at zero implying that the corresponding kernel functions can be 'pruned'. While space here precludes a detailed explanation, this occurs because there is an 'Occam' penalty to be paid for smaller values of ai, due to their appearance in the determinant in the marginal likelihood (7). For some ai, a lesser penalty can be paid by explaining the data with increased noise 0'2, in which case those ai -+ 00. The Relevance Vector Machine 655 3 Examples of Relevance Vector Regression 3.1 Synthetic example: the 'sine' function The function sinc(x) = Ixl-1 sin Ixl is commonly used to illustrate support vector regression [8], where in place of the classification margin, the f.-insensitive region is introduced, a 'tube' of ±f. around the function within which errors are not penalised. In this case, the support vectors lie on the edge of, or outside, this region. For example, using linear spline kernels and with f. = 0.01, the approximation ofsinc(x) based on 100 uniformly-spaced noise-free samples in [-10, 10J utilises 39 support vectors [8]. By comparison, we approximate the same function with a relevance vector model utilising the same kernel. In this case the noise variance is fixed at 0.012 and 0 alone re-estimated. The approximating function is plotted in Figure 1 (left), and requires only 9 relevance vectors. The largest error is 0.0087, compared to 0.01 in the SV case. Figure 1 (right) illustrates the case where Gaussian noise of standard deviation 0.2 is added to the targets. The approximation uses 6 relevance vectors, and the noise is automatically estimated, using (10), as (7 = 0.189. 1.2 0.8 0.6 0.4 .. 0.2 '. . -, . 0 -0.2 -0.4 -10 -5 5 10 10 Figure 1: Relevance vector approximation to sinc(x): noise-free data (left), and with added Gaussian noise of (]" = 0.2 (right). The estimated functions are drawn as solid lines with relevance vectors shown circled, and in the added-noise case (right) the true function is shown dashed. 3.2 Some benchmarks The table below illustrates regression performance on some popular benchmark datasets Friedman's three synthetic functions (results averaged over 100 randomly generated training sets of size 240 with a lOOO-example test set) and the 'Boston housing' dataset (averaged over 100 randomised 481/25 train/test splits). The prediction error obtained and the number of kernel functions required for both support vector regression (SVR) and relevance vector regression (RVR) are given. Dataset Friedman #1 Friedman #2 Friedman #3 Boston Housing _ errors_ SVR RVR 2.92 2.80 4140 3505 0.0202 0.0164 8.04 7.46 _ kernels _ SVR RVR 116.6 59.4 110.3 6.9 106.5 11.5 142.8 39.0 656 M E. TIpping 4 Relevance Vector Classification We now extend the relevance vector approach to the case of classification Le. where it is desired to predict the posterior probability of class membership given the input x. We generalise the linear model by applying the logistic sigmoid function a(y) = 1/(1 + e-Y ) to y(x) and writing the likelihood as N P(tlw) = II a{y(xn)}tn [1 - a{Y(Xn)}]l-tn . (11) n==l However, we cannot integrate out the weights to obtain the marginal likelihood analytically, and so utilise an iterative procedure based on that of MacKay [3]: 1. For the current, fixed, values of a we find the most probable weights WMP (the location of the posterior mode). This is equivalent to a standard optimisation of a regularised logistic model, and we use the efficient iterativelyreweighted least-squares algorithm [5] to find the maximum. 2. We compute the Hessian at WMP: \7\7logp(t, wla)1 = _(<)TB<) + A), WMP (12) where Bnn = a{y(xn)} [1 - a{y(xn)}], and this is negated and inverted to give the covariance I: for a Gaussian approximation to the posterior over weights, and from that the hyperparameters a are updated using (9). Note that there is no 'noise' variance a 2 here. This procedure is repeated until some suitable convergence criteria are satisfied. Note that in the Bayesian treatment of multilayer neural networks, the Gaussian approximation is considered a weakness of the method if the posterior mode is unrepresentative of the overall probability mass. However, for the RVM, we note that p(t, wla) is log-concave (i.e. the Hessian is negative-definite everywhere), which gives us considerably more confidence in the Gaussian approximation. 5 Examples of RVM Classification 5.1 Synthetic example: Gaussian mixture data We first utilise artificially generated data in two dimensions in order to illustrate graphically the selection of relevance vectors. Class 1 (denoted by 'x ') was sampled from a single Gaussian, and overlaps to a small degree class 2 ('.'), sampled from a mixture of two Gaussians. A relevance vector classifier was compared to its support vector counterpart, using the same Gaussian kernel. A value of C for the SVM was selected using 5-fold crossvalidation on the training set. The results for a typical dataset of 200 examples are given in Figure 2. The test errors for the RVM (9.32%) and SVM (9.48%) are comparable, but the remarkable feature of contrast is the complexity of the classifiers. The support vector machine utilises 44 kernel functions compared to just 3 for the relevance vector method. It is also notable that the relevance vectors are some distance from the decision boundary (in x-space). Given further analysis, this observation can be seen to be consistent with the hyperparameter update equations. A more qualitative explanation is that the output of a basis function lying on or near the decision boundary is a poor indicator of class membership, and such basis functions are naturally 'penalised' under the Bayesian framework. The Relevance Vector Machine SVM: error=9.48% vectors=44 , x x \ @. \ . ~ . . " • II • , , .. \ . • X X x ®\ • 'C' • x x x x x ~ •• _~ • x ~x x x~ .. x.":X~x x "I. x ~ x :xx • xx'f. xxx x x x , , , , , \ RVM: error=9.32% vectors=3 I I • 657 Figure 2: Results of training functionally identical SVM (left) and RVM (right) classifiers on a typical synthetic dataset. The decision boundary is shown dashed, and relevance/support vectors are shown circled to emphasise the dramatic reduction in complexity of the RVM model. 5.2 Real examples In the table below we give error and complexity results for the 'Pima Indian diabetes' and the 'U.S.P.S. handwritten digit' datasets. The former task has been recently used to illustrate Bayesian classification with the related Gaussian Process (GP) technique [9], and we utilised those authors' split of the data into 200 training and 332 test examples and quote their result for the GP case. The latter dataset is a popular support vector benchmark, comprising 7291 training examples along with a 2007-example test set, and the SVM result is quoted from [7]. Dataset Pima Indians U.S.P.S. ___ errors __ _ SVM GP RVM 67 68 65 4.4% 5.1% __ kernels __ SVM GP RVM 109 2540 200 4 316 In terms of prediction accuracy, the RVM is marginally superior on the Pima set, but outperformed by the SVM on the digit data. However, consistent with other examples in this paper, the RVM classifiers utilise many fewer kernel functions. Most strikingly, the RVM achieves state-of-the-art performance on the diabetes dataset with only 4 kernels. It should be noted that reduced set methods exist for subsequently pruning support vector models to reduce the required number of kernels at the expense of some increase in error (e.g. see [7] for some example results on the U.S.P.S. data). 6 Discussion Examples in this paper have effectively demonstrated that the relevance vector machine can attain a comparable (and for regression, apparently superior) level of generalisation accuracy as the well-established support vector approach, while at the same time utilising dramatically fewer kernel functions implying a considerable 658 ME. Tipping saving in memory and computation in a practical implementation. Importantly, we also benefit from the absence of any additional nuisance parameters to set, apart from the need to choose the type of kernel and any associated parameters. In fact, for the case of kernel parameters, we have obtained improved (both in terms of accuracy and sparsity) results for all the benchmarks given in Section 3.2 when optimising the marginal likelihood with respect to multiple input scale parameters in Gaussian kernels (q. v. [9]). Furthermore, we may also exploit the Bayesian formalism to guide the choice of kernel itself [2], and it should be noted that the presented methodology is applicable to arbitrary basis functions, so we are not limited, for example, to the use of 'Mercer' kernels as in the SVM. A further advantage of the RVM classifier is its standard formulation as a probabilistic generalised linear model. This implies that it can be extended to the multiple-class case in a straightforward and principled manner, without the need to train and heuristically combine multiple dichotomous classifiers as is standard practice for the SVM. Furthermore, the estimation of posterior probabilities of class membership is a major benefit, as these convey a principled measure of uncertainty of prediction, and are essential if we wish to allow adaptation for varying class priors, along with incorporation of asymmetric misclassification costs. However, it must be noted that the principal disadvantage of relevance vector methods is in the complexity of the training phase, as it is necessary to repeatedly compute and invert the Hessian matrix, requiring O(N2) storage and O(N3) computation. For large datasets, this makes training considerably slower than for the SVM. Currently, memory constraints limit us to training on no more than 5,000 examples, but we have developed approximation methods for handling larger datasets which were employed on the U.S.P.S. handwritten digit database'. We note that while the case for Bayesian methods is generally strongest when data is scarce, the sparseness of the resulting classifier induced by the Bayesian framework presented here is a compelling motivation to apply relevance vector techniques to larger datasets. Acknowledgements The author wishes to thank Chris Bishop, John Platt and Bernhard Sch5lkopf for helpful discussions, and JP again for his Sequential Minimal Optimisation code. References [1) J. O. Berger. Statistical decision theory and Bayesian analysis. Springer, New York, second edition, 1985. [2) D. J. C. Mackay. Bayesian interpolation. Neural Computation, 4(3):415-447, 1992. [3) D. J. C. Mackay. The evidence framework applied to classification networks. Neural Computation, 4(5):720-736, 1992. (4) D. J. C. Mackay. Bayesian non-linear modelling for the prediction competition. In ASHRAE Transactions, vol. 100, pages 1053- 1062. ASHRAE, Atlanta, Georgia, 1994. (5) 1. T. Nabney. Efficient training of RBF networks for classification. In Proceedings of ICANN99, pages 210-215, London, 1999. lEE. [6) R. M. Neal. Bayesian Learning for Neural Networks. Springer, New York, 1996. (7) B. Sch6lkopf, S. Mika, C. J. C. Burges, P. Knirsch, K.-R. Miiller, G. Ratsch, and A. J. Smola. Input space versus feature space in kernel-based methods. IEEE Transactions on Neural Networks, 10(5):1000- 1017, 1999. [8) V. N. Vapnik. Statistical Learning Theory. Wiley, New York, 1998. [9) C. K. 1. Williams and D. Barber. Bayesian classification with Gaussian processes. IEEE Trans. Pattern Analysis and Machine Intelligence, 20(12) :1342-1351, 1998.
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Dual Estimation and the Unscented Transformation EricA. Wan ericwan@ece.ogi.edu Rudolph van der Merwe rudmerwe@ece.ogi.edu Alex T. Nelson atneison@ece.ogi.edu Oregon Graduate Institute of Science & Technology Department of Electrical and Computer Engineering 20000 N.W. Walker Rd., Beaverton, Oregon 97006 Abstract Dual estimation refers to the problem of simultaneously estimating the state of a dynamic system and the model which gives rise to the dynamics. Algorithms include expectation-maximization (EM), dual Kalman filtering, and joint Kalman methods. These methods have recently been explored in the context of nonlinear modeling, where a neural network is used as the functional form of the unknown model. Typically, an extended Kalman filter (EKF) or smoother is used for the part of the algorithm that estimates the clean state given the current estimated model. An EKF may also be used to estimate the weights of the network. This paper points out the flaws in using the EKF, and proposes an improvement based on a new approach called the unscented transformation (UT) [3]. A substantial performance gain is achieved with the same order of computational complexity as that of the standard EKF. The approach is illustrated on several dual estimation methods. 1 Introduction We consider the problem of learning both the hidden states Xk and parameters w of a discrete-time nonlinear dynamic system, F(Xk , Vk, w) H(xk, nk, w), (1) (2) where Yk is the only observed signal. The process noise Vk drives the dynamic system, and the observation noise is given by nk. Note that we are not assuming additivity of the noise sources. A number of approaches have been proposed for this problem. The dual EKF algorithm uses two separate EKFs: one for signal estimation, and one for model estimation. The states are estimated given the current weights and the weights are estimated given the current states. In the joint EKF, the state and model parameters are concatenated within a combined state vector, and a single EKF is used to estimate both quantities simultaneously. The EM algorithm uses an extended Kalman smoother for the E-step, in which forward and Dual Estimation and the Unscented Transformation 667 backward passes are made through the data to estimate the signal. The model is updated during a separate M-step. For a more thorough treatment and a theoretical basis on how these algorithms relate, see Nelson [6]. Rather than provide a comprehensive comparison between the different algorithms, the goal of this paper is to point out the assumptions and flaws in the EKF (Section 2), and offer a improvement based on the unscented transformation/filter (Section 3). The unscented filter has recently been proposed as a substitute for the EKF in nonlinear control problems (known dynamic model) [3]. This paper presents new research on the use of the UF within the dual estimation framework for both state and weight estimation. In the case of weight estimation, the UF represents a new efficient "second-order" method for training neural networks in general. 2 Flaws in the EKF Assume for now that we know the model (weight parameters) for the dynamic system in Equations 1 and 2. Given the noisy observation Yk, a recursive estimation for Xk can be expressed in the form, Xk = (optimal prediction ofxk) + Gk x [Yk - (optimal prediction ofYk)] (3) This recursion provides the optimal MMSE estimate for Xk assuming the prior estimate Xk and current observation Yk are Gaussian. We need not assume linearity of the model. The optimal terms in this recursion are given by yl: = E[H(xl:, nk)], (4) where the optimal prediction xl: is the expectation of a nonlinear function of the random variables Xk-l and Vk-l (similar interpretation for the optimal prediction of Yk). The optimal gain term is expressed as a function of posterior covariance matrices (with Yk = Yk - Yl:)· Note these terms also require taking expectations of a nonlinear function of the prior state estimates. The Kalman filter calculates these quantities exactly in the linear case. For nonlinear models, however, the extended KF approximates these as: YI: = H(xl:,fl), (5) where predictions are approximated as simply the function of the prior mean value for estimates (no expectation taken). The covariance are determined by linearizing the dynamic equations (Xk+l ~ AXk + BVk, Yk ~ CXk + Dnk), and then determining the posterior covariance matrices analytically for the linear system. As such, the EKF can be viewed as providing "first-order" approximations to the optimal terms (in the sense that expressions are approximated using a first-order Taylor series expansion of the nonlinear terms around the mean values). While "second-order" versions of the EKF exist, their increased implementation and computational complexity tend to prohibit their use. 3 The Unscented TransformationIFilter The unscented transformation (UT) is a method for calculating the statistics of a random variable which undergoes a nonlinear transformation [3]. Consider propagating a random variable a (dimension L) through a nonlinear function, (3 = g( a). Assume a has mean ct and covariance P Q. To calculate the statistics of {3, we form a matrix X of 2L + 1 sigma vectors Xi, where the first vector (Xo) corresponds to ct, and the rest are computed from the mean (+ )plus and (-)minus each column of the matrix square-root of P Q. These sigma 668 E. A. Wan, R. v. d. Merwe and A. T. Nelson vectors are propagated through the nonlinear function, and the mean and covariance for [3 are approximated using a weighted sample mean and covariance, /3 ~ -£ 1 {~g(xo) + -21 I:9(Xi)} , (6) + ~ i=l Pp" L:~ {~[g(XO) - il][g(Xo) - il)T + ~ ~[g(X') - il)[g(X,) - ilf} (7) where ~ is a scaling factor. Note that this method differs substantially from general "sampling" methods (e.g., Monte-Carlo methods and particle filters [1]) which require orders of magnitude more sample points in an attempt to propagate an accurate (possibly nonGaussian) distribution of the state. The UT approximations are accurate to the third order for Gaussian inputs for all nonlinearities. For non-Gaussian inputs, approximations are accurate to at least the second-order, with the accuracy determined by the choice of ~ [3]. A simple example is shown in Figure 1 for a 2-dimensional system: the left plots shows the true mean and covariance propagation using Monte-Carlo sampling; the center plots show the performance of the UT (note only 5 sigma points are required); the right plots show the results using a linearization approach as would be done in the EKF. The superior performance of the UT is clear. Actual (sampling) UT Linearized (EKF) mean .11"=- 0 I - I (3 = g(o) (3 = g(o) Yi = g(Xi ) P(:! = ATPaA 1 1 l Figure 1: Example of the UT for mean and covariance propagation. a) actual, b) UT, c) first-order linear (EKF). The unscented filter (UF) [3] is a straightforward extension of the UT to the recursive estimation in Equation 3, where we set 0: = Xk, and denote the corresponding sigma matrix as X(klk). The UF equations are given on the next page. It is interesting to note that no explicit calculation of lacobians or Hessians are necessary to implement this algorithm. The total number of computations is only order £2 as compared to £3 for the EKF. I 4 Application to Dual Estimation This section shows the use of the UF within several dual estimation approaches. As an application domain for comparison, we consider modeling a noisy time-series as a nonlinear INote that a matrix square-root using the Cholesky factorization is of order L3 /6. However, the covariance matrices are expressed recursively, and thus the square-root can be computed in only order L2 by performing a recursive update to the Cholesky factorization. Dual Estimation and the Unscented Transformation UF Equations Wo = K/(L + K) , WI . .. W2L = 1/2(L + fl.) X(klk - 1) = F[X(k - 11k - 1), P!lv2] x"k = 2:~!o WiXi(klk - 1) P"k = 2:~!o WdXi(klk - 1) - x"k][Xi(klk - 1) - x"kf Y(klk - 1) = H[X(klk - 1), P~;] Y"k = 2:~!o WiYi(klk - 1) Phh = 2::!o Wi[Yi(klk - 1) - Y"kJ[Yi(klk - 1) - Yk"f P XkYk = 2::!o Wi[Xi(klk - 1) - x"k][Yi(klk - 1) - Y"kf Xk = x"k + PXkYkP~:h (n - Y"k) Pk = P"k PX"Yk(P~:yJTP!'kYk autoregression: Xk = f(Xk-l, ... Xk-M, w) + Vk Yk = Xk + nk, Vk E {l. . . N} 669 (8) The underlying clean signal Xk is a nonlinear function of its past M values, driven by white Gaussian process noise Vk with variance 11;. The observed data point Yk includes the additive noise nk, which is assumed to be Gaussian with variance 11;. The corresponding state-space representation for the signal Xk is given by: + B· Vk-I (9) Yk = [1 0 (10) In this context, the dual estimation problem consists of simultaneously estimating the clean signal Xk and the model parameters w from the noisy data Yk. 4.1 Dual EKF I Dual UF One dual estimation approach is the dual extended Kalman filter developed in [8, 6]. The dual EKF requires separate state-space representation for the signal and the weights. A state-space representation for the weights is generated by considering them to be a stationary process with an identity state transition matrix, driven by process noise Uk: Wk = Wk-l + Uk Yk = f(Xk-I,Wk) +Vk +nk· (11) (12) The noisy measurement Yk has been rewritten as an observation on w. This allows the use of an EKF for weight estimation (representing a "second-order" optimization procedure) [7]. Two EKFs can now be run simultaneously for signal and weight estimation. At every time-step, the current estimate of the weights is used in the signal-filter, and the current estimate of the signal-state is used in the weight-filter. 670 E. A. Wan, R. v. d. Merwe and A. T Nelson The dual UFIEKF algorithm is formed by simply replacing the EKF for state-estimation with the UF while still using an EKF for weight-estimation. In the dual UF algorithm both state- and weight-estimation are done with the UF. Note that the state-transition is linear in the weight filter, so the nonlinearity is restricted to the measurement equation. Here, the UF gives a more exact measurement-update phase of estimation. The use of the UF for weight estimation in general is discussed in further detail in Section 5. 4.2 Joint EKF I Joint UF An alternative approach to dual estimation is provided by the joint extended Kalman filter [4,5]. In this framework the signal-state and weight vector are concatenated into a single, joint state vector: Zk = [xf wfV. The estimation of Zk can be done recursively by writing the state-space equations for the joint state as: (13) and running an EKF on the joint state-space to produce simultaneous estimates of the states Xk and w . As discussed in [6], the joint EKF provides approximate MAP estimates by maximizing the joint density of the signal and weights given the noisy data. Again, our approach in this paper is to use the UF instead of the EKF to provide more accurate estimation of the state, resulting in the joint UF algorithm. 4.3 EM - Unscented Smoothing A somewhat different iterative approach to dual estimation is given by the expectationmaximization (EM) algorithm applied to nonlinear dynamic systems [2]. In each iteration, the conditional expectation of the signal is computed, given the data and the current estimate of the model (E-step). Then the model is found that maximizes a function of this conditional mean (M-step). For linear models, the M-step can be solved in closed form. The E-step is computed with a Kalman smoother, which combines the forward-time estimated mean and covariance (x{ ,pt) of the signal given past data, with the backward-time predicted mean and covariance (xf ,pf) given the future data, producing the following smoothed statistics given all the data: (14) (15) When a MLP neural network model is used, the M-step can no longer be computed in closed-form, and a gradient-based approach is used instead. The resulting algorithm is usually referred to as generalized EM (GEM) 2. The E-step is typically approximated by an extended Kalman smoother, wherein a linearization of the model is used for backward propagation of the state estimates. We propose improving the E-step of the EM algorithm for nonlinear models by using a UP instead of an EKF to compute both the forward and backward passes in the Kalman smoother. Rather than linearize the model for the backward pass, as in [2], a neural network is trained on the backward dynamics (as well as the forward dynamics). This allows for a more exact backward estimation phase using the UF, and enables the development of an unscented smoother (US). 2 An exact M-step is possible using RBF networks [2]. Dual Estimation and the Unscented Transformation 671 4.4 Experiments We present results on two simple time-series to provide a clear illustration of the use of the UP over the EKE The first series is the Mackey-Glass chaotic series with additive WGN (SNR ~ 3dB). The second time series (also chaotic) comes from an autoregressive neural network with random weights driven by Gaussian process noise and also corrupted by additive WGN (SNR ~ 3dB). A standard 5-3-1 MLP with tanh hidden activation functions and a linear output layer was used in all the filters. The process and measurement noise variances were assumed to be known. Results on training and testing data, as well as training curves for the different dual estimation methods are shown below. The quoted numbers are normalized (clean signal variance) mean-square estimation and prediction errors. The superior performance of the UT based algorithms (especially the dual UF) are clear. Note also the more stable learning curves using the UF approaches. These improvements have been found to be consistent and statistically significant on a number of additional experiments. Mackey-Glass Train Test Chaotic AR-NN Train Test Algorithm Est. Pred. Est. Pred. Algorithm Est. Pred. Est. Pred. Dual EKF 0.20 0.50 0.21 0.54 . Dual EKF 0.32 0.62 0.36 0.69 Dual UF/EKF 0.19 0.50 0.19 0.53 Dual UF/EKF 0.26 0.58 0.28 0.69 Dual UF 0.15 0.45 0.14 0.48 Dual UF 0.23 0.55 0.27 0.63 Joint EKF 0.22 0.53 0.22 0.56 Joint EKF 0.29 0.58 0.34 0.72 Joint UF 0.19 0.50 0.18 0.53 Joint UF 0.25 0.55 0.30 0.67 Mackey-Glass Chaotic AR-NN o. . Dual EKF . Dual UFIEKF 0.' 0 Dual UF 0 JointEKF " ...... UF 01 ~O8 W en :::;; :::;; o. ]05 " " ~ .. '" E E035 go" 5 C 0 3 03 02 0 25 0.1 0 2 0 5 • 10 11 0 10 15 20 2S '" iteration ~eration The final table below compares smoother performance used for the E-step in the EM algorithm. In this case, the network models are trained on the clean time-series, and then tested on the noisy data using either the standard Kalman smoother with linearized backward model (EKS 1), a Kalman smoother with a second nonlinear backward model (EKS2), and the unscented smoother (US). The forward (F), backward (B), and smoothed (S) estimation errors are reported. Again the performance benefits of the unscented approach is clear. Mackey-Glass Norm. MSE Chaotic AR-NN Norm. MSE Algorithm F B S Algorithm F B S EKSI 0.20 0.70 0.27 EKSI 0.35 0.32 0.28 EKS2 0.20 0.3] 0.19 EKS2 0.35 0.22 0.23 US 0.]0 0.24 0.08 US 0.23 0.2] 0.16 5 UF Neural Network Training As part of the dual UF algorithm, we introduced the use of the UF for weight estimation. The approach can also be seen as a new method for the general problem of training neural networks (i.e., for regression or classification problems where the input x is observed and 672 E. A. Wan. R. v. d. Merwe and A. T. Nelson no state-estimation is required). The advantage of the UF over the EKF in this case is not as obvious, as the state-transition function is linear (See Equation 11). However, as pointed out earlier, the observation is nonlinear. Effectively, the EKF builds up an approximation to the expected Hessian by taking outer products of the gradient. The UF, however, may provide a more accurate estimate through direct approximation of the expectation of the Hessian. We have performed a number of preliminary experiments on standard benchmark data. The figure below shows the mean and std. oflearning curves (computed over 100 experiments with different initial weights) for the Mackay Robot Arm Mapping dataset. Note the faster convergence, lower variance, and lower final MSE performance of the UF weight training. While these results are encouraging, further study is still necessary to fully contrast differences between UF and EKF weight training. W O.06n---.-----,"L"le;O;anmW ln;:;r,g1"it.;'"luwrv:v:ec.s,..---r= I::;;U~F (;:=m.== an7=>l) I r UF(. ld) 0.05 : : ~~~ ~~~n) ~ 0.004 c: ~ 0,03 '\ E 0.02 ~_. __ -.. ~ __ -.. ______________ • _ _ 0.01 "---~ ...... - .. .. 6 Conclusions The EKF has been widely accepted as a standard tool in the machine learning community. In this paper we have presented an alternative to the EKF using the unscented filter. The UF consistently achieves a better level of accuracy than the EKF at a comparable level of complexity. We demonstrated this performance gain on a number of dual estimation methods as well as standard regression modeling. Acknowledgements This work was sponsored in part by the NSF under grant IRI-9712346. References [1] J. F. G. de Freitas, M. Niranjan, A. H. Gee, and A. Doucet. Sequential Monte Carlo methods for optimisation of neural network models. Technical Report TR-328, Cambridge University Engineering Department, Cambridge, England, November 1998. [2] Z. Ghahramani and S. T. Roweis. Learning nonlinear dynamical systems using an EM algorithm. In M. J. Keams, S. A. Solla, and D. A. Cohn, editors, Advances in Neural Information Processing Systems II: Proceedings of the 1998 Conference. MIT Press, 1999. [3] S. J. Julier and J. K. Uhlmann. A New Extension of the Kalman Filter to Nonlinear Systems. In Proc. of AeroSense: The 11th International Symposium on Aerospace/Defence Sensing. Simulation and Controls. Orlando. Florida., 1997. [4] R. E. Kopp and R. J. Orford. Linear regression applied to system identification for adaptive control systems. AlAA 1., I :2300-06, October 1963. [5] M. B. Matthews and G. S. Moschytz. Neural-network nonlinear adaptive filtering using the extended Kalman filter algorithm. In INNC, pages 115-8, 1990. [6] A. T. Nelson. Nonlinear Estimation and Modeling of Noisy Time-Series by Dual Kalman Filtering Methods. PhD thesis, Oregon Graduate Institute, 1999. In preparation. [7] S. Singhal and L. Wu. Training multilayer perceptrons with the extended Kalman filter. In Advances in Neural Information Processing Systems 1, pages 133-140, San Mateo, CA, 1989. Morgan Kauffman. [8] E. A. Wan and A. T. Nelson. Dual Kalman filtering methods for nonlinear prediction, estimation, and smoothing. In Advances in Neural Information Processing Systems 9, 1997.
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Learning Statistically Neutral Tasks without Expert Guidance Ton Weijters Information Technology, Eindhoven University, The Netherlands Antal van den Bosch ILK, Tilburg University, The Netherlands Abstract Eric Postma Computer Science, Universiteit Maastricht, The Netherlands In this paper, we question the necessity of levels of expert-guided abstraction in learning hard, statistically neutral classification tasks. We focus on two tasks, date calculation and parity-12, that are claimed to require intermediate levels of abstraction that must be defined by a human expert. We challenge this claim by demonstrating empirically that a single hidden-layer BP-SOM network can learn both tasks without guidance. Moreover, we analyze the network's solution for the parity-12 task and show that its solution makes use of an elegant intermediary checksum computation. 1 Introduction Breaking up a complex task into many smaller and simpler subtasks facilitates its solution. Such task decomposition has proved to be a successful technique in developing algorithms and in building theories of cognition. In their study and modeling of the human problem-solving process, Newell and Simon [1] employed protocol analysis to determine the subtasks human subjects employ in solving a complex task. Even nowadays, many cognitive scientists take task decomposition, Le., the necessity of explicit levels of abstraction, as a fundamental property of human problem solving. Dennis Norris' [2] modeling study on the problem-solving capacity of autistic savants is a case in point. In the study, Norris focuses on the date-calculation task (Le., to calculate the day of the week a given date fell on), which some autistic savants have been reported to perform flawlessly [3]. In an attempt to train a multi-layer neural network on the task, Norris failed to get a satisfactory level of generalization performance. Only by decomposing the task into three sub-tasks, and training the separate networks on each of the sub-tasks, the date-calculation task could be learned. Norris concluded that the date-calculation task is solvable (learnable) only when it is decomposed into intermediary steps using human assistance [2]. The date-calculation task is a very hard task for inductive learning algorithms, because it is a statistically neutral task: all conditional output probabilities on any input feature have chance values. Solving the task implies decomposing it, if possible, into subtasks that are not statistically neutral. The only suggested decomposition of the date-calculation task known to date involves explicit assistance 74 MFN T. Weijters, A. v. d. Bosch and E. Postma SOM • - class A elements o - class B elements I!I - unlabelled element Figure 1: An example BP-SOM network. from a human supervisor [2J. This paper challenges the decomposition assumption by showing that the date-calculation task can be learned in a single step with a appropriately constrained single hidden-layer neural network. In addition, another statistically neutral task, called the parity-n task (given an n-Iength bit string of 1 's and O's, calculate whether the number of 1 's is even or odd) is investigated. In an experimental study by Dehaene, Bossini, and Giraux [4], it is claimed that humans decompose the parity-n task by first counting over the input string, and then perform the even/odd decision. In our study, parity-12 is shown to be learnable by a network with a single hidden layer. 2 BP-SOM Below we give a brief characterization of the functioning of BP-SOM. For details we refer to [5J. The aim of the BP-SOM learning algorithm is to establish a cooperation between BP learning and SOM learning in order to find adequately constrained hidden-layer representations for learning classification tasks. To achieve this aim, the traditional MFN architecture [6J is combined with SOMS [7]: each hidden layer of the MFN is associated with one SOM (See Figure 1). During training of the weights in the MFN, the corresponding SOM is trained on the hidden-unit activation patterns. After a number of training cycles of BP-SOM learning, each SOM develops a twodimensional representation, that is translated into classification information, i.e., each SOM element is provided with a class label (one of the output classes of the task). For example, let the BP-SOM network displayed in Figure 1 be trained on a classification task which maps instances to either output class A or B. Three types of elements can be distinguished in the SOM: elements labelled with class A, elements labelled with class B, and unlabelled elements (no winning class could be found). The two-dimensional representation of the SOM is used as an addition to the standard BP learning rule [6J. Classification and reliability information from the SOMS is included when updating the connection weights of the MFN. The error of a hidden-layer vector is an accumulation of the error computed by the BP learning rule, and a SOM-error. The SOM-error is the difference between the hidden-unit activation vector and the vector of its best-matching element associated with the same class on the SOM. An important effect of including SOM information in the error signals is that clusters of hidden-unit activation vectors of instances associated with the same class tend to become increasingly similar to each other. On top of this effect, individual hidden-unit activations tend to become more streamlined, and often end up having activations near one of a limited number of discrete values. Learning Statistically Neutral Tasks without Expert Guidance 75 3 The date-calculation task The first statistically neutral calculation task we consider is the date-calculation task: determining the day of the week on which a given date fell. (For instance, October 24, 1997 fell on a Friday.) Solving the task requires an algorithmic approach that is typically hard for human calculators and requires one or more intermediate steps. It is generally assumed that the identity of these intermediate steps follows from the algorithmic solution, although variations exist in the steps as reportedly used by human experts [2]. We will show that such explicit abstraction is not needed, after reviewing the case for the necessity of "human assistance" in learning the task. 3.1 Date calculation with expert-based abstraction Norris [2] attempted to model autistic savant date calculators using a multi-layer feedforward network (MFN) and the back-propagation learning rule [6]. He intended to build a model mimicking the behavior of the autistic savant without the need either to develop arithmetical skills or to encode explicit knowledge about regularities in the structure of dates. A standard multilayer network trained with backpropagation [6] was not able to solve the date-calculation task. Although the network was able to learn the examples used for training, it did not manage to generalize to novel date-day combinations. In a second attempt Norris split up the date-calculation task in three simpler subtasks and networks. U sing the three-stage learning strategy Norris obtained a nearly perfect performance on the training material and a performance of over 90% on the test material (errors are almost exclusively made on dates falling in January or February in leap years). He concludes with the observation that "The only reason that the network was able to learn so well was because it had some human assistance." [2, p.285]. In addition, Norris claims that "even if the [backpropagation] net did have the right number of layers there would be no way for the net to distribute its learning throughout the net such that each layer learned the appropriate step in computation." [2, p. 290]. 3.2 Date calculation without expert-based abstraction We demonstrate that with the BP-SOM learning rule, a single hidden-layer feedforward network can become a successful date calculator. Our experiment compares three types of learning: standard backpropagation learning (BP, [6]), backpropagation learning with weight decay (BPWD, [8]), and BP-SOM learning. Norris used BP learning in his experiment which leads to overfitting [2] (a considerably lower generalization accuracy on new material as compared to reproduction accuracy on training material); BPWD learning was included to avoid overfitting. The parameter values for BP (including the number of hidden units for each task) were optimized by performing pilot experiments with BP. The optimal learning-rate and momentum values were 0.15 and 0.4, respectively. BP, BPWD, and BP-SOM were trained for a fixed number of cycles m = 2000. Early stopping, a common method to prevent overfitting, was used in all experiments with BP, BPWD, and BP-SOM [9]. In our experiments with BP-SOM, we used the same interval of dates as used by Norris, i.e., training and test dates ranged from January 1, 1950 to December 31, 1999. We generated two training sets, each consisting of 3,653 randomly selected instances, i.e., one-fifth of all dates. We also generated two corresponding test sets and two validation sets (with 1,000 instances each) of new dates within the same 50-year period. In all our experiments, the training set, test set, and validation set 76 T. Weijters, A. v. d. Bosch and E. Postma Table 1: Average generalization performances (plus standard deviation, after '±'; averaged over ten experiments) in terms of incorrectly-processed training and test instances, of BP, BPWD, and BP-SOM, trained on the date-calculation task and the parity-12 task. had empty intersections. We partitioned the input into three fields, representing the day of the month (31 units), the month (12 units) and the year (50 units). The output is represented by 7 units, one for each day of the week. The MFN contained one hidden layer with 12 hidden units for BP, and 25 hidden units for BPWD and BP-SOM. The SOM of the BP-SOM network contained 12 x 12 elements. Each of the three learning types was tested on two different data sets. Five runs with different random weight initializations were performed on each set, yielding ten runs per learning type. The averaged classification errors on the test material are reported in Table 1. From Table 1 it follows that the average classification error of BP is high: on test instances BP yields a classification error of 28.8%, while the classification error of BP on training instances is 20.8%. Compared to the classification error of BP, the classification errors on both training and test material of BPWD and BP-SOM are much lower. However, BPWD'S generalization performance on the test material is considerably worse than its performance on the training material: a clear indication of overfitting. We note in passing that the results of BPWD contrast with Norris' [2J claim that BP is unable to learn the date-calculation task when it is not decomposed into subtasks. The inclusion of weight decay in BP is sufficient for a good approximation of the performance results of Norris' decomposed network. The results in Table 1 also show that the performance of BP-SOM on test material is significantly better than that of BPWD (t(19)=7.39, p<O.OOl); BP-SOM has learned the date-calculation task at a level well beyond the average of human date calculators as reported by Norris [2J. In contrast with Norris' pre-structured network, BP-SOM does not rely on expert-based levels of abstraction for learning the date-calculation task. 4 The parity-12 task The parity-n problem, starting from the XOR problem (parity-2), continues to be a relevant topic on the agenda of many neural network and machine learning researchers. Its definition is simple (determine whether there is an odd or even number of l's in an n-Iength bit string of l's and O's), but established state-of-the-art algorithms such as C4.5 [1OJ and backpropagation [6J cannot learn it even with small n, i.e., backpropagation fails with n 2: 4 [l1J. That is, these algorithms are unable to generalize from learning instances of a parity-n task to unseen new instances of the same task. As with date calculation, this is due to the statistical neutrality of the task. The solution of the problem must lie in having some comprehensive overview over all input values at an intermediary step before the odd/even decision is made. Indeed, humans appear to follow this strategy [4J. Learning Statistically Neutral Tasks without Expert Guidance 77 BP BPWD BP-SOM Figure 2: Graphic representation of a 7 x 7 SOM associated with a BP-trained MFN (left) and a BPwD-trained MFN (middle), and a 7 x 7 SOM associated with a BP-SOM network (right), all trained on the parity-12 task_ Analogous to our study of the date-calculation task presented in Section 3, we apply BP, BPWD, and BP-SOM to the parity-n task_ We have selected n to be 12_ The training set contained 1,000 different instances selected at random out of the set of 4,096 possible bit strings. The test set and the validation set contained 100 new instances each. The hidden layer of the MFN in all three algorithms contained 20 hidden units, and the SOM in BP-SOM contained 7 x 7 elements. The algorithms were run with 10 different random weight initializations. Table 1 displays the classification errors on training instances and test instances. Analysis of the results shows that BP-SOM performs significantly better than BP and BPWD on test material (t(19)=3.42, p<O.Ol and t(19)=2.42, p<0.05, respectively). (The average error of 6.2% made by BP-SOM stems from a single experiment out of the ten performing at chance level, and the remaining nine yielding about 1 % error). BP-SOM is able to learn the parity-12 task quite accurately; BP and BPWD fail relatively, which is consistent with other findings [11]. As an additional analysis, we have investigated the differences in hidden unit activations after training with the three learning algorithms. To visualize the differences between the representations developed at the hidden layers of the MFNS trained with BP, BPWD, and BP-SOM, we also trained SOMs with the hidden layer activities of the trained BP and BPWD networks. The left part of Figure 2 visualizes the class labelling of the SOM attached to the BP-trained MFN after training; the middle part visualizes the SOM of the BpwD-trained MFN, and the right part displays the SOM of the BP-SOM network after training on the same material. The SOM of the BP-SOM network is much more organized and clustered than that of the SOMs corresponding with the BP-trained and BpwD-trained MFNS. The reliability values of the elements of all three SOMS are represented by the width of the black and white squares. It can be seen that the overall reliability and the degree of clusteredness of the SOM of the BP-SOM network is considerably higher than that of the SOM of the BP-trained and BpwD-trained MFNS. 5 How parity-12 is learned Given the hardness of the task and the supposed necessity of expert guidance, and given BP-SOM'S success in learning parity-12 in contrast, it is relevant to analyze what solution was found in the BP-SOM learning process. In this subsection we provide such an analysis, and show that the trained network performs an elegant checksum calculation at the hidden layer as the intermediary step_ All elements of SOMS of BP-SOM networks trained on the paritY-12 task are either the prototype for training instances that are all labeled with the same class, or 78 T. Weijters, A. v. d. Bosch and E. Postma Table 2: List of some training instances of the parity-12 task associated with SOM elements (1,1), (2,4), and (3,3) of a trained BP-SOM network. SOM (1,1), class-even, reliability 1.0 inl in2 in3 in4 in5 in6 in7 inS in9 inl0 inll in12 checksum 1 1 0 0 0 0 0 0 0 0 0 0 -2 0 0 1 0 0 0 1 0 1 1 0 0 -2 1 1 0 1 0 0 0 1 0 0 0 0 -2 SOM 12,4), class-odd, reliability 1.0 inl in2 in3 in4 in5 in6 in7 inS in9 inl0 inll in12 checksum 0 1 1 1 1 0 1 1 0 1 0 0 -1 1 1 1 0 1 1 1 0 1 1 0 1 -1 1 0 1 1 0 1 0 1 1 0 1 0 -1 :OM (3,3), class-even, reliability 1.0 inl in2 in3 in4 in5 in6 in7 inS in9 inl0 inll in12 checksum 0 0 1 1 0 0 1 1 0 1 0 1 0 1 1 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 1 1 0 1 1 0 0 1 0 II - 1+ + + I - I + + + II II prototype of no instances at all. Non-empty elements (the black and white squares in the right part of Figure 2) can thus be seen as containers of homogeneouslylabeled subsets of the training set (i.e., fully reliable elements). The first step of our analysis consists of collecting, after training, for each non-empty SOM element all training instances clustered at that SOM element. As an illustration, Table 2 lists some training instances clustered at the SOM elements at coordinates (1,1), (2,4), and (3,3). At first sight the only common property of instances associated with the same SOM element is the class to which they belong; e.g., all instances of SOM element (1,1) are even, all instances of SOM element (2,4) are odd, and all instances of SOM element (3,3) are again even. The second step of our analysis focuses on the sign of the weights of the connections between input and hidden units. Surprisingly, we find that the connections of each individual input unit to all hidden units have the same sign; each input unit can therefore be labeled with a sign marker (as displayed at the bottom of Table 2). This allows the clustering on the SOM to become interpretable. All weights from input unit 1,2,3, 7,8, and 9 to all units of the hidden layer are negative, all weights from input unit 4,5,6, 10, 11, and 12 to all units of the hidden layer are positive. At the hidden layer, this information is gathered as if a checksum is computed; each SOM element contains instances that add up to an identical checksum. This can already be seen using only the sign information rather than the specific weights. For instance, all instances clustered at SOM element (1,1) lead to a checksum of -2 when a sum is taken of the product of all input values with all weight signs. Analogously, all instances of cluster (2,4) count up to -1 and the instances of cluster (3,3) to zero. The same regularity is present in the instances of the other SOM elements. In sum, the BP-SOM solution to the parity-12 task can be interpreted as to transform it at the hidden layer into the mapping of different, approximately discrete, checksums to either class 'even' or 'odd'. Learning Statistically Neutral Tasks without Expert Guidance 79 6 Conclusions We have performed two learning experiments in which the BP-SOM learning algorithm was trained on the date-calculation task and on the parity-12 task. Both tasks are hard to learn because they are statistically neutral, but can be learned adequately and without expert guidance by the BP-SOM learning algorithm. The effect of the SOM part in BP-SOM (adequately constrained hidden-layer vectors, reliable clustering of vectors on the SOM, and streamlined hidden-unit activations) clearly contributes to this success. From the results of the experiments on the date-calculation task, we conclude that Norris' claim that, without human assistance, a backpropagation net would never learn the date-calculation task is inaccurate. While BP with weight decay performs at Norris' target level of accuracy, BP-SOM performs even better. Apparently BPSOM is able to distribute its learning throughout the net such that the two parts of the network (from input layer to hidden layer, and from hidden layer to output layer) perform the mapping with an appropriate intermediary step. The parity-12 experiment exemplified that such a discovered intermediary step can be quite elegant; it consists of the computation of a checksum via the connection weights between the input and hidden layers. Unfortunately, a similar elegant simplicity was not found in the connection weights and SOM clustering of the date calculation task; future research will be aimed at developing more generic analyses for trained BP-SOM networks, so that automatically-discovered intermediary steps may be made understandably explicit. References [1] Newell, A. and Simon, H.A. (1972) Human problem solving. Engelwood Cliffs, NJ: Prentice-Hall. [2] Norris, D. (1989). How to build a connectionist idiot (savant). Cognition, 35, 277-291. [3] Hill, A. L. (1975). An investigation of calendar calculating by an idiot savant. American Journal of Psychiatry, 132, 557- 560. [4] Dehaene, P., Bossini, S., and Giraux, P. (1993) . The mental representation of parity and numerical magnitude. Journal of Experimental Psychology: General, 122, 371396. [5] Weijters, A., Van den Bosch, A., Van den Herik, H. J. (1997). Behavioural Aspects of Combining Backpropagation Learning and Self-organizing Maps. Connection Science, 9,235-252. [6] Rumelhart, D. E., Hinton, G. E., and Williams, R. J. (1986). Learning internal representations by error propagation. In D. E. Rumelhart and J. L. McClelland (Eds.), Parallel Distributed Processing: Explorations in the Microstructure of Cognition, volume 1: Foundations (pp. 318-362). Cambridge, MA: The MIT Press. [7] Kohonen, T. (1989). Self-organisation and Associative Memory. Berlin: Springer Verlag. [8] Hinton, G. E. (1986). Learning distributed representations of concepts. In Proceedings of the Eighth Annual Conference of the Cognitive Science Society, 1-12. Hillsdale, NJ: Erlbaum. [9] Prechelt, L. (1994). Probenl: A set of neural network benchmark problems and benchmarking rules. Technical Report 24/94, Fakultat fUr Informatik, Universitat Karlsruhe, Germany. [10] Quinlan, J. R. (1993). C4.5: Programs for Machine Learning. San Mateo, CA: Morgan Kaufmann. [11] Thornton, C. (1996). Parity: the problem that won't go away. In G. McCalla (Ed.), Proceeding of AI-96, Toronto, Canada (pp. 362-374). Berlin: Springer Verlag.
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Correctness of belief propagation in Gaussian graphical models of arbitrary topology Yair Weiss Computer Science Division UC Berkeley, 485 Soda Hall Berkeley, CA 94720-1776 Phone: 510-642-5029 yweiss@cs.berkeley.edu William T. Freeman Mitsubishi Electric Research Lab 201 Broadway Cambridge, MA 02139 Phone: 617-621-7527 freeman @merl.com Abstract Local "belief propagation" rules of the sort proposed by Pearl [15] are guaranteed to converge to the correct posterior probabilities in singly connected graphical models. Recently, a number of researchers have empirically demonstrated good performance of "loopy belief propagation"using these same rules on graphs with loops. Perhaps the most dramatic instance is the near Shannon-limit performance of "Turbo codes", whose decoding algorithm is equivalent to loopy belief propagation. Except for the case of graphs with a single loop, there has been little theoretical understanding of the performance of loopy propagation. Here we analyze belief propagation in networks with arbitrary topologies when the nodes in the graph describe jointly Gaussian random variables. We give an analytical formula relating the true posterior probabilities with those calculated using loopy propagation. We give sufficient conditions for convergence and show that when belief propagation converges it gives the correct posterior means for all graph topologies, not just networks with a single loop. The related "max-product" belief propagation algorithm finds the maximum posterior probability estimate for singly connected networks. We show that, even for non-Gaussian probability distributions, the convergence points of the max-product algorithm in loopy networks are maxima over a particular large local neighborhood of the posterior probability. These results help clarify the empirical performance results and motivate using the powerful belief propagation algorithm in a broader class of networks. Problems involving probabilistic belief propagation arise in a wide variety of applications, including error correcting codes, speech recognition and medical diagnosis. If the graph is singly connected, there exist local message-passing schemes to calculate the posterior probability of an unobserved variable given the observed variables. Pearl [15] derived such a scheme for singly connected Bayesian networks and showed that this "belief propagation" algorithm is guaranteed to converge to the correct posterior probabilities (or "beliefs"). Several groups have recently reported excellent experimental results by running algorithms 674 Y. Weiss and W T. Freeman equivalent to Pearl's algorithm on networks with loops [8, 13, 6]. Perhaps the most dramatic instance of this performance is for "Turbo code" [2] error correcting codes. These codes have been described as "the most exciting and potentially important development in coding theory in many years" [12] and have recently been shown [10, 11] to utilize an algorithm equivalent to belief propagation in a network with loops. Progress in the analysis of loopy belief propagation has been made for the case of networks with a single loop [17, 18, 4, 1]. For these networks, it can be shown that (1) unless all the compatabilities are deterministic, loopy belief propagation will converge. (2) The difference between the loopy beliefs and the true beliefs is related to the convergence rate of the messages the faster the convergence the more exact the approximation and (3) If the hidden nodes are binary, then the loopy beliefs and the true beliefs are both maximized by the same assignments, although the confidence in that assignment is wrong for the loopy beliefs. In this paper we analyze belief propagation in graphs of arbitrary topology, for nodes describing jointly Gaussian random variables. We give an exact formula relating the correct marginal posterior probabilities with the ones calculated using loopy belief propagation. We show that if belief propagation converges, then it will give the correct posterior means for all graph topologies, not just networks with a single loop. We show that the covariance estimates will generally be incorrect but present a relationship between the error in the covariance estimates and the convergence speed. For Gaussian or non-Gaussian variables, we show that the "max-product" algorithm, which calculates the MAP estimate in singly connected networks, only converges to points that are maxima over a particular large neighborhood of the posterior probability of loopy networks. 1 Analysis To simplify the notation, we assume the graphical model has been preprocessed into an undirected graphical model with pairwise potentials. Any graphical model can be converted into this form, and running belief propagation on the pairwise graph is equivalent to running belief propagation on the original graph [18]. We assume each node X i has a local observation Yi . In each iteration of belief propagation, each node X i sends a message to each neighboring X j that is based on the messages it received from the other neighbors, its local observation Yl and the pairwise potentials Wij(Xi , Xj) and Wii(Xi, Yi) . We assume the message-passing occurs in parallel. The idea behind the analysis is to build an unwrapped tree. The unwrapped tree is the graphical model which belief propagation is solving exactly when one applies the belief propagation rules in a loopy network [9, 20, 18]. It is constructed by maintaining the same local neighborhood structure as the loopy network but nodes are replicated so there are no loops. The potentials and the observations are replicated from the loopy graph. Figure 1 (a) shows an unwrapped tree for the diamond shaped graph in (b). By construction, the belief at the root node X-I is identical to that at node Xl in the loopy graph after four iterations of belief propagation. Each node has a shaded observed node attached to it, omitted here for clarity. Because the original network represents jointly Gaussian variables, so will the unwrapped tree. Since it is a tree, belief propagation is guaranteed to give the correct answer for the unwrapped graph. We can thus use Gaussian marginalization formulae to calculate the true mean and variances in both the original and the unwrapped networks. In this way, we calculate the accuracy of belief propagation for Gaussian networks of arbitrary topology. We assume that the joint mean is zero (the means can be added-in later). The joint distriCorrectness of Belief Propagation 675 Figure 1: Left: A Markov network with mUltiple loops. Right: The unwrapped network corresponding to this structure. bution of z = ( : ) is given by P(z) = ae-!zTVz, where V = (~:: ~::) . It is straightforward to construct the inverse covariance matrix V of the joint Gaussian that describes a given Gaussian graphical model [3]. Writing out the exponent of the joint and completing the square shows that the mean I-' of x, given the observations y, is given by: (1) and the covariance matrix C~IY of x given y is: C~IY = V~-;l. We will denote by C~dY the ith row of C~IY so the marginal posterior variance of Xi given the data is (72 (i) = C~i Iy (i). We will use - for unwrapped quantities. We scan the tree in breadth first order and denote by x the vector of values in the hidden nodes of the tree when so scanned. Simlarly, we denote by y the observed nodes scanned in the same order and Vn , V~y the inverse covariance matrices. Since we are scanning in breadth first order the last nodes are the leaf nodes and we denote by L the number of leaf nodes. By the nature of unwrapping, tL(1) is the mean of the belief at node Xl after t iterations of belief propagation, where t is the number of unwrappings. Similarly 0-2 (1) = 6~1Iy(1) is the variance of the belief at node Xl after t iterations. Because the data is replicated we can write y = Oy where O(i, j) = 1 if Yi is a replica of Yj and 0 otherwise. Since the potentials W(Xi' Yi) are replicated, we can write V~yO = OV~y. Since the W (Xi, X j) are also replicated and all non-leaf Xi have the same connectivity as the corresponding Xi, we can write V~~O = OVzz + E where E is zero in all but the last L rows. When these relationships between the loopy and unwrapped inverse covariance matrices are substituted into the loopy and unwrapped versions of equation I, one obtains the following expression, true for any iteration [19]: (2) where e is a vector that is zero everywhere but the last L components (corresponding to the leaf nodes). Our choice of the node for the root of the tree is arbitrary, so this applies to all nodes of the loopy network. This formula relates, for any node of a network with loops, the means calculated at each iteration by belief propagation with the true posterior means. Similarly when the relationship between the loopy and unwrapped inverse covariance matrices is substituted into the loopy and unwrapped definitions of C~IY we can relate the 676 Y Weiss and W T Freeman 0.5 0.4 ~ 0.3 ~ .~ 0.2 n; ~ 0.1 8 "t:> § 0 -0.1 -0.2 0 20 40 60 80 100 node Figure 2: The conditional correlation between the root node and all other nodes in the unwrapped tree of Fig. 1 after eight iterations. Potentials were chosen randomly. Nodes are presented in breadth first order so the last elements are the correlations between the root node and the leaf nodes. We show that if this correlation goes to zero, belief propagation converges and the loopy means are exact. Symbols plotted with a star denote correlations with nodes that correspond to the node Xl in the loopy graph. The sum of these correlations gives the correct variance of node Xl while loopy propagation uses only the first correlation. marginalized covariances calculated by belief propagation to the true ones [19]: -2 2 a (1) = a (1) + CZllyel - Czt/ye2 (3) where el is a vector that is zero everywhere but the last L components while e2 is equal to 1 for all nodes in the unwrapped tree that are replicas of Xl except for Xl. All other components of e2 are zero, Figure 2 shows Cz1lY for the diamond network in Fig. 1. We generated random potential functions and observations and calculated the conditional correlations in the unwrapped tree. Note that the conditional correlation decreases with distance in the tree we are scanning in breadth first order so the last L components correspond to the leaf nodes. As the number of iterations of loopy propagation is increased the size of the unwrapped tree increases and the conditional correlation between the leaf nodes and the root node decreases. From equations 2-3 it is clear that if the conditional correlation between the leaf nodes and the root nodes are zero for all sufficiently large unwrappings then (1) belief propagation converges (2) the means are exact and (3) the variances may be incorrect. In practice the conditional correlations will not actually be equal to zero for any finite unwrapping. In [19] we give a more precise statement: if the conditional correlation of the root node and the leaf nodes decreases rapidly enough then (1) belief propagation converges (2) the means are exact and (3) the variances may be incorrect. We also show sufficient conditions on the potentials III (Xi, X j) for the correlation to decrease rapidly enough: the rate at which the correlation decreases is determined by the ratio of off-diagonal and diagonal components in the quadratic fonn defining the potentials [19]. How wrong will the variances be? The tenn CZllye2 in equation 3 is simply the sum of many components of Cz11y. Figure 2 shows these components. The correct variance is the sum of all the components witHe the belief propagation variance approximates this sum with the first (and dominant) tenn. Whenever there is a positive correlation between the root node and other replicas of Xl the loopy variance is strictly less than the true variance the loopy estimate is overconfident. Correctness of Belief Propagation (a) ~07 e iDO.6 " ., ;;;05 fr ~04 '" ~03 0.2 0.1 677 SOR 20 30 40 50 60 iterations (b) Figure 3: (a) 25 x 25 graphical model for simulation. The unobserved nodes (unfilled) were connected to their four nearest neighbors and to an observation node (filled). (b) The error of the estimates of loopy propagation and successive over-relaxation (SOR) as a function of iteration. Note that belief propagation converges much faster than SOR. Note that when the conditional correlation decreases rapidly to zero two things happen. First, the convergence is faster (because CZdyel approaches zero faster). Second, the approximation error of the variances is smaller (because CZ1 /y e2 is smaller). Thus we have shown, as in the single loop case, quick convergence is correlated with good approximation. 2 Simulations We ran belief propagation on the 25 x 25 2D grid of Fig. 3 a. The joint probability was: (4) where Wij = 0 if nodes Xi, Xj are not neighbors and 0.01 otherwise and Wii was randomly selected to be 0 or 1 for all i with probability of 1 set to 0.2. The observations Yi were chosen randomly. This problem corresponds to an approximation problem from sparse data where only 20% of the points are visible. We found the exact posterior by solving equation 1. We also ran belief propagation and found that when it converged, the calculated means were identical to the true means up to machine precision. Also, as predicted by the theory, the calculated variances were too small the belief propagation estimate was overconfident. In many applications, the solution of equation 1 by matrix inversion is intractable and iterative methods are used. Figure 3 compares the error in the means as a function of iterations for loopy propagation and successive-over-relaxation (SOR), considered one of the best relaxation methods [16]. Note that after essentially five iterations loopy propagation gives the right answer while SOR requires many more. As expected by the fast convergence, the approximation error in the variances was quite small. The median error was 0.018. For comparison the true variances ranged from 0.01 to 0.94 with a mean of 0.322. Also, the nodes for which the approximation error was worse were indeed the nodes that converged slower. 678 Y. Weiss and W T Freeman 3 Discussion Independently, two other groups have recently analyzed special cases of Gaussian graphical models. Frey [7] analyzed the graphical model corresponding to factor analysis and gave conditions for the existence of a stable fixed-point. Rusmevichientong and Van Roy [14] analyzed a graphical model with the topology of turbo decoding but a Gaussian joint density. For this specific graph they gave sufficient conditions for convergence and showed that the means are exact. Our main interest in the Gaussian case is to understand the performance of belief propagation in general networks with multiple loops. We are struck by the similarity of our results for Gaussians in arbitrary networks and the results for single loops of arbitrary distributions [18]. First, in single loop networks with binary nodes, loopy belief at a node and the true belief at a node are maximized by the same assignment while the confidence in that assignment is incorrect. In Gaussian networks with multiple loops, the mean at each node is correct but the confidence around that mean may be incorrect. Second, for both singleloop and Gaussian networks, fast belief propagation convergence correlates with accurate beliefs. Third, in both Gaussians and discrete valued single loop networks, the statistical dependence between root and leaf nodes governs the convergence rate and accuracy. The two models are quite different. Mean field approximations are exact for Gaussian MRFs while they work poorly in sparsely connected discrete networks with a single loop. The results for the Gaussian and single-loop cases lead us to believe that similar results may hold for a larger class of networks. Can our analysis be extended to non-Gaussian distributions? The basic idea applies to arbitrary graphs and arbitrary potentials: belief propagation is performing exact inference on a tree that has the same local neighbor structure as the loopy graph. However, the linear algebra that we used to calculate exact expressions for the error in belief propagation at any iteration holds only for Gaussian variables. We have used a similar approach to analyze the related "max-product" belief propagation algorithm on arbitrary graphs with arbitrary distributions [5] (both discrete and continuous valued nodes). We show that if the max-product algorithm converges, the max-product assignment has greater posterior probability then any assignment in a particular large region around that assignment. While this is a weaker condition than a global maximum, it is much stronger than a simple local maximum of the posterior probability. The sum-product and max-product belief propagation algorithms are fast and parallelizable. Due to the well known hardness of probabilistic inference in graphical models, belief propagation will obviously not work for arbitrary networks and distributions. Nevertheless, a growing body of empirical evidence shows its success in many networks with loops. Our results justify applying belief propagation in certain networks with mUltiple loops. This may enable fast, approximate probabilistic inference in a range of new applications. References [1] S.M. Aji, G.B. Hom, and R.J. McEliece. On the convergence of iterative decoding on graphs with a single cycle. In Proc. 1998 ISIT, 1998. [2] c. Berrou, A. Glavieux, and P. Thitimajshima. Near Shannon limit error-correcting coding and decoding: Turbo codes. In Proc. IEEE International Communications Conference '93, 1993. [3] R. Cowell. Advanced inference in Bayesian networks. In M.1. Jordan, editor, Learning in Graphical Models. MIT Press, 1998. [4] G.D. Forney, F.R. Kschischang, and B. Marcus. Iterative decoding of tail-biting trellisses. preprint presented at 1998 Information Theory Workshop in San Diego, 1998. Correctness of Belief Propagation 679 [5] W. T. Freeman and Y. Weiss. On the fixed points of the max-product algorithm. Technical Report 99-39, MERL, 201 Broadway, Cambridge, MA 02139, 1999. [6] W.T. Freeman and E.C. Pasztor. Learning to estimate scenes from images. In M.S. Kearns, S.A. SoUa, and D.A. Cohn, editors, Adv. Neural Information Processing Systems I I. MIT Press, 1999. [7] B.J. Frey. Turbo factor analysis. In Adv. Neural Information Processing Systems 12. 2000. to appear. [8) Brendan J. Frey. Bayesian Networksfor Pattern Classification, Data Compression and Channel Coding. MIT Press, 1998. [9) R.G. Gallager. Low Density Parity Check Codes. MIT Press, 1963. [10) F. R. Kschischang and B. J. Frey. Iterative decoding of compound codes by probability propagation in graphical models. IEEE Journal on Selected Areas in Communication , 16(2):219-230, 1998. [11] R.J. McEliece, D.J.C. MackKay, and J.F. Cheng. Turbo decoding as as an instance of Pearl's 'belief propagation' algorithm. IEEE Journal on Selected Areas in Communication, 16(2): 140152,1998. [12J R.J. McEliece, E. Rodemich, and J.F. Cheng. The Turbo decision algorithm. In Proc. 33rd Allerton Conference on Communications, Control and Computing, pages 366-379, Monticello, IL, 1995. [I3J K.P. Murphy, Y. Weiss, and M.1. Jordan. Loopy belief propagation for approximate inference: an empirical study. In Proceedings of Uncertainty in AI, 1999. [14] Rusmevichientong P. and Van Roy B. An analysis of Turbo decoding with Gaussian densities. In Adv. Neural Information Processing Systems I2 . 2000. to appear. [15) Judea Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, 1988. [16J Gilbert Strang. Introduction to Applied Mathel1Ultics. Wellesley-Cambridge, 1986. [I7J Y. Weiss. Belief propagation and revision in networks with loops. Technical Report 1616, MIT AI lab, 1997. [18J Y. Weiss. Correctness of local probability propagation in graphical models with loops. Neural Computation, to appear, 2000. [19] Y. Weiss and W. T. Freeman. Loopy propagation gives the correct posterior means for Gaussians. Technical Report UCB.CSD-99-1046, Berkeley Computer Science Dept., 1999. www.cs.berkeley.edu yweiss/. [20J N. Wiberg. Codes and decoding on general graphs. PhD thesis, Department of Electrical Engineering, U. Linkoping, Sweden, 1996. Gaussian Fields for Approximate Inference in Layered Sigmoid Belief Networks David Barber'" Stichting Neurale Netwerken Medical Physics and Biophysics Nijmegen University, The Netherlands barberdOaston.ac.uk Peter Sollich Department of Mathematics King's College, University of London London WC2R 2LS, U.K. peter.sollichOkcl.ac.uk Abstract Layered Sigmoid Belief Networks are directed graphical models in which the local conditional probabilities are parameterised by weighted sums of parental states. Learning and inference in such networks are generally intractable, and approximations need to be considered. Progress in learning these networks has been made by using variational procedures. We demonstrate, however, that variational procedures can be inappropriate for the equally important issue of inference - that is,· calculating marginals of the network. We introduce an alternative procedure, based on assuming that the weighted input to a node is approximately Gaussian distributed. Our approach goes beyond previous Gaussian field assumptions in that we take into account correlations between parents of nodes. This procedure is specialized for calculating marginals and is significantly faster and simpler than the variational procedure. 1 Introduction Layered Sigmoid Belief Networks [1] are directed graphical models [2] in which the local conditional probabilities are parameterised by weighted sums of parental states, see fig ( 1). This is a graphical representation of a distribution over a set of binary variables Si E {a, I}. Typically, one supposes that the states of the nodes at the bottom of the network are generated by states in previous layers. Whilst, in principle, there is no restriction on the number of nodes in any layer, typically, one considers structures similar to the "fan out" in fig(l) in which higher level layers provide an "explanation" for patterns generated in lower layers. Such graphical models are attractive since they correspond to layers of information processors, of potentially increasing complexity. Unfortunately, learning and inference in such networks is generally intractable, and approximations need to be considered. Progress in learning has been made by using variational procedures [3,4, 5]. However, another crucial aspect remains inference [2]. That is, given some evidence ( or none), calculate the marginal of a variable, conditional on this evidence. This assumes that we have found a suitable network from some learning procedure, and now wish ·Present Address: NCRG, Aston University, Birmingham B4 7ET, U.K. 394 D. Barber and P. Sollich to query this network. Whilst the variational procedure is attractive for learning, since it generally provides a bound on the likelihood of the visible units, we demonstrate that it may not always be equally appropriate for the inference problem. A directed graphical model defines a distribution over a set of variables s = (S1 ... sn) that factorises into the local conditional distributions, n p(S1 . .. sn) = IIp(silll'i) (1) i=1 where lI'i denotes the parent nodes of node i. In a layered network, these are the nodes in the proceeding layer that feed into node i. In a sigmoid belief network the local probabilities are defined as Figure 1: A Layered Sigmoid Belief Network P (s; = ll~;) = " ( ~ W;jSj + 0;) =" (h;) (2) where the "field" at node i is defined as hi = 2:j WijSj + fh and er(h) = 1/(1 + e- h ). Wij is the strength of the connection between node i and its parent node j; if j is not a parent of i we set Wij = O. Oi is a bias term that gives a parent-independent bias to the state of node i. We are interested in inference - in particular, calculating marginals of the network for cases with and without evidential nodes. In section (2) we describe how to approximate the quantities p(Si = 1) and discuss in section (2.1) why our method can improve on the standard variational mean field theory. Conditional marginals, such as p(Si = IISj = 1, Sk = 0) are considered in section (3). 2 Gaussian Field Distributions Under the 0/1 coding for the variables Si, the mean of a variable, mi is given by the probability that it is in state 1. Using the fact from (2) that the local conditional distribution of node i is dependent on its parents only through its field hi, we have (3) where we use the notation «(-)p to denote an average with respect to the distribution p. If there are many parents of node i, a reasonable assumption is that the distribution of the field hi will be Gaussian, p(hi) ~ N (J,Li,er[). Under this Gaussian Field (GF) assumption, we need to work out the mean and variance, which are given by (4) j j err = ((Llhi)2) = L WijWikRjk (5) j,k where Rjk = (LlSjLlsk). We use the notation Ll (-) == (-) - «(.) . The diagonal terms of the node covariance matrix are ~i = mi (1- mi)' In contrast to previous studies, we include off diagonal terms in the calculation of R [4] . From Gaussian Fields for Approximate Inference 395 (5) we only need to find correlations between parents i and j of a node. These are easy to calculate in the layered networks that we are considering, because neither i nor j is a descendant of the other: Rjj = p(Sj = 1, Sj = 1) - mjmj = J p(Si = Ilhj)p(Sj = Ilhj)p(hj, hj)dh - mimj = (0" (hd 0" (hj ) (h h) - mjmj P J, J (6) (7) (8) Assuming that the joint distribution p( hj , hj ) is Gaussian, we again need its mean and covariance, given by ~ij = (D.hjD.hj) = L WjkWjl (D.skD.SI) = L WikWjlRkl (10) kl kl Under this scheme, we have a closed set of equations, (4,5,8,10) for the means mj and covariance matrix Rij which can be solved by forward propagation of the equations. That is, we start from nodes without parents, and then consider the next layer of nodes, repeating the procedure until a full sweep through the network has been completed. The one and two dimensional field averages, equations (3) and (8), are computed using Gaussian Quadrature. This results in an extremely fast procedure for approximating the marginals mi, requiring only a single sweep through the network. Our approach is related to that of [6] by the common motivating assumption that each node has a large number of parents. This is used in [6] to obtain actual bounds on quantities of interest such as joint marginals. Our approach does not give bounds. Its advantage, however, is that it allows fluctuations in the fields hi, which are effectively excluded in [6] by the assumed scaling of the weights Wij with the number of parents per node. 2.1 Relation to Variational Mean Field Theory In the variational approach, one fits a tractable approximating distribution Q to the SBN. Taking Q factorised, Q(s) = Dj m:' (1 - md l - 3 • we have the bound In p (Sl ... sn) 2: L {-mj In mj - (1 - md In (1 - md} i The final term in (11) causes some difficulty even in the case in which Q is a factorised model. Formally, this is because this term does not have the same graphical structure as the tractable model Q. One way around around this difficulty is to employ a further bound, with associated variational parameters [7]. Another approach is to make the Gaussian assumption for the field hi as in section (2). Because Q is factorised, corresponding to a diagonal correlation matrix R, this gives [4] (12) 396 D. Barber and P Sollich where Pi = ~j Wijmj + Oi and (1[ = ~j w[jmj(l - mj). Note that this is a one dimensional integral of a smooth function. In contrast to [4] we therefore evaluate this quantity using Gaussian Quadrature. This has the advantage that no extra variational parameters need to be introduced. Technically, the assumption of a Gaussian field distribution means that (11) is no longer a bound. Nevertheless, in practice it is found that this has little effect on the quality of the resulting solution. In our implementation of the variational approach, we find the optimal parameters mi by maximising the above equation for each component mi separately, cycling through the nodes until the parameters mi do not change by more than 10- 1°. This is repeated 5 times, and the solution with the highest bound score is chosen. Note that these equations cannot be solved by forward propagation alone since the final term contains contributions from all the nodes in the network. This is in contrast to the GF approach of section (2) . Finding appropriate parameters mi by the variational approach is therefore rather slower than using the GF method. In arriving at the above equations, we have made two assumptions. The first is that the intractable distribution is well approximated by a factorised model. The second is that the field distribution is Gaussian. The first step is necessary in order to obtain a bound on the likelihood of the model (although this is slightly compromised by the Gaussian field assumption). In the GF approach we dispense with this assumption of an effectively factorised network (partially because if we are only interested in inference, a bound on the model likelihood is less relevant). The GF method may therefore prove useful for a broader class of networks than the variational approach. 2.2 Results for unconditional marginals We compared three procedures for estimating the conditional values p(Si = 1) for all the nodes in the network, namely the variational theory, as described in section (2.1), the diagonal Gaussian field theory, and the non-diagonal Gaussian field theory which includes correlation effects between parents. Results for small weight values Wij are shown in fig(2). In this case, all three methods perform reasonably well, although there is a significant improvement in using the GF methods over the variational procedure; parental correlations are not important (compare figs(2b) and (2c)) . In fig(3) the weights and biases are chosen such that the exact mean variables mi are roughly 0.5 with non-trivial correlation effects between parents. Note that the variational mean field theory now provides a poor solution, whereas the GF methods are relatively accurate. The effect of using the non-diagonal R terms is beneficial, although not dramatically so. 3 Calculating Conditional Marginals We consider now how to calculate conditional marginals, given some evidential nodes. (In contrast to [6], any set of nodes in the network, not just output nodes, can be considered evidential.) We write the evidence in the following manner E = {SCi = SCi' . . . Sc" = SC,.} = {ECl ... Ec,.} The quantities that we are interested in are conditional marginals which, from Bayes rule are related to the joint distribution by P (Si = liE) = P (Si = 1, E) P (Si = 0, E) + P (Si = 1, E) (13) That is, provided that we have a procedure for estimating joint marginals, we can obtain conditional marginals too. Without loss of generality, we therefore consider Gaussian Fields for Approximate Inference Em>ruoing1_ model 1ft 2Or--~ (a) Mean error = 0.0377 Em>r using Ga_ Fiaid. Ooagonal ooyariance (b) Mean error = 0.0018 397 cowuiance O<Xll 001 (c) Mean error = 0.0017 Figure 2: Error in approximating p(Si = 1) for the network in fig(l), averaged over all the nodes in the network. In each of 100 trials, weights were drawn from a zero mean, unit variance Gaussian; biases were set to O. Note the different scale in (b) and (c). In (a) we use the variational procedure with a factorised Q, as in section (2.1). In (b) we use the Gaussian field equations, assuming a diagonal covariance matrix R. This procedure was repeated in (c) including correlations between parents. E+ = E U {Si = I}, which then contains n + 1 "evidential" variables. That is, the desired marginal variable is absorbed into the evidence set. For convenience, we then split the nodes into two sets, those containing the evidential or "clamped" nodes, C, and the remaining "free" nodes F . The joint evidence is then given by (14) 8F = I:p (ECllll'~l) ... p (En+llll'~"+l) p (sh 11I'jJ ... p (Sfm 11I'jJ 8F (15) where 11'; are the parents of node i, with any evidential parental nodes set to their values as specified in E+. In the sigmoid belief network if i is an evidential node otherwise (16) p(Eklll'Z) is therefore determined by the distribution of the field hZ = Li WkiS; +Ok . Examining (15), we see that the product over the "free" nodes defines a SBN in which the local probability distributions are given by those of the original network, but with any evidential parental nodes clamped to their evidence values. Therefore, (17) Consistent with our previous assumptions, we assume that the distribution of the fields h+ = (h~l'" h~"+l) is jointly Gaussian. We can then find the mean and covariance matrix for the distribution of h+ by repeating the calculation of section (2) in which evidential nodes have been clamped to their evidence values. Once this Gaussian has been determined, it can be used in (17) to determine p( E+). Gaussian averages of products of sigmoids are calculated by drawing 1000 samples from the Gaussian over which we wish to integrate1 . Note that if there are evidential nodes lIn one and two dimensions (n = 0, 1), or n = 1, we use Gaussian Quadrature. 398 Error uoing lado_ model 1M l00r-~--~----~~--~ eo (a) Mean error = 0.4188 D. Barber and P Sollich Error uoing Ga_ian Field. Diegorel """arianee Em:>< uoing Ga ..... n Field. Non Diagonal ""w"ianee EO 70,---~------~--------, 50 30 20 10 EO 50 40 30 20 10 II.. o 1 0.2 0.3 0.4 0.5 06 00 0 1 02 0.3 0.4 0.5 0.6 (b) Mean error = 0.0253 (c) Mean error = 0.0198 Figure 3: All weights are set to uniformly from 0 to 50. Biases are set to -0.5 of the summed parental weights plus a uniform random number from -2.5 to 2.5. The root node is set to be 1 with probability 0.5. This has the effect of making all the nodes in the exact network roughly 0.5 in mean, with non-negligible correlations between parental nodes. 160 simulations were made. in different layers, we require the correlations between their fields h to evaluate (17). Such 'inter-layer' correlations were not required in section (2) , and to be able to use the same calculational scheme we simply neglect them. (We leave a study of the effects of this assumption for future work.) The average in (17) then factors into groups, where each group contains evidential terms in a particular layer. The conditional marginal for node i is obtained from repeating the above procedure in which the desired marginal node is clamped to its opposite value, and then using these results in (13). The above procedure is repeated for each conditional marginal that we are interested in. Although this may seem computationally expensive, the marginal for each node is computed quickly, since the equations are solved by one forward propagation sweep only. Error uoing Gauosian Field, Diago".1 covarIanee Em:>< uoing Gau"ian Field. Non Diagonal """ariance (a) Mean error = 0.1534 (b) Mean error = 0.0931 (c) Mean error = 0.0865 Figure 4: Estimating the conditional marginal of the top node being in state 1, given that the four bottom nodes are in state 1. Weights were drawn from a zero mean Gaussian with variance 5, with biases set to -0.5 the summed parental weights plus a uniform random number from -2.5 to 2.5. Results of 160 simulations. 3.1 Results for conditional marginals We used the same structure as in the previous experiments, as shown in fig(I). We are interested here in calculating the probability that the top node is in state 1, Gaussian Fields for Approximate Inference 399 given that the four bottom nodes are in state 1. Weights were chosen from a zero mean Gaussian with variance 5. Biases were set to negative half of the summed parent weights, plus a uniform random value from -2.5 to 2.5. Correlation effects in these networks are not as strong as in the experiments in section (2.2), although the improvement of the G F theory over the variational theory seen in fig ( 4) remains clear. The improvement from the off diagonal terms in R is minimal. 4 Conclusion Despite their appropriateness for learning, variational methods may not be equally suited to inference, making more tailored methods attractive. We have considered an approximation procedure that is based on assuming that the distribution of the weighted input to a node is approximately Gaussian. Correlation effects between parents of a node were taken into account to improve the Gaussian theory, although in our examples this gave only relatively modest improvements. The variational mean field theory performs poorly in networks with strong correlation effects between nodes. On the other hand, one may conjecture that the Gaussian Field approach will not generally perform catastrophically worse than the factorised variational mean field theory. One advantage of the variational theory is the presence of an objective function against which competing solutions can be compared. However, finding an optimum solution for the mean parameters mj from this function is numerically complex. Since the Gaussian Field theory is extremely fast to solve, an interesting compromise might be to prime the variational solution with the results from the Gaussian Field theory. Acknowledgments DB would like to thank Bert Kappen and Wim Wiegerinck for stimulating and helpful discussions. PS thanks the Royal Society for financial support. [1] R. Neal. Connectionist learning of Belief Networks. Artificial Intelligence, 56:71-113, 1992. [2] E. Castillo, J. M. Gutierrez, and A. S. Radi. Expert Systems and Probabilistic Network Models. Springer, 1997. [3] M. I. Jordan, Z. Gharamani, T. S. Jaakola, and L. K. Saul. An Introduction to Variational Methods for Graphical Models. In M. I. Jordan, editor, Learning in Graphical Models, pages 105-161. Kluwer, 1998. [4] L. Saul and M. I. Jordan. A mean field learning algorithm for unsupervised neural networks. In M. I. Jordan, editor, Learning in Graphical Models, 1998. [5] D. Barber and W Wiegerinck. Tractable variational structures for approximating graphical models. In M.S. Kearns, S.A. Solla, and D.A. Cohn, editors, Advances in Neural Information Processing Systems NIPS 11. MIT Press, 1999. [6] M. Kearns and 1. Saul. Inference in Multilayer Networks via Large Deviation Bounds. In Advances in Neural Information Processing Systems NIPS 11, 1999. [7] L. K. Saul, T. Jaakkola, and M. I. Jordan. Mean Field Theory for Sigmoid Belief Networks. Journal of Artificial Intelligence Research, 4:61-76, 1996.
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A Multi-class Linear Learning Algorithm Related to Winnow Chris Mesterhann* Rutgers Computer Science Department 110 Frelinghuysen Road Piscataway, NJ 08854 mesterha@paul.rutgers.edu Abstract In this paper, we present Committee, a new multi-class learning algorithm related to the Winnow family of algorithms. Committee is an algorithm for combining the predictions of a set of sub-experts in the online mistake-bounded model oflearning. A sub-expert is a special type of attribute that predicts with a distribution over a finite number of classes. Committee learns a linear function of sub-experts and uses this function to make class predictions. We provide bounds for Committee that show it performs well when the target can be represented by a few relevant sub-experts. We also show how Committee can be used to solve more traditional problems composed of attributes. This leads to a natural extension that learns on multi-class problems that contain both traditional attributes and sub-experts. 1 Introduction In this paper, we present a new multi-class learning algorithm called Committee. Committee learns a k class target function by combining information from a large set of sub-experts. A sub-expert is a special type of attribute that predicts with a distribution over the target classes. The target space of functions are linear-max functions. We define these as functions that take a linear combination of sub-expert predictions and return the class with maximum value. It may be useful to think of the sub-experts as individual classifying functions that are attempting to predict the target function. Even though the individual sub-experts may not be perfect, Committee attempts to learn a linear-max function that represents the target function. In truth, this picture is not quite accurate. The reason we call them sub-experts and not experts is because even though a individual sub-expert might be poor at prediction, it may be useful when used in a linear-max function. For example, some sub-experts might be used to add constant weights to the linear-max function. The algorithm is analyzed for the on-line mistake-bounded model oflearning [Lit89]. This is a useful model for a type of incremental learning where an algorithm can use feedback about its current hypothesis to improve its performance. In this model, the algorithm goes through a series of learning trials. A trial is composed of three steps. First, the algorithm ·Part of this work was supported by NEe Research Institute, Princeton, NJ. 520 C. Mesterharm receives an instance, in this case, the predictions of the sub-experts. Second, the algorithm predicts a label for the instance; this is the global prediction of Committee. And last, the algorithm receives the true label of the instance; Committee uses this information to update its estimate of the target. The goal of the algorithm is to minimize the total number of prediction mistakes the algorithm makes while learning the target. The analysis and performance of Committee is similar to another learning algorithm, Winnow [Lit89]. Winnow is an algorithm for learning a linear-threshold function that maps attributes in [0, 1] to a binary target. It is an algorithm that is effective when the concept can be represented with a few relevant attributes, irrespective of the behavior of the other attributes. Committee is similar but deals with learning a target that contains only a few relevant sub-experts. While learning with sub-experts is interesting in it's own right, it turns out the distinction between the two tasks is not significant. We will show in section 5 how to transform attributes from [0, 1] into sub-experts. Using particular transformations, Committee is identical to the Winnow algorithms, Balanced and WMA [Lit89]. Furthermore, we can generalize these transformations to handle attribute problems with multi-class targets. These transformations naturally lead to a hybrid algorithm that allows a combination of sub-experts and attributes for multi-class learning problems. This opens up a range of new practical problems that did not easily fit into the previous framework of [0, 1 J attributes and binary classification. 2 Previous work Many people have successfully tried the Winnow algorithms on real-world tasks. In the course of their work, they have made modifications to the algorithms to fit certain aspects of their problem. These modifications include multi-class extensions. For example, [DKR97] use Winnow algorithms on text classification problems. This multiclass problem has a special form; a document can belong to more than one class. Because of this property, it makes sense to learn a different binary classifier for each class. The linear functions are allowed, even desired, to overlap. However, this paper is concerned with cases where this is not possible. For example, in [GR96] the correct spelling of a word must be selected from a set of many possibilities. In this setting, it is more desirable to have the algorithm select a single word. The work in [GR96] presents many interesting ideas and modifications of the Winnow algorithms. At a minimum, these modification are useful for improving the performance of Winnow on those particular problems. Part of that work also extends the Winnow algorithm to general multi-class problems. While the results are favorable, the contribution ofthis paper is to give a different algorithm that has a stronger theoretical foundation for customizing a particular multi-class problem. Blum also works with multi-class Winnow algorithms on the calendar scheduling problem of [MCF+94]. In [Blu95], a modified Winnow is given with theoretical arguments for good performance on certain types of multi-class disjunctions. In this paper, these results are extended, with the new algorithm Committee, to cover a wider range of multi-class linear functions. Other related theoretical work on multi-class problems includes the regression algorithm EG±. In [KW97], Kivinen and Warmuth introduce EG±, an algorithm related to Winnow but used on regression problems. In general, while regression is a useful framework for many multi-class problems, it is not straightforward how to extend regression to the concepts learned by Committee. A particular problem is the inability of current regression techniques to handle 0-1 loss. A Multi-class Linear Learning Algorithm Related to Winnow 521 3 Algorithm This section of the paper describes the details of Committee. Near the end of the section, we will give a formal statement of the algorithm. 3.1 Prediction scheme Assume there are n sub-experts. Each sub-expert has a positive weight that is used to vote for k different classes; let Wi be the weight of sub-expert i. A sub-expert can vote for several classes by spreading its weight with a prediction distribution. For example, if k = 3, a sub-expert may give 3/5 of its weight to class 1, 1/5 of its weight to class 2, and 1/5 of its weight to class 3. Let Xi represent this prediction distribution, where x~ is the fraction of the weight sub-expert i gives to class j . The vote for class j is L~=I WiX~. Committee predicts the class that has the highest vote. (On ties, the algorithm picks one of the classes involved in the tie.) We call the function computed by this prediction scheme a linear-max function, since it is the maximum class value taken from a linear combination of the SUb-expert predictions. 3.2 Target function The goal of Committee is to mInimIZe the number of mistakes by quickly learning sub-expert weights that correctly classify the target function. Assume there exists fL, a vector of nonnegative weights that correctly classifies the target. Notice that fL can be multiplied by any constant without changing the target. To remove this confusion, we will normalize the weights to sum to 1, i.e., L~=Il-1i = 1. Let ((j) be the target's vote for class j. n ((j) = L l-1iX~ t=I Part of the difficulty of the learning problem is hidden in the target weights. Intuitively, a target function will be more difficult to learn ifthere is a small difference between the (votes of the correct and incorrect classes. We measure this difficulty by looking at the minimum difference, over all trials, of the vote of the correct label and the vote of the other labels. Assume for trial t that Pt is the correct label. 8= min (min(((pt)-((j))) tETnals rlpt Because these are the weights of the target, and the target always makes the correct prediction, 8 > o. One problem with the above assumptions is that they do not allow noise (cases where 8 <::; 0). However, there are variations of the analysis that allow for limited amounts of noise [Lit89, Lit91]. Also experimental work [Lit95, LM] shows the family of Winnow algorithms to be much more robust to noise than the theory would predict. Based on the similarity of the algorithm and analysis, and some preliminary experiments, Committee should be able to tolerate some noise. 3.3 Updates Committee only updates on mistakes using multiplicative updates. The algorithm starts by initializing all weights to 1 In. During the trials, let P be the correct label and .x be the predicted label of Committee. When .x =1= P the weight of each sub-expert i is multiplied by aX; -x;. This corresponds to increasing the weights of the sub-experts who predicted the 522 C. Mesterharm correct label instead of the label Committee predicted. The value of 0' is initialized at the start of the algorithm. The optimal value of 0' for the bounds depends on 6. Often 6 is not known in advance, but experiments on Winnow algorithms suggest that these algorithms are more flexible, often performing well with a wider range of 0' values [LM). Last, the weights are renormalize to sum to 1. While this is not strictly necessary, normalizing has several advantages including reducing the likelyhood of underflow/overflow errors. 3.4 Committee code Initialization Trials 'Vi E {l, . .. , n} Wi:= lin. Set 0' > 1. Instance sub-experts (Xl , . .. , xn). Prediction >. is the first class c such that for all other classes J, "n c > "n j L...-i =1 W i X i L...-i=1 W i X t • Update Let p be the correct label. If mistake (>' # p) fori:=l ton p >. W i := O'X i -x, Wi . Normalize weights, L:~= l W t = 1 3.5 Mistake bound We do not have the space to give the proof for the mistake bound of Committee, but the technique is similar to the proof of the Winnow algorithm, Balanced, given in [Lit89). For the complete proof, the reader can refer to [Mes99). Theorem 1 Committee makes at most 2ln (n) 162mistakes when the target conditions in section 3.2 are satisfied and 0' is set to (1 - 6) - 1/2. Surprisingly, this bound does not refer to the number of classes. The effects of larger values of k show up indirectly in the 6 value. While it is not obvious, this bound shows that Committee performs well when the target can be represented by a small fraction of the sub-experts. Call the sub-experts in the target the relevant sub-experts. Since 6 is a function of the target, 6 only depends on the relevant sub-experts. On the other hand, the remaining sub-experts have a small effect on the bound since they are only represented in the In( n) factor. This means that the mistake bound of Committee is fairly stable even when adding a large number of additional sub-experts. In truth, this doesn't mean that the algorithm will have a good bound when there are few relevant sub-experts. In some cases, a small number of sub-experts can give an arbitrarily small 6 value. (This is a general problem with all the Winnow algorithms.) What it does mean is that, given any problem, increasing the number of irrelevant sub-experts will only have a logarithmic effect on the mistake bound. 4 Attributes to sub-experts Often there are no obvious sub-experts to use in solving a learning problem. Many times the only information available is a set of attributes. For attributes in [0,1]' we will show how to use Committee to learn a natural kind of k class target function, a linear machine. To learn this target, we will transform each attribute into k separate sub-experts. We will use some of the same notion as Committee to help understand the transformation. A Multi-class Linear Learning Algorithm Related to Winnow 523 4.1 Attribute target (linear machine) A linear machine [DH73] is a prediction function that divides the feature space into disjoint convex regions where each class corresponds to one region. The predictions are made by a comparing the value of k different linear functions where each function corresponds to a class. More formally, assume there are m - 1 attributes and k classes. Let Zi E [0,1] be attribute i. Assume the target function is represented using k linear functions of the attributes. Let ((j) = L::II-L{ Zi be the linear function for class j where I-Li is the weight of attribute i in class j. Notice that we have added one extra attribute. This attribute is set to 1 and is needed for the constant portion of the linear functions. The target function labels an instance with the class of the largest ( function. (Ties are not defined.) Therefore, ((j) is similar to the voting function for class j used in Committee. 4.2 Transforming the target One difficulty with these linear functions is that they may have negative weights. Since Committee only allows targets with nonnegative weights, we need transform to an equivalent problem that has nonnegative weights. This is not difficult. Since we are only concerned with the relative difference between the ( functions, we are allowed to add any function to the (functions as long as we add it to all (functions. This gives us a simple procedure to remove negative weights. For example, if ((1) = 3Z1 2Z2 + 1z3 -4, we can add 2Z2 +4 to every ( function to remove the negative weights from ((1). It is straightforward to extend this and remove all negative weights. We also need to normalize the weights. Again, since only the relative difference between the ( functions matter, we can divide all the ( functions by any constant. We normalize the weights to sum to 1, i.e., L:~=1L:~11-L{ = 1. At this point, without loss of generality, assume that the original ( functions are nonnegative and normalized. The last step is to identify a 8 value. We use the same definition of 8 as Committee substituting the corresponding ( functions of the linear machine. Assume for trial t that Pt is the correct label. 8 = min (min( ((Pt) - ((j))) tETrwls ji-P, 4.3 Transforming the attributes The transformation works as follows: convert attribute Zi into k sub-experts. Each sub-expert will always vote for one of the k classes with value Zi. The target weight for each of these sub-experts is the corresponding target weight of the attribute, label pair in the ( functions. Do this for every attribute. Notice that we are not using distributions for the sub-expert predictions. A sub-expert's prediction can be converted to a distribution by adding a constant amount to each class prediction. For example, a sub-expert that predicts Zl = .7, Z2 = 0, Z3 = ° can be changed to Zl = .8, Z2 = .1, Z3 = .1 by adding .1 to each class. This conversion does not affect the predicting or updating of Committee. 524 C. Mesterharm Theorem 2 Committee makes at most 2In(mk)/82mistakes on a linear machine, as defined in this section, when 0 is set to (1 - 8)-1/2. Proof: The above target transformation creates mk normalized target sub-experts that vote with the same ( functions as the linear machine. Therefore, this set of sub-experts has the same 8 value. Plugging these values into the bound for Committee gives the result. This transformation provides a simple procedure for solving linear machine problems. While the details of the transformation may look cumbersome, the actual implementation of the algorithm is relatively simple. There is no need to explicitly keep track of the sub-experts. Instead, the algorithm can use a linear machine type representation. Each class keeps a vector of weights, one weight for each attribute. During an update, only the correct class weights and the predicted class weights are changed. The correct class weights are multiplied by O Zi; the predicted class weights are multiplied by o -z' . The above procedure is very similar to the Balanced algorithm from [Lit89], in fact, for k = 2, it is identical. A similar transformation duplicates the behavior of the linear-threshold learning version ofWMA as given in [Lit89]. While this transformation shows some advantages for k = 2, more research is needed to determine the proper way to generalize to the multi-class case. For both of these transformations, the bounds given in this paper are equivalent (except for a superficial adjustment in the 8 notation of WMA) to the original bounds given in [Lit89]. 4.4 Combining attributes and sub-experts These transformations suggest the proper way to do a hybrid algorithm that combines sub-experts and attributes: use the transformations to create new sub-experts from the attributes and combine them with the original sub-experts when running Committee. It may even be desirable to break original sub-experts into attributes and use both in the algorithm because some sub-experts may perform better on certain classes. For example, if it is felt that a sub-expert is particularly good at class 1, we can perform the following transformation. Now, instead of using one weight for the whole sub-expert, Committee can also learn based on the sub-expert's performance for the first class. Even if a good target is representable only with the original sub-experts, these additional sub-experts will not have a large effect because of the logarithmic bound. In the same vein, it may be useful to add constant attributes to a set of sub-experts. These add only k extra SUb-experts, but allow the algorithm to represent a larger set of target functions. 5 Conclusion In this paper, we have introduced Committee, a multi-class learning algorithm. We feel that this algori thm will be important in practice, extending the range of problems that can be handled by the Winnow family of algorithms. With a solid theoretical foundation, researchers can customize Winnow algorithms to handle various multi-class problems. A Multi-class Linear Learning Algorithm Related to Winnow 525 Part of this customization includes feature transformations. We show how Committee can handle general linear machine problems by transforming attributes into sub-experts. This suggests a way to do a hybrid learning algorithm that allows a combination of sub-experts and attributes. This same techniques can also be used to add to the representational power on a standard sub-expert problem. In the future, we plan to empirically test Committee and the feature transformations on real world problems. Part of this testing will include modifying the algorithm to use extra information, that is related to the proof technique [Mes99), in an attempt to lower the number of mistakes. We speculate that adjusting the multiplier to increase the change in progress per trial will be useful for certain types of multi-class problems. Acknowledgments We thank Nick Littlestone for stimulating this work by suggesting techniques for converting the Balanced algorithm to multi-class targets. Also we thank Haym Hirsh, Nick Littlestone and Warren Smith for providing valuable comments and corrections. References [Blu95] Avrim Blum. Empirical support for winnow and weighted-majority algorithms: results on a calendar scheduling domain. In ML-95, pages 64-72, 1995. [DH73) R. O. Duda and P. Hart. Pattern Classification and Scene Analysis. Wiley, New York,1973. [DKR97] I. Dagan, Y. Karov, and D. Roth. Mistake-driven learning in text categorization. In EMNLP-97, pages 55-63,1997. [GR96) A. R. Golding and D. Roth. Applying winnow to context-sensitive spelling correction. In ML-96, 1996. [KW97) Jyrki Kivinen and Manfred K. Warmuth. Additive versus exponentiated gradient updates for linear prediction. Information and Computation, 132(1): 1-64, 1997. [Lit89] Nick Littlestone. Mistake bounds and linear-threshold learning algorithms. PhD thesis, University of California, Santa Cruz, 1989. Technical Report UCSC-CRL-89-11. [Lit91) Nick Littlestone. Redundant noisy attributes, attribute errors, and linearthreshold learning using winnow. In COLT-91 , pages 147-156,1991. [Lit95] Nick Littlestone. Comparing several linear-threshold learning algorithms on tasks involving superfluous attributes. In ML-95, pages 353-361 , 1995. [LM) Nick Littlestone and Chris Mesterharm. A simulation study of winnow and related algorithms. Work in progress. [MCF+94) T. Mitchell, R. Caruana, D. Freitag, 1. McDermott, and D. Zabowski. Experience with a personal learning assistant. CACM, 37(7):81-91, 1994. [Mes99) Chris Mesterharm. A multi-class linear learning algorithm related to winnow with proof. Technical report, Rutgers University, 1999.
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Churn Reduction in the Wireless Industry Michael C. Mozer*+, Richard Wolniewicz*, David B. Grimes*+, Eric Johnson * , Howard Kaushansky* * Athene Software + Department of Computer Science 2060 Broadway, Suite 300 University of Colorado Boulder, CO 80302 Boulder, CO 80309-0430 Abstract Competition in the wireless telecommunications industry is rampant. To maintain profitability, wireless carriers must control chum, the loss of subscribers who switch from one carrier to another. We explore statistical techniques for chum prediction and, based on these predictions. an optimal policy for identifying customers to whom incentives should be offered to increase retention. Our experiments are based on a data base of nearly 47,000 U.S. domestic subscribers, and includes information about their usage, billing, credit, application, and complaint history. We show that under a wide variety of assumptions concerning the cost of intervention and the retention rate resulting from intervention, chum prediction and remediation can yield significant savings to a carrier. We also show the importance of a data representation crafted by domain experts. Competition in the wireless telecommunications industry is rampant. As many as seven competing carriers operate in each market. The industry is extremely dynamic, with new services, technologies, and carriers constantly altering the landscape. Carriers announce new rates and incentives weekly, hoping to entice new subscribers and to lure subscribers away from the competition. The extent of rivalry is reflected in the deluge of advertisements for wireless service in the daily newspaper and other mass media. The United States had 69 million wireless subscribers in 1998, roughly 25% of the population. Some markets are further developed; for example, the subscription rate in Finland is 53%. Industry forecasts are for a U.S. penetration rate of 48% by 2003. Although there is significant room for growth in most markets, the industry growth rate is declining and competition is rising. Consequently, it has become crucial for wireless carriers to control chum-the loss of customers who switch from one carrier to another. At present, domestic monthly chum rates are 2-3% of the customer base. At an average cost of $400 to acquire a subscriber, churn cost the industry nearly $6.3 bilIion in 1998; the total annual loss rose to nearly $9.6 billion when lost monthly revenue from subscriber cancellations is considered (Luna, 1998). It costs roughly five times as much to sign on a new subscriber as to retain an existing one. Consequently, for a carrier with 1.5 milIion subscribers, reducing the monthly churn' rate from 2% to 1 % would yield an increase in annual earnings of at least $54 milIion, and an increase in shareholder value of approximately $150 million. (Estimates are even higher when lost monthly revenue is considered; see Fowlkes, Madan, Andrew, & Jensen, 1999; Luna, 1998.) The goal of our research is to evaluate the benefits of predicting churn using techniques from statistical machine learning. We designed models that predict the probability 936 M. C. Mozer, R. Wolniewicz. D. B. Grimes. E. Johnson and H. Kaushansky of a subscriber churning within a short time window, and we evaluated how well these predictions could be used for decision making by estimating potential cost savings to the wireless carrier under a variety of assumptions concerning subscriber behavior. 1 THE FRAMEWORK Figure 1 shows a framework for churn prediction and profitability maximization. Data from a subscriber-on which we elaborate in the next section-is fed into three components which estimate: the likelihood that the subscriber will churn, the profitability (expected monthly revenue) of the subscriber, and the subscriber's credit risk. Profitability and credit risk determine how valuable the subscriber is to the carrier, and hence influences how much the carrier should be willing to spend to retain the subscriber. Based on the predictions of subscriber behavior, a decision making component determines an intervention strategy-whether a subscriber should be contacted, and if so, what incentives should be offered to appease them. We adopt a decision-theoretic approach which aims to maximize the expected profit to the carrier. In the present work, we focus on churn prediction and utilize simple measures of subscriber profitability and credit risk. However, current modeling efforts are directed at more intelligent models of profitability and credit risk. 2 DATASET The subscriber data used for our experiments was provided by a major wireless carrier. The carrier does not want to be identified, as churn rates are confidential. The carrier provided a data base of 46,744 primarily business subscribers, all of whom had multiple services. (Each service corresponds to a cellular telephone or to some other service, such as voice messaging or beeper capability.) All subscribers were from the same region of the United States, about 20% in major metropolitan areas and 80% more geographically distributed. The total revenue for all subscribers in the data base was $14 million in October 1998. The average revenue per subscriber was $234. We focused on multi-service subscribers, because they provide significantly more revenue than do typical single-service subscribers. When subscribers are on extended contracts, churn prediction is relatively easy: it seldom occurs during the contract period, and often occurs when the contract comes to an end. Consequently, all subscribers in our data base were month-to-month, requiring the use of more subtle features than contract termination date to anticipate churn. The subscriber data was extracted from the time interval October through December, 1998. Based on these data, the task was to predict whether a subscriber would churn in January or February 1999. The carrier provided their internal definition of churn, which was based on the closing of all services held by a subscriber. From this definition, 2,876 of the subscribers active in October through December churned-6.2% of the data base. subscriber data .. ---.. subscriber churn prediction subscriber profitability estimation subscriber credit risk estimation .. decision -making ... . --intervention strategy FIGURE 1. The framework for churn prediction and profitability maximization Churn Reduction in the Wireless Industry 937 2.1 INPUT FEATURES Ultimately, churn occurs because subscribers are dissatisfied with the price or quality of service, usually as compared to a competing carrier. The main reasons for subscriber dissatisfaction vary by region and over time. Table 1 lists important factors that influence subscriber satisfaction, as well as the relative importance of the factors (J. D. Power and Associates, 1998). In the third column, we list the type of information required for determining whether a particular factor is likely to be influencing a subscriber. We categorize the types of information as follows. Network. Call detail records (date, time, duration, and location of all calls), dropped cans (calls lost due to lack of coverage or available bandwidth), and quality of service data (interference, poor coverage). Billing. Financial information appearing on a subscriber's bill (monthly fee, additional charges for roaming and additional minutes beyond monthly prepaid limit). Customer Service. Cans to the customer service department and their resolutions. Application for Service. Information from the initial application for service, including contract details, rate plan, handset type, and credit report. Market. Details of rate plans offered by carrier and its competitors, recent entry of competitors into market, advertising campaigns, etc. Demographics. Geographic and population data of a given region. A subset of these information sources were used in the present study. Most notably, we did not utilize market information, because the study was conducted over a fairly short time interval during which the market did not change significantly. More important, the market forces were fairly uniform in the various geographic regions from which our subscribers were selected. Also, we were unable to obtain information about the subscriber equipment (age and type of handset used). The information sources listed above were distributed over three distinct data bases maintained by the carrier. The data bases contained thousands of fields, from which we identified 134 variables associated with each subscriber which we conjectured might be linked to churn. The variables included: subscriber location, credit classification, customer classification (e.g., corporate versus retail), number of active services of various types, beginning and termination dates of various services, avenue through which services were activated, monthly charges and usage, number, dates and nature of customer service calls, number of cans made, and number of abnormany terminated cans. 2.2 DATA REPRESENTATION As all statisticians and artificial intelligence researchers appreciate, representation is key. A significant portion of our effort involved working with domain experts in the wireless telecommunications industry to develop a representation of the data that highlights and makes explicit those features which-in the expert's judgement-were highly related to churn. To evaluate the benefit of carefuny constructing the representation, we performed TABLE 1. Factors influencing subscriber satisfaction Factor Importance Nature of data required for prediction call quality 21% network pricing options 18% market, billing corporate capability 17% market, customer service customer service 17% customer service credibility I customer communications 10% market, customer service roaming I coverage 7% network nandset 4V/o application olillng 3% billing cost of roaming 3"10 marKet, billing 938 M C. Mozer. R. Wo/niewicz. D. B. Grimes. E. Johnson and H. Kaushansky studies using both naive and a sophisticated representations. The naive representation mapped the 134 variables to a vector of 148 elements in a straightforward manner. Numerical variables, such as the length of time a subscriber had been with the carrier, were translated to an element of the representational vector which was linearly related to the variable value. We imposed lower and upper limits on the variables, so as to suppress irrelevant variation and so as not to mask relevant variation by too large a dynamic range; vector elements were restricted to lie between --4 and +4 standard deviations of the variable. One-of-n discrete variables, such as credit classification, were translated into an n-dimensional subvector with one nonzero element. The sophisticated representation incorporated the domain knowledge of our experts to produce a 73-element vector encoding attributes of the subscriber. This representation collapsed across some of the variables which, in the judgement of the experts, could be lumped together (e.g., different types of calls to the customer service department), and expanded on others (e.g., translating the scalar length-of-time-with-carrier to a multidimensional basis-function representation, where the receptive-field centers of the basis functions were suggested by the domain experts), and performed transformations of other variables (e.g., ratios of two variables, or time-series regression parameters). 3 PREDICTORS The task is to predict the probability of churn from the vector encoding attributes of the subscriber. We compared the churn-prediction performance of two classes of models: logit regression and a nonlinear neural network with a single hidden layer and weight decay (Bishop, 1995). The neural network model class was parameterized by the number of units in the hidden layer and the weight decay coefficient. We originally anticipated that we would require some model selection procedure, but it turned out that the results were remarkably insensitive to the choice of the two neural network parameters; weight decay up to a point seemed to have little effect, and beyond that point it was harmful, and varying the number of hidden units from 5 to 40 yielded nearly identical performance. We likely were not in a situation where overfitting was an issue, due to the large quantity of data available; hence increasing the model complexity (either by increasing the number of hidden units or decreasing weight decay) had little cost. Rather than selecting a single neural network model, we averaged the predictions of an ensemble of models which varied in the two model parameters. The average was uniformly weighted. 4 METHODOLOGY We constructed four predictors by combining each of the two model classes (logit regression and neural network) with each of the two subscriber representations (naive and sophisticated). For each predictor, we performed a ten-fold cross validation study, utilizing the same splits across predictors. In each split of the data, the ratio of churn to no churn examples in the training and validation sets was the same as in the overall data set. For the neural net models, the input variables were centered by subtracting the means and scaled by dividing by their standard deviation. Input values were restricted to lie in the range [--4, +4]. Networks were trained until they reached a local minimum in error. 5 RESULTS AND DISCUSSION 5.1 CHURN PREDICTION For each of the four predictors, we obtain a predicted probability of churn for each subscriber in the data set by merging the test sets from the ten data splits. Because decision making ultimately requires a "churn" or "no churn" prediction. the continuous probability measure must be thresholded to obtain a discrete predicted outcome. Chum Reduction in the Wireless Industry 939 For a given threshold, we determine the proportion of churners who are correctly identified as churners (the hit rate), and the proportion of nonchurners who are correctly identified as nonchurners (the rejection rate). Plotting the hit rate against the rejection rate for various thresholds, we obtain an ROC curve (Green & Swets, 1966). In Figure 2, the closer a curve comes to the upper right corner of the graph-lOO% correct prediction of churn and 100% correct prediction of nonchurn-the better is the predictor at discriminating churn from nonchurn. The dotted diagonal line indicates no discriminability: If a predictor randomly classifies x% of cases as churn, it is expected to obtain a hit rate of x% and a rejection rate of (lOO--x)%. As the Figure indicates, discriminability is clearly higher for the sophisticated representation than for the naive representation. Further, for the sophisticated representation at least, the nonlinear neural net outperforms the logit regression. It appears that the neural net can better exploit nonlinear structure in the sophisticated representation than in the naive representation, perhaps due to the basis-function representation of key variables. Although the four predictors appear to yield similar curves, they produce large differences in estimated cost savings. We describe how we estimate cost savings next. 5.2 DECISION MAKING Based on a subscriber's predicted churn probability, we must decide whether to offer the subscriber some incentive to remain with the carrier, which will presumably reduce the likelihood of churn. The incentive will be offered to any subscriber whose churn probability is above a certain threshold. The threshold will be selected to maximize the expected cost savings to the carrier; we will refer to this as the optimal decision-making policy. The cost savings will depend not only on the discriminative ability of the predictor, but also on: the cost to the carrier of providing the incentive, denoted Ci (the cost to the carrier may be much lower than the value to the subscriber, e.g., when air time is offered); the time horizon over which the incentive has an effect on the subscriber's behavior; the reduction in probability that the subscriber will leave within the time horizon as a result of the incentive, Pi; and the lost-revenue cost that results when a subscriber churns, Ct. 100 90 CD Cii 80 .... C 0 :u 70 Q) ·CD .!:;.. 60 "0 Q) ~ 50 E Q) ~ 40 C .... :::J ~ 30 U C 0 20 C ~ 0 10 0 0 neural net I sophisticated logit regression lsophisticated neural net I naive log~ regression I naive " """ , " , " , " \ '\ " . \ \ " \ \. \ , \ \, \ '\ \ \ -.. \ \ I \ 1 \ 1 \\ \I \\ 10 20 30 40 50 60 70 80 90 100 % chur~ identified (hit rate) FIGURE 2. Test-set performance for the four predictors. Each curve shows, for various thresholds, the ability of a predictor to correctly identify churn (x axis) and nonchum (y axis). The more bowed a curve, the better able a predictor is at discriminating churn from nonchurn. 940 M. C. Mozer; R. Wo[niewicz, D. B. Grimes, E. Johnson and H. Kaushansky We assume a time horizon of six months. We also assume that the lost revenue as a result of churn is the average subscriber bill over the time horizon, along with a fixed cost of $500 to acquire a replacement subscriber. (This acquisition cost is higher than the typical cost we stated earlier because subscribers in this data base are high valued, and often must be replaced with multiple low-value subscribers to achieve the same revenue.) To estimate cost savings, the parameters Ci' Pi' and C, are combined with four statistics obtained from a predictor: N(pL,aL): N(pS,aL): N(pL,aS): N(pS,aS): number of subscribers who are predicted to leave (churn) and who actually leave barring intervention number of subscribers who are predicted to stay (nonchurn) and who actually leave barring intervention number of subscribers who are predicted to leave and who actually stay number of subscribers who are predicted to stay and who actually stay Given these statistics, the net cost to the carrier of performing no intervention is: net(no intervention) = [ N(pL,aL) + N(pS,aL) ] C, This equation says that whether or not churn is predicted, the subscriber will leave, and the cost per subscriber will be C,. The net cost of providing an incentive to all subscribers whom are predicted to churn can also be estimated: net(incentive) = [N(pL,aL) + N(pL,aS)] q + [Pi N(pL,aL) + N(pS,aL)] C, This equation says that the cost of offering the incentive, Ci' is incurred for all subscribers for who are predicted to churn, but the lost revenue cost will decrease by a fraction Pi for the subscribers who are correctly predicted to churn. The savings to the carrier as a result of offering incentives based on the churn predictor is then savings per churnable subscriber = [ net(no intervention) - net(incentive)] / [N(pL,aL) + N(pS,aL)] The contour plots in Figure 3 show expected savings per churnable subscriber, for a range of values of q, Pi, and C" based on the optimal policy and the sophisticated neuralnet predictor. Each plot assumes a different subscriber retention rate (= I-Pi) given intervention. The "25% retention rate" graph supposes that 25% of the churning subscribers who are offered an incentive wiII decide to remain with the carrier over the time horizon of six months. For each plot, the cost of intervention (q) is varied along the x-axis, and the average monthly bill is varied along the y-axis. (The average monthly biII is converted to lost revenue, C" by computing the total biII within the time horizon and adding the subscriber acquisition cost.) The shading of a region in the plot indicates the expected savings assuming the specified retention rate is achieved by offering the incentive. The grey-level bar to the right of each plot translates the shading into dollar savings per subscriber who will churn barring intervention. Because the cost of the incentive is factored into the savings estimate, the estimate is actually the net return to the carrier. The white region in the lower right portion of each graph is the region in which no cost savings will be obtained. As the graphs clearly show, if the cost of the incentive needed to achieve a certain retention rate is low and the cost of lost revenue is high, significant per-subscriber savings can be obtained. As one might suspect in examining the plots, what's important for determining persubscriber savings is the ratio of the incentive cost to the average monthly bill. The plots clearly show that for a wide range of assumptions concerning the average monthly bill, incentive cost, and retention rate, a significant cost savings is realized. The plots assume that all subscribers identified by the predictor can be contacted and offered the incentive. If only some fraction F of aII subscribers are contacted, then the estimated savings indicated by the plot should be multiplied by F. To pin down a likely scenario, it is reasonable to assume that 50% of subscribers can be contacted, 35% of whom will be retained by offering an incentive that costs the carrier Chum Reduction in the Wireless Industry 941 25% retention rate 35% retention rate 400 600 250 350 250 500 ~2oo 300 ~2oo :0 250 :0 400 ~150 200 ~150 C § 0 300 E 100 150 E 100 0> 0> 1; 100 > <II 200 50 50 50 0 0 50 100 150 200 50 100 150 200 intervention cost ($) intervention cost ($) 50% retention rate 75% retention rate 900 BOO 250 700 ~2oo 600 :s 500 ~150 400 C 0 0> 300 ;, 100 > > <II 200 <II 50 100 0 50 100 150 200 intervention cost ($) intervention cost ($) FIGURE 3. Expected savings to the carrier per chumable subscriber, under a variety of assumptions concerning intervention cost, average monthly bill of subscriber, and retention rate that will be achieved by offering an incentive to a churnable subscriber. $75, and in our data base, the average monthly bill is $234. Under this scenario, the expected savings-above and beyond recovering the incentive cost-to the carrier is $93 based on the sophisticated neural net predictor. In contrast, the expected savings is only $47 based on the naive neural net predictor, and $81 based on the sophisticated logistic regression model. As we originally conjectured, both the nonlinearity of the neural net and the bias provided by the sophisticated representation are adding value to the predictions. Our ongoing research involves extending these initial results in a several directions. First, we have confirmed our positive results with data from a different time window, and for test data from a later time window than the training data (as would be necessary in real-world usage). Second, we have further tuned and augmented our sophisticated representation to obtain higher prediction accuracy, and are now awaiting additional data to ensure the result replicates. Third, we are applying a variety of techniques, including sensitivity analysis and committee and boosting techniques, to further improve prediction accuracy. And fourth, we have begun to explore the consequences of iterating the decision making process and evaluating savings over an extended time period. Regardless of these directions for future work, the results presented here show the promise of data mining in the domain of wireless telecommunications. As is often the case for decision-making systems, the predictor need not be a perfect discriminator to realize significant savings. 6 REFERENCES Bishop, C. (1995). Neural networks for pattern recognition. New York: Oxford University Press. Fowlkes, A. J., Madan, A., Andrew, J., and Jensen, C.(1999). The effect of churn on value: An industry advisory. Green, D. M., & Swets, J. A. (1966). Signal detection theory and psychophysics. New York: Wiley. Luna, L. (1998). Churn is epidemic. Radio Communications Report, December 14, 1998. Power, J. D., & Associates (1998). 1998 Residential Wireless Customer Satisfaction Survey. September 22, 1998.
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Variational Inference for Bayesian Mixtures of Factor Analysers Zoubin Ghahramani and Matthew J. Beal Gatsby Computational Neuroscience Unit University College London 17 Queen Square, London WC1N 3AR, England {zoubin,m.beal}Ggatsby.ucl.ac.uk Abstract We present an algorithm that infers the model structure of a mixture of factor analysers using an efficient and deterministic variational approximation to full Bayesian integration over model parameters. This procedure can automatically determine the optimal number of components and the local dimensionality of each component (Le. the number of factors in each factor analyser). Alternatively it can be used to infer posterior distributions over number of components and dimensionalities. Since all parameters are integrated out the method is not prone to overfitting. Using a stochastic procedure for adding components it is possible to perform the variational optimisation incrementally and to avoid local maxima. Results show that the method works very well in practice and correctly infers the number and dimensionality of nontrivial synthetic examples. By importance sampling from the variational approximation we show how to obtain unbiased estimates of the true evidence, the exact predictive density, and the KL divergence between the variational posterior and the true posterior, not only in this model but for variational approximations in general. 1 Introduction Factor analysis (FA) is a method for modelling correlations in multidimensional data. The model assumes that each p-dimensional data vector y was generated by first linearly transforming a k < p dimensional vector of unobserved independent zero-mean unit-variance Gaussian sources, x, and then adding a p-dimensional zeromean Gaussian noise vector, n, with diagonal covariance matrix \}!: i.e. y = Ax+n. Integrating out x and n, the marginal density of y is Gaussian with zero mean and covariance AA T + \}!. The matrix A is known as the factor loading matrix. Given data with a sample covariance matrix I:, factor analysis finds the A and \}! that optimally fit I: in the maximum likelihood sense. Since k < p, a single factor analyser can be seen as a reduced parametrisation of a full-covariance Gaussian. 1 IFactor analysis and its relationship to principal components analysis (peA) and mixture models is reviewed in (10). 450 Z. Ghahramani and M. J. Heal A mixture of factor analysers (MFA) models the density for y as a weighted average of factor analyser densities s P(yjA, q,,7r) = LP(sj7r)P(yjs,AS, '11), (1) s=1 where 7r is the vector of mixing proportions, s is a discrete indicator variable, and A S is the factor loading matrix for factor analyser s which includes a mean vector for y. By exploiting the factor analysis parameterisation of covariance matrices, a mixture of factor analysers can be used to fit a mixture of Gaussians to correlated high dimensional data without requiring O(P2) parameters or undesirable compromises such as axis-aligned covariance matrices. In an MFA each Gaussian cluster has intrinsic dimensionality k (or ks if the dimensions are allowed to vary across clusters). Consequently, the mixture of factor analysers simultaneously addresses the problems of clustering and local dimensionality reduction. When '11 is a multiple of the identity the model becomes a mixture of probabilistic PCAs. Tractable maximum likelihood procedure for fitting MFA and MPCA models can be derived from the Expectation Maximisation algorithm [4, 11]. The maximum likelihood (ML) approach to MFA can easily get caught in local maxima.2 Ueda et al. [12] provide an effective deterministic procedure for avoiding local maxima by considering splitting a factor analyser in one part of space and merging two in a another part. But splits and merges have to be considered simultaneously because the number of factor analysers has to stay the same since adding a factor analyser is always expected to increase the training likelihood. A fundamental problem with maximum likelihood approaches is that they fail to take into account model complexity (Le. the cost of coding the model parameters). So more complex models are not penalised, which leads to overfitting and the inability to determine the best model size and structure (or distributions thereof) without resorting to costly cross-validation procedures. Bayesian approaches overcome these problems by treating the parameters 0 as unknown random variables and averaging over the ensemble of models they define: P(Y) = / dO P(YjO)P(O). (2) P(Y) is the evidence for a data set Y = {yl, .. . ,yN}. Integrating out parameters penalises models with more degrees of freedom since these models can a priori model a larger range of data sets. All information inferred from the data about the parameters is captured by the posterior distribution P(OjY) rather than the ML point estimate 0.3 While Bayesian theory deals with the problems of overfitting and model selection/averaging, in practice it is often computationally and analytically intractable to perform the required integrals. For Gaussian mixture models Markov chain Monte Carlo (MCMC) methods have been developed to approximate these integrals by sampling [8, 7]. The main criticism of MCMC methods is that they are slow and 2 Technically, the log likelihood is not bounded above if no constraints are put on the determinant of the component covariances. So the real ML objective for MFA is to find the highest finite local maximum of the likelihood. 3We sometimes use () to refer to the parameters and sometimes to all the unknown quantities (parameters and hidden variables). Formally the only difference between the two is that the number of hidden variables grows with N, whereas the number of parameters usually does not. Variational Inference for Bayesian Mixtures of Factor Analysers 451 it is usually difficult to assess convergence. Furthermore, the posterior density over parameters is stored as a set of samples, which can be inefficient. Another approach to Bayesian integration for Gaussian mixtures [9] is the Laplace approximation which makes a local Gaussian approximation around a maximum a posteriori parameter estimate. These approximations are based on large data limits and can be poor, particularly for small data sets (for which, in principle, the advantages of Bayesian integration over ML are largest). Local Gaussian approximations are also poorly suited to bounded or positive parameters such as the mixing proportions of the mixture model. Finally, it is difficult to see how this approach can be applied to online incremental changes to model structure. In this paper we employ a third approach to Bayesian inference: variational approximation. We form a lower bound on the log evidence using Jensen's inequality: 1: == In P(Y) = In / dO P(Y, 0) ~ / dO Q(O) In P6~~~) == F, (3) which we seek to maximise. Maximising F is equivalent to minimising the KLdivergence between Q(O) and P(OIY), so a tractable Q can be used as an approximation to the intractable posterior. This approach draws its roots from one way of deriving mean field approximations in physics, and has been used recently for Bayesian inference [13, 5, 1]. The variational method has several advantages over MCMC and Laplace approximations. Unlike MCMC, convergence can be assessed easily by monitoring F. The approximate posterior is encoded efficiently in Q(O). Unlike Laplace approximations, the form of Q can be tailored to each parameter (in fact the optimal form of Q for each parameter falls out of the optimisation), the approximation is global, and Q optimises an objective function. Variational methods are generally fast, F is guaranteed to increase monotonically and transparently incorporates model complexity. To our knowledge, no one has done a full Bayesian analysis of mixtures of factor analysers. Of course, vis-a-vis MCMC, the main disadvantage of variational approximations is that they are not guaranteed to find the exact posterior in the limit. However, with a straightforward application of sampling, it is possible to take the result of the variational optimisation and use it to sample from the exact posterior and exact predictive density. This is described in section 5. In the remainder of this paper we first describe the mixture of factor analysers in more detail (section 2). We then derive the variational approximation (section 3). We show empirically that the model can infer both the number of components and their intrinsic dimensionalities, and is not prone to overfitting (section 6). Finally, we conclude in section 7. 2 The Model Starting from (1), the evidence for the Bayesian MFA is obtained by averaging the likelihood under priors for the parameters (which have their own hyperparameters): P(Y) / d7rP(7rIa:) / dvP(vla,b) / dA P(Alv), g [.t, P(s·I1r) J dx·P(xn)p(ynlx·,sn,A', q;)]. (4) 452 Z. Ghahramani and M. J. Beal Here {a, a, b, "Ill} are hyperparameters4 , v are precision parameters (Le. inverse variances) for the columns of A. The conditional independence relations between the variables in this model are shown graphically in the usual belief network representation in Figure 1. :''!:~",~ .................. 1 Figure 1: Generative model for variational Bayesian mixture of factor analysers. Circles denote random variables, solid rectangles denote hyperparameters, and the dashed rectangle shows the plate (i.e. repetitions) over the data. While arbitrary choices could be made for the priors on the first line of (4), choosing priors that are conjugate to the likelihood terms on the second line of (4) greatly simplifies inference and interpretability.5 So we choose P(7rJa) to be symmetric Dirichlet, which is conjugate to the multinomial P(sJ7r). The prior for the factor loading matrix plays a key role in this model. Each component of the mixture has a Gaussian prior P(ABJVB), where each element of the vector VB is the precision of a column of A. IT one of these precisions vi -t 00, then the outgoing weights for factor Xl will go to zero, which allows the model to reduce the intrinsic dimensionality of X if the data does not warrant this added dimension. This method of intrinsic dimensionality reduction has been used by Bishop [2] for Bayesian peA, and is closely related to MacKay and Neal's method for automatic relevance determination (ARD) for inputs to a neural network [6]. To avoid overfitting it is important to integrate out all parameters whose cardinality scales with model complexity (Le. number of components and their dimensionalities). We therefore also integrate out the precisions using Gamma priors, P(vJa, b). 3 The Variational Approximation Applying Jensen's inequality repeatedly to the log evidence (4) we lower bound it using the following factorisation of the distribution of parameters and hidden variables: Q(A)Q(7r, v)Q(s, x). Given this factorisation several additional factorisations fallout of the conditional independencies in the model resulting in the variational objective function: F= jd-n;Q(-n;) In PJ7;~) + t, j dv'Q(v') lIn P6;~~) b) + jdA'Q(A') In P6~~~') 1 + t, .t, Q(s") [j d-n; Q(-n;) In Pci~:~~) + j dx"Q(x"Js") In Q~~:~") + jdABQ(AB) j dxnQ(xnJsn)lnp(ynJxn,sn,AB, "Ill)] (5) The variational posteriors Q('), as given in the Appendix, are derived by performing a free-form extremisation of F w.r.t. Q. It is not difficult to show that these extrema are indeed maxima of F. The optimal posteriors Q are of the same conjugate forms as the priors. The model hyperparameters which govern the priors can be estimated in the same fashion (see the Appendix). 4We currently do not integrate out 1lJ', although this can also be done. 5Conjugate priors have the same effect as pseudo-observations. Variational lriference for Bayesian Mixtures of Factor Analysers 453 4 Birth and Death When optimising F , occasionally one finds that for some s: Ln Q(sn) = O. These zero responsibility components are the result of there being insufficient support from the local data to overcome the dimensional complexity prior on the factor loading matrices. So components of the mixture die of natural causes when they are no longer needed. Removing these redundant components increases F . Component birth does not happen spontaneously, so we introduce a heuristic. Whenever F has stabilised we pick a parent-component stochastically with probability proportional to e-f3F• and attempt to split it into two; Fa is the s-specific contribution to F with the last bracketed term in (5) normalised by Ln Q(sn). This works better than both cycling through components and picking them at random as it concentrates attempted births on components that are faring poorly. The parameter distributions of the two Gaussians created from the split are initialised by partitioning the responsibilities for the data, Q(sn), along a direction sampled from the parent's distribution. This usually causes F to decrease, so by monitoring the future progress of F we can reject this attempted birth if F does not recover. Although it is perfectly possible to start the model with many components and let them die, it is computationally more efficient to start with one component and allow it to spawn more when necessary. 5 Exact Predictive Density, True Evidence, and KL By importance sampling from the variational approximation we can obtain unbiased estimates of three important quantities: the exact predictive density, the true log evidence [" and the KL divergence between the variational posterior and the true posterior. Letting 0 = {A, 7r}, we sample Oi '" Q (0). Each such sample is an instance of a mixture of factor analysers with predictive density given by (1). We weight these predictive densities by the importance weights Wi = P(Oi, Y)/Q(Oi), which are easy to evaluate. This results in a mixture of mixtures of factor analysers, and will converge to the exact predictive density, P(ylY), as long as Q(O) > 0 wherever P(OIY) > O. The true log evidence can be similarly estimated by [, = In(w), where (.) denotes averaging over the importance samples. Finally, the KL divergence is given by: KL(Q(O)IIP(OIY)) = In(w) - (In w). This procedure has three significant properties. First, the same importance weights can be used to estimate all three quantities. Second, while importance sampling can work very poorly in high dimensions for ad hoc proposal distributions, here the variational optimisation is used in a principled manner to pick Q to be a good approximation to P and therefore hopefully a good proposal distribution. Third, this procedure can be applied to any variational approximation. A detailed exposition can be found in [3]. 6 Results Experiment 1: Discovering the number of components. We tested the model on synthetic data generated from a mixture of 18 Gaussians with 50 points per cluster (Figure 2, top left). The variational algorithm has little difficulty finding the correct number of components and the birth heuristics are successful at avoiding local maxima. After finding the 18 Gaussians repeated splits are attempted and rejected. Finding a distribution over number of components using F is also simple. Experiment 2: The shrinking spiral. We used the dataset of 800 data points from a shrinking spiral from [12] as another test of how well the algorithm could 454 Z. Ghahramani and M. J. Beal Figure 2: (top) Exp 1: The frames from left to right are the data, and the 2 S.D. Gaussian ellipses after 7, 14, 16 and 22 accepted births. (bottom) Exp 2: Shrinking spiral data and 1 S.D. Gaussian ellipses after 6, 9, 12, and 17 accepted births. Note that the number of Gaussians increases from left to right. number intrinsic dlmensionalnies of points per cluster 7 4 3 2 2 8 I 2 8 I 1 2 16 1 I 4 2 32 1 6 3 3 2 2 64 1 7 4 3 2 2 -7600 128 1 7 4 3 2 2 - 76OOQ 500 1000 1500 2000 Figure 3: (left) Exp 2: :F as function of iteration for the spiral problem on a typical run. Drops in :F constitute component births. Thick lines are accepted attempts, thin lines are rejected attempts. (middle) Exp 3: Means of the factor loading matrices. These results are analogous to those given by Bishop [2] for Bayesian peA. (right) Exp 3: Table with learned number of Gaussians and dimension ali ties as training set size increases. Boxes represent model components that capture several of the clusters. escape local maxima and how robust it was to initial conditions (Figure 2, bottom). Again local maxima did not pose a problem and the algorithm always found between 12-14 Gaussians regardless of whether it was initialised with 0 or 200. These runs took about 3-4 minutes on a 500MHz Alpha EV6 processor. A plot of:F shows that most of the compute time is spent on accepted moves (Figure 3, left). Experiment 3: Discovering the local dimensionalities. We generated a synthetic data set of 300 data points in each of 6 Gaussians with intrinsic dimensionalities (7432 2 1) embedded in 10 dimensions. The variational Bayesian approach correctly inferred both the number of Gaussians and their intrinsic dimensionalities (Figure 3, middle). We varied the number of data points and found that as expected with fewer points the data could not provide evidence for as many components and intrinsic dimensions (Figure 3, right). 7 Discussion Search over model structures for MFAs is computationally intractable if each factor analyser is allowed to have different intrinsic dimensionalities. In this paper we have shown that the variational Bayesian approach can be used to efficiently infer this model structure while avoiding overfitting and other deficiencies of ML approaches. One attraction of our variational method, which can be exploited in other models, is that once a factorisation of Q is assumed all inference is automatic and exact. We can also use :F to get a distribution over structures if desired. Finally we derive Variational Inference for Bayesian Mixtures of Factor Analysers 455 a generally applicable importance sampler that gives us unbiased estimates of the true evidence, the exact predictive density, and the KL divergence between the variational posterior and the true posterior. Encouraged by the results on synthetic data, we have applied the Bayesian mixture of factor analysers to a real-world unsupervised digit classification problem. We will report the results of these experiments in a separate article. Appendix: Optimal Q Distributions and Hyperparameters Q(xnlsn) '" N(xn,", E S ) Q(A~) ""' N(X:, Eq,S) Q(vl) ""' Q(ai,bl) Q(1r) '" D(wu) 1 In Q(sn) = [1jJ(wus ) -1jJ(w)] + 2 In IEs I + (In p(ynlxn, sn, AS , 'IT)) + c xn,s=EsxsT'lT-lyn, X:= ['IT-l"tQ(sn)ynxn'STEq,S] , ai=a+~, bi=b+~:t(A~12) n=l q q=l N N Es - 1= (AST 'IT -1 AS) + I, Eq ,S -~ 'IT;ql L Q(sn)(xnxnT) +diag(vS), wUs = ~ + L Q(sn) n=l n=l where {N, Q, D} denote Normal, Gamma and Dirichlet distributions respectively, (-) denotes expectation under the variational posterior, and 1jJ (x) is the digamma function 1jJ(x) == tx lnr(x). Note that the optimal distributions Q(AS) have block diagonal covariance structure; even though each AS is a p x q matrix, its covariance only has O(pq2) parameters. Differentiating:F with respect to the parameters, a and b, of the precision prior we get fixed point equations 1jJ(a) = (In v)+lnb and b = a/(v). Similarly the fixed point for the parameters of the Dirichlet prior is 1jJ(a) -1jJ(a/S) + 2: [1jJ(wus ) -1jJ(w)]/S = o. References [1] H. Attias. Inferring parameters and structure of latent variable models by variational Bayes. In Proc. 15th Conf. on Uncertainty in Artificial Intelligence, 1999. [2] C.M. Bishop. Variational PCA. In Proc. Ninth Int. Conf. on Artificial Neural Networks. ICANN, 1999. [3] Z. Ghahramani, H. Attias, and M.J. Beal. Learning model structure. Technical Report GCNU-TR-1999-006, (in prep.) Gatsby Unit, Univ. College London, 1999. [4] Z. Ghahramani and G.E. Hinton. The EM algorithm for mixtures of factor analyzers. Technical Report CRG-TR-96-1 [http://~.gatsby . ucl. ac. uk/ ~zoubin/papers/tr-96-1.ps.gz], Dept. of Compo Sci., Univ. of Toronto, 1996. [5] D.J.C. MacKay. Ensemble learning for hidden Markov models. Technical report, Cavendish Laboratory, University of Cambridge, 1997. [6] R.M. Neal. Assessing relevance determination methods using DELVE. In C.M. Bishop, editor, Neural Networks and Machine Learning, 97-129. Springer-Verlag, 1998. [7] C.E. Rasmussen. The infinite gaussian mixture model. In Adv. Neur. Inf. Pmc. Sys. 12. MIT Press, 2000. [8] S. Richardson and P.J. Green. On Bayesian analysis of mixtures with an unknown number of components. J. Roy. Stat. Soc.-Ser. B, 59(4):731-758, 1997. [9] S.J. Roberts, D. Husmeier, 1. Rezek, and W. Penny. Bayesian approaches to Gaussian mixture modeling. IEEE PAMI, 20(11):1133- 1142, 1998. [10] S. T . Roweis and Z. Ghahramani. A unifying review of linear Gaussian models. Neural Computation, 11(2):305- 345, 1999. [11] M.E. Tipping and C.M. Bishop. Mixtures of probabilistic principal component analyzers. Neural Computation, 11(2):443- 482, 1999. [12] N. Ueda, R. Nakano, Z. Ghahramani, and G.E. Hinton. SMEM algorithm for mixture models. In Adv. Neur. Inf. Proc. Sys. 11. MIT Press, 1999. [13] S. Waterhouse, D.J.C. Mackay, and T. Robinson. Bayesian methods for mixtures of experts. In Adv. Neur. 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Recognizing Evoked Potentials in a Virtual Environment * Jessica D. Bayliss and Dana H. Ballard Department of Computer Science University of Rochester Rochester, NY 14627 {bayliss,dana}@cs.rochester.edu Abstract Virtual reality (VR) provides immersive and controllable experimental environments. It expands the bounds of possible evoked potential (EP) experiments by providing complex, dynamic environments in order to study cognition without sacrificing environmental control. VR also serves as a safe dynamic testbed for brain-computer .interface (BCl) research. However, there has been some concern about detecting EP signals in a complex VR environment. This paper shows that EPs exist at red, green, and yellow stop lights in a virtual driving environment. Experimental results show the existence of the P3 EP at "go" and "stop" lights and the contingent negative variation (CNY) EP at "slow down" lights. In order to test the feasibility of on-line recognition in VR, we looked at recognizing the P3 EP at red stop tights and the absence of this signal at yellow slow down lights. Recognition results show that the P3 may successfully be used to control the brakes of a VR car at stop lights. 1 Introduction The controllability of VR makes it an excellent candidate for use in studying cognition. It expands the bounds of possible evoked potential (EP) experiments by providing complex, dynamic environments in order to study decision making in cognition without sacrificing environmental control. We have created a flexible system for real-time EEG collection and analysis from within virtual environments. The ability of our system to give quick feedback enables it to be used in brain-computer interface (BCl) research, which is aimed at helping individuals with severe motor deficits to become more independent. Recent BCl work has shown the feasibility of on-line averaging and biofeedback methods in order to choose characters or move a cursor on a computer screen with up to 95% accuracy while sitting still and concentrating on the screen [McFarland et aI., 1993; Pfurtscheller et al., 1996; Vaughn et al., 1996; Farwell and Donchin, 1988]. Our focus is to dramatically extend the BCl by allowing evoked potentials to propel the user through alternate virtual environments. For example, a *This research was supported by NIHIPHS grantl-P41-RR09283. It was also facilitated in part by a National Physical Science Consortium Fellowship and by stipend support from NASA Goddard Space Flight Center. 4 J. D. Bayliss and D. H. Ballard Figure 1: (Left) An individual demonstrates driving in the modified go cart. (Right) A typical stoplight scene in the virtual environment. user could choose a virtual living room from a menu of rooms, navigate to the living room automatically in the head-mounted display, and then choose to turn on the stereo. As shown in [Farwell and Donchin, 1988], the P3 EP may be used for a brain-computer interface that picks characters on a computer monitor. Discovered by [Chapman and Bragdon, 1964; Sutton et aI., 1965] and extensively studied (see [Polich, 1998] for a literature review), the P3 is a positive waveform occurring approximately 300-500 ms after an infrequent task-relevant stimulus. We show that requiring subjects to stop or go at virtual traffic lights elicits this EP. The contingent negative variation (CNV), an EP that happens preceding an expected stimulus, occurs at slow down lights. In order to test the feasibility of on-line recognition in the noisy VR environment, we recognized the P3 EP at red stop lights and the lack of this signal at yellow slow down lights. Results using a robust Kalman filter for off-line recognition indicate that the car may be stopped reliably with an average accuracy of 84.5% while the on-line average for car halting is 83%. 2 The Stoplight Experiments The first experiment we performed in the virtual driving environment shows that a P3 EP is obtained when subjects stop or go at a virtual light and that a CNV occurs when subjects see a slow down light. Since all subjects received the same light colors for the slow down, go, and stop conditions we then performed a second experiment with different light colors in order to disambiguate light color from the occurrence of the P3 and CNV. Previous P3 research has concentrated primarily on static environments such as the continuous performance task [Rosvold et aI., 1956]. In the visual continuous performance task (VCPT), static images are flashed on a screen and the subject is told to press a button when a rare stimulus occurs or to count the number of occurrences of a rare stimulus. This makes the stimulus both rare and task relevant in order to evoke a P3. As an example, given red and yellow stoplight pictures, a P3 should occur if the red picture is less frequent than the yellow and subjects are told to press a mouse button only during the red light. We assumed a similar response would occur in a VR driving world if certain lights were infrequent and subjects were told to stop or go at them. This differs from the VCPT in two important ways: 1. In the VCPT subjects sit passively and respond to stimuli. In the driving task, Recognizing Evoked Potentials in a Virtual Environment 5 subjects control when the stimuli appear by where they drive in the virtual world. 2. Since subjects are actively involved and fully immersed in the virtual world, they make more eye and head movements. The movement amount can be reduced by a particular experimental paradigm, but it can not be eliminated. The first difference makes the VR environment a more natural experimental environment. The second difference means that subjects create more data artifacts with extra movement. We handled these artifacts by first manipulating the experimental environment to reduce movements where important stimulus events occurred. This meant that all stoplights were placed at the end of straight stretches of road in order to avoid the artifacts caused by turning a corner. For our on-line recognition, we then used the eye movement reduction technique described in [Semlitsch et al., 1986] in order to subtract a combination of the remaining eye and head movement artifact. 2.1 Experimental Setup All subjects used a modified go cart in order to control the virtual car (see Figure 1). The virtual reality interface is rendered on a Silicon Graphics Onyx machine with 4 processors and an Infinite Reality Graphics Engine. The environment is presented to the subject through a head-mounted display (HMD). Since scalp EEG recordings are measured in microvolts, electrical signals may easily interfere during an experiment. We tested the effects of wearing a VR4 HMD containing an ISCAN eye tracker and discovered that the noise levels inside of the VR helmet were comparable to noise levels while watching a laptop screen [Bayliss and Ballard, 1998]. A trigger pulse containing information about the color of the light was sent to the EEG acquisition system whenever a light changed. While an epoch size from -100 ms to 1 sec was specified, the data was recorded continuously. Information about head position as well as gas, braking, and steering position were saved to an external file. Eight electrodes sites (FZ, CZ, CPZ, PZ, P3, P4, as well as 2 vertical EOG channels) were arranged on the heads of seven subjects with a linked mastoid reference. Electrode impedances were between 2 and 5 kohms for all subjects. Subjects ranged in age from 19 to 52 and most had no previous experiences in a virtual environment. The EEG signal was amplified using Grass amplifiers with an analog bandwidth from 0.1 to 100 Hz. Signals were then digitized at a rate of 500 Hz and stored to a computer. 2.2 Ordinary Traffic Light Color Experiment Five subjects were instructed to slow down on yellow lights, stop for red lights, and go for green lights. These are normal traffic light colors. Subjects were allowed to drive in the environment before the experiment to get used to driving in VR. In order to make slow down lights more frequent, all stoplights turned to the slow down color when subjects were further than 30 meters aways from them. When the subject drove closer than 30 meters the light then turned to either the go or stop color with equal probability. The rest of the light sequence followed normal stoplights with the stop light turning to the go light after 3 seconds and the go light not changing. We calculated the grand averages over red, green, and yellow light trials (see Figure 2a). Epochs affected by artifact were ignored in the averages in order to make sure that any existing movements were not causing a P3-like signal. Results show that a P3 EP occurs for both red and green lights. Back averaging from the green/red lights to the yellow light shows the existence of a CNV starting at approximately 2 seconds before the light changes to red or green. 6 Stop Light ~ 1\ i .~'"\ " .v'l,,/AI \ \ I \, I \ ! I \ I \ I i: I) .E bO ;J u ~ ~ \"~'~ :"' b) ~ ; " t) ,: .~ > '::1 j , ".; '-lOOms t<iooms' Go Light ,~t r/'" '\ f I".j" • , I "\ ..... \ A f f...j Vv '-lOOms lOOOms I J. D. Bayliss and D. H Ballard Slow Down Light -5 uv +lOuv -8 uv h'-3~000ii1S;;:::::::::==::;;2;;OO~ms~1 + 12 uv Figure 2: a) Grand averages for the red stop, green go, and yellow slow down lights. b) Grand averages for the yellow stop, red go, and green slow down lights. All slow down lights have been back-averaged from the occurrence of the go/stop light in order to show the existence of a CNY. 2.3 Alternative Traffic Light Colors The P3 is related to task relevance and should not be related to color, but color needed to be disambiguated as the source of the P3 in the experiment. We had two subjects slow down at green lights, stop at yellow lights, and go at red lights. In order to get used to this combination of colors, subjects were allowed to drive in the town before the experiment. The grand averages for each light color were calculated in the same manner as the averages above and are shown in Figure 2b. As expected, a P3 signal existed for the stop condition and a CNV for the slow down condition. The go condition P3 was much noisier for these two subjects, although a slight P3-like signal is still visible. 3 Single Trial Recognition Results While averages show the existence of the P3 EP at red stop lights and the absence of such at yellow slow down lights, we needed to discover if the signal was clean enough for single trial recognition as the quick feedback needed by a BCI depends on quick recognition. While there were three light conditions to recognize, there were only two distinct kinds of evoked potentials. We chose to recognize the difference between the P3 and the CNV since their averages are very different. Recognizing the difference between two kinds of EPs gives us the ability to use a BCI in any task that can be performed using a series of binary decisions. We tried three methods for classification of the P3 EP: correlation, independent component analysis (ICA), and a robust Kalman filter. Approximately, 90 slow down yellow light and 45 stop red light trials from each subject were classified. The reason we allowed a yellow light bias to enter recognition is because the yellow light currently represents an unimportant event in the environment. In a real BCI unimportant events are likely to occur more than user-directed actions, making this bias justifiable. Recognizing Evoked Potentials in a Virtual Environment 7 Table 1: Recognition Results (p < 0.01) Correlation %Correct ICA %Correct Robust Kalman Filter %Correct Subjects Red Yel Total Red Yel Total Red Yel Total S1 81 51 64 76 77 77 55 86 77 S2 95 63 73 86 88 87 82 94 90 S3 89 56 66 72 87 82 74 85 81 S4 81 60 67 73 69 71 65 91 82 S5 63 66 65 65 79 74 78 92 87 Table 2: Recognition Results for Return Subjects Robust K-Filter % Correct Subjects Red Yel Total S4 73 90 85 S5 67 87 80 As expected, the data obtained while driving contained artifacts, but in an on-line BCI these artifacts must be reduced in order to make sure that what the recognition algorithm is recognizing is not an artifact such as eye movement. In order to reduce these artifacts, we performed the on-line linear regression technique described in [Semlitsch et aI. , 1986] in order to subtract a combination of eye and head movement artifact. In order to create a baseline from which to compare the performance of other algorithms, we calculated the correlation of all sample trials with the red and yellow light averages from each subject's maximal P3 electrode site using the following formula: correlation = (sample * aveT)/(11 sample II * II ave II) (1) where sample and ave are both 1 x 500 vectors representing the trial epochs and light averages (respectively). We used the whole trial epoch for recognition because it yielded better recognition than just the time area around the P3. If the highest correlation of a trial epoch with the red and yellow averages was greater than 0.0, then the signal was classified as that type of signal. If both averages correlated negatively with the single trial, then the trial was counted as a yellow light signal. As can be seen in Table 1, the correct signal identification of red lights was extremely high while the yellow light identification pulled the results down. This may be explained by the greater variance of the yellow light epochs. Correlations in general were poor with typical correlations around 0.25. ICA has successfully been used in order to minimize artifacts in EEG data [Jung et at., 1997; Vigario, 1997] and has also proven useful in separating P3 component data from an averaged waveform [Makeig et aI., 1997]. The next experiment used ICA in order to try to separate the background EEG signal from the P3 signal. Independent component analysis (lCA) assumes that n EEG data channels x are a linear combination of n statistically independent signals s: x= As (2) where x and s are n x 1 vectors. We used the matlab package mentioned in [Makeig et aI. , 1997] with default learning values, which finds a matrix W by stochastic gradient descent. 8 J D. Bayliss and D. H. Ballard This matrix W performs component separation. All data was sphered in order to speed convergence time. After training the W matrix, the source channel showing the closest P3-like signal (using correlation with the average) for the red light average data was chosen as the signal with which to correlate individual epochs. The trained W matrix was also used to find the sources of the yellow light average. The red and yellow light responses were then correlated with individual epoch sources in the manner of the first experiment. The third experiment used the robust Kalman filter framework formulated by Rao [Rao, 1998]. The Kalman filter assumes a linear model similar to the one ofICA in equation 2, but assumes the EEG output x is the observable output of a generative or measurement matrix A and an internal state vector s of Gaussian sources. The output may also have an additional noise component n, a Gaussian stochastic noise process with mean zero and a covariance matrix given by ~ = E[nn Tj, leading to the model expression: x = As + n. In order to find the most optimal value of s, a weighted least-squares criterion is formulated: (3) where s follows a Gaussian distribution with mean s and covariance M. Minimizing this criterion by setting ~; = 0 and using the substitution N = (AT~-lU + M-1)-1 yields the Kalman filter equation, which is basically equal to the old estimate plus the Kalman gain times the residual error. (4) In an analogous manner, the measurement matrix A may be estimated (learned) if one assumes the physical relationships encoded by the measurement matrix are relatively stable. The learning rule for the measurement matrix may be derived in a manner similar to the rule for the internal state vector. In addition, a decay term is often needed in order to avoid overfitting the data set. See [Rao, 1998] for details. In our experiments both the internal state matrix s and the measurement matrix A were learned by training them on the average red light signal and the average yellow light signal. The signal is measured from the start of the trial which is known since it is triggered by the light change. We used a Kalman gain of 0.6 and a decay of 0.3. After training, the signal estimate for each epoch is correlated with the red and yellow light signal estimates in the manner of experiment 1. We made the Kalman filter statistically robust by ignoring parts of the EEG signal that fell outside a standard deviation of 1.0 from the training signals. The overall recognition results in Table 1 suggest that both the robust Kalman filter and ICA have a statistically significant advantage over correlation (p < 0.01). The robust Kalman filter has a very small advantage over ICA (not statistically significant). In order to look at the reliability of the best algorithm and its ability to be used on-line two of the Subjects (S4 and SS) returned for another VR driving session. In these sessions the brakes of the driving simulator were controlled by the robust Kalman filter recognition algorithm for red stop and yellow slow down lights. Green lights were ignored. The results of this session using the Robust Kalman Filter trained on the first session are shown in Table 2. The recognition numbers for red and yellow lights between the two sessions were compared using correlation. Red light scores between the sessions correlated fairly highly - 0.82 for S4 and 0.69 for SS. The yellow light scores between sessions correlated poorly with both S4 and SS at approximately -0.1. This indicates that the yellow light epochs tend to correlate poorly with each other due to the lack of a large component such as the P3 to tie them together. Recognizing Evoked Potentials in a Virtual Environment 9 4 Future Work This paper showed the viability of recognizing the P3 EP in a VR environment. We plan to allow the P3 EP to propel the user through alternate virtual rooms through the use of various binary decisions. In order to improve recognition for the BCI we need to experiment with a wider and more complex variety of recognition algorithms. Our most recent work has shown a dependence between the human computer interface used in the BCI and recognition. We would like to explore this dependence in order to improve recognition as much as possible. References [Bayliss and Ballard, 1998) lD. Bayliss and D.H. Ballard, ''The Effects of Eye Tracking in a VR Helmet on EEG Recording," TR 685, University of Rochester National Resource Laboratory for the Study of Brain and Behavior, May 1998. [Chapman and Bragdon, 1964) R.M. Chapman and H.R. Bragdon, "Evoked responses to numerical and non-numerical visual stimuli while problem solving.," Nature, 203: 1155-1157, 1964. [Farwell and Donchin, 1988) L. A. Farwell and E. Donchin, "Talking off the top of your head: toward a mental prosthesis utilizing event-related brain potentials," Electroenceph. Clin. Neurophysiol., pages 510-523, 1988. [Jung et al., 1997) 1'.P. Jung, C. Humphries,1'. Lee, S. Makeig, M.J. McKeown, Y. lragui, and 1'.l Sejnowski, "Extended ICA Removes Artifacts from Electroencephalographic Recordings," to Appear in Advances in Neural Information Processing Systems, 10, 1997. [Makeig et al., 1997) S. Makeig, 1'. Jung, A.J. Bell, D. Ghahremani, and 1'.J. Sejnowski, "Blind Separation of Auditory Event-related Brain Responses into Independent Components," Proc. Nat'l Acad. Sci. USA , 94:10979-10984, 1997. [McFarland et al., 1993) D.l McFarland, G.w. Neat, R.F. Read, and J.R. Wolpaw, "An EEG-based method for graded cursor control," Psychobiology, 21(1):77-81, 1993. [Pfurtscheller et al. , 1996) G. Pfurtscheller, D. Flotzinger, M. Pregenzer, J. Wolpaw, and D. McFarland, "EEG-based Brain Computer Interface (BCI)," Medical Progress through Technology, 21:111-121, 1996. [Polich, 1998] J. Polich, "P300 Clinical Utility and Control of Variability," J. of Clinical NeurophYSiology, 15(1): 14-33, 1998. [Rao, 1998] R. P.N. Rao, "Visual Attention during Recognition," Advances in Neural Information Processing Systems, 10, 1998. [Rosvold et al., 1956] H.E. Rosvold, A.F. Mirsky, I. Sarason, E.D. Bransome Jr., and L.H. Beck, "A Continuous Performance Test of Brain Damage," 1. Consult. Psychol., 20, 1956. [SemIitschetal., 1986) H.Y. SemIitsch, P. Anderer, P Schuster, and O. Presslich, "A solution for reliable and valid reduction of ocular artifacts applied to the P300 ERP;' Psychophys., 23:695703,1986. [Sutton et al., 1965) S. Sutton, M. Braren, J. Zublin, and E. John, "Evoked potential correlates of stimulus uncertainty," Science, 150: 1187-1188, 1965. [Vaughn et al., 1996) 1'.M. Vaughn, J.R. Wolpaw, and E. Donchin, "EEG-Based Communication: Prospects and Problems," IEEE Trans. on Rehabilitation Engineering, 4(4):425-430, 1996. [Vigario, 1997) R. Vigario, "Extraction of ocular artifacts from eeg using independent component analysis," Electroenceph. Clin. Neurophysiol., 103:395-404, 1997.
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Topographic Transformation as a Discrete Latent Variable Nebojsa Jojic Beckman Institute University of Illinois at Urbana www.ifp.uiuc.edu/",jojic Brendan J. Frey Computer Science University of Waterloo www.cs.uwaterloo.ca/ ... frey Abstract Invariance to topographic transformations such as translation and shearing in an image has been successfully incorporated into feedforward mechanisms, e.g., "convolutional neural networks", "tangent propagation". We describe a way to add transformation invariance to a generative density model by approximating the nonlinear transformation manifold by a discrete set of transformations. An EM algorithm for the original model can be extended to the new model by computing expectations over the set of transformations. We show how to add a discrete transformation variable to Gaussian mixture modeling, factor analysis and mixtures of factor analysis. We give results on filtering microscopy images, face and facial pose clustering, and handwritten digit modeling and recognition. 1 Introduction Imagine what happens to the point in the N-dimensional space corresponding to an N-pixel image of an object, while the object is deformed by shearing. A very small amount of shearing will move the point only slightly, so deforming the object by shearing will trace a continuous curve in the space of pixel intensities. As illustrated in Fig. la, extensive levels of shearing will produce a highly nonlinear curve (consider shearing a thin vertical line ), although the curve can be approximated by a straight line locally. Linear approximations of the transformation manifold have been used to significantly improve the performance of feedforward discriminative classifiers such as nearest neighbors (Simard et al., 1993) and multilayer perceptrons (Simard et al., 1992). Linear generative models (factor analysis, mixtures of factor analysis) have also been modified using linear approximations of the transformation manifold to build in some degree of transformation invariance (Hinton et al., 1997). In general, the linear approximation is accurate for transformations that couple neighboring pixels, but is inaccurate for transformations that couple nonneighboring pixels. In some applications (e.g., handwritten digit recognition), the input can be blurred so that the linear approximation becomes more robust. For significant levels of transformation, the nonlinear manifold can be better modeled using a discrete approximation. For example, the curve in Fig. 1a can be 478 N. Jojic and B. J. Frey (b) (c) (d) (e) p(z) ~ Figure 1: (a) An N-pixel greyscale image is represented by a point (unfilled disc) in an Ndimensional space. When the object being imaged is deformed by shearing. the point moves along a continuous curve. Locally. the curve is linear. but high levels of shearing produce a highly nonlinear curve. which we approximate by discrete points (filled discs) indexed bye. (b) A graphical model showing how a discrete transformation variable e can be added to a density model p(z) for a latent image z to model the observed image x . The Gaussian pdf p(xle, z) captures the eth transformation plus a small amount of pixel noise. (We use a box to represent variables that have Gaussian conditional pdfs.) We have explored (c) transformed mixtures of Gaussians. where c is a discrete cluster index; (d) transformed component analysis (TeA). where y is a vector of Gaussian factors. some of which may model locally linear transformation perturbations; and (e) mixtures of transformed component analyzers. or transformed mixtures of factor analyzers. represented by a set of points (filled discs). In this approach, a discrete set of possible transformations is specified beforehand and parameters are learned so that the model is invariant to the set of transformations. This approach has been used to design "convolutional neural networks" that are invariant to translation (Le Cun et al., 1998) and to develop a general purpose learning algorithm for generative topographic maps (Bishop et al., 1998). We describe how invariance to a discrete set of known transformations (like translation) can be built into a generative density model and we show how an EM algorithm for the original density model can be extended to the new model by computing expectations over the set of transformations. We give results for 5 different types of experiment involving translation and shearing. 2 Transformation as a Discrete Latent Variable We represent transformation f by a sparse transformation generating matrix Ge that operates on a vector of pixel intensities. For example, integer-pixel translations of an image can be represented by permutation matrices. Although other types of transformation matrix may not be accurately represented by permutation matrices, many useful types of transformation can be represented by sparse transformation matrices. For example, rotation and blurring can be represented by matrices that have a small number of nonzero elements per row (e.g., at most 6 for rotations). The observed image x is linked to the nontransformed latent image z and the transformation index f E {I, ... , L} as follows: p(xlf, z) = N(x; Gez, w), (1) where W is a diagonal matrix of pixel noise variances. Since the probability of a transformation may depend on the latent image, the joint distribution over the latent image z, the transformation index f and the observed image x is p(x, f, z) = N(x; Gez, w)P(flz)p(z). (2) The corresponding graphical model is shown in Fig. lb. For example, to model noisy transformed images of just one shape, we choose p(z) to be a Gaussian distribution. Topographic Transformation as a Discrete Latent Variable 479 2.1 Transformed mixtures of Gaussians (TMG). Fig. lc shows the graphical model for a TMG, where different clusters may havp. different transformation probabilities. Cluster c has mixing proportion 7rc , mean /-tc and diagonal covariance matrix ~ c. The joint distribution is (3) where the probability of transformation f for cluster c is Plc. Marginalizing over the latent image gives the cluster/transformation conditional likelihood, (4) which can be used to compute p(x) and the cluster/transformation responsibility P(f, clx). This likelihood looks like the likelihood for a mixture of factor analyzers (Ghahramani and Hinton, 1997). However, whereas the likelihood computation for N latent pixels takes order N 3 time in a mixture of factor analyzers, it takes linear time, order N, in a TMG, because Gl~cG'I + W is sparse. 2.2 Transformed component analysis (TCA). Fig. Id shows the graphical model for TCA (or "transformed factor analysis"). The latent image is modeled using linearly combined Gaussian factors, y. The joint distribution is p(x, f, z, y) = N(x; Glz, w)N(z; /-t + Ay, ~ )N(y; 0, I)Pl, (5) where /-t is the mean of the latent image, A is a matrix of latent image components (the factor loading matrix) and ~ is a diagonal noise covariance matrix for the latent image. Marginalizing over the factors and the latent image gives the transformation conditional likelihood, p(xlf) = N(x; Gl/-t, Gl(AA T + ~)G'I + w), (6) which can be used to compute p(x) and the transformation responsibility p(flx). Gl(AA T + ~)G'I is not sparse, so computing this likelihood exactly takes N 3 time. However, the likelihood can be computed in linear time if we assume IGl(AA T + f))G'I + wi ~ IGl(AAT + ~)G'II, which corresponds to assuming that the observed noise is smaller than the variation due to the latent image, or that the observed noise is accounted for by the latent noise model, ~. In our experiments, this approximation did not lead to degenerate behavior and produced useful models. By setting columns of A equal to the derivatives of /-t with respect to continuous transformation parameters, a TCA can accommodate both a local linear approximation and a discrete approximation to the transformation manifold. 2.3 Mixtures of transformed component analyzers (MTCA). A combination of a TMG and a TCA can be used to jointly model clusters, linear components and transformations. Alternatively, a mixture of Gaussians that is invariant to a discrete set of transformations and locally linear transformations can be obtained by combining a TMG with a TCA whose components are all set equal to transformation derivatives. The joint distribution for the combined model in Fig. Ie is p(x, f, z, c, y) = N(x; GlZ, w)N(z; /-tc + AcY, ~c)N(y; 0, I)Plc7rc. (7) The cluster/transformation likelihood is p(xlf,c) = N(X;Gl/-tc,Gl(AcA; + ~c)G'I + w), which can be approximated in linear time as for TCA. 480 N. Jojic and B. J. Frey 3 Mixed Transformed Component Analysis (MTCA) We present an EM algorithm for MTCA; EM algorithms for TMG or TCA emerge by setting the number of factors to 0 or setting the number of clusters to 1. Let 0 represent a parameter in the generative model. For Li.d. data, the derivative of the log-likelihood of a training set Xl, ... ,XT with respect to 0 can be written T 8 10gp(XI , ... ,XT) '"' [8 IJ)I ] 80 = ~ E 80 logp(xt, c, (., Z, Y Xt , t=l (8) where the expectation is taken over p(c, f, z, ylxt). The EM algorithm iteratively solves for a new set of parameters using the old parameters to compute the expectations. This procedure consistently increases the likelihood of the training data. By setting (8) to 0 and solving for the new parameter values, we obtain update equations based on the expectations given in the Appendix. Notation: (-) = .!. Ei=l (.) is a sufficient statistic computed by averaging over the training set; diag(A) gives a vector containing the diagonal elements of matrix A; diag(a) gives a diagonal matrix whose diagonal contains the elements of vector a; and a 0 h gives the element-wise product of vectors a and h. Denoting the updated parameters by "-", we have ire = (P(cJXt)), he = (P(flxt, c)), (9) _ (P(clxt)E[z - AeyIXt,c]) J.£e = (P(cIXt)) , diag( (P( cIXt)E[(z - J.£e - AeY) 0 (z - J.£e - Aey)IXt, cD) ~e = (P(cIXt)) , ~ = diag( (E[(Xt -GiZ)O(Xt - Giz)IXtD), Ae = (P(cJxdE[(z - J.£e)yTlxtl)(P(cIXt)E[yyTlxtD-I. (10) (11) (12) (13) To reduce the number of parameters, we will sometimes assume Pic does not depend on c or even that Pic is held constant at a uniform distribution. 4 Experiments 4.1 Filtering Images from a Scanning Electron Microscope (SEM). SEM images (e.g., Fig. 2a) can have a very low signal to noise ratio due to a high variance in electron emission rate and modulation of this variance by the imaged material (Golem and Cohen, 1998). To reduce noise, multiple images are usually averaged and the pixel variances can be used to estimate certainty in rendered structures. Fig. 2b shows the estimated means and variances of the pixels from 230 140 x 56 SEM images like the ones in Fig. 2a. In fact, averaging images does not take into account spatial uncertainties and filtering in the imaging process introduced by the electron detectors and the high-speed electrical circuits. We trained a single-cluster TMG with 5 horizontal shifts and 5 vertical shifts on the 230 SEM images using 30 iterations of EM. To keep the number of parameters almost equal to the number of parameters estimated using simple averaging, the transformation probabilities were not learned and the pixel variances in the observed image were set equal after each M step. So, TMG had 1 more parameter. Fig. 2c shows the mean and variance learned by the TMG. Compared to simple averaging, the TMG finds sharper, more detailed structure. The variances are significantly lower, indicating that the TMG produces a more confident estimate of the image. Topographic Transformation as a Discrete Latent Variable 481 (a) (b) (e) Figure 2: (a) 140 x 56 pixel SEM images. (b) The mean and variance of the image pixels. (c) The mean and variance found by a TMG reveal more structure and less uncertainty. (a) f f= l~ t -. f" I I\:~ , (d) ...-" y: ..', OJ ' .. . . ~ ,,~: .. ., .. ", .~ ,. .. --,-,;I - -, !. iI' . t., ~I '.j ,. .~ . . ,.?:" . ~ r; ,.,io .. - --.- .. -i ~ -.. .... VI ... ~ .. . ,.". , , ijIt .. FO .<t ~. .:p. ~;'''; . ~ -~_1~_. "'t~ ~ ~ . '-«, --.,j , . .l ~ ! . .,. .jIo f .. ~ r <,.\ Sf' ....... :or ... . ....... ~ ..... ~ :s... "iIIio ~'-- .,f~ ... !~V ~ ,fll" .. .~. , .,. ...... ;.'~. ....# ... ,:.~ • . ~ L --.~ ,.. ..,,,, . tr ' .. ~ .. -' I. ~- r . (e) (b) (f) (e) (g) Figure 3: (a) Frontal face images of two people. (b) Cluster means learned by a TMG and (c) a mixture of Gaussians. (d) Images of one person with different poses. (e) Cluster means learned by a TMG. (f) Less detailed cluster means learned by a mixture of Gaussians. (g) Mean and first 4 principal components of the data. which mostly model lighting and translation. 4.2 Clustering Faces and Poses. Fig. 3a shows examples from a training set of 400 jerky images of two people walking across a cluttered background. We trained a TMG with 4 clusters, 11 horizontal shifts and 11 vertical shifts using 15 iterations of EM after initializing the weights to small, random values. The loop-rich MATLAB script executed in 40 minutes on a 500MHz Pentium processor. Fig. 3b shows the cluster means, which include two sharp representations of each person's face, with the background clutter suppressed. Fig. 3c shows the much blurrier means for a mixture of Gaussians trained using 15 iterations of EM. Fig. 3d shows examples from a training set of 400 jerky images of one person with different poses. We trained a TMG with 5 clusters, 11 horizontal shifts and 11 vertical shifts using 40 iterations of EM. Fig. 3e shows the cluster means, which capture 4 poses and mostly suppress the background clutter. The mean for cluster 4 includes part of the background, but this cluster also has a low mixing proportion of 0.1. A traditional mixture of Gaussians trained using 40 iterations of EM finds blurrier means, as shown in Fig. 3f. The first 4 principal components mostly try to account for lighting and translation, as shown in Fig. 3g. 482 N Jojic and B. J. Frey (d) (e) (f) Figure 4: Modeling handwritten digits. (a) Means and components and (b) the sheared + translated means (dimmed transformations have low probability) for each of 10 TCA models trained on 200 examples of each digit. (c) Means and components of 10 FA models trained on the same data. (d) Digits generated from the 10 TCA models and (e) the 10 FA models. (f) The means for a mixture of 10 Gaussians, a mixture of 10 factor analyzers and a 10-ciuster TMG trained on all 2000 digits. In each case, the best of 10 experiments was selected. 4.3 Modeling Handwritten Digits. We performed both supervised and unsupervised learning experiments on 8 x 8 greyscale versions of 2000 digits from the CEDAR CDROM (Hull, 1994). Although the preprocessed images fit snugly in the 8 x 8 window, there is wide variation in "writing angle" (e.g., the vertical stroke of the 7 is at different angles). So, we produced a set of 29 shearing+translation transformations (see the top row of Fig. 4b) to use in transformed density models. In our supervised learning experiments, we trained one 10-component TCA on each class of digit using 30 iterations of EM. Fig. 4a shows the mean and 10 components for each of the 10 models. The lower 10 rows of images in Fig. 4b show the sheared and translated means. In cases where the transformation probability is below 1%, the image is dimmed. We also trained one lO-component factor analyzer on each class of digit using 30 iterations of EM. The means and components are shown in Fig. 4c. The means found by TCA are sharper and whereas the components found by factor analysis often account for writing angle (e.g., see the components for 7) the components found by TCA tend to account for line thickness and arc size. Fig. 4d and e show digits that were randomly generated from the TCAs and the factor analyzers. Since different components in the factor analyzers account for different stroke angles, the simulated digits often have an extra stroke, whereas digits simulated from the TCAs contain fewer spurious strokes. To test recognition performance, we trained 10-component factor analyzers and TCAs on 200 examples of each digit using 50 iterations of EM. Each set of models used Bayes rule to classify 1000 test patterns and while factor analysis gave an error rate of 3.2%, TCA gave an error rate of only 2.7%. In our unsupervised learning experiments, we fit 10-cluster mixture models to the entire set of 2000 digits to see which models could identify all 10 digits. We tried a mixture of 10 Gaussians, a mixture of 10 factor analyzers and a lO-cluster TMG. In each case, 10 models were trained using 100 iterations of EM and the model with Topographic Transformation as a Discrete Latent Variable 483 the highest likelihood was selected and is shown in Fig. 4f. Compared to the TMG, the first two methods found blurred and repeated classes. After identifying each cluster with its most prevalent class of digit, we found that the first two methods had error rates of 53% and 49%, but the TMG had a much lower error rate of 26%. 5 Summary In many learning applications, we know beforehand that the data includes transformations of an easily specified nature (e.g., shearing of digit images). If a generative density model is learned from the data, the model must extract a model of both the transformations and the more interesting and potentially useful structure. We described a way to add transformation invariance to a generative density model by approximating the transformation manifold with a discrete set of points. This releases the generative model from needing to model the transformations. 5 different types of experiment show that the method is effective and quite efficient. Although the time needed by this method scales exponentially with the dimensionality of the transformation manifold, we believe that it will be useful in many practical applications and that it illustrates what is possible with a generative model that incorporates a latent transformation variable. We are exploring the performance of a faster variational learning method and extending the model to time series. Acknowledgements. We used CITO, NSERC, NSF and Beckman Foundation grants. References C. M. Bishop, M. Svensen and C. K. I. Williams 1998. GTM: The generative topographic mapping. Neural Computation 10:1, 215- 235. G. E. Hinton, P. Dayan and M. Revow 1997. Modeling the manifolds of images of handwritten digits. IEEE 1rans. on Neural Networks 8, 65- 74. Z. Ghahramani and G. E. Hinton 1997. The EM algorithm for mixtures of factor analyzers. University of Toronto Technical Report CRG-TR-96-1. Available at www.gatsby.ucl.ac.uk/ ... zoubin. R. Golem and I. Cohen 1998. Scanning electron microscope image enhancement. School of Computer and Electrical Engineering project report, Ben-Gurion University. J. J . Hull 1994. A database for handwritten text recognition research. IEEE 1rans. on Pattern Analysis and Machine Intelligence 16:5, 550-554. Y. Le Cun, L. Bottou, Y. Bengio and P. Haffner 1998. Gradient-based learning applied to document recognition. Proceedings of the IEEE 86:11, November, 2278-2324. P. Y. Simard, B. Victorri, Y. Le Cun and J. Denker 1992. Tangent Prop - A formalism for specifying selected invariances in an adaptive network. In Advances in Neural Information Processing Systems 4, Morgan Kaufmann, San Mateo, CA. P. Y. Simard, Y. Le Cun and J. Denker 1993. Efficient pattern recognition using a new transformation distance. In S. J. Hanson, J . D. Cowan and C. L. Giles, Advances in Neural Information Processing Systems 5, Morgan Kaufmann, San Mateo, CA. Appendix: The Sufficient Statistics Found in the E-Step The sufficient statistics for the M-Step are computed in the E-Step using sparse linear algebra during a single pass through the training set. Before making this pass, the following matrices are computed: Ot,c = COV(zlx,y,l,c) = (~;l +G~-¥-lGt)-\ (3t,c = COV(ylx,l,c) = (I+A~~;JAc A~~;lOt.c~;lAc)-l. For each case in the training set, P(c,llxt) is first computed for each combination of c, l, before computing E[ylxt,l, c] = {3l,cA~ ~;;-l [Ot,cGi-¥-lXt - (I-Ot,c~;;-l )I-'c]' E[zlxt, l, c] = I-'c +Ot,cG~-¥-l (Xt - Gtl-'c) +Ot~-l Ac{3t,cA~ ~;;-lOt , cG~-¥-J (Xt - Gtl-'c), E[(2J-I.'c}:(2J-I.'c) IXt ,l, c] = (E[zlxt ,l,cH.&c}c(E[zlxt ,l,cH.&c)+diag(Ot,c)+diag(Ol , c~;l Ac{3t,cA~ ~;lOt,c), E[(z-l-'c)y'lxt , l, c] = (E[zlxt,l, c]-l-'c)E[Ylxt,l, c]' + Ot,c~;l A c{3t ,c' The expectations needed in (10)-(13) are then computed from P(clxt}E[z - AcylXt, c] = Et P(c, lIXt)(E[zIXt, l, c] - AcE[Ylxt,l, cD, P(clxt}E[(z-l-'cAcy)o(z-l-'c-AcY) IXt, c] = Et P(c, llxt} {E[(z-l-'c)o(z-I-'C>lxt,l, c] +diag(Acl3t,cA~) - 2diag(AcE[(zl-'C>y'lxt, l, en +(AcE[Ylxt, e, c])o(AcE[ylxt, e, cD}, E[(Xt-Gtz)o(Xt-Gtz)lxtl = Ec,t P(c,lIXt) {(XtGtE[zlxt, l, c]) 0 (Xt - GtE[zIXt, l, c]) + diag(GtOt,cGi) + diag(GtOt,c~;l Ac{3t,cA~~;lOt,cG~)}, P(clxt}E[(z-l-')y'lxt, c] = El P(c,llxt}E[(z-l-')y'lxt, l, c], P(clxt)E[yy'lxt, c] = Et P(c,llxt}{3t,c + Et P(c, llxt)E[ylxt.e, c]E[ylxt.l, c]'.
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Channel Noise in Excitable Neuronal Membranes Amit Manwani; Peter N. Steinmetz and Christof Koch Computation and Neural Systems Program, M-S 139-74 California Institute of Technology Pasadena, CA 91125 { quixote,peter,koch } @klab.caltech.edu Abstract Stochastic fluctuations of voltage-gated ion channels generate current and voltage noise in neuronal membranes. This noise may be a critical determinant of the efficacy of information processing within neural systems. Using Monte-Carlo simulations, we carry out a systematic investigation of the relationship between channel kinetics and the resulting membrane voltage noise using a stochastic Markov version of the Mainen-Sejnowski model of dendritic excitability in cortical neurons. Our simulations show that kinetic parameters which lead to an increase in membrane excitability (increasing channel densities, decreasing temperature) also lead to an increase in the magnitude of the sub-threshold voltage noise. Noise also increases as the membrane is depolarized from rest towards threshold. This suggests that channel fluctuations may interfere with a neuron's ability to function as an integrator of its synaptic inputs and may limit the reliability and precision of neural information processing. 1 Introduction Voltage-gated ion channels undergo random transitions between different conformational states due to thermal agitation. Generally, these states differ in their ionic permeabilities and the stochastic transitions between them give rise to conductance fluctuations which are a source of membrane noise [1]. In excitable cells, voltage-gated channel noise can contribute to the generation of spontaneous action potentials [2, 3], and the variability of spike timing [4]. Channel fluctuations can also give rise to bursting and chaotic spiking dynamics in neurons [5, 6]. Our interest in studying membrane noise is based on the thesis that noise ultimately limits the ability of neurons to transmit and process information. To study this problem, we combine methods from information theory, membrane biophysics and compartmental neuronal modeling to evaluate ability of different biophysical components of a neuron, such as the synapse, the dendritic tree, the soma and so on, to transmit information [7, 8, 9]. These neuronal components differ in the type, density, and kinetic properties of their constituent ion channels. Thus, measuring the impact of these differences on membrane noise rep• http://www.klab.caltech.edwquixote 144 A. Manwani, P. N. Steinmetz and C. Koch resents a fundamental step in our overall program of evaluating information transmission within and between neurons. Although information in the nervous system is mostly communicated in the form of action potentials, we first direct our attention to the study of sub-threshold voltage fluctuations for three reasons. Firstly, voltage fluctuations near threshold can cause variability in spike timing and thus directly influence the reliability and precision of neuronal activity. Secondly, many computations putatively performed in the dendritic tree (coincidence detection, multiplication, synaptic integration and so on) occur in the sub-threshold regime and thus are likely to be influenced by sub-threshold voltage noise. Lastly, several sensory neurons in vertebrates and invertebrates are non-spiking and an analysis of voltage fluctuations can be used to study information processing in these systems as well. Extensive investigations of channel noise were carried out prior to the advent of the patchclamp technique in order to provide indirect evidence for the existence of single ion channels (see [1] for an excellentreview). More recently, theoretical studies have focused on the effect of random channel fluctuations on spike timing and reliability of individual neurons [4], as well as their effect on the dynamics of interconnected networks of neurons [5, 6). In this paper, we determine the effect of varying the kinetic parameters, such as channel density and the rate of channel transitions, on the magnitude of sub-threshold voltage noise in an iso-potential membrane patches containing stochastic voltage-gated ion channels using Monte-Carlo simulations. The simulations are based on the Mainen-Sejnowski (MS) kinetic model of active channels in the dendrites of cortical pyramidal neurons [10). By varying two model parameters (channel densities and temperature), we investigate the relationship between excitability and noise in neuronal membranes. By linearizing the channel kinetics, we derive analytical expressions which provide closed-form estimates of noise magnitudes; we contrast the results of the simulations with the linearized expressions to determine the parameter range over which they can be used. 2 Monte-Carlo Simulations Consider an iso-potential membrane patch containing voltage-gated K+and Na+channels and leak channels, dVrn ) ( -c dt = 9K (Vm - EK) + 9Na (Vm - ENa + 9L Vm - Ed + Iinj (1) where C is the membrane capacitance and 9K (9Na, 9L) and EK (ENa, EL) denote the K+(Na+, leak) conductance and the K+(Na+, leak) reversal potential respectively. Current injected into the patch is denoted by Iinj , with the convention that inward current is negative. The channels which give rise to potassium and sodium conductances switch randomly between different conformational states with voltage-dependent transition rates. Thus,9K and 9Na are voltage-dependent random processes and eq. 1 is a non-linear stochastic differential equation. Generally, ion channel transitions are assumed to be Markovian [11] and the stochastic dynamics of eq. 1 can be studied using Monte-Carlo simulations of finitestate Markov models of channel kinetics. Earlier studies have carried out simulations of stochastic versions of the classical HodgkinHuxley kinetic model [12] to study the effect of conductance fluctuations on neuronal spiking [13, 2, 4]. Since we are interested in sub-threshold voltage noise, we consider a stochastic Markov version of a less excitable kinetic model used to describe dendrites of cortical neurons [10]. We shall refer to it as the Mainen-Sejnowski (MS) kinetic scheme. The K+conductance is modeled by a single activation sub-unit (denoted by n) whereas the Na+conductance is comprised of three identical activation sub-units (denoted by m) and one inactivation sub-unit (denoted by h). Thus, the stochastic discrete-state Markov models of the K+and Na+channel have 2 and 8 states respectively (shown in Fig. 1). The Channel Noise in Excitable Neural Membranes 145 single channel conductances and the densities of the ion channels (K+ ,Na+) are denoted by (,K,''(Na) and ('TJK,'f)Na) respectively. Thus, 9K and 9Na) are given by the products of the respective single channel conductances and the corresponding numbers of channels in the conducting states. A Figure I: Kinetic scheme for the voltage-gated Mainen-Sejnowski K+(A) and Na+(B) channels. no and nl represent the closed and open states of K+channel. mO-2hl represent the 3 closed states, mO-3ho the four inactivated states and m3hl the open state of the N a + channel. We performed Monte-Carlo simulations of the MS kinetic scheme using a fixed time step of i).t = 10 J.tsec. During each step, the number of sub-units undergoing transitions between states i and j was determined by drawing a pseudorandom binomial deviate (bnldev subroutine [14] driven by the ran2 subroutine of the 2nd edition) with N equal to the number of sub-units in state i and p given by the conditional probability of the transition between i and j. After updating the number of channels in each state, eq. 1 was integrated using fourth order Runge-Kutta integration with adaptive step size control [14]. During each step, the channel conductances were held at the fixed value corresponding to the new numbers of open channels. (See [4] for details ofthis procedure). 6 4 -4 -6 _8L-~------~----~------~~ -70 -60V (mV)50 -40 m Figure 2: Steady-state I-V curves for different multiples (f\,Na) of the nominal MS Na+channel density. Circles indicate locations of fixed-points in the absence of current injection. Due to random channel transitions, the membrane voltage fluctuates around the steady-state resting membrane voltage Vrest . By varying the magnitude of the constant injected current linj, the steady-state voltage can be varied over a broad range, which depends on the channel densities. The current required to maintain the membrane at a holding voltage Vhold can be determined from the steady-state I-V curve of the system, as shown in Fig. 2. Voltages for which the slope of the I-V curve is negative cannot be maintained as steady-states. By injecting an external current to offset the total membrane current, a fixed point in the negative slope region can be obtained but since the fixed point is unstable, any perturbation, such as a stochastic ion channel opening or closing, causes the system to be driven to the closest stable fixed point. We measured sub-threshold voltage noise only for stable steady-state holding voltages. A typical voltage trace from our simulations is shown in Fig. 3. To estimate the standard deviation of the voltage noise accurately, simulations were performed for a total of 492 seconds, divided into 60 blocks of 8.2 seconds each, for each steady-state value. 146 A. Manwani, P. N. Steinmetz and C. Koch 5r---~----~--~----~---. -~ Figure 3: Monte-Carlo simulations of a 1000 j.Lm2 membrane patch with stochastic Na+ and deterministic K+ channels with MS kinetics. Bottom record shows the number of open Na+ channels as a function of time. Top trace shows the corresponding fluctuations of the membrane voltage. Summary of nominal MS parameters: em = 0.75 j.LF/cm2 , 11K = 1.5 channels/j.Lm2 , 11Na = 2 channelslj.Lm2 , EK = -90 mY, ENa = 60 mY, EL = -70 mY, gL = 0.25 pSlj.Lm2 , "IK = "INa = 20 pS. '" ~ 4 ffi 1:; o ~ 3 ! '0 2 ~ E :0 Z 1 100 200 300 400 Time (msec) -65 :> .§. ~ -66 ~ Q) c: ~ D -67 ~ 3 Linearized Analysis The non-linear stochastic differential equation (eq. 1) cannot be solved analytically. However, one can linearize it by expressing the ionic conductances and the membrane voltage as small perturbations (8) around their steady-state values: -c d~~m = (9~ + 9Na + 9L) 8Vm + (V~ - EK) 89K + (V~ - ENa) 89Na (2) where 9~ and 9Na denote the values of the ionic conductances at the steady-state voltage va. G = 9K + 9N a + 9 L is the total steady-state patch conductance. Since the leak channel conductance is constant , 89 L = o. On the other hand, 89 K and 89 N a depend on 8V and t. It is known that, to first order, the voltage- and time-dependence of active ion channels can be modeled as phenomenological impedances [15, 16]. Fig. 4 shows the linearized equivalent circuit of a membrane patch, given by the parallel combination of the capacitance C, the conductance G and three series RL branches representing phenomenological models of K+activation, Na+activation and Na+inactivation. In = 9K(EK V~) + 9Na(ENa V~) (3) represents the current noise due to fluctuations in the channel conductances (denoted by 9K and 9Na) at the membrane voltage V~ (also referred to as holding voltage Vhald). The details of the linearization are provided [16]. The complex admittance (inverse of the impedance) of Fig. 4 is given by, I I I Y(J) = G + j27r fC + . + . + . (4) Tn + J27rf ln Tm + J27rf lm Th + J27rfl h The variance of the voltage fluctuations O"~ can be computed as, 2 / 00 SIn(J) O"v = -00 df IY(J)12 (5) where the power spectral density of In is given by the sum of the individual channel current noise spectra, SIn(J) = SIK(J) + SINa(J). For the MS scheme, the autocovariance of the K+ current noise for patch of area A, clamped at a voltage V~, can be derived using [1, 11], CIK (t) = A 'f/K "Ik ( V~ - EK)2 noo (1 - noo) e-Itl/rn (6) where noo and Tn respectively denote the steady-state probability and time constant of the K+ activation sub-unit at V~ . The power spectral density of the K+ current noise S I K (J) can be obtained from the Fourier transform of C I K (t), S (f) = 2 A 'f/K "Ik (V~ - EK )2noo Tn (7) IK 1 + (21ffTn)2 Channel Noise in Excitable Neural Membranes c G 147 Figure 4: Linearized circuit of the membrane patch containing stochastic voltage-gated ion channels. C denotes the membrane capacitance, G is the sum of the steady-state conductances of the channels and the leak. ri's and li'S denote the phenomenological resistances and inductances due to the voltage- and time-dependent ionic conductances. Thus, SIK(J) is a single Lorentzian spectrum with cut-off frequency determined by Tn. Similarly, the auto-covariance of the MS Na+ current noise can be written as [1], CINa(t) = A rJNa ,iva (V~ - ENa)2 m~ hoo [m 3(t) h(t) - m~ hoo] (8) where m(t) = moo + (1 - moo) e-t / Tm , h(t) = hoo + (1 - hoo) e-t / Th (9) As before, moo (hoo) and Tm (Th) are the open probability and the time constant of Na+activation (inactivation) sub-unit. The Na+current noise spectrum SINa(J) can be expressed as a sum of Lorentzian spectra with cut-off frequencies corresponding to the seven time constants T m, Th, 2 T m, 3 T m, T m + Th, 2 T m + Th and 3 T m + Th. The details of the derivations can be found in [8]. A 5 4 1 0~~~--~~----~4~0----~-20 Vh01d(mV) B 3 + + 1 o~o:s:-=----::-:---~:------! -60 -40 -20 VhOId(mv) Figure 5: Standard deviation of the voltage noise av in a 1000 f..£m 2 patch as a function of the holding voltage Vho1d . Circles denote results of the Monte-Carlo simulations for the nominal MS parameter values (see Fig. 3). The solid curve corresponds to the theoretical expression obtained by linearizing the channel kinetics. (A) Effect of increasing the sodium channel density by a factor (compared to the nominal value) of 2 (pluses), 3 (asterisks) and 4 (squares) on the magnitude of voltage noise. (B) Effect of increasing both the sodium and potassium channel densities by a factor of two (pluses). 4 Effect of Varying Channel Densities Fig. 5 shows the voltage noise for a 1000 J.im2 patch as a function of the holding voltage for different values of the channel densities. Noise increases as the membrane is depolarized from rest towards -50 mV and the rate of increase is higher for higher Na+densities. The range of Vho1d for sub-threshold behavior extends up to -20 m V for nominal densities, 148 A. Manwani, P N Steinmetz and C. Koch but does not exceed -60 m V for higher N a + densities. For moderate levels of depolarization, an increase in the magnitude of the ionic current noise with voltage is the dominant factor which leads to an increase in voltage noise; for higher voltages phenomenological impedances are large and shunt away the current noise. Increasing Na+density increases voltage noise, whereas, increasing K+density causes a decrease in noise magnitude (compare Fig. SA and SB). We linearized closed-form expressions provide accurate estimates of the noise magnitudes when the noise is small (of the order 3 m V). 5 Effect of Varying Temperature Fig. 6 shows that voltage noise decreases with temperature. To model the effect of temperature, transition rates were scaled by a factor Q':oT/lO (QlO = 2.3 for K+, QlO = 3 for Na+). Temperature increases the rates of channel transitions and thus the bandwidth of the ionic current noise fluctuations. The magnitude of the current noise, on the other hand, is independent of temperature. Since the membrane acts as a low-pass RC filter (at moderately depolarized voltages, the phenomenological inductances are small), increasing the bandwidth of the noise results in lower voltage noise as the high frequency components are filtered out. 6 Conclusions 1.5 .---~---~---...-, 0_5 o~--~---~----~ 20 25 30 35 T (CelsiuS) Figure 6: ay as a function of temperature for a 1000 J-Lm2 patch with MS kinetics (V hold = -60 m V). Circles denote Monte-Carlo simulations. solid curve denotes linearized approximation. We studied sub-threshold voltage noise due to stochastic ion channel fluctuations in an isopotential membrane patch with Mainen-Sejnowski kinetics. For the MS kinetic scheme, noise increases as the membrane is depolarized from rest, up to the point where the phenomenological impedances due to the voltage- and time-dependence of the ion channels become large and shunt away the noise. Increasing Na+channel density increases both the magnitude of the noise and its rate of increase with membrane voltage. On the other hand, increasing the rates of channel transitions by increasing temperature, leads to a decrease in noise. It has previously been shown that neural excitability increases with Na+channel density [17] and decreases with temperature [IS] . Thus, our findings suggest that an increase in membrane excitability is inevitably accompanied by an increase in the magnitude of sub-threshold voltage noise fluctuations. The magnitude and the rapid increase of voltage noise with depolarization suggests that channel fluctuations can contribute significantly to the variability in spike timing [4] and the stochastic nature of ion channels may have a significant impact on information processing within individual neurons. It also potentially argues against the conventional role of a neuron as integrator of synaptic inputs [18], as the the slow depolarization associated with integration of small synaptic inputs would be accompanied by noise. making the membrane voltage a very unreliable indicator of the integrated inputs. We are actively investigating this issue more carefully. When the magnitudes of the noise and the phenomenological impedances are small, the non-linear kinetic schemes are well-modeled by their linearized approximations. We have found this to be valid for other kinetic schemes as well [19]. These analytical approximations can be used to study noise in more sophisticated neuronal models incorporating realistic dendritic geometries, where Monte-Carlo simulations may be too computationally intensive to use. Channel Noise in Excitable Neural Membranes 149 Acknowledgments This work was funded by NSF, NIMH and the Sloan Center for Theoretical Neuroscience. We thank our collaborators Michael London, Idan Segev and YosefYarom for their invaluable suggestions. References [1] DeFelice LJ. (1981). Introduction to Membrane Noise. Plenum Press: New York, New York. [2] Strassberg A.F. & DeFelice LJ. (1993). Limitations of the Hodgkin-Huxley formalism: effect of single channel kinetics on transmembrane voltage dynamics. Neural Computation, 5:843855. [3] Chow C. & White l (1996). Spontaneous action potentials due to channel fluctuations. Biophy. 1.,71:3013-3021. [4] Schneidman E., Freedman B. & Segev I. (1998). Ion-channel stochasticity may be critical in determining the reliability and precision of spike timing. Neural Computation, 10:1679-1703. [5] DeFelice LJ. & Isaac A. (1992). Chaotic states in a random world. 1. Stat. Phys., 70:339-352. [6] White lA., Budde T. & Kay A.R. (1995). A bifurcation analysis of neuronal subthreshold oscillations. Biophy. J., 69:1203-1217. [7] Manwani A. & Koch C. (1998). Synaptic transmission: An information-theoretic perspective. In: Jordan M., Kearns M.S. & SoBa S.A., eds., Advances in Neural Information Processing Systems 10. pp 201-207. MIT Press: Cambridge, Massachusetts. [8] Manwani A. & Koch C. (1999). Detecting and estimating signals in noisy cable structures: I. Neuronal noise sources. Neural Computation. In press. [9] Manwani A. & Koch C. (1999). Detecting and estimating signals in noisy cable structures: II. Information-theoretic analysis. Neural Computation. In press. [10] Mainen Z.F. & Sejnowski TJ. (1995). Reliability of spike timing in neocortical neurons. Science, 268: 1503-1506. [11] Johnston D. & Wu S.M. (1995). Foundations of Cellular Neurophysiology. MIT Press: Cambridge, Massachusetts. [12] Hodgkin A.L. & Huxley A.F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. 1. Physiol. (London), 117: 500-544. [13] Skaugen E. & Wallre L. (1979). Firing behavior in a stochastic nerve membrane model based upon the Hodgkin-Huxley equations. Acta Physiol. Scand., 107:343-363. [14] Press W.H., Teukolsky S.A., Vetterling w.T. & Flannery B.P. (1992). Numerical Recipes in C: The An of Scientific Computing. Cambridge University Press, second edn. [15] Mauro A., Conti F., Dodge F. & Schor R. (1970). Subthreshold behavior and phenomenological impedance of the squid giant axon. 1. Gen. Physiol., 55:497-523. [16] Koch C. (1984). Cable theory in neurons with active, linearized membranes. BioI. Cybem., 50:15-33. [17] Sabah N.H. & Leibovic K.N. (1972). The effect of membrane parameters on the properties of the nerve impulse. Biophys. 1., 12:1132-44. [18] Shadlen M.N. & Newsome w.T. (1998). The variable discharge of cortical neurons: implications for connectivity, computation, and information coding. 1. Neurosci., 18:3870-3896. [19] P. N. Steinmetz A. Manwani M.L. & Koch C. (1999). Sub-threshold voltage noise due to channel fluctuations in active neuronal membranes. In preparation.
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Efficient Approaches to Gaussian Process Classification Lehel Csato, Ernest Fokoue, Manfred Opper, Bernhard Schottky Neural Computing Research Group School of Engineering and Applied Sciences Aston University Birmingham B4 7ET, UK. {opperm,csatol}~aston.ac.uk Ole Winther Theoretical Physics II, Lund University, Solvegatan 14 A, S-223 62 Lund, Sweden winther~thep.lu.se Abstract We present three simple approximations for the calculation of the posterior mean in Gaussian Process classification. The first two methods are related to mean field ideas known in Statistical Physics. The third approach is based on Bayesian online approach which was motivated by recent results in the Statistical Mechanics of Neural Networks. We present simulation results showing: 1. that the mean field Bayesian evidence may be used for hyperparameter tuning and 2. that the online approach may achieve a low training error fast. 1 Introduction Gaussian processes provide promising non-parametric Bayesian approaches to regression and classification [2, 1]. In these statistical models, it is assumed that the likelihood of an output or target variable y for a given input x E RN can be written as P(Yla(x)) where a : RN --+ R are functions which have a Gaussian prior distribution, i.e. a is (a priori) assumed to be a Gaussian random field. This means that any finite set of field variables a(xi), i = 1, ... ,l are jointly Gaussian distributed with a given covariance E[a(xi)a(xj)] = K(Xi' Xj) (we will also assume a zero mean throughout the paper). Predictions on a(x) for novel inputs x, when a set D of m training examples (Xi, Yi) i = 1, . . . , m , is given, can be computed from the posterior distribution of the m + 1 variables a(x) and a(xd, ... ,a(xm). A major technical problem of the Gaussian process models is the difficulty of computing posterior averages as high dimensional integrals, when the likelihood is not Gaussian. This happens for example in classification problems. So far, a variety of approximation techniques have been discussed: Monte Carlo sampling [2], the MAP approach [4], bounds on the likelihood [3] and a TAP mean field approach [5]. In this paper, we will introduce three different novel methods for approximating the posterior mean of the random field a(x), which we think are simple enough to be used in practical applications. Two of the techniques 252 L. Csato, E. Fokoue, M Opper, B. Schottky and 0. Winther are based on mean field ideas from Statistical Mechanics, which in contrast to the previously developed TAP approach are easier to implement. They also yield simple approximations to the total likelihood of the data (the evidence) which can be used to tune the hyperparameters in the covariance kernel K (The Bayesian evidence (or MLII) framework aims at maximizing the likelihood of the data). We specialize to the case of a binary classification problem, where for simplicity, the class label Y = ± 1 is assumed to be noise free and the likelihood is chosen as P(Yla) = 8(ya) , (1) where 8(x) is the unit step function, which equals 1 for x > 0 and zero else. We are interested in computing efficient approximations to the posterior mean (a(x)), which we will use for a prediction of the labels via Y = sign(a(x)), where ( .. . ) denotes the posterior expectation. If the posterior distribution of a(x) is symmetric around its mean, this will give the Bayes optimal prediction. Before starting, let us add two comments on the likelihood (1). First, the MAP approach (i.e. predicting with the fields a that maximize the posterior) would not be applicable, because it gives the trivial result a(x) = O. Second, noise can be easily introduced within a probit model [2], all subsequent calculations will only be slightly altered. Moreover, the Gaussian average involved in the definition of the probit likelihood can always be shifted from the likelihood into the Gaussian process prior, by a redefinition of the fields a (which does not change the prediction), leaving us with the simple likelihood (1) and a modified process covariance [5]. 2 Exact Results At first glance, it may seem that in order to calculate (a(x)) we have to deal with the joint posterior of the fields ai = a(xi)' i = 1, ... , m together with the field at the test point a(x) . This would imply that for any test point, a different new m + 1 dimensional average has to be performed. Actually, we will show that this is not the case. As above let E denote the expectation over the Gaussian prior. The posterior expectation at any point, say x E [a (x) TI7=l P(Yj laj)] (a(x)) = --=-~---~--"E [TI7=l P(Yj laj)] can by integration by parts-for any likelihood-be written as (2) )) ~ ) and ",.=y . (CHnp(Yj1aj)) (3) (a(x = L...J K(x, Xj ajYj .... J J aa . j J showing that aj is not dependent on the test point x. It is therefore not necessary to compute a m + 1 dimensional average for every prediction. We have chosen the specific definition (3) in order to stress the similarity to predictions with Support Vector Machines (for the likelihood (1), the aj will come out nonnegative). In the next sections we will develop three approaches for an approximate computation of the aj. 3 Mean Field Method I: Ensemble Learning Our first goal is to approximate the true posterior distribution 1 1 1 T -1 m p(alD ) - e-"2 a K a II P(Y ·la·) m - Z J(27r)m detK j=l J J (4) Efficient Approaches to Gaussian Process Classification 253 of a == (al,"" am) by a simpler, tractable distribution q. Here, K denotes the covariance matrix with elements Kij = K (Xi, Xj). In the variational mean field approach-known as ensemble learning in the Neural Computation Community,the relative entropy distance K L(q,p) = J da q(a) In :~:~ is minimized in the family of product distributions q(a) = TI,7=l qj(aj). This is in contrast to [3], where a variational bound on the likelihood is computed. We get KL(q,p) = ! qi(ai) ~ daiqi(ai) In P(Yilai) + , ~ L [K-1Lj (ai)O(aj)o + ~ L [K-1Li (a~)o i,j,i#j i where ( .. ')0 denotes expectation w.r.t. q. By setting the functional derivative of KL(q,p) with respect to qi(a) equal to zero, we find that the best product distribution is a Gaussian prior times the original Likelihood: 1 (o-"'i)2 qi(a) ex: P(Yila) ~ehi (5) where mi = -Ai :Ej,#i(K-l)ij(aj)o and Ai = [K-l]:l. Using this specific form for the approximated posterior q(a), replacing the average over the true posterior in (3) by the approximation (5), we get (using the likelihood (I)) a set of m nonlinear equations in the unknowns aj: where D(z) = e- z2 / 2/..f2i and <I>(z) = J~oo dt D(t). As a useful byproduct of the variational approximation, an upper bound on the Bayesian evidence P(D) = J da 7r(a)P{Dla) can be derived. (71' denotes the Gaussian process prior and P{Dla) = TI,7=l P(Yjlaj)). The bound can be written in terms of the mean field 'free energy' as -lnP(D) < Eqlnq(a) -Eqln[7r(a)P(Dla)] - '''In<l> (Y' m j ) + ~ "'y·a ·(K· · -8" A·)y ·a· (7) ~ , JXi 2 ~ " '3 'J' 3 3 , ~ which can be used as a yardstick for selecting appropriate hyperparameters in the covariance kernel. The ensemble learning approach has the little drawback, that it requires inversion of the covariance matrix K and, for the free energy (7) one must compute a determinant. A second, simpler approximation avoids these computations. 4 Mean Field Theory II: A 'Naive' Approach The second mean field theory aims at working directly with the variables aj. As a starting point, we consider the partition function (evidence), Z = P(D) = ! dze-tzTKz IT P(Yjlzj) , (8) ;=1 254 L. Csato. E. Fokoue. M. Opper. B. Schottky and O. Winther which follows from (4) by a standard Gaussian integration, introducing the Fourier transform of the Likelihood .P(Ylz) = I g~ eiaz P(Yla) with i being the imaginary unit. It is tempting to view (8) as a normalizing partition function for a Gaussian process Zi having covariance matrix K-l and likelihood P. Unfortunately, P is not a real number and precludes a proper probabilistic interpretation. N evertheless, dealing formally with the complex measure defined by (8), integration by parts shows that one has YjCl!j = -i(zj)., where the brackets ( ... ). denote a average over the complex measure. This suggests a simple approximation for calculating the Cl!j. One may think of trying a saddle-point (or steepest descent) approximation to (8) and replace (Zj). by the value of Zj (in the complex Z plane) which makes the integrand stationary thereby neglecting the fluctuations of the Zj. Hence, this approximation would treat expectations of products as (ZiZj). as (Zi).(Zj)*, which may be reasonable for i i= j, but definitely not for the self-correlation i = j. According to the general formalism of mean field theories (outlined e.g. in [6]), one can improve on that idea, by treating the 'self-interactions' z; separately. This can be done by replacing all Zi (except in the form zi) by a new variable J.Li by inserting a Dirac 8 function representation 8(z J.L) = J ~r;:e-im(z-J1.) into (8) and integrate over the Z and a variables exactly (the integral factorizes), and finally perform a saddle-point integration over the m and J.L variables. The details of this calculation will be given elsewhere. Within the saddle-point approximation, we get the system of nonlinear equations i,i=l=j i,i=l=j which is of the same form as (6) with Aj replaced by the simpler K jj . These equations have also been derived by us in [5] using a Callen identity, but our present derivation allows also for an approximation to the evidence. By plugging the saddlepoint values back into the partition function, we get -lnP(D) ~ - ~ln~ (Y'~) + ~ ~Y'Cl!'(K" - 8· ·K .. )y·Cl!· ~ t V Kii 2 ~ t t tJ tJ 11 J J t ~ which is also simpler to compute than (7) but does not give a bound on the true evidence. 5 A sequential Approach Both previous algorithms do not give an explicit expression for the posterior mean, but require the solution of a set of nonlinear equations. These must be obtained by an iterative procedure. We now present a different approach for an approximate computation of the posterior mean, which is based on a single sequential sweep through the whole dataset giving an explicit update of the posterior. The algorithm is based on a recently proposed Bayesian approach to online learning (see [8] and the articles of Opper and Winther& Solla in [9]). Its basic idea applied to the Gaussian process scenario, is as follows: Suppose, that qt is a Gaussian approximation to the posterior after having seen t examples. This means that we approximate the posterior process by a Gaussian process with mean (a(x))t and covariance Kt(x, y), starting with (a(x))o = 0 and Ko(x, y) = K(x, y). After a new data point Yt+l is observed, the posterior is updated according to Bayes rule. The new non-Gaussian posterior qt is projected back into the family of Gaussians by choosing the closest Gaussian qt+l minimizing the relative entropy K L(qt, qt+d Efficient Approaches to Gaussian Process Classification 255 in order to keep the loss of information small. This projection is equivalent to a matching of the first two moments of lit and qt+1' E.g., for the first moment we get (a(x) P(Yt+1la(Xt+d))t (a(x))t+l = (P( I ( ))) = (a(x))t + Kl (t)Kt(x, Xt+l) Yt+l a Xt+l t where the second line follows again from an integration by parts and Kt{t) 1!:!±!. ~ with z Yt±l (a(Xt±l)t and ~2(t) K (x x ) This recursion and u tJ>(zt} t u(t) t t+l, t+l . the corresponding one for K t can be solved by the ansatz (a(x))t t L K(x, xj)Yjaj(t) j=l L K(x, Xi)Cij (t)K(x, Xj) + K(x, y) i,j (10) (11) where the vector a(t) = (al,"" at, 0, 0, ... ) and the matrix C(t) (which has also only txt nonzero elements) are updated as a(t + 1) C(t + 1) a(t) + Kl(t) (C(t)kt+l + et+l) @y C(t) + K2(t) (C(t)kt+l + et+1) (C(t)kt+1 + et+1f (12) where K2 (t) = ~ { ~~~:») - (~gt\) r}, k t is the vector with elements K tj , j = 1 ... , t and @ denotes the element-wise product between vectors. The sequential algorithm defined by (10)-(12) has the advantage of not requiring any matrix inversions. There is also no need to solve a numerical optimization problem at each time as in the approach of [11] where a different update of a Gaussian posterior approximation was proposed. Since we do not require a linearization of the likelihood, the method is not equivalent to the extended Kalman Filter approach. Since it is possible to compute the evidence of the new datapoint P(Yt+1) = (P(Yt+1lat+d)t based on the old posterior, we can compute a further approximation to the log evidence for m data via In P(Dm) = 2:~lln(P(Yt+llat+1)k 6 Simulations We present two sets of simulations for the mean field approaches. In the first, we test the Bayesian evidence framework for tuning the hyperparameters of the covariance function (kernel). In the second, we test the ability of the sequential approach to achieve low training error and a stable test error for fixed hyperparameters. For the evidence framework, we give simulation results for both mean field free energies (7) and (10) on a single data set, 'Pima Indian Diabetes (with 200/332 training/test-examples and input dimensionality d = 7) [7]. The results should therefore not be taken as a conclusive evidence for the merits of these approaches, but simply as an indication that they may give reasonable results. We use the radial basis function covariance function K(x,x') = exp (-~ 2:~WI(XI - XD2) . A diagonal term v is added to the covariance matrix corresponding to a Gaussian noise added to the fields with variance v [5]. The free energy, -lnP(D) is minimized by gradient descent with respect to v and the lengthscale parameters WI, • .. , Wd and the mean field equations for aj are solved by iteration before each update of the hyperparameters (further details will be given elsewhere). Figure 1 shows the evolution of the naive mean free energy and the test error starting from uniform 256 L. Csat6, E. Fokoue, M. Opper, B. &hottky and 0. Winther ws. It typically requires of the order of 10 iteration steps of the a;-equations between each hyperparameter update. We also used hybrid approaches, where the free energy was minimized by one mean field algorithm and the hyperparameters used in the other. As it may be seen from table 1, the naive mean field theory can overestimate the free energy (since the ensemble free energy is an upper bound to the free energy). The overestimation is not nearly as severe at the minimum of the naive mean field free energy. Another interesting observation is that as long as the same hyperparameters are used the actual performance (as measured by the test error) is not very sensitive to the algorithm used. This also seems to be the case for the TAP mean field approach and Support Vector Machines [5]. 115..-----~----~-------,-----__, 0114 \\ Ii:" ' .E 1.113 >. ~ Q) c: ~112 ~ u... 111 , , \ , '. "'----74..-----~----~-------,-----__, 70 g W68 ~ 1-66 64 62 o 20 ~ 60 ~ 0~----2~0----~~~--~~~--~~ Iterations Iterations Figure 1: Hyperparameter optimization for the Pima Indians data set using the naive mean field free energy. Left figure: The free energy as a function of the number of hyperparameter updates. Right figure: The test error count (out of 332) as a function of the number of hyperparameter updates. Table 1: Pima Indians dataset. Hyperparameters found by free energy minimization. Left column gives the free energy -lnP(D) used in hyperparameter optimization. Test error counts in range 63- 75 have previously been reported [5] Ensemble MF Naive MF Free Energy minimization Error -lnP(D) Error -lnP(D) Ensemble Mean Field, eq. (7) 72 100.6 70 183.2 Naive Mean Field, eq. (10) 62 107.0 62 110.9 For the sequential algorithm, we have studied the sonar [10] and crab [7] datasets. Since we have not computed an approximation to the evidence so far, a simple fixed polynomial kernel was used. Although a probabilistic justification of the algorithm is only valid, when a single sweep through the data is used (the independence of the data is assumed), it is tempting to reuse the same data and iterate the procedure as a heuristic. The two plots show that in this way, only a small improvement is obtained, and it seems that the method is rather efficient in extracting the information from the data in a single presentation. For the sonar dataset, a single sweep is enough to achieve zero training error. Acknowledgements: BS would like to thank the Leverhulme Trust for their support (F /250/K). The work was also supported by EPSRC Grant GR/L52093. Efficient Approaches to Gaussian Process Classification 257 so Training Error 45 - - Test Error 40 35 35 30 30 ... 25 g25 w 20 I 20 1,~ 15 15 \ ,- - .. - - .. 10 10 1'\ .. " \ ___ 5 -" 00 20 40 60 80 140 160 °0L----5~0---F=::::lIQ.-A.1SO'::-----::-2~OO Heration. Figure 2: Training and test errors during learning for the sonar (left) and crab dataset (right). The vertical dash-dotted line marks the end of the training set and the starting point of reusing of it. The kernel function used is K(x, x') (1 + x . x' jm)k with order k = 2 (m is the dimension of inputs). References [1] Williams C.K.I. and Rasmussen C.E., Gaussian Processes for Regression, in Neural Information Processing Systems 8, Touretzky D.s, Mozer M.C. and Hasselmo M.E. (eds.), 514-520, MIT Press (1996). [2] Neal R.M, Monte Carlo Implementation of Gaussian Process Models for Bayesian Regression and Classification, Technical Report 9702, Department of Statistics, University of Toronto (1997). [3] Gibbs M.N. and Mackay D.J.C., Variational Gaussian Process Classifiers, Preprint Cambridge University (1997). [4] Williams C.K.I. and Barber D, Bayesian Classification with Gaussian Processes, IEEE Trans Pattern Analysis and Machine Intelligence, 20 1342-1351 (1998). [5] Opper M. and Winther O. Gaussian Processes for Classification: Mean Field Algorithms, Submitted to Neural Computation, http://www.thep.lu.se /tf2/staff/winther/ (1999). [6] Zinn-Justin J, Quantum Field Theory and Critical Phenomena, Clarendon Press Oxford (1990) . [7] Ripley B.D, Pattern Recognition and Neural Networks, Cambridge University Press (1996). [8] Opper M., Online versus Offline Learning from Random Examples: General Results, Phys. Rev. Lett. 77, 4671 (1996). [9] Online Learning in Neural Networks, Cambridge University Press, D. Saad (ed.) (1998). [10] Gorman R.P and Sejnowski T.J, Analysis of hidden units in a layered network trained to classify sonar targets, Neural Networks 1, (1988). [11] Jaakkola T. and Haussler D. Probabilistic kernel regression, In Online Proceedings of 7-th Int. Workshop on AI and Statistics (1999), http://uncertainty99.microsoft.com/proceedings.htm.
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Optimal sizes of dendritic and axonal arbors Dmitri B. Chklovskii Sloan Center for Theoretical Neurobiology The Salk Institute, La Jolla, CA 92037 mitya@salk.edu Abstract I consider a topographic projection between two neuronal layers with different densities of neurons. Given the number of output neurons connected to each input neuron (divergence or fan-out) and the number of input neurons synapsing on each output neuron (convergence or fan-in) I determine the widths of axonal and dendritic arbors which minimize the total volume ofaxons and dendrites. My analytical results can be summarized qualitatively in the following rule: neurons of the sparser layer should have arbors wider than those of the denser layer. This agrees with the anatomical data from retinal and cerebellar neurons whose morphology and connectivity are known. The rule may be used to infer connectivity of neurons from their morphology. 1 Introduction Understanding brain function requires knowing connections between neurons. However, experimental studies of inter-neuronal connectivity are difficult and the connectivity data is scarce. At the same time neuroanatomists possess much data on cellular morphology and have powerful techniques to image neuronal shapes. This suggests using morphological data to infer inter-neuronal connections. Such inference must rely on rules which relate shapes of neurons to their connectivity. The purpose of this paper is to derive such rule for a frequently encountered feature in the brain organization: a topographic projection. Two layers of neurons are said to form a topographic projection if adjacent neurons of the input layer connect to adjacent neurons of the output layer, Figure 1. As a result, output neurons form an orderly map of the input layer. I characterize inter-neuronal connectivity for a topographic projection by divergence and convergence factors defined as follows, Figure 1. Divergence, D, of the projection is the number of output neurons which receive connections from an input neuron. Convergence, C, of the projection is the number of input neurons which connect with an output neuron. I assume that these numbers are the same for each neuron in a given layer. Furthermore, each neuron makes the required connections with the nearest neurons of the other layer. In most cases, this completely specifies the wiring diagram. A typical topographic wiring diagram shown in Figure 1 misses an important biological detail. In real brains, connections between cell bodies are implemented by neuronal processes: axons which carry nerve pulses away from the cell bodies and dendrites which carry signals Optimal Sizes of Dendritic and Axonal Arbors 109 n J n 2 Figure 1: Wiring diagram of a topographic projection between input (circles) and output (squares) layers of neurons. Divergence, D, is the number of outgoing connections (here, D = 2) from an input neuron (wavey lines). Convergence, C, is the number of connections incoming (here, C = 4) to an output neuron (bold lines). Arrow shows the direction of signal propagation. a) ... ~~ ... wiring diagram b) ... QQQQQQQQQQQQ ... [5 [5 Type I c) "'9~?9~?'" Type II Figure 2: Two different arrangements implement the same wiring diagram. (a) Topographic wiring diagram with C = 6 and D = 1. (b) Arrangement with wide dendritic arbors and no axonal arbors (Type I) (c) Arrangement with wide axonal arbors and no dendritic arbors (Type II). Because convergence exceeds divergence type I has shorter wiring than type II. towards cell bodies.[I] Therefore each connection is interrupted by a synapse which separates an axon of one neuron from a dendrite of another. Both axons and dendrites branch away from cell bodies fonning arbors. In general, a topographic projection with given divergence and convergence may be implemented by axonal and dendritic arbors of different sizes, which depend on the locations of synapses. For example, consider a wiring diagram with D = 1 and C = 6, Figure 2a. Narrow axonal arbors may synapse onto wide dendritic arbors, Figure 2b or wide axonal arbors may synapse onto narrow dendritic arbors, Figure 2c. I call these arrangements type I and type II, correspondingly. The question is: which arbor sizes are preferred? I propose a rule which specifies the sizes of axonal arbors of input neurons and dendritic arbors of output neurons in a topographic projection: High divergence/convergence ratio favors wide axonal and narrow dendritic arbors while low divergence/convergence ratio favors narrow axonal arbors and wide dendritic arbors. Alternatively, this rule may be formulated in tenns of neuronal densities in the two layers: Sparser layer has wider arbors. In the above example, divergence/convergence (and neuronal density) ratio is 116 and, according to the rule, type I arrangement, Figure 2b, is preferred. In this paper I derive a quantitative version of this rule from the principle of wiring economy which can be summarized as follows. [2, 3, 4, 5, 6] Space constraints require keeping the brain volume to a minimum. Because wiring (axons and dendrites) takes up a significant fraction of the volume, evolution has probably designed axonal and dendritic arbors in a way that minimizes their total volume. Therefore we may understand the existing arbor sizes as a result of wiring optimization. 110 D. B. Chklovsldi To obtain the rule I formulate and solve a wiring optimization problem. The goal is to find the sizes ofaxons and dendrites which minimize the total volume of wiring in a topographic wiring diagram for fixed locations of neurons. I specify the wiring diagram with divergence and convergence factors. Throughout most of the paper I assume that the cross-sectional area of dendrites and axons are constant and equal. Therefore, the problem reduces to the wire length minimization. Extension to unequal fiber diameters is given below. 2 Topographic projection in two dimensions Consider two parallel layers of neurons with densities nl and n2. The topographic wiring diagram has divergence and convergence factors, D and C, requiring each input neuron to connect with D nearest output neurons and each output neuron with C nearest input neurons. Again, the problem is to find the arrangement of arbors which minimizes the total length of axons and dendrites. For different arrangements I compare the wirelength per unit area, L. I assume that the two layers are close to each other and include only those parts of the wiring which are parallel to the layers. I start with a special case where each input neuron connects with only one output neuron (D = 1). Consider an example with C = 16 and neurons arranged on a square grid in each layer, Figure 3a. Two extreme arrangements satisfy the wiring diagram: type I has wide dendritic arbors and no axonal arbors, Figure 3b; type II has wide axonal arbors and no dendritic arbors, Figure 3c. I take the branching angles equal to 120°, an optimal value for constant crossectional area. [ 4] Assuming "point" neurons the ratio of wire length for type I and type II arrangements: Lr -L ~0.57. II (1) Thus, the type I arrangement with wide dendritic arbors has shorter wire length. This conclusion holds for other convergence values much greater than one, provided D = 1. However, there are other arrangements with non-zero axonal arbors that give the same wire length. One of them is shown in Figure 3d. Degenerate arrangements have axonal arbor width 0 < Sa < 1/ vnI where the upper bound is given by the approximate inter-neuronal distance. This means that the optimal arbor size ratio for D = 1 (2) By using the symmetry in respect to the direction of signal propagation I adapt this result for the C = 1 case. For D > 1, arrangements with wide axonal arbors and narrow dendritic arbors (0 < Sd < 1/ vnv have minimal wirelength. The arbor size ratio is (3) Next, I consider the case when both divergence and convergence are greater than one. Due to complexity of the problem I study the limit of large divergence and convergence (D, C » 1). I find analytically the optimal layout which minimizes the total length of axons and dendrites. Notice that two neurons may form a synapse only if the axonal arbor of the input neuron overlaps with the dendritic arbor of the output neuron in a two-dimensional projection, Figure 4. Thus the goal is to design optimal dendritic and axonal arbors so that each dendritic arbor intersects C axonal arbors and each axonal arbor intersects D dendritic arbors. To be specific, I consider a wiring diagram with convergence exceeding divergence, C > D (the argument can be readily adapted for the opposite case). I make an assumption, to be Optimal Sizes of Dendritic and Axonal Arbors 111 a) b) wiring diagram Type I c) Type II Type I' Figure 3: Different arrangements implement the same wiring diagram in two dimensions. (a) Topographic wiring diagram with D = 1 and C = 16. (b) Arrangement with wide dendritic arbors and no axonal arbors, Type I. (c) Arrangement with wide axonal arbors and no dendritic arbors, Type II. Because convergence exceeds divergence type I has shorter wiring than type II. (d) Intermediate arrangement which has the same wire length as type I. Figure 4: Topographic projection between the layers of input (circles) and output (squares) neurons. For clarity, out of the many input and output neurons with overlapping arbors only few are shown. The number of input neurons is greater than the number of output neurons (C / D > 1). Input neurons have narrow axonal arbors of width Sa connected to the wide but sparse dendritic arbors of width Sd. Sparseness of the dendritic arbor is given by Sa because all the input neurons spanned by the dendritic arbor have to be connected. 112 D. B. Chklovskii verified later, that dendritic arbor diameter Sd is greater than axonal one, Sa. In this regime each output neuron's dendritic arbor forms a sparse mesh covering the area from which signals are collected, Figure 4. Each axonal arbor in that area must intersect the dendritic arbor mesh to satisfy the wiring diagram. This requires setting mesh size equal to the axonal arbor diameter. By using this requirement I express the total length of axonal and dendritic arbors as a function of only the axonal arbor size, Sa. Then I find the axonal arbor size which minimizes the total wirelength. Details of the calculation will be published elsewhere. Here, I give an intuitive argument for why in the optimal layout both axonal and dendritic size are non-zero. Consider two extreme layouts. In the first one, dendritic arbors have zero width, type II. In this arrangement axons have to reach out to every output neuron. For large convergence, C » 1, this is a redundant arrangement because of the many parallel axonal wires whose signals are eventually merged. In the second layout, axonal arbors are absent and dendrites have to reach out to every input neuron. Again, because each input neuron connects to many output neurons (large divergence, D » 1) many dendrites run in parallel inefficiently carrying the same signal. A non-zero axonal arbor rectifies this inefficiency by carrying signals to several dendrites along one wire. I find that the optimal ratio of dendritic and axonal arbor diameters equals to the square root of the convergenceldivergenceratio, or, alternatively, to the square root of the neuronal density ratio: (4) Since I considered the case with C > D this result also justifies the assumption about axonal arbors being smaller than dendritic ones. For arbitrary axonal and dendritic cross-sectional areas, ha and hd, expressions ofthis Section are modified. The wiring economy principle requires minimizing the total volume occupied by axons and dendrites resulting in the following relation for the optimal arrangement: (5) Notice that in the optimal arrangement the total axonal volume of input neurons is equal to the total dendritic volume of the output neurons. 3 Discussion 3.1 Comparison of the theory with anatomical data This theory predicts a relationship between the con-/divergence ratio and the sizes of axonal and dendritic arbors. I test these predictions on several cases of topographic projection in two dimensions. The predictions depend on whether divergence and convergence are both greater than one or not. Therefore, I consider the two regimes separately. First, I focus on topographic projections of retinal neurons whose divergence factor is equal or close to one. Because retinal neurons use mostly graded potentials the difference between axons and dendrites is small and I assume that their cross-sectional areas are equal. The theory predicts that the ratio of dendritic and axonal arbor sizes must be greater than the square root of the input/output neuronal density ratio, Sd/ Sa > (ndn2)1/2 (Eq.2). I represent the data on the plot of the relative arbor diameter, Sd/ Sa, vs. the square root of the relative densities, (ndn2)1/2, (Figure 5). Because neurons located in the same layer may belong to different classes, each having different arbor size and connectivity, I plot data Optimal Sizes of Dendritic and Axonal Arbors 113 s~/s •. .--_______ ...,--______ --." 50 I. A C o .. I U5 uz C=l 5 I. ZO Figure 5: Anatomical data for several pairs of retinal cell classes which form topographic projections with D = 1. All the data points fall in the triangle above the Sd/ Sa = (ndn2)1/2 line in agreement with the theoretical prediction, Eq.2. The following data has been used: 0 - midget bipolar -+ midget ganglion,[7, 8, II]; U - diffuse bipolar -+ parasol ganglion,[7, 9]; 'V - rods -+ rod bipolar,[lO]; b. - cones -+ HI horizontals.[12]; 0 - rods -+ telodendritic arbors of HI horizontals,[ 13]. from different classes separately. All the data points lie above the Sd/ Sa = (ndn2)1/2line in agreement with the prediction. Second, I apply the theory to cerebellar neurons whose divergence and convergence are both greater than one. I consider a projection from granule cell axons (parallel fibers) onto Purkinje cells. Ratio of granule cells to Purkinje cells is 33()(),[14], indicating a high convergence/divergence ratio. This predicts a ratio of dendritic and axonal arbor sizes of 58. This is qualitatively in agreement with wide dendritic arbors of Purkinje cells and no axonal arbors on parallel fibers. Quantitative comparison is complicated because the projection is not strictly twodimensional: Purkinje dendrites stacked next to each other add up to a significant third dimension. Naively, given that the dendritic arbor size is about 400ILm Eq.4 predicts axonal arbor of about 7 ILm. This is close to the distance between two adjacent Purkinje cell arbors of about 9 ILm. Because the length of parallel fibers is greater than 7 ILm absence of axonal arbors comes as no surprise. 3.2 Other factors affecting arbor sizes One may argue that dendrites and axons have functions other than linking cell bodies to synapses and, therefore, the size of the arbors may be dictated by other considerations. Although I can not rule out this possibility, the primary function ofaxons and dendrites is to connect cell bodies to synapses in order to conduct nerve pulses between them. Indeed, if neurons were not connected more sophisticated effects such as non-linear interactions between different dendritic inputs could not take place. Hence the most basic parameters of axonal and dendritic arbors such as their size should follow from considerations of connectivity. Another possibility is that the size of dendritic arbors is dictated by the surface area needed J14 D. B. Chklovskii to arrange all the synapses. This argument does not specify the arbor size, however: a compact dendrite of elaborate shape can have the same surface area as a wide dendritic arbor. Finally, agreement of the predictions with the existing anatomical data suggests that the rule is based on correct principles. Further extensive testing of the rule is desirable. Violation of the rule in some system would suggest the presence of other overriding considerations in the design of that system, which is also interesting. Acknowledgements I benefited from helpful discussions with E.M. Callaway, EJ. Chichilnisky, H.J. Karten, C.P. Stevens and TJ. Sejnowski and especially with A.A. Koulakov. I thank G.D. Brown for suggesting that the size of axonal and dendritic arbors may be related to con-/divergence. References [1] Cajal, S.R.y. (1995a). Histology of the nervous system p.95 (Oxford University Press, NewYork). [2] Cajal, S.R.y. ibid. p.1l6. [3] Mitchison, G. (1991). Neuronal branching patterns and the economy of cortical wiring. Proc R Soc Lond B Bioi Sci 245, 151-8. [4] Chemiak, C. (1992). Local optimization of neuron arbors, Bioi Cybem 66,503-510. [5] Young, M.P. (1992). Objective analysis of the topological organization of the primate cortical visual system Nature 358, 152-5. [6] Chklovskii, D.B. & Stevens, c.F. (1999). Wiring the brain optimally, submitted Nature Neuroscience. [7] Watanabe, M. & Rodieck, R. W. (1989). Parasol and midget ganglion cells of the primate retina. J Comp Neurol 289, 434-54. [8] Milam, A.H., Dacey, D.M. & Dizhoor, A.M. (1993). Recoverin immunoreactivity in mammalian cone bipolar cells. Vis Neurosci 10, 1-12. [9] Grunert, U., Martin, P.R. & Wassle H. (1994). Immunocytochemical analysis of bipolar cells in the macaque monkey retina. J Comp Neuro1348, 607-27. [10] Grunert, U. & Martin, P.R. (1991). Rod bipolar cells in the macaque monkey retina: immunoreactivity and connectivity. J Neurosci 11,2742-58. [11] Dacey, D.M. (1993). The mosaic of midget ganglion cells in the human retina. J Neurosci 13, 5334-55. [12] Wassle, H., Boycott, B.B. & Rohrenbeck, J. (1989). Horizontal cells in the monkey retina: cone connections and dendritic network. Eur J Neurosci I, 421-435. [13] Rodieck, R.W. (1989) The First Steps in Seeing (Sinauer Associates, Sunderland, MA). [14] Andersen, B.B., Korbo, L. & Pakkenberg, B. (1992). A quantitative study of the human cerebellum with unbiased stereological techniques. J Comp Neurol 326, 549-60. [15] Peters A., Payne B.R. & Budd, J. (1994). A numerical analysis of the geniculocortical input to striate cortex in the monkey. Cereb Cortex 4, 215-229. [16] Blasdel, G.G. & Lund, J.S. (1983) Termination of afferent axons in macaque striate cortex. J Neurosci 3, 1389-1413. [17] Wiser, A.K. & Callaway, E.M. (1996). Contributions of individual layer 6 pyramidal neurons to local circuitry in macaque primary visual cortex. J Neurosci 16,2724-2739.
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An MEG Study of Response Latency and Variability in the Human Visual System During a Visual-Motor Integration Task Akaysha C. Tang Dept. of Psychology University of New Mexico Albuquerque, NM 87131 akaysha@unm.edu Barak A. Pearlmutter Dept. of Computer Science University of New Mexico Albuquerque, NM 87131 bap@cs. unm. edu Tim A. Hely Santa Fe Institute 1399 Hyde Park Road Santa Fe, NM 87501 timhely@santafe. edu Michael Zibulevsky Dept. of Computer Science University of New Mexico Albuquerque, NM 87131 michael@cs.unm.edu Michael P. Weisend VA Medical Center 1501 San Pedro SE Albuquerque, NM 87108 mweisend@unm.edu Abstract Human reaction times during sensory-motor tasks vary considerably. To begin to understand how this variability arises, we examined neuronal populational response time variability at early versus late visual processing stages. The conventional view is that precise temporal information is gradually lost as information is passed through a layered network of mean-rate "units." We tested in humans whether neuronal populations at different processing stages behave like mean-rate "units". A blind source separation algorithm was applied to MEG signals from sensory-motor integration tasks. Response time latency and variability for multiple visual sources were estimated by detecting single-trial stimulus-locked events for each source. In two subjects tested on four visual reaction time tasks, we reliably identified sources belonging to early and late visual processing stages. The standard deviation of response latency was smaller for early rather than late processing stages. This supports the hypothesis that human populational response time variability increases from early to late visual processing stages. 1 Introduction In many situations, precise timing of a motor output is essential for successful task completion. Somehow the reliability in the output timing is related to the reliability of the underlying neural systems associated with different stages of processing. Recent literature from animal studies suggests that individual neurons from different brain regions and different species can be surprising reliable [1, 2, 5, 7-9, 14, 17, 18], 186 A. C. Tang, B. A. Pearlmutter. T. A. Hely, M. Zibulevsky and M P. Weisend on the order of a few milliseconds. Due to the low spatial resolution of electroencephalography (EEG) and the requirement of signal averaging due to noisiness of magnetoencephalography (MEG), in vivo measurement of human populational response time variability from different processing stages has not been available. In four visual reaction time (RT) tasks, we estimated neuronal response time variability at different visual processing stages using MEG. One major obstacle that has prevented the analysis of response timing variability using MEG before is the relative weakness of the brain's magnetic signals (lOOf!') compared to noise in a shielded environment (magnetized lung contaminants: 106f!'j abdominal currents lO5f!'j cardiogram and oculogram: 104f!'j epileptic and spontaneous activity: lO3f!') and in the sensors (10fT) [13]. Consequently, neuronal responses evoked during cognitive tasks often require signal averaging across many trials, making analysis of singletrial response times unfeasible. Recently, Bell-Sejnowski Infomax [1995] and Fast ICA [10] algorithms have been used successfully to isolate and remove major artifacts from EEG and MEG data [11, 15, 20]. These methods greatly increase the effective signal-to-noise ratio and make single-trial analysis of EEG data feasible [12]. Here, we applied a SecondOrder Blind Identification algorithm (SOBI) [4] (another blind source separation, or BSS, algorithm) to MEG data to find out whether populational response variability changes from early to late visual processing stages. 2 Methods 2.1 Experimental Design Two volunteer normal subjects (females, right handed) with normal or correctedto-normal visual acuity and binocular vision participated in four different visual RT tasks. Subjects gave informed consent prior to the experimental procedure. During each task we recorded continuous MEG signals at a 300Hz sampling rate with a band-pass filter of I-100Hz using a 122 channel Neuromag-122. In all four tasks, the subject was presented with a pair of abstract color patterns, one in the left and the other in the right visual field. One of the two patterns was a target pattern. The subject pressed either a left or right mouse button to indicate on which side the target pattern was presented. When a correct response was given, a low or high frequency tone was presented binaurally following respectively a correct or wrong response. The definition of the target pattern varied in the four tasks and was used to control task difficulty which ranged from easy (task 1) to more difficult (task 4) with increasing RTs. (The specific differences among the four tasks are not important for the analysis which follows and are not discussed further.) In this study we focus on the one element that all tasks have in common, Le. activation of multiple visual areas along the visual pathways. Our goal is to identify visual neuronal sources activated in all four visual RT tasks and to measure and compare response time variability between neuronal sources associated with early and later visual processing stages. Specifically, we test the hypothesis that populational neuronal response times increase from early to later visual processing stages. 2.2 Source Separation Using SOBI In MEG, magnetic activity from different neuronal populations is observed by many senSOrs arranged around the subject's head. Each sensor responds to a mixture of the signals emitted by multiple sources. We used the Second-Order Blind IdentiMEG Study of Response Latency and Variability 187 fication algorithm (SOBI) [4] (a BSS algorithm) to simultaneously separate neuromagnetic responses from different neuronal populations associated with different stages of visual processing. Responses from different neuronal populations will be referred to as source responses and the neuronal populations that give rise to these responses will be referred to as neuronal sources or simply sources. These neuronal sources often, but not always, consist of a spatially contiguous population of neurons. BSS separates the measured sensor signals into maximally independent components, each having its own spatial map. Previously we have shown that some of these BSS separated components correspond to noise sources, and many others correspond to neuronal sources [19]. To establish the identity of the components, we analyzed both temporal and spatial properties of the BSS separated components. Their temporal properties are displayed using MEG images, similar to the ERP images described by [12] but without smoothing across trials. These MEG images show stimulus or response locked responses across many trials in a map, from which response latencies across all displayed trials can be observed with a glance. The spatial properties of the separated components are displayed using a field map that shows the sensor projection of a given component. The intensity at each point on the field map indicates how strongly this component influences the sensor at this location. The correspondence between the separated components and neuronal populational responses at different visual processing stages were established by considering both spatial and temporal properties of the separated components [19]. For example, a component was identified as an early visual neuronal source if and only if (1) the field pattern, or the sensor projection, of the separated component showed a focal response over the occipital lobe, and (2) the ERP image showed visual stimulus locked responses with latencies shorter than all other visual components and falling within the range of early visual responses reported in studies using other methods. Only those components consistent both spatially and temporally with known neurophysiology and neuroanatomy were identified as neuronal sources. 2.3 Single Event Detection and Response Latency Estimation For all established visual components we calculated the single-trial response latency as follows. First, a detection window was defined using the stimulus-triggered average (STA). The beginning of the detection window was defined by the time at which the STA first exceeded the range of baseline fluctuation. Baseline fluctuation was estimated from the time of stimulus onset for approximately 50ms (the visual response occurred no earlier than 60ms after stimulus onset.) The detection window ended when the STA first returned to the same level as when the detection window began. The detection threshold was determined using a control window with the same width as the detection window, but immediately preceding the detection window. The threshold was adjusted until no more than five false detections occurred within the control window for each ninety trials. We estimated RTs using the leading edge of the response, rather than the time of the peak as this is more robust against noise. 3 Results In both subjects across all four visual RT tasks, SOBI generated components that corresponded to neuronal populational responses associated with early and late stages of visual processing. In both subjects, we identified a single component with a sensor projection at the occipital lobe whose latency was the shortest among all 188 1 2 3 4 1 2 3 4 A. C. Tang, B. A. Pearlmutter, T. A. Hely, M Zibulevsky and M P. Weisend e ••• e ••• e ••• e ••• e ••• e ••• e ••• e ••• late source • ••• e ••• -. a • • ••• e ••• e ••• e ••• e ••• Figure 1: MEG images and field maps for an early and a late source from each task, for subject 1 (top) and subject 2 (bottom). MEG image pixels are brightnesscoded source strength. Each row of a bitmap is one trial, running 1170ms from left to right. Vertical bars mark stimulus onset, and 333ms of pre-stimulus activity is shown. Each panel contains 90 trials. Field map brightness indicates the strength with which a source activates each of the 61 sensor pairs. visual stimulus locked components within task and subject (Fig. 1 left). We identified multiple components that had sensor projections either at occipital-parietal, occipital-temporal, or temporal lobes, and whose response latencies are longer than early-stage components within task and subject (Fig. 1 right). Fig. 2a shows examples of detected single-trial responses for one early and one late visual component (left: early; right: late) from one task. To minimize false positives, the detection threshold was set high (allowing 5 false detections out of 90 trials) at the expense of a low detection rate (15%- 67%.) When Gaussian filters were applied to the raw separated data, the detection rates were increased to 22- 91% (similar results hold but not shown). Fig. 2b shows such detected response time histograms superimposed on the stimulus triggered average using raw separated data. One early (top row) and two late visual components (middle and bottom rows) are plotted for each of the four experiments in subject one. The histogram width is smallest for early visual components (short mean response latency) and larger for late visual components (longer latency.) We computed the standard deviation of component response times as a measure of response variability. Fig. 2c shows the response variability as a function of mean response latency for subject one. Early components (solid boxes, shorter mean latency) have smaller variability (height of the boxes) while late components (dashed boxes, longer mean latency) have larger variability (height of the boxes). Multiple MEG Study of Response Latency and Variability 189 :bd:Qlld~~ a 100 200 300 40Q 0 100 200 300 40Q a 100 200 300 400 0 100 200 300 400 ~-~~ -~ '~[d llJ l~J l~J o 100 200 300 40CI C 100 200 300 40CI 0 100 200 300 400 0 100 200 300 .wo ----l~J ~~ I!] ':~ o 100 200 300 400 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 40 ---. -. 100 ,------, i 1 1. ____ 1 150 latency (ms) \ i ! : : : .-----------: 200 Figure 2: (a, left) Response onset was estimated for each trial via threshold crossing within a window of eligibility. (b, top right) The stimulus-locked averages for a number of sources overlaid on histograms of response onset times. (c, bottom right) Scatter plot of visual components from all experiments on subject 1 showing the standard deviation of the latency (y axis) versus the mean latency (x axis), with the error bars in each direction indicating one standard error in the respective measurement. Lines connect sources from each task. visual components from each task are connected by a line. Four tasks were shown here. There is a general trend of increasing standard deviation of response times as a function of early-to-late processing stages (increasing mean latency from left to right). For the early visual components the standard deviation ranges from 6.6±0.63ms to 13.4±1.23ms, and for the late visual components, from 9.9±0.86ms to 38.8±3.73ms (t = 3.565, p = 0.005.) 4 Discussion By applying SOBl to MEG data from four visual RT tasks, we separated components corresponding to neuronal populational responses associated with early and 190 A. C. Tang, B. A. Pearlmutter; T. A. Hely, M Zibulevsky and M P. Weisend later stage visual processing in both subjects across all tasks. We performed singletrial RT detection on these early- and late-stage components and estimated both the mean and stdev of their response latency. We found that variability of the populational response latency increased from early to late processing stages. These results contrast with single neuron recordings obtained previously. In early and late visual processing stages, the rise time of mean firing rate in single units remained constant, suggesting an invariance in RT variability [16]. Characterizing the precise relationship between single neuron and populational response reliability is difficult without careful simulations or simultaneous single unit and MEG recording. However, some major differences exist between the two types of studies. While MEG is more likely to sample a larger neuronal population, single unit studies are more likely to be selective to those neurons that are already highly reliable in their responses to stimulus presentation. It is possible that the most reliable neurons at both the early and late processing stages are equally reliable while large differences exist between the early and late stages for the low reliability neurons. Previously, ICA algorithms have been used successfully to separate out various noise and neuronal sources in MEG data [19, 20J. Here we show that SOBI can also be used to separate different neuronal sources, particularly those associated with different processing stages. The SOBI algorithm assumes that the components are independent across multiple time scales and attempts to minimize the temporal correlation at these time scales. Although neuronal sources at different stages of processing are not completely independent as assumed in SOBl's derivation, BSS algorithms of this sort are quite robust even when the underlying assumptions are not fully met [6J, i.e. the goodness of the separation is not significantly affected. The ultimate reality check should come from satisfying physiological and anatomical constraints derived from prior knowledge of the neural system under study. This was carried out for our analysis. Firstly, the average response latencies of the separated components fell within the range of latencies reported in MEG studies using conventional source modeling methods. Secondly, the spatial patterns of sensor responses to these separated components are consistent with the known functional anatomy of the visual system. We have attempted to rule out many confounding factors. Our observed results cannot be accounted for by a higher signal to noise ratio in the early visual responses. The increase in measured onset response time variability from early to late visual processing stages was actually accompanied by an slightly lower signalto-noise ratio among the early components. The number of events detected for the later components were also slightly greater than the earlier components. The higher signal-to-noise ratio at later components should reduce noise-induced variability in the later components, which would bias against the hypothesis that later visual responses have greater response time variability. We also found that response duration and detection window size cannot account for the observed differential variabilities. Later visual responses also had gentler onset slopes (as measured by the stimulustriggered average). Sensor noise unavoidably introduces noise into the response onset detection process. We cannot rule out the possibility that the interaction of the noise with the response onset profiles might give rise to the observed differential variabilities. Similarly, we cannot rule out the possibility that even greater control of the experimental situation, such as better fixation and more effective head restraints, would differentially reduce the observed variabilities. In general, all measured variabilities can only be upper bounds, subject to downward revision as improved instrumentation and experiments become available. It is with this caution in mind that we conclude that response time variability of neuronal populations increases from early to late processing stages in the human visual system. MEG Study of Response Latency and Variability 191 Acknowledgments This research was supported by NSF CAREER award 97-02-311, and by the National Foundation for Functional Brain Imaging. References [1] M. Abeles, H. Bergman, E. Margalit, and E Vaadia. Spatiotemporal firing patterns in the frontal cortex of behaving monkeys. J. Neurophys., 70:1629-1638, 1993. [2] W. Bair and C. Koch. Temporal precision of spike trains in extrastriate cortex of the behaving macaque monkey. Neural Computation, 8(6):1184-1202, 1996. [3] A. J. Bell and T. J. Sejnowski. An information-maximization approach to blind separation and blind deconvolution. Neural Computation, 7(6):1129-1159, 1995. [4] A. Belouchrani, K. A. Meraim, J.-F. Cardoso, and E. Moulines. Second-order blind separation of correlated sources. In Proc. Int. ConJ. on Digital Sig. Proc., pages 346-351, Cyprus, 1993. [5) M. J. Berry, W. K. Warland, and M. Meister. The structure and precision of retinal spike trains. Proc. Natl. Acad. Sci. USA, 94:5411-5416, 1997. [6] J.-F. Cardoso. Blind signal separation: statistical principles. Proceedings of the IEEE, 9(10):2009-2025, October 1998. [7] R. R. de Ruyter van Steveninck, G. D. Lewen, S. P. Strong, R. Koberle, and W. Bialek. Reproducibility and variability in neural spike trains. Science, 275:1805-1808, 1997. [8] R. C. deCharms and M. M. Merzenich. Primary cortical representation of sounds by the coordination of action-potential timing. Nature, 381:610-3, 1996. [9] M. Gur, A. Beylin, and D. M. Snodderly. Response variability of neurons in primary visual cortex (VI) of alert monkeys. J. Neurosci., 17(8):2914-2920, 1997. [10] A. Hyvarinen and E. Oja. A fast fixed-point algorithm for independent component analysis. Neural Computation, 9(7), October 1997. [11] T.-P. Jung, C. Humphries, T.-W. Lee, M. J. McKeown, V. Iragui, S. Makeig, and T. J. Sejnowski. Removing electroencephalographic artifacts by blind source separation. Psychophysiology, 1999. In Press. [12] T.-P. Jung, S. Makeig, M. Westerfield, J. Townsend, E. Courchesne, and T. J. Sejnowski. Analyzing and visualizing single-trial event-related potentials. In Advances in Neural Infonnation Processing Systems 11, pages 118-124. MIT Press, 1999. [13] J. D. Lewine and W. W. Orrison, II. Magnetoencephalography and magnetic source imaging. In Functional Brain Imaging, pages 369-417. Mosby, St. Louis, 1995. [14] Z. F. Mainen and T. J. Sejnowski. Reliability of spike timing in neocortical neurons. Science, 268:1503-1506, 1995. [15] S. Makeig, T.-P. Jung, A. J. Bell, D. Ghahremani, and T. J. Sejnowski. Blind separation of auditory event-related brain responses into independent components. Proc. Nat. A cad. Sci., 94:10979-84, 1997. [16] P. Marsalek, C. Koch, and J. Maunsell. On the relationship between synaptic input and spike output jitter in individual neurons. Proc. Natl. Acad. Sci., 94:735-40, 1997. [17] D. S. Reich, J. D. Victor, B. W. Knight, and T. Ozaki. Response variability and timing precision of neuronal spike trains in vivo. J. Neurophys., 77:2836-2841, 1997. [18] A. C. Tang, A. M. Bartels, and T. J. Sejnowksi. Effects of cholinergic modulation on responses of neocortical neurons to fluctuating inputs. Cereb. Cortex, 7:502-9, 1997. [19] A. C. Tang, B. A. Pearlmutter, M. Zibulevsky, and R. Loring. Response time variability in the human sensory and motor systems. In Computational Neuroscience, 1999. To appear as a special issue of Neurocomputing. [20] R. Vigario, V. Jousmaki, M. Hamruainen, R. Hari, and E. Oja. Independent component analysis for identification of artifacts in magnetoencephalographic recordings. In Advances in Neural Infonnation Processing Systems 10. MIT Press, 1998.
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v-Arc: Ensemble Learning in the Presence of Outliers G. Ratscht , B. Scholkopf1, A. Smola", K.-R. Miillert, T. Onodatt , and S. Mikat t GMD FIRST, Rudower Chaussee 5,12489 Berlin, Germany t Microsoft Research, 1 Guildhall Street, Cambridge CB2 3NH, UK * Dep. of Engineering, ANU, Canberra ACT 0200, Australia tt CRIEPI, 2-11-1, Iwado Kita, Komae-shi, Tokyo, Japan {raetsch, klaus, mika}~first.gmd.de,bsc~microsoft.com, Alex.Smola~anu.edu.au,onoda~criepi.denken.or.jp Abstract AdaBoost and other ensemble methods have successfully been applied to a number of classification tasks, seemingly defying problems of overfitting. AdaBoost performs gradient descent in an error function with respect to the margin, asymptotically concentrating on the patterns which are hardest to learn. For very noisy problems, however, this can be disadvantageous. Indeed, theoretical analysis has shown that the margin distribution, as opposed to just the minimal margin, plays a crucial role in understanding this phenomenon. Loosely speaking, some outliers should be tolerated if this has the benefit of substantially increasing the margin on the remaining points. We propose a new boosting algorithm which allows for the possibility of a pre-specified fraction of points to lie in the margin area Or even on the wrong side of the decision boundary. 1 Introduction Boosting and related Ensemble learning methods have been recently used with great success in applications such as Optical Character Recognition (e.g. [8, 16]). The idea of a large minimum margin [17] explains the good generalization performance of AdaBoost in the low noise regime. However, AdaBoost performs worse on noisy tasks [10, 11], such as the iris and the breast cancer benchmark data sets [1]. On the latter tasks, a large margin on all training points cannot be achieved without adverse effects on the generalization error. This experimental observation was supported by the study of [13] where the generalization error of ensemble methods was bounded by the sum of the fraction of training points which have a margin smaller than some value p, say, plus a complexity term depending on the base hypotheses and p. While this bound can only capture part of what is going on in practice, it nevertheless already conveys the message that in some cases it pays to allow for some points which have a small margin, or are misclassified, if this leads to a larger overall margin on the remaining points. To cope with this problem, it was mandatory to construct regularized variants of AdaBoost, which traded off the number of margin errors and the size of the margin 562 G. Riitsch, B. Sch6lkopf, A. J. Smola, K.-R. Muller, T. Onoda and S. Mika [9, 11]. This goal, however, had so far been achieved in a heuristic way by introducing regularization parameters which have no immediate interpretation and which cannot be adjusted easily. The present paper addresses this problem in two ways. Primarily, it makes an algorithmic contribution to the problem of constructing regularized boosting algorithms. However, compared to the previous efforts, it parameterizes the above trade-off in a much more intuitive way: its only free parameter directly determines the fraction of margin errors. This, in turn, is also appealing from a theoretical point of view, since it involves a parameter which controls a quantity that plays a crucial role in the generalization error bounds (cf. also [9, 13]). Furthermore, it allows the user to roughly specify this parameter once a reasonable estimate of the expected error (possibly from other studies) can be obtained, thus reducing the training time. 2 Boosting and the Linear Programming Solution Before deriving a new algorithm, we briefly discuss the properties of the solution generated by standard AdaBoost and, closely related, Arc-GV (2], and show the relation to a linear programming (LP) solution over the class of base hypotheses G. Let {gt(x) : t = 1, ... ,T} be a sequence of hypotheses and a = [al ... aT] their weights satisfying at ~ O. The hypotheses gt are elements of a hypotheses class G = {g: x 14 [-1, In, which is defined by a base learning algorithm. The ensemble generates the label which is the weighted majority of the votes by sign(f(x)) where f(x) = ~ lI:ill gt(x). (1) In order to express that f and therefore also the margin p depend on a and for ease of notation we define p(z, a) := yf(x) where z := (x, y) and f is defined as in (1). (2) Likewise we use the normalized margin: p(a):= min P(Zi, a) , (3) l~t~m Ensemble learning methods have to find both, the hypotheses gt E G used for the combination and their weights a. In the following we will consider only AdaBoost algorithms (including Arcing). For more details see e.g. (4, 2]. The main idea of AdaBoost is to introduce weights Wt(Zi) on the training patterns. They are used to control the importance of each single pattern in learning a new hypothesis (Le. while repeatedly running the base algorithm). Training patterns that are difficult to learn (which are misclassified repeatedly) become more important. The minimization objective of AdaBoost can be expressed in terms of margins as m (4) i=1 In every iteration, AdaBoost tries to minimize this error by a stepwise maximization of the margin. It is widely believed that AdaBoost tries to maximize the smallest margin on the training set [2, 5, 3, 13, 11]. Strictly speaking, however, a general proof is missing. It would imply that AdaBoost asymptotically approximates (up to scaling) the solution of the following linear programming problem over the complete hypothesis set G (cf. [7], assuming a finite number of basis hypotheses): maximize subject to p p( Zi, a) ~ p for all 1 < i < m at, P ~ 0 for all 1 ~ t ~ IGI lIalil = 1 (5) v-Arc: Ensemble Learning in the Presence o/Outliers 563 Since such a linear program cannot be solved exactly for a infinite hypothesis set in general, it is interesting to analyze approximation algorithms for this kind of problems. Breiman [2] proposed a modification of AdaBoost - Arc-GV - making it possible to show the asymptotic convergence of p(at ) to the global solution pIP: Theorem 1 (Breiman [2]). Choose at in each iteration as at := argmin Lexp [-llatlll (p(Zi' at) - p(at- I ))], (6) aE[O,I] i and assume that the base learner always finds the hypothesis 9 E G which minimizes the weighted training error with respect to the weights. Then lim p( at) = pIp. t-HX> Note that the algorithm above can be derived from the modified error function 9gv(at):= Lexp [-llatlll (p(Zi' at) - p(at - I ))]. (7) The question one might ask now is whether to use AdaBoost or rather Arc-GV in practice. Does Arc-GV converge fast enough to benefit from its asymptotic properties? In [12] we conducted experiments to investigate this question. We empirically found that (a) AdaBoost has problems finding the optimal combination if pIp < 0, (b) Arc-GV's convergence does not depend on pIp, and (c) for pIp> 0, AdaBoost usually converges to the maximum margin solution slightly faster than Arc-GV. Observation (a) becomes clear from (4): 9(a) will not converge to ° and lIal11 can be bounded by some value. Thus the asymptotic case cannot be reached, whereas for Arc-GV the optimum is always found. Moreover, the number of iterations necessary to converge to a good solution seems to be reasonable, but for a near optimal solution the number of iterations is rather high. This implies that for real world hypothesis sets, the number of iterations needed to find an almost optimal solution can become prohibitive, but we conjecture that in practice a reasonably good approximation to the optimum is provided by both AdaBoost and Arc-GV. 3 v-Algorithms For the LP-AdaBoost [7] approach it has been shown for noisy problems that the generalization performance is usually not as good as the one of AdaBoost [7, 2, 11]. From Theorem 5 in [13] (cf. Theorem 3 on page 5) this fact becomes clear, as the minimum of the right hand side of inequality (cf. (13)) need not necessarily be achieved with a maximum margin. We now propose an algorithm to directly control the number of margin errors and therefore also the contribution of both terms in the inequality separately (cf. Theorem 3). We first consider a small hypothesis class and end up with a linear program - v-LP-AdaBoost. In subsection 3.2 we then combine this algorithm with the ideas from section 2 and get a new algorithm v-Arc - which approximates the v-LP solution. 3.1 v-LP-AdaBoost Let us consider the case where we are given a (finite) set G = {g: x I-t [-1, 1n ofT hypotheses. To find the coefficients a for the combined hypothesis f(x) we extend the LP-AdaBoost algorithm [7, 11] by incorporating the parameter v [15] and solve the following linear optimization problem: maximize subject to p - v!n E::'I ~i P(Zi' a) ::::: p ~i for all 1 ~ i ~ m ~i' at, P ::::: ° for all 1 ~ t ~ T and 1 ~ i ~ m lIalh = 1 (8) 564 G. Riitsch, B. SchOlkopf, A. J. Smola, K.-R. Muller, T. Onoda and S. Mika This algorithm does not force all margins to be beyond zero and we get a soft margin classification (cf. SVMs) with a regularization constant v!n. The following proposition shows that v has an immediate interpretation: Proposition 2 (Ratsch et al. [12]). Suppose we run the algorithm given in (8) on some data with the resulting optimal P > o. Then 1. v upper bounds the fraction of margin errors. 2. 1 - v upper bounds the fraction of patterns with margin larger than p. Since the slack variables ~i only enter the cost function linearly, their absolute size is not important. Loosely speaking, this is due to the fact that for the optimum of the primal objective function, only derivatives wrt. the primal variables matter, and the derivative of a linear function is constant. In the case of SVMs [14], where the hypotheses can be thought of as vectors in some feature space, this statement can be translated into a precise rule for distorting training patterns without changing the solution: we can move them locally orthogonal to a separating hyperplane. This yields a desirable robustness property. Note that the algorithm essentially depends on the number of outliers, not on the size of the error [15]. 3.2 The v-Arc Algorithm Suppose we have a very large (but finite) base hypothesis class G. Then it is difficult to solve (8) as (5) directly. To this end, we propose a new algorithm - v-Arc - that approximates the solution of (8). The optimal p for fixed margins P(Zi' a) in (8) can be written as pv(a) := argmax (p - _1 f)p - p(Zi' a»+) . (9) pE[O,I] vm i=1 where (~)+ := max(~, 0). Setting ~i := (pv(a) - P(Zi' a»+ and subtracting v!n I:~l ~i from the resulting inequality on both sides yields (for all 1 ~ i ~ m) 1 m 1 m P(Zi' a) + ~i L~i ~ pv(a) - L~i . (10) vm i=1 vm i=1 Two more substitutions are needed to transform the problem into one which can be solved by the AdaBoost algorithm. In particular we have to get rid of the slack variables ~i again by absorbing them into quantities similar to P(Zi' a) and p(a). This works as follows: on the right hand side of (10) we have the objective function (cf. (8» and on the left hand side a term that depends nonlinearly on a. Defining _ 1 m _ 1 m pv(a) := pv(a) - L ~i and Pv(Zi' a) := P(Zi' a) + ~i '"' ~i, (11) vm. vm~ ,=1 i=l which we substitute for p(a) and p(z,a) in (5), respectively, we obtain a new optimization problem. Note that ,ov (a) and ,ov (Zi' a) play the role of a corrected or virtual margin. We obtain a nonlinear min-max problem maximize subject to ,o(a) ,o( Zi, a) ~ ,o( a) at > 0 lIallt ~ 1 for all 1 ~ i ~ m for all 1 ~ t ~ T ' (12) which Arc-GV can solve approximately (cf. section 2). Hence, by replacing the margin p(Z, a) by ,o(z,a) in equation (4) and the other formulas for Arc-GV (cf. [2]), v-Arc: Ensemble Learning in the Presence o/Outliers 565 we obtain a new algorithm which we refer to as v-Arc. We can now state interesting properties for v-Arc by using Theorem 5 of [13] that bounds the generalization error R(f) for ensemble methods. In our case Rp(f) ~ v by construction (i.e. the number of patterns with a margin smaller than p, cf. Proposition 2), thus we get the following simple reformulation of this bound: Theorem 3. Let p(x, y) be a distribution over X x [-1,1]' and let X be a sample of m examples chosen iid according to p. Suppose the base-hypothesis space G has VC dimension h, and let [) > 0. Then with probability at least 1 [) over the random choice of the training set X, Y, every function f generated by v-Arc, where v E (0,1) and pv > 0, satisfies the following bound: R(f) ~ v + ~ (hIOg2 (ml h) I (!)) 2 + og ~ . m Pv u (13) So, for minimizing the right hand side we can tradeoff between the first and the second term by controlling an easily interpretable regularization parameter v. 4 Experiments We show a set of toy experiments to illustrate the general behavior of v-Arc. As base hypothesis class G we use the RBF networks of [11], and as data a two-class problem generated from several 2D Gauss blobs (cf. Banana shape dataset from http://www.first.gmd.derdata/banana.html.). We obtain the following results: • v-Arc leads to approximately vm patterns that are effectively used in the training of the base learner: Figure 1 (left) shows the fraction of patterns that have high average weights during the learning process (i.e. Ei=l Wt(Zi) > 112m). We find that the number of the latter increases (almost) linearly with v. This follows from (11) as the (soft) margin of patterns with p(z, a) < Pv is set to pv and the weight of those patterns will be the same. • The (estimated) test error, averaged over 10 training sets, exhibits a rather flat minimum in v (Figure 1 (lower)). This indicates that just as for vSVMs, where corresponding results have been obtained, v is a well-behaved parameter in the sense that a slight misadjustment it is not harmful. • v-Arc leads to the fraction v of margin errors (cf. dashed line in Figure 1) exactly as predicted in Proposition 2. • Finally, a good value of v can already be inferred from prior knowledge of the expected error. Setting it to a value similar to the latter provides a good starting point for further optimization (cf. Theorem 3). Note that for v = 1, we recover the Bagging algorithm (if we used bootstrap samples), as the weights of all patterns will be the same (Wt(Zi) = 11m for all i = 1, . . . ,m) and also the hypothesis weights will be constant (at'" liT for all t = 1, .. . ,T). Finally, we present a small comparison on ten benchmark data sets obtained from the VCI [1] benchmark repository (cf. http://ida.first.gmd.de/-raetsch/data/benchmarks.html). We analyze the performance of single RBF networks, AdaBoost, v-Arc and RBF-SVMs. For AdaBoost and v-Arc we use RBF networks [11] as base hypothesis. The model parameters of RBF (number of centers etc.), v-Arc (v) and SVMs (0', C) are optimized using 5-fold cross-validation. More details on the experimental setup can 566 G. Riitsch, B. SchO/kopf, A. J. Smola, K.-R. Muller, T. Onoda and S. Mika ~ ~ 0.8 number of important panems ~ 0.6 "'0 c ~ 0.4 ~ training error o o. I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.16 Arc-GV 0.15 ~ w 0.'4 Bagging 0.12 0.11 o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 1: Toy experim~nt (0' = 0): the left graph shows the average Yfraction of important patterns, the avo fraction of margin errors and the avo training error for different values of the regularization constant v for v-Arc. The right graph shows the corresponding generalization error. In both cases, the parameter v allows us to reduce the test errors to values much lower than for the hard margin algorithm (for v = 0 we recover ArcGV / AdaBoost, and for v = 1 we get Bagging.) be found in [11]. Fig. 1 shows the generalization error estimates (after averaging over 100 realizations of the data sets) and the confidence interval. The results of the best classifier and the classifiers that are not significantly worse are set in bold face. To test the significance, we used at-test (p = 80%). On eight out of the ten data sets, v-Arc performs significantly better than AdaBoost. This clearly shows the superior performance of v-Arc for noisy data sets and supports this soft margin approach for AdaBoost. Furthermore, we find comparable performances for v-Arc and SVMs. In three cases the SVM performs better and in two cases v-Arc performs best. Summarizing, AdaBoost is useful for low noise cases, where the classes are separable. v-Arc extends the applicability of boosting to problems that are difficult to separate and should be applied if the data are noisy. 5 Conclusion We analyzed the AdaBoost algorithm and found that Arc-GV and AdaBoost are efficient for approximating the solution of non-linear min-max problems over huge hypothesis classes. We re-parameterized the LP Reg-AdaBoost algorithm (cf. [7, 11]) and introduced a new regularization constant v that controls the fraction of patterns inside the margin area. The new parameter is highly intuitive and has to be optimized only on a fixed interval [0,1]. Using the fact that Arc-GV can approximately solve min-max problems, we found a formulation of Arc-G V - v-Arc - that implements the v-idea for Boosting by defining an appropriate soft margin. The present paper extends previous work on regularizing boosting (DOOM [9], AdaBoostReg [11]) and shows the utility and flexibility of the soft margin approach for AdaBoost. RBF AB v-Arc SVM Banana 10.8 ± 0.06 12.3 ± 0.07 10.6 ± 0.05 11.5 ± 0.07 B.Cancer 27.6 ± 0.47 30.4 ± 0.47 25.8 ± 0.46 26.0 ± 0.47 Diabetes 24.3 ± 0.19 26.5 ± 0.23 23.7 ± 0.20 23.5 ± 0.17 German 24.7 ± 0.24 27.5 ± 0.25 24.4 ± 0.22 23.6 ± 0.21 Heart 17.6 ± 0.33 20.3 ± 0.34 16.5 ± 0.36 16.0 ± 0.33 Ringnorm 1.7 ± 0.02 1.9 ± 0.03 1.7 ± 0.02 1.7 ± 0.01 F.Sonar 34.4 ± 0.20 35.7 ± 0.18 34.4 ± 0.19 32.4 ± 0.18 Thyroid 4.5 ± 0.21 4.4 ± 0.22 4.4 ± 0.22 4.8 ± 0.22 Titanic 23.3 ± 0.13 22.6 ± 0.12 23.0 ± 0.14 22.4 ± 0.10 Waveform 10.7 ± 0.11 10.8 ± 0.06 10.0 ± 0.07 9.9 ± 0.04 Table 1: Generalization error estimates and confidence intervals. The best classifiers for a particular data set are marked in bold face (see text). v-Arc: Ensemble Learning in the Presence of Outliers 567 We found empirically that the generalization performance in v-Arc depends only slightly on the choice of the regularization constant. This makes model selection (e.g. via cross-validation) easier and faster. Future work will study the detailed regularization properties of the regularized versions of AdaBoost, in particular in comparison to v-LP Support Vector Machines. Acknowledgments: Partial funding from DFG grant (Ja 379/52) is gratefully acknowledged. This work was done while AS and BS were at GMD FIRST. References [1] C. Blake, E. Keogh, and C. J. Merz. UCI repository of machine learning databases, 1998. http://www.ics. uci.edu/ ",mlearn/MLRepository.html. [2] L. Breiman. Prediction games and arcing algorithms. Technical Report 504, Statistics Department, University of California, December 1997. [3] M. Frean and T. Downs. A simple cost function for boosting. Technical report, Dept. of Computer Science and Electrical Eng., University of Queensland, 1998. [4] Y. Freund and R. E. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. In Computational Learning Theory: Eurocolt '95, pages 23-37. Springer-Verlag, 1995. [5] Y. Freund and R. E. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. J. of Compo fj Syst. Sc. , 55(1):119- 139, 1997. [6] J . Friedman, T. Hastie, and R. Tibshirani. Additive logistic regression: a statistical view of boosting. Technical report, Stanford University, 1998. [7) A. Grove and D. Schuurmans. Boosting in the limit: Maximizing the margin of learned ensembles. In Proc. of the 15th Nat. Conf. on AI, pages 692- 699, 1998. [8] Y. LeCun, L. D. Jackel, L. Bottou, C. Cortes, J. S. Denker, H. Drucker, I. Guyon, U. A. Muller, E. Sackinger, P. Simard, and V. Vapnik. Learning algorithms for classification: A comparison on handwritten digit recognition. Neural Networks, pages 261-276, 1995. [9) L. Mason, P. L. Bartlett, and J. Baxter. Improved generalization through explicit optimization of margins. Machine Learning, 1999. to appear. (10) J. R. Quinlan. Boosting first-order learning (invited lecture). Lecture Notes in Computer Science, 1160:143, 1996. (11) G. Ratsch, T. Onoda, and K.-R. Muller. Soft margins for AdaBoost. Technical Report NC-TR-1998-021, Department of Computer Science, Royal Holloway, University of London, Egham, UK, 1998. To appear in Machine Learning. (12) G. Ratsch, B. Schokopf, A. Smola, S. Mika, T. Onoda, and K.-R. Muller. Robust ensemble learning. In A.J. Smola, P.L. Bartlett, B. Scholkopf, and D. Schuurmans, editors, Advances in LMC, pages 207-219. MIT Press, Cambridge, MA, 1999. [13] R. Schapire, Y. Freund, P. L. Bartlett, and W . Sun Lee. Boosting the margin: A new explanation for the effectiveness of voting methods. Annals of Statistics, 1998. (Earlier appeared in: D. H. Fisher, Jr. (ed.), Proc. ICML97, M. Kaufmann). [14] B. Scholkopf, C. J. C. Burges, and A. J. Smola. Advances in Kernel Methods Support Vector Learning. MIT Press, Cambridge, MA, 1999. (15) B. Scholkopf, A. Smola, R. C. Williamson, and P. L. Bartlett. New support vector algorithms. Neural Computation, 12:1083 - 1121, 2000. (16) H. Schwenk and Y. Bengio. Training methods for adaptive boosting of neural networks. In Michael I. Jordan, Michael J. Kearns, and Sara A. Solla, editors, Advances in Neural Inf. Processing Systems, volume 10. The MIT Press, 1998. [17) V. Vapnik. The Nature of Statistical Learning Theory. Springer Verlag, New York, 1995.
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Monte Carlo POMDPs Sebastian Thrun School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Abstract We present a Monte Carlo algorithm for learning to act in partially observable Markov decision processes (POMDPs) with real-valued state and action spaces. Our approach uses importance sampling for representing beliefs, and Monte Carlo approximation for belief propagation. A reinforcement learning algorithm, value iteration, is employed to learn value functions over belief states. Finally, a samplebased version of nearest neighbor is used to generalize across states. Initial empirical results suggest that our approach works well in practical applications. 1 Introduction POMDPs address the problem of acting optimally in partially observable dynamic environment [6]. In POMDPs, a learner interacts with a stochastic environment whose state is only partially observable. Actions change the state of the environment and lead to numerical penalties/rewards, which may be observed with an unknown temporal delay. The learner's goal is to devise a policy for action selection that maximizes the reward. Obviously, the POMDP framework embraces a large range of practical problems. Past work has predominately studied POMDPs in discrete worlds [1]. Discrete worlds have the advantage that distributions over states (so-called "belief states") can be represented exactly, using one parameter per state. The optimal value function (for finite planning horizon) has been shown to be convex and piecewise linear [lO, 14], which makes it possible to derive exact solutions for discrete POMDPs. Here we are interested in POMDPs with continuous state and action spaces, paying tribute to the fact that a large number of real-world problems are continuous in nature. In general, such POMDPs are not solvable exactly, and little is known about special cases that can be solved. This paper proposes an approximate approach, the MC-POMDP algorithm, which can accommodate real-valued spaces and models. The central idea is to use Monte Carlo sampling for belief representation and propagation. Reinforcement learning in belief space is employed to learn value functions, using a sample-based version of nearest neighbor for generalization. Empirical results illustrate that our approach finds to close-to-optimal solutions efficiently. 2 Monte Carlo POMDPs 2.1 Preliminaries POMDPs address the problem of selection actions in stationary, partially observable, controllable Markov chains. To establish the basic vocabulary, let us define: • State. At any point in time, the world is in a specific state, denoted by x. Monte Carlo POMDPs 1065 • Action. The agent can execute actions, denoted a. • Observation. Through its sensors, the agent can observe a (noisy) projection of the world's state. We use 0 to denote observations. • Reward. Additionally, the agent receives rewards/penalties, denoted R E ~. To simplify the notation, we assume that the reward is part of the observation. More specifically, we will use R( 0) to denote the function that "extracts" the reward from the observation. Throughout this paper, we use the subscript t to refer to a specific point in time (e.g., St refers to the state at time t). POMDPs are characterized by three probability distributions: 1. The initial distribution, 7r( x) := Pr( xo), specifies the initial distribution of states at time t = O. 2. The next state distribution, p(x' I a,x) := Pr(xt = x' I at-I = a,Xt-l = x), describes the likelihood that action a, when executed at state x, leads to state x'. 3. The perceptual distribution, v( 0 Ix) := Pr( 0t = 0 I Xt = x), describes the likelihood of observing 0 when the world is in state x. A history is a sequence of states and observations. For simplicity, we assume that actions and observations are alternated. We use dt to denote the history leading up to time t: dt {Ot,at-l,Ot-l,at-2, ... ,ao,00} (1) The fundamental problem in POMDPs is to devise a policy for action selection that maximizes reward. A policy, denoted (T : d--+a (2) is a mapping from histories to actions. Assuming that actions are chosen by a policy (T, each policy induces an expected cumulative (and possibly discounted by a discount factor , :::; 1) reward, defined as 00 J<7 = L E [,T R(OT)] (3) T=O Here E[ ] denotes the mathematical expectation. The POMDP problem is, thus, to find a policy (T* that maximizes r, i.e., (T* = argmax J<7 (4) <7 2.2 Belief States To avoid the difficulty of learning a function with unbounded input (the history can be arbitrarily long), it is common practice to map histories into belief states, and learn a mapping from belief states to actions instead [10]. Formally, a belief state (denoted e) is a probability distribution over states conditioned on past actions and observations: et Pr(xt I dt} = Pr(xt lOt, at-I,"" 00) (5) Belief are computed incrementally, using knowledge of the POMDP's defining distributions 7r, p, and v. Initially eo = 7r (6) For t ~ 0, we obtain Bt+1 Pr(xt+1 I Ot+l, at,···, 00) (7) 0' Pr(Ot+1 I Xt+I,···, 00) Pr(Xt+l I at,···, 00) (8) 0' Pr(ot+1 I Xt+l) J Pr(Xt+l I at,···, 00, xt} Pr(xt I at,···, 00) dXt (9) 0' Pr(Ot+l I Xt+d J Pr(xt+1 I at, Xt) et dXt (10) 1066 0.2 0.1 11. ___ •••••••• 1111. 1.11111. II I I 2 • 10 12 S. Thrun '''-', I , I \ I \ 9 / \ I \ I \ I , I \ I , I \ / \ '-, ...• I II III. HI "I I •••... _ ___ ....... 4 10 12 Figure 1: Sampling: (a) Likelihood-weighted sampling and (b) importance sampling. At the bottom of each graph, samples are shown that approximate the function f shown at the top. The height of the samples illustrates their importance/actors. Here a denotes a constant normalizer. The derivations of (8) and (10) follow directly from the fact that the environment is a stationary Markov chain, for which future states and observations are conditionally independent from past ones given knowledge of the state. Equation (9) is obtained using the theorem of total probability. Armed with the notion of belief states, the policy is now a mapping from belief states (instead of histories) to actions: (j : 0 -+ a (11) The legitimacy of conditioning a on 0, instead of d, follows directly from the fact that the environment is Markov, which implies that 0 is all one needs to know about the past to make optimal decisions. 2.3 Sample Representations Thus far, we intentionally left open how belief states 0 are represented. In prior work, state spaces have been discrete. In discrete worlds, beliefs can be represented by a collection of probabilities (one for each state), hence, beliefs can be represented exactly. Here were are interested in real-valued state spaces. In general, probability distributions over realvalued spaces possess infinitely many dimensions, hence cannot be represented on a digital computer. The key idea is to represent belief states by sets of (weighted) samples drawn from the belief distribution. Figure 1 illustrates two popular schemes for sample-based approximation: likelihood-weighted sampling, in which samples (shown at the bottom of Figure la) are drawn directly from the target distribution (labeled f in Figure la), and importance sampling, where samples are drawn from some other distribution, such as the curve labeled 9 in Figure 1 b. In the latter case, samples x are annotated by a numerical importance factor p(x) f(x) g(x) (12) to account for the difference in the sampling distribution, g, and the target distribution f (the height of the bars in Figure 1 b illustrates the importance factors). Importance sampling requires that f > 0 -+ 9 > 0, which will be the case throughout this paper. Obviously, both sampling methods generate approximations only. Under mild assumptions, they converge to the target distribution at a rate of -j;;, with N denoting the sample set size [16]. In the context of POMDPs, the use of sample-based representations gives rise to the following algorithm for approximate belief propagation (c.f., Equation (10»: Algorithm particleJilter(Ot , at, 0t+l): Ot+l = 0 doN times: draw random state Xt from Ot Monte Carlo POMDPs sample Xt+1 according to p(Xt+1 I at, xt} set importance factorp(xt+J) = V(Ot+1 I xt+d add (Xt+l,p(Xt+I)) toBt+1 normalize all p(Xt+d E Bt+1 so that LP(Xt+d = 1 return Bt+1 1067 This algorithm converges to (10) for arbitrary models p, v, and 11" and arbitrary belief distributions B, defined over discrete, continuous, or mixed continuous-discrete state and action spaces. It has, with minor modifications, been proposed under names like particle filters [131. condensation algorithm [5], survival of the fittest [8], and, in the context of robotics, Monte Carlo localization [4]. 2.4 Projection In conventional planning, the result of applying an action at at a state Xt is a distribution Pr(Xt+l, Rt+1 I at, xt} over states Xt+1 and rewards Rt+1 at the next time step. This operation is called projection. In POMDPs, the state Xt is unknown. Instead, one has to compute the result of applying action at to a belief state Bt. The result is a distribution Pr(Bt+I' Rt+ 1 I at, Bt ) over belief states Bt+1 and rewards Rt+ I. Since belief states themselves are distributions, the result of a projection in POMDPs is, technically, a distribution over distributions. The projection algorithm is derived as follows. Using total probability, we obtain: Pr(Bt+l, Rt+1 I at,Bd Pr(Bt+I,Rt+11 at,dt} (13) = J !r(Bt+l , Rt+: I Ot+l, at, dt), !r(ot+I,,1 at, dt}, dOt+1 (14) (*) (**) The term (*) has already been derived in the previous section (c.f., Equation (10», under the observation that the reward Rt+1 is trivially computed from the observation 0t+l. The second term, (**), is obtained by integrating out the unknown variables, Xt+1 and Xt. and by once again exploiting the Markov property: Pr(Ot+l I at, dt} J Pr(Ot+1 I Xt+d Pr(xt+1 I at. dt} dXt+1 (15) J Pr(Ot+1 I Xt+l) J Pr(xt+1 I Xt, at} Pr(xt I dt} dXt dXt-t616) J V(Ot+1 I Xt+d J p(Xt+1 I Xt, at} Bt(xt) dXt dXt+1 (17) This leads to the following approximate algorithm for projecting belief state. In the spirit of this paper, our approach uses Monte Carlo integration instead of exact integration. It represents distributions (and distributions over distributions) by samples drawn from such distributions. Algorithm particle_projection(Bt, at): 8 t = 0 doN times: draw random state Xt from Bt sample a next state Xt+1 accordingtop(xt+1 I at,xt) sample an observation Ot+1 according to V(Ot+1 I Xt+d compute Bt+1 = partic1e_filter(Bt. at. Ot+l) add (Bt+I,R(ot+J)) t08t return8t The result of this algorithm, 8 t, is a sample set of belief states Bt+1 and rewards Rt+I, drawn from the desired distribution Pr( Bt+ I, Rt+ 1 I Bt , at}. As N ~ 00, at converges with probability 1 to the true posterior [16]. 1068 S. Thrun 2.5 Learning Value Functions Following the rich literature on reinforcement learning [7, 15], our approach solves the POMDP problem by value iteration in belief space. More specifically, our approach recursively learns a value function Q over belief states and action, by backing up values from subsequent belief states: Q(Ot,at} ~ E[R(ot+t}+,m:xQ(Ot+l,a)] (18) Leaving open (for a moment) how Q is represented, it is easy to be seen how the algorithm particle_projection can be applied to compute a Monte Carlo approximation of the right hand-side expression: Given a belief state Ot and an action at, particle_projection computes a sample of R( 0t+ I) and Ot+ I, from which the expected value on the right hand side of (18) can be approximated. It has been shown [2] that if both sides of (18) are equal, the greedy policy (1'Q(O) = argmaxQ(O,a) (19) a is optimal, i.e., (1'* = (1'Q. Furthermore, it has been shown (for the discrete case!) that repetitive application of (18) leads to an optimal value function and, thus, to the optimal policy [17, 3]. Our approach essentially performs model-based reinforcement learning in belief space using approximate sample-based representations. This makes it possible to apply a rich bag of tricks found in the literature on MDPs. In our experiments below, we use online reinforcement learning with counter-based exploration and experience replay [9] to determine the order in which belief states are updated. 2.6 Nearest Neighbor We now return to the issue how to represent Q. Since we are operating in real-valued spaces, some sort of function approximation method is called for. However, recall that Q accepts a probability distribution (a sample set) as an input. This makes most existing function approximators (e.g., neural networks) inapplicable. In our current implementation, nearest neighbor [11] is applied to represent Q. More specifically, our algorithm maintains a set of sample sets 0 (belief states) annotated by an action a and a Q-value Q(O, a). When a new belief state Of is encountered, its Q-value is obtained by finding the k nearest neighbors in the database, and linearly averaging their Q-values. If there aren't sufficiently many neighbors (within a pre-specified maximum distance), Of is added to the database; hence, the database grows over time. Our approach uses KL divergence (relative entropy) as a distance function I. Technically, the KL-divergence between two continuous distributions is well-defined. When applied to sample sets, however, it cannot be computed. Hence, when evaluating the distance between two different sample sets, our approach maps them into continuous-valued densities using Gaussian kernels, and uses Monte Carlo sampling to approximate the KL divergence between them. This algorithm is fairly generic an extension of nearest neighbors to function approximation in density space, where densities are represented by samples. Space limitations preclude us from providing further detail (see [11, 12]). 3 Experimental Results Preliminary results have been obtained in a world shown in two domains, one synthetic and one using a simulator of a RWI B21 robot. In the synthetic environment (Figure 2a), the agents starts at the lower left comer. Its objective is to reach "heaven" which is either at the upper left comer or the lower right 1 Strictly speaking, KL divergence is not a distance metric, but this is ignored here. Monte Carlo POMDPs 1069 ... (a .... ) ,.-__ ...,.._ (~"----~~,-~--~-v----v.n-"'" I=) \on... 50 P '-, 1M 25 ·25 -50 t.S:: t .. .....,.· ·75 ·100 0 20 40 60 80 10 15 20 25 30 Figure 2: (a) The environment, schematically. (b) Average perfonnance (reward) as a function of training episodes. The black graph corresponds to the smaller environment (25 steps min), the grey graph to the larger environment (50 steps min). (c) Same results, plotted as a function of number of backups (in thousands). comer. The opposite location is "hell." The agent does not know the location of heaven, but it can ask a "priest" who is located in the upper right comer. Thus, an optimal solution requires the agent to go first to the priest, and then head to heaven. The state space contains a real-valued (coordinates of the agent) and discrete (location of heaven) component. Both are unobservable: In addition to not knowing the location of heaven, the agent also cannot sense its (real-valued) coordinates. 5% random motion noise is injected at each move. When an agent hits a boundary, it is penalized, but it is also told which boundary it hit (which makes it possible to infer its coordinates along one axis). However, notice that the initial coordinates of the agent are known. The optimal solution takes approximately 25 steps; thus, a successful POMDP planner must be capable of looking 25 steps ahead. We will use the term "successful policy" to refer to a policy that always leads to heaven, even if the path is suboptimal. For a policy to be successful, the agent must have learned to first move to the priest (information gathering), and then proceed to the right target location. Figures 2b&c show performance results, averaged over 13 experiments. The solid (black) curve in both diagrams plots the average cumulative reward J as a function of the number of training episodes (Figure 2b), and as a function of the number of backups (Figure 2c). A successful policy was consistently found after 17 episodes (or 6,150 backups), in all 13 experiments. In our current implementation, 6,150 backups require approximately 29 minutes on a Pentium Pc. In some experiments, a successful policy was identified in 6 episodes (less than 1,500 backups or 7 minutes). After a successful policy is found, further learning gradually optimizes the path. To investigate scaling, we doubled the size of the environment (quadrupling the size of the state space), making the optimal sol uti on 50 steps long. The results are depicted by the gray curves in Figures 2b&c. Here a successful policy is consistently found after 33 episodes (10,250 backups, 58 minutes). In some runs, a successful policy is identified after only 14 episodes. We also applied MC-POMDPs to a robotic locate-and-retrieve task. Here a robot (Figure 3a) is to find and grasp an object somewhere in its vicinity (at floor or table height). The robot's task is to grasp the object using its gripper. It is rewarded for successfully grasping the object, and penalized for unsuccessful grasps or for moving too far away from the object. The state space is continuous in x and y coordinates, and discrete in the object's height. The robot uses a mono-camera system for object detection; hence, viewing the object from a single location is insufficient for its 3D localization. Moreover, initially the object might not be in sight of the robot's camera, so that the robot must look around first. In our simulation, we assume 30% general detection error (false-positive and false-negative), with additional Gaussian noise if the object is detected correctly. The robot's actions include turns (by a variable angle), translations (by a variable distance), and grasps (at one of two legal heights). Robot control is erroneous with a variance of20% (in x-y-space) and 5% (in rotational space). Typical belief states range from uniformly distributed sample sets (initial belief) to samples narrowly focused on a specific x-y-z location. 1070 , \ \ \ , (b) L C (c) % success 1 OB 0.6 0.4 2000 S. Thrun 4000 6000 BOOO iteration Figure 3: Find and fetch task: (a) The mobile robot with gripper and camera, holding the target object (experiments are carried out in simulation!), (b) three successful runs (trajectory projected into 2D), and (c) success rate as a function of number of planning steps. Figure 3c shows the rate of successful grasps as a function of iterations (actions). While initially, the robot fails to grasp the object, after approximately 4,000 iterations its performance surpasses 80%. Here the planning time is in the order of 2 hours. However, the robot fails to reach 100%. This is in part because certain initial configurations make it impossible to succeed (e.g., when the object is too close to the maximum allowed distance), in part because the robot occasionally misses the object by a few centimeters. Figure 3b depicts three successful example trajectories. In all three, the robot initially searches the object, then moves towards it and grasps it successfully. 4 Discussion We have presented a Monte Carlo approach for learning how to act in partially observable Markov decision processes (POMDPs). Our approach represents all belief distributions using samples drawn from these distributions. Reinforcement learning in belief space is applied to learn optimal policies, using a sample-based version of nearest neighbor for generalization. Backups are performed using Monte Carlo sampling. Initial experimental results demonstrate that our approach is applicable to real-valued domains, and that it yields good performance results in environments that are-by POMDP standards-relatively large. References [1] AAAI Fall symposium on POMDPs. 1998. See http://www.cs.duke.edu/ ... mlittman/talks/ pomdp-symposiurn.html [2] R E. Bellman. Dynamic Programming. Princeton University Press, 1957. [3] P. Dayan and T. 1. Sejnowski. ID('>') converges with probability 1. 1993. [4] D. Fox, W. Burgard, F. Dellaert, and S. Thrun. Monte carlo localization: Efficient position estimation for mobile robots. AAAI-99. [5] M. lsard and A. Blake. Condensation: conditional density propagationforvisual tracking.lnternationalJoumalofComputer Vision, 1998. [6] L.P. Kaelbling, M.L. Littman, and A.R Cassandra. Planning and acting in partially observable stochastic domains. Submitted for publication, 1997. [7] L.P. Kaelbling, M.L. Littman, and A. W. Moore. Reinforcement learning: A survey. lAIR,4, 1996. [8] K Kanazawa, D. Koller, and S.l. Russell. Stochastic simulation algorithms for dynamic probabilistic networks. UAI-95. [9] L.-l. Lin. Self-improving reactive agents based on reinforcement learning, planning and teaching. Machine Learning, 8, 1992. [10] M.L. Littman, A.R Cassandra, and L.P. KaeJbling. Learning poliCies for partially observable environments: Scaling up. ICML-95. [11] A.w. Moore, C.G. Atkeson, and S.A. Schaal. Locally weighted learning for control. AI Review, II, 1997. [12] D. Ormoneit and S. Sen. Kernel-based reinforcernentlearning. TR 1999-8, Statistics, Stanford University, 1999. [13] M. Pitt and N. Shephard. Filtering via simulation: auxiliary particle filter. lournal of the American Statistical Association, 1999. [14] E. Sondik. The Optimal Control of Partially Observable Markov Processes. PhD thesis, Stanford, 1971. [I 5] R.S. Sutton and A.G. Barto. Reinforcement Learning: An Introduction. MIT Press, 1998. [16] M.A. Tanner. ToolsforStatistical Inference. Springer Verlag, 1993. [17] C. 1. C. H. Watkins. Learningfrom Delayed Rewards. PhD thesis, King's College, Cambridge, 1989.
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A recurrent model of the interaction between Prefrontal and Inferotemporal cortex in delay tasks ALFONSO RENART, NESTOR PARGA Departamento de F{sica Te6rica Universidad Aut6noma de Madrid Canto Blanco, 28049 Madrid, Spain http://www.ft.uam.es/neurociencialGRUPO/grup0.1!nglish.html and EDMUND T. ROLLS Oxford University Department of Experimental Psychology South Parks Road, Oxford OX] 3UD, England Abstract A very simple model of two reciprocally connected attractor neural networks is studied analytically in situations similar to those encountered in delay match-to-sample tasks with intervening stimuli and in tasks of memory guided attention. The model qualitatively reproduces many of the experimental data on these types of tasks and provides a framework for the understanding of the experimental observations in the context of the attractor neural network scenario. 1 Introduction Working memory is usually defined as the capability to actively hold information in memory for short periods of time. In primates, visual working memory is usually studied in experiments in which, after the presentation of a given visual stimulus, the monkey has to withhold its response during a certain delay period in which no specific visual stimulus is shown. After the delay, another stimulus is presented and the monkey has to make a response which depends on the interaction between the two stimuli. In order to bridge the temporal gap between the stimuli, the first one has to be held in memory during the delay. Electrophysiological recordings in primates during the performance of this type of tasks has revealed that some populations of neurons in different brain areas such as prefrontal (PF), inferotemporal (IT) or posterior parietal (PP) cortex, maintain approximately constant firing rates during the delay periods (for a review see [1]) and this delay activity states have been postulated as the internal representations of the stimuli provoking them [2]. Although up to now most of the modeling effort regarding the operation of networks able to support stable delay activity states has been put in the study of un i-modular (homogeneous) networks, there is evidence that in order for the monkey to solve the tasks satisfactorily, the interaction of several different neural structures is needed. A number of studies of delay match-to-sample tasks with intervening stimuli in primates performed by Desimone and 172 A. Renart, N Parga and E. TRolls colleagues has revealed that although IT cortex supports delay activity states and shows memory related effects (differential responses to the same, fixed stimulus depending on its status on the trial, e.g. whether it matches or not the sample), it cannot, by itself, provide the information necessary to solve the task, as the delay activity states elicited by each of the stimuli in a sequence are disrupted by the input information associated with each new stimulus presented [3, 4, 5]. Another structure is therefore needed to store the information for the whole duration of the trial. PF cortex is a candidate, since it shows selective delay activity maintained through entire trials even with intervening stimuli [6]. A series of parallel experiments by the same group on memory guided attention [7, 8] have also shown differential firing of IT neurons in response to the same visual stimulus shown after a delay (an array of figures), depending on previous information shown before the delay (one of the figures in the array working as a target stimulus). This evidence suggests a distributed memory system as the proper scenario to study working memory tasks as those described above. Taking into account that both IT and PF cortex are known to be able to support delay activity states, and that they are bi-directionally connected, in this paper we propose a simple model consisting of two reciprocally connected attractor neural networks to be identified with IT and PF cortex. Despite its simplicity, the model is able to qualitatively reproduce the behavior of IT and PF cortex during delay match-to-sample tasks with intervening stimuli, the behavior of IT cells during memory guided attention tasks, and to provide an unified picture of these experimental data in the context of associative memory and attractor neural networks. 2 Model and dynamics The model network consists of a large number of (excitatory) neurons arranged in two modules. Following [9, 10], each neuron is assumed to be a dynamical element which transforms an incoming afferent current into an output spike rate according to a given transduction function. A given afferent current Iai to neuron i (i = 1, ... ,N) in module a (a = IT, PF) decays with a characteristic time constant T but increases proportionally to the spike rates Vbj of the rest of the neurons in the network (both from inside and outside its module) connected to it, the contribution of each presynaptic neuron, e.g. neuron j from module b, being proportional to the synaptic efficacy Jt/ between the two. This can be expressed through the following equation d1ai(t) = _ Iai(t) + '" J~~,b) . + h(~xt) dt T ~ ~J VbJ a~ . bj (1) An external current h~~xt) from outside the network, representing the stimuli, can also be imposed on every neuron. Selective stimuli are modeled as proportional to the stored patterns, i.e. h~~ezt) = haTJ~i' where ha is the intensity of the external current to module a. The transduction function of the neurons transforming currents into rates has been chosen as a threshold hyperbolic tangent of gain G and threshold O. The synaptic efficacies between the neurons of each module and between the neurons in different modules are respectively [11, 12] p J(a,a) Jo "'( I-' I) (I-' I) i.../- J' a = IT,PF (2) ij 1(1 _ J)Nt ~ TJai TJaj r p j (a,b) 9 "'( I-' I) (I-' I) \.J" .../- b (3) ij 1(1 _ J)Nt ~ TJai TJbj v ~,J a r . Recurrent Model of IT-PF Interactions in Delay Tasks 173 The intra-modular connections express the learning of P binary patterns {17~i = 0,1, f.L = 1, ... , P} by each module, each of them signaling which neurons are active in each of the sustained activity configurations. Each variable Tl~i is supposed to take the values 1 and 0 with probabilities f and (1 - f) respectively, independently across neurons and across patterns. The inter-modular connections reflect the temporal associations between the sustained activity states of each module. In this way, every stored pattern f.L in the IT module has an associated pattern in the PF module which is labelled by the same index. The normalization constant Nt = N(Jo + g) has been chosen so that the sum of the magnitudes of the inter- and the intra-modular connections remains constant and equal to 1 while their relative values are varied. When this constraint is imposed the strength of the connections can be expressed in terms of a single independent parameter 9 measuring the relative intensity of the inter- vs. the intra-modular connections (Jo can be set equal to 1 everywhere). We will limit our study to the case where the number of stored patterns per module P does not increase proportionally to the size of the modules N since a large number of stored patterns does not seem necessary to describe the phenomenology of the delay match-to-sample experiments. Since the number of neurons in a typical network one may be interested in is very large, e.g. '" 105 - 106, the analytical treatment of the set of coupled differential equations (1) becomes intractable. On the other hand, when the number of neurons is large, a reliable description of the asymptotic solutions of these equations can be found using the techniques of statistical mechanics [13, 9]. In this framework, instead of characterizing the states of the system by the state of every neuron, this characterization is performed in terms of macroscopic quantities called order parameters which measure and quantify some global properties of the network as a whole. The relevant order parameters appearing in the description of our system are the overlaps of the state of each module with each of the stored patterns m~, defined as: 1 m~ = N« 2)17~i - f)Vai »1/ , X i (4) where the symbol « ... »1/ stands for an average over the stored patterns. Using the free energy per neuron of the system at zero temperature :F (which we do not write explicitly to reduce the technicalities to a minimum) we have modeled the experiments by giving the order parameters the following dynamics: (5) This dynamics ensures that the stationary solutions, corresponding to the values of the order parameters at the attractors, correspond also to minima of the free energy, and that, as the system evolves, the free energy is always minimized through its gradient. The time constant of the macroscopic dynamics is a free parameter which has been chosen equal to the time constant of the individual neurons, reflecting the assumption that neurons operate in parallel. Its value has been set to T = 10 ms. Equations (5) have been solved by a simple discretizing procedure (first order Runge-Kutta method). Since not all neurons in the network receive the same inputs, not all of them behave in the same way, i.e. have the same firing rates. In fact, the neurons in each of the module can be split into different sub-populations according to their state of activity in each of the stored patterns. The mean firing rate of the neurons in each SUb-population depends on the particular state realized by the network (characterized by the values of the order parameters). Associated to each pattern there are two larger sub-populations, to be denoted as foreground (all active neurons) and background (all inactive neurons) of that pattern. 174 A. Renart, N. Parga and E. T. Rolls The overlap with a given pattern can be expressed as the difference between the mean firing rate of the neurons in its foreground and its background. The average is performed over all other sub-populations to which each neuron in the foreground (background) may belong to, where the probability of a given sub-population is equal to the fraction of neurons in the module belonging to it (determined by the probability distribution of the stored patterns as given above). This partition of the neurons into sub-populations is appealing since, in experiments, cells are usually classified in terms of their response properties to a set of fixed stimuli, i.e. whether each stimulus is effective or ineffective in driving their response. The modeling of the different experiments proceeded according to the macroscopic dynamics (5), where each stimulus was implemented as an extra current for a desired period of time. 3 Sequence with intervening stimuli In order to study delay match-to-sample tasks with intervening stimuli [5, 6], the module to be identified with IT was sequentially stimulated with external currents proportional to some of the stored patterns with a delay between them. To take into account the large fraction of PF neurons with non-selective responses to the visual stimuli (which may be involved in other aspects of the task different from the identification of the stimuli), and since the neurons in our modules are, by definition, stimulus selective (although they are probably connected to the non-selective neurons) a constant, non-selective current of the same intensity as the selective input to the IT module was applied (during the same time) equally to all sub-populations of the PF module. The external current to the IT module was stimulus selective because the fraction of IT neurons with non-selective responses to the visual stimuli is very small [6]. The results can be seen in Figure 1 where the sequence ABA with A as the sample stimulus and B as a non-matching stimulus has been studied. The values of the model parameters are listed in the caption. In Figure 1 a, the mean firing rates of the foreground populations of patterns AIT and BIT of the IT module have been plotted as a function of time. The main result is that, as observed in the experiments, the delay activity in the IT module is determined by the last stimulus presented. The delay activity provoked by a given stimulus is disrupted by the next, unless it corresponds to the same stimulus, in which case the effect of the stimulus is to increase the firing rate of the neurons in its foreground. We have checked that no noticeable effects occur if more nonmatching stimuli are presented (they are all equivalent with respect to the sample) or if a non-match stimulus is repeated. If the coupling g between the modules is weak enough [12] the behavior in the PF module is different. This can be seen in Figure 1 b, where the time evolution of the mean firing rates of the foreground of the two associated patterns ApF and BpF stored in the PF module are shown. In agreement with the findings of Desimone and colleagues, the neurons in the PF module remain correlated with the sample for the whole trial, despite the non-selective signal received by all PF neurons (not only those in the foreground of the sample) and the fact that the selective current from the IT module tends to activate the pattern associated with the current stimulus. Desimone and colleagues [5, 6] report that the response of some neurons (not necessarily those with sample selective delay activity or with stimulus selective responses) in both IT and PF cortex to some stimuli, is larger if those stimuli are matches in their trials than if the same stimuli are non-matches. This has been denoted as match enhancement. In the present scenario the explanation is straightforward: when a stimulus is a non-match, IT and PF are in different states and therefore send inconsistent signals to each other. The firing rate of the neurons of each module is maintained in that case solely by the contribution to the total current coming from the recurrent collaterals. On the other hand, when the stimulus is the match, both modules find themselves in states associated in the synapses Recurrent Model of IT-PF Interactions in Delay Tasks (a) 0.8 0.7 Ir' ''I r! i 0.6 ! ' i \ ! , ., 0.5 i a; I a: I C> 0.4 \ c: \ .'" Ii: 0.3 "'---._ .. _-0.2 0.1 r.~ .... .. _ .. _ .-J ~ ___ --,·L .. __ O'---'--~~-~~~-"""""'-'---' o 2 3 4 56789 Time (s) ., a; a: C> c: ·c Ii: 0.7 0.6 0.5 0.4 0.3 0.2 \. (b) \ . fL... i\..... •• 0.1 "l .... ____ J \. ___ . ___ ._.J l. __ ._. ___ . OL--'--~~-~~~-........... -,---, o 234 5 6 7 8 9 Time (s) 175 Figure 1: (a) Mean rates in the foreground of patterns AIT (solid line) and BIT (dashed line) in the IT module as a function of time. (b) Same but for patterns ApF and BpF of the PF module. Model parameters are G = 1.3, () = 10-3, f = 0.2, 9 = 10-2, h = 0.13. Stimuli are presented during 500 ms at seconds 0, 3, and 6 following the sequence ABA. between the neurons connecting them, PF because it has remained that way the whole trial, and IT because it is driven by the current stimulus. When this happens, the contribution to the total current from the recurrent collaterals and from the long range afferents add up consistently, and the firing rate increases. In order for this explanation to hold there should be a correlation between the top-down input from PF and the sensory bottom-up signal to IT. Indeed, experimental evidence for such a correlation has very recently been found [14]. This is an important experimental finding which supports our theory. Looking at Figure 1, one sees that the effect is not evident in the model during the time of stimulus presentation, which is the period where it has been reported. The effect is, in fact, present, although its magnitude is too small to be noticeable in the figure. We would argue, however, that this quantitative difference is an artifact of the model. This is because the enhancement effect is very noticeable on the delay periods, where essentially the same neurons are active as during the stimulus presentations (i.e., where the same correlations between the top-down and bottom-up signals exist) but with lower firing rates. During stimulus presentations the firing rates are closer to the saturation regime, and therefore the dynamical response range of the neurons is largely reduced. 4 Memory guided attention To test the differential response of cells as a function of the contents of memory, we have followed [7, 8] and studied a sub-population of IT cells which are simultaneously in the foreground of one of the patterns (AIT) and in the background of another (BIT) in the same conditions as the previous section (same model parameters). In Figure 2a the response of this sub-population as a function of time has been plotted in two different situations. In the first one, the effective stimulus AIT was shown first (throughout this section non selective stimulation of PF proceeded as in the last section) and after a delay, a stimulus array equal to the sum of AIT and BIT was presented. The second situation is exactly equal, except for the fact that the cue stimulus shown first was the ineffective stimulus BIT. The response of the same sub-population to the same stimulus array is totally different and determined by the cue stimulus: If the sub-population is in the background of the cue, its response is null during the trial except for the initial period of the presentation of the array. In accordance with the experimental observations [7, 8], its response grows initially (as one would expect, since during the array presentation time, stimulation is symmetric with 176 A. Renart, N. Parga and E. T. Rolls respect of A and B) but is later suppressed by the top-down signal being sent by the PF module. This suppression provides a clear example of a situation in which the contents of memory (in the form of an active PF activity state) are explicitly gating the access of sensory information to IT, implementing a non-spatial attentional mechanism. Q) iii cr: '" c ." u: (a) 0.8 .---~~~-~-~-~---, 0.7 rr 0.6 0.5 0.4 0.3 0.2 0.1 r\ 0 ! \ _ .. _--_ .. -----_ .... _.... "-.. _----_ .. _2 3 Time (s) 4 5 6 * cr: '" c ·c u: 0.7 0.6 0.5 0.4 0.3 0.2 0.1 r--\ 0 -0.1 0 2 (b) r-·1 I ! i i • I l.-\ l_ .. ___ .. __ ._._ .. 3 Time (s) 4 5 6 Figure 2: (a) Mean rates as a function of time in IT neurons which are both in the foreground AIT and in the background of BIT when the cue stimulus is AIT (solid line) or BIT (dashed line). (b) Mean rates of the same neurons when CIT is the cue stimulus and the array is AIT alone (long dashed line), BIT alone (short dashed line) or the sum of AIT and BIT (solid line). Cue present until 500 ms. Array present from 3000 ms to 3500 ms. Model parameters as in Figure 1 In the model, the PF module remains in a state correlated with the cue during the whole trial (to our knowledge there are no measurements of PF activity during memory guided attention tasks) and therefore provides a persistent signal 'in the direction' ofthe cue which biases the competition between AIT and BIT established at the onset of the array. This is how the gating mechanism is implemented. The competitive interactions between the stimuli in the array are studied in Figure 2b, which is an emulation of the target-absent trials of [8]. In this figure, the same sub-population is studied under situations in which the cue stimulus is not present in the array (another one of the stored patterns, i.e. CIT) The three curves correspond to different arrays: The effective stimulus alone, the ineffective stimulus alone, and a sum of the two as in the previous experiment. In all three, the PF module remains in a sustained activity state correlated with CIT the whole trial and therefore, since the patterns are independent, the signal it sends to IT is symmetric with respect of A and B. Thus, the response of the sub-population during the array is in this case unbiased, and the effect of the competitive interactions can be isolated. The result is that, as observed experimentally, the response to the complex array is intermediate between the one to the effective stimulus alone and the one to the ineffective stimulus alone. The nature of the competition in an attractor network like the one under study here is based on the fact that complex stimulus combinations are not stored in the recurrent collaterals of each module. These connections tend to stabilize the individual patterns which, being independent, tend to cancel each other when presented together. After the array is presented, the state of the IT module, which is correlated with CIT in the initial delay, becomes correlated with AIT or BIT if they are presented alone. When the array contains both of them in a symmetric fashion, since the sum of the patterns is not a stored pattern itself, the IT module remains correlated with pattern CIT due to the signal from the PF module. Recurrent Model of IT-PF Interactions in Delay Tasks 177 5 Discussion We have proposed a toy model consisting of two reciprocally connected attractor modules which reproduces nicely experimental observations regarding intra-trial data in delay match-to-sample and memory guided attention experiments in which the interaction between IT and PF cortex is relevant. Several important issues are taken into account in the model: a complex interaction between the PF and IT modules resultant from the association of frequent patterns of activity in both modules, delay activity states in each module which exert mutually modulatory influences on each other, and a common substrate (we emphasize that the results on Sections 3 and 4 where obtained with exactly the same model parameters, just by changing the type of task) for the explanation of apparently diverse phenomena. Perception is clearly an active process which results from the complex interactions between past experience and incoming sensory information. The main goal of this model was to show that a very simple associational (Hebbian) pattern of connectivity between a perceptual module and a 'working memory' module can provide the basic ingredients needed to explain coherently different experimentally found neural mechanisms related to this process. The model has clear limitations in terms of 'biological realism' which will have to be improved in order to use it to make quantitative predictions and comparisons, and does not provide a complete an exhaustive account of the very complex and diverse phenomena in which temporo-frontal interactions are relevant (there is, for example, the issue of how to reset PF activity in between trials [15]). However, it is precisely the simplicity of the mechanism it provides and the fact that it captures the essential features of the experiments, despite being so simple, what makes it likely that it will remain relevant after being refined. Acknowledgements This work was funded by a Spanish grant PB96-0047. We acknowledge the Max Planck Institute for Physics of Complex Systems in Dresden, Germany, for the hospitality received by A.R. and N.P. during the meeting held there from March 1 to 26, 1999. References [1] J. M. Fuster. Memory in the cerebral cortex. Cambridge, MA: MIT Press (1995) [2] D. J. Amit. Belulvioral and Brain Sciences 18, 617-657 (1995) [3] G. C. Baylis & E. T. Rolls. Exp. Brain Res. 65,614-622 (1987) [4] E. K. Miller, L. Li & R. Desimone. 1. Neurosci. 13, 1460-1478 (1993) [5] E. K. Miller & R. Desimone. Science 263,520-522 (1994) [6] E. K. Miller, C. A. Erickson & R. Desimone. 1. Neurosci. 16,5154-5167 (1996) [7] L. Chelazzi, E. K. Miller, J. Duncan & R. Desimone. Nature 363,345-347 (1993) [8] L. Chelazzi, J. Duncan, E. K. Miller & R. Desimone. 1. Neurophysiol. 80,2918-2940 (1998) [9] R. Kuhn. In Statistical Meclulnics of Neural Networks. (ed. L. Garrido), 19-32. Berlin: Springer-Verlag (1990) [10] D. J. Amit & M. V. Tsodyks. Network 2,259-273 (1991) [11] A. Renart, N. Parga & E. T. Rolls. Neural Computation 11, 1349-1388 (1999). [12] A. Renart, N. Parga & E. T. Rolls. Network 10,237-255 (1999). [13] M. Mezard, G. Parisi & M. Virasoro. Spin glass theory and beyond. Singapore: World Scientific (1987) [14] H. Tomita, M Ohbayashi, K. Nakahara, I. Hasegawa & Y. Miyashita. Nature 401, 699-703 (1999) [15] D. Durstewitz, M. Kelc & o. Giintiirkiin. 1. Neurosci. 19,2807-2822 (1999)
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Information Factorization in Connectionist Models of Perception Javier R. Movellan Department of Cognitive Science Institute for Neural Computation University of California San Diego James L. McClelland Center for the Neural Bases of Cognition Department of Psychology Carnegie Mellon University Abstract We examine a psychophysical law that describes the influence of stimulus and context on perception. According to this law choice probability ratios factorize into components independently controlled by stimulus and context. It has been argued that this pattern of results is incompatible with feedback models of perception. In this paper we examine this claim using neural network models defined via stochastic differential equations. We show that the law is related to a condition named channel separability and has little to do with the existence of feedback connections. In essence, channels are separable if they converge into the response units without direct lateral connections to other channels and if their sensors are not directly contaminated by external inputs to the other channels. Implications of the analysis for cognitive and computational neurosicence are discussed. 1 Introduction We examine a psychophysical law, named the Morton-Massaro law, and its implications to connectionist models of perception and neural information processing. For an example of the type of experiments covered by the Morton-Massaro law consider an experiment by Massaro and Cohen (1983) in which subjects had to identify synthetic consonant sounds presented in the context of other phonemes. There were two response alternatives, seven stimulus conditions, and four context conditions. The response alternatives were /1/ and /r/, the stimuli were synthetic sounds generated by varying the onset frequency of the third formant, followed by the vowel /i/. Each of the 7 stimuli was placed after each offour different context consonants, /v/, /s/, /p/, and /t/. Morton (1969) and Massaro independently showed that in a remarkable range of experiments of this type, the influence of stimulus and context on response probabilities can be accounted for with a factorized version of Luce's strength model (Luce, 1959) P(R = k I S = i, C = j) Tls(i, k) Tlc(j, k) . . 2: C l) C l)' for (l,,),k) E S x ex 'R. (1) I TIs 1" TIc), Here S, C and R are random variables representing the stimulus, context and the subject's response, S, C and 'R are the set of stimulus, context and response al46 J. R. Movellan and J. L. McClelland ternatives, l1s(i, k) > 0 represents the support of stimulus i for response k, and l1c(j, k) > 0 the support of context j for response k. Assuming no strength parameter is exactly zero, (1) is equivalent to P(R = k I S = i,e = j) = '(l1S(i,k)) (l1c(j,k)) , for all (i,j,k) E S x ex R. P(R = II S = i, e = j) l1s(i, l) l1c(j, l) (2) This says that response probability ratios factorize into two components, one which is affected by the stimulus but unaffected by the context and one affected by the context but unaffected by the stimulus. 2 Diffusion Models of Perception Massaro (1989) conjectured that the Morton-Massaro law may be incompatible with feedback models of perception. This conjecture was based on the idea that in networks with feedback connections the stimulus can have an effect on the context units and the context can have an effect on the stimulus units making it impossible to factorize the influence of information sources. In this paper we analyze such a conjecture and show that, surprisiQ.gly, the Morton-Massaro law has little to do with the existence of feedback and lateral connections. We ground our analysis on continuous stochastic versions of recurrent neural networks 1. We call these models diffusion (neural) networks for they are stochastic diffusion processes defined by adding Brownian motion to the standard recurrent neural network dynamics. Diffusion networks are defined by the following stochastic differential equation dYi(t) = JLi(Y(t), X) dt + (J dBi(t) for i E {I, ... , n}, (3) where Yi(t) is a random variable representing the internal potential at time t of the ith unit, Y(t) = (Yl(t),··· ,Yn(t))', X represents the external input, which consists of stimulus and context, and Bi is Brownian motion, which acts as a stochastic driving term. The constant (J > 0, known as the dispersion, controls the amount of noise injected onto each unit. The function JLi, known as the drift, determines the average instantaneous change of activation and is borrowed from the standard recurrent neural network literature: this change is modulated by a matrix w of connections between units, and a matrix v that controls the influence of the external inputs onto each unit. 1 JLi(Yi(t), X) = ~i(Yi(t)) (Yi(t) - Yi(t)), for all i E {I,··· , n}, (4) where 1/ ~i is a positive function, named the capacitance, controlling the speed of processing and Yi(t) = L Wi,j Zj(t) + LVi,kXk, for alli E {I, .. · ,n}, (5) j k Zj(t) = CPi(}j(t)) = CP(O!i }j(t)) = 1/(1 + e-a • Y;(t)). (6) Here Wi,j, an element of the connection matrix w, is the weight from unit j to unit i, Vi,k is an element of the matrix v, cP is the logistic activation function and the O!i > 0 terms are gain parameters, that control the sharpness of the activation functions. For large values of O!i the activation function of unit i converges to a step function. The variable Zj(t) represents a short-time mean firing rate (the activation) of unit lFor an analysis grounded on discrete time networks with binary states see McClelland (1991). Information Factorization 47 j scaled in the (0,1) range. Intuition for equation (4) can be achieved by thinking of it as a the limit of a discrete time difference equation, in such case Y(t + ~t) = Yi(t) + J.'i (Yi (t), X)~t + (rli5:tNi (t), (7) where the Ni(t) are independent standard Gaussian random variables. For a fixed state at time t there are two forces controlling the change in activation: the drift, which is deterministic, and the dispersion which is stochastic. This results in a distribution of states at time t + ~t. As ~t goes to zero, the solution to the difference equation (7) converges to the diffusion process defined in (4). In this paper we focus on the behavior of diffusion networks at stochastic equilibrium, i.e., we assume the network is given enough time to approximate stochastic equilibrium before its response is sampled. 3 Channel Separability In this section we show that the Morton-Massaro is related to an architectural constraint named channel separability, which has nothing to do with the existence of feedback connections. In order to define channel separability it is useful to characterize the function of different units using the following categories: 1) Response specification units: A unit is a response specification unit, if, when the state of all the other units in the network is fixed, changing the state of this unit affects the probability distribution of overt responses. 2) Stimulus units: A unit belongs to the stimulus channel if: a) it is not a response unit, and b) when the state of the response units is fixed, the probability distribution of the activations of this unit is affected by the stimulus. 3) Context units: A unit belongs to the context channel if: a) it is not a response unit, and b) when the states of the response units are fixed, the probability distribution of the activations of this unit can be affected by the context. Given the above definitions, we say that a network has separable stimulus and context channels if the stimulus and context units are disjoint: no unit simultaneously belongs to the stimulus and context channels. In essence, channels are structurally separable if they converge into the response units without direct lateral connections to other channels and if their sensors are not directly contaminated by external inputs to the other channels (see Figure 1). In the rest of the paper we show that if a diffusion network is structurally separable the Morton-Massaro law can be approximated with arbitrary precision regardless of the existence of feedback connections. For simplicity we examine the case in which the weight matrix is symmetric. In such case, each state has an associated goodness function that greatly simplifies the analysis. In a later section we discuss how the results generalize to the non-symmetric case. Let y E IRn represent the internal potential of a diffusion network. Let Zi = cp(aiYi) for i = 1,··· , n represent the firing rates corresponding to y. Let zS, ZC and zr represent the components of z for the units in the stimulus channel, context channel and response specification module. Let x be a vector representing an input and let x S , XC be the components of x for the external stimulus and context. Let a = (a1,··· , an) be a fixed gain vector and ZO/(t) a random vector representing the firing rates at time t of a network with gain vector a. Let za = limt-+oo za (t), represent the firing rates at stochastic equilibrium. In Movellan (1998) it is shown that if the weights are symmetric i.e., W = w' and l/Ki(x) = dcpi(X)/dx then the equilibrium probability density of za is as follows PZQlx(zs,zc,zr I XS,X C ) = K ( 1 ) exp((2/0'2) Ga(zs,zr I XS,X C )) , (8) a Xs,Xc 48 J. R. Movellan and J. L. McClelland SdmU~ /CoDtut Input Figure 1: A network with separable context and stimulus processing channels. The stimulus sensor and stimulus relay units make up the stimulus channel units, and the context sensor and context channel units make up the context channel units. Note that any of the modules can be empty except the response module. where Ka(xs, xc) = / exp((2/(72) Ga(z I Xs, xc)) dz, (9) n Ga(z I x) = H(z I x) - L Sa; (Zi), (10) i=l H(z I x) = z' w z/2 + z' V x, (11) Sa; (Zi) = ai (IOg(Zi) + log(1 - Zi)) + ~i (Zi log(zi) + (1 - Zi) log(1 - Zi)) . (12) Without loss of generality hereafter we set (72 = 2. When there are no direct connections between the stimulus and context units there are no terms in the goodness function in which XS or ZS occur jointly with XC or ZC. Because of this, the goodness can be separated into three additive terms, that depend on x S , XC and a third term which depends on the response units: Ga(z\zc,zr I XS,XC) = G~(zs,zr I XS ) + G~(zr,zc I XC) + G~(zr) , (13) where G~(ZS, zr I XS) = (zs),ws,szs/2 + (zS)'ws,rzr + (ZS)'vs,sxs + (zr),vr,sxs - L S(zt) , i (14) G~(ZC, zr I XS ) = (ZC)'wc,czc /2 + (zc),wc,rzr + (zc),vc,cxc + (zr),vr,cxc - L S(zf) , i (15) (16) Information Factorization 49 where ws,r is a submatrix of w connecting the stimulus and response units. Similar notation is used for the other submatrices of wand v. It follows that we can write the ratio of the jOint probability density' of two states z and z as follows: PZ .. lx(zs,zc,zr I XS,X C ) exp(G~(zS,zr I xs) + G~(zc,zr I XC) + G~(zr» (17) pZ .. lx(zS,zC,zrlxs,xc) exp(G~(zS,zrlxs)+G~(zC,zrlxC)+G~(zr»' which factorizes as desired. To get probability densities for the response units, we integrate over the states of all the other units PZ;;IX(zr I XS,XC) = / / PZ .. lx(zs,zc,zr I XS,XC) dzs dzc , and after rearranging terms pZ;;IX(zr I XS,XC) = Kcr(:s,xC) (/ exp( Gz(zs,zr I XS) + Gr(zr» dZS) (/ exp( G c(ZC, zr I xc» dZC) , (18) (19) which also factorizes. All is left is mapping continuous states of the response units to discrete external responses. To do so we partition the space of the response specification units into discrete regions. The probability of a response becomes the integral of the probability density over the region corresponding to that response. The problem is that the integral of probability densities does not necessarily factorize even though the densities factorize at every point. Fortunately there are two important cases for which the law holds, at least as a good approximation. The first case is when the response regions are small and thus we can approximate the integral over that region by the density at a point times the volume of the region. In such a case the ratio of the integrals can be approximated by the ratio of the probability densities of those individual states. The second case applies to models, like McClelland and Rumelhart's (1981) interactive activation model, in which each response is associated with a distinct response unit. These models typically have negative connections amongst the response units so that at equilibrium one unit tends to be active while the others are inactive. In such a case a common response policy picks the response corresponding to the active unit. We now show that such a policy can approximate the Morton-Massaro law to an arbitrary level of precision as the gain parameter of the response units is increased. Let z represent the joint state of a network and let the first r components of z be the states of the response specification units. Let z(1) = (1,0,0, ... ,0)', Z(2) = (0,1,0,··· ,0)' be two r-dimensional vectors representing states of the response specification units. For i E {1,2} and ~ E (0,1) let z~) = (1 Z(i»~ + (z(i»(l ~), (20) R~) = {x E IRr : Xj E ((1- ~)Z~i), ~ + (1 ~)Z~i», for j = 1,··· , r}. (21) The sets R~) and R~) are regions of the [O,l]r space mapping into two distinct external responses. We now investigate the convergence of the probability ratio of these two responses as we let ~ 4 0, i.e., as the response regions collapse into corners of [0, l]r. p(zr E R(2) I X = x) JR(2) PzrlX(u I x)du lim cr A = lim t." = (22) A-+O P(Z~ E R~) I X = x) A-+O JR~) PZ;;lx(u I x)du . ~rpZ;;IX(z~) I x) . J J eG~(z~),z·,ze I z}dz8 dzc hm (1) = hm (1) • (23) A-+O ~rpZ;;IX(zA I x) A-+O J J eG .. (zt. ,Z',ze I z)dzs dzc 50 J. R. Movellan and J. L. McClelland Table 1: Predictions by the Morton-Massaro law (left side) versus diffusion network (square brackets) for subject 7 of Massaro and Cohen (1983) Experiment 2. Each prediction of the diffusion network is based on 100 random samples. Context Stimulus V S P T 0 0.0017 0.01 0.0000 0.00 0.0152 0.03 0.9000 0.91 1 0.0126 0.00 0.0000 0.00 0.1008 0.10 0.9849 0.97 2 0.1105 0.19 0.0008 0.00 0.5208 0.45 0.9984 1.00 3 0.5463 0.54 0.0079 0.00 0.9133 0.91 0.9998 1.00 4 0.9827 1.00 0.2756 0.30 0.9980 1.00 0.9999 1.00 5 0.9999 1.00 0.9924 0.99 0.9999 1.00 1.0000 1.00 6 0.9999 1.00 0.9924 1.00 0.9999 1.00 1.0000 1.00 Now note that r Go(z~), Z8, ZC I x) = H(z~), Z8, ZC I x) - L So; (Z~!i) - L So; (zt) - L SOj (zj), i=1 i j (24) and since E~=l So; (Z~!i) = E;=1 So; (Z~!i)' it follows that . P(Z~ E R~) I X = x) _ J J eH(z~),z·,ze I x)-E;Sa;(zi)-Ej Saj(zj)dz 8 dzc hm (1) (1) • ~-+o P(Z~ E R~ I X = x) J J eH(za.. ,Z',ze I x)-E; Sai(zt)-E j Saj(Zj)dz 8 dzc (25) It is easy to show that this ratio factorizes. Moreover, for all .6. > 0 if we let 0:1 = ... = O:r = 0:, where 0: > 0 then lim P(Z~ E [.6.,1 - .6.t) = 0, 0-+00 (26) since as the gain of the response units increases So; decreases very fast at the corners of (0, 1 y. Thus as 0: -4 00 the random variable Z~ converges in distribution to a discrete random variable with mass at the corner of the [0, It hypercube and with factorized probability ratios as expressed on (25). Since the indexing ofthe response units is arbitrary the argument applies to all the responses. o 4 Discussion Our analysis establishes that in diffusion networks the Morton-Massaro law is not incompatible with the presence of feedback and lateral connections. Surprisingly, even though in diffusion networks with feedback connections stimulus and context units are interdependent, it is still possible to factorize the effect of stimulus and context on response probabilities. The analysis shows that the Morton-Massaro can be arbitrarily approximated as the sharpness of the response units is increased. In practice we have found very good approximations with relatively small values of the sharpness parameter (see Table 1 for an example). The analysis assumed that the weights were symmetric. Mathematical analysis of the general case with non-symmetric weights is difficult. Information Factorization 51 However useful approximations exist (Movellan & McClelland, 1995) showing that if the noise parameter (7 is relatively small or if the activation function c.p is approximately linear, symmetric weights are not needed to exhibit the Morton-Massaro law. The analysis presented here has potential applications to investigate models of perception and the functional architecture of the brain. For example the interactive activation model of word perception has a separable architecture and thus, diffusion versions of it adhere to the Morton Massaro law. The analysis also points to potential applications in computational neuroscience. It would be of interest to study whether the Morton-Massaro holds at the level of neural responses. For example, we may excite a neuron with two different sources of information and observe its short term average response to combination of stimuli. If the observed distribution of responses exhibits the Morton-Massaro law, this would be consistent with the existence of separable channels converging into that neuron. Otherwise, it would indicate that the channels from the two input areas to the response may not be structurally separable. References Luce, R. D. (1959). Individual choice behavior. New York: Wiley. Massaro, D. W. (1989). Testing between the TRACE Model and the fuzzy logical model of speech perception. Cognitive Psychology, 21, 398-42l. Massaro, D. W. (1998). Perceiving Talking Faces. Cambridge, Massachusetts: MIT Press. Massaro, D. W. & Cohen, M. M. (1983a). Phonological constraints in speech perception. Perception and Psychophysics, 94, 338-348. McClelland, J. L. (1991). Stochastic interactive activation and the effect of context on perception. Cognitive Psychology, 29, 1-44. Morton, J. (1969). The interaction of information in word recognition. Psychological Review, 76, 165-178. Movellan, J. R. (1998). A Learning Theorem for Networks at Detailed Stochastic Equilibrium. Neural Computation, 10(5), 1157-1178. Movellan, J. R. & McClelland, J. L. (1995) . Stochastic interactive processing, channel separability and optimal perceptual inference: an examination of Morton's law. Technical Report PDP.CNS.95A, Available at http://cnbc.cmu.edu, Carnegie Mellon University.
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Hierarchical Image Probability (HIP) Models Clay D. Spence and Lucas Parra Sarnoff Corporation CN5300 Princeton, NJ 08543-5300 { cspence, lparra} @samoff.com Abstract We formulate a model for probability distributions on image spaces. We show that any distribution of images can be factored exactly into conditional distributions of feature vectors at one resolution (pyramid level) conditioned on the image information at lower resolutions. We would like to factor this over positions in the pyramid levels to make it tractable, but such factoring may miss long-range dependencies. To fix this, we introduce hidden class labels at each pixel in the pyramid. The result is a hierarchical mixture of conditional probabilities, similar to a hidden Markov model on a tree. The model parameters can be found with maximum likelihood estimation using the EM algorithm. We have obtained encouraging preliminary results on the problems of detecting various objects in SAR images and target recognition in optical aerial images. 1 Introduction Many approaches to object recognition in images estimate Pr(class I image). By contrast, a model of the probability distribution of images, Pr(image), has many attractive features. We could use this for object recognition in the usual way by training a distribution for each object class and using Bayes' rule to get Pr(class I image) = Pr(image I class) Pr(class)J Pr{image). Clearly there are many other benefits of having a model of the distribution of images, since any kind of data analysis task can be approached using knowledge of the distribution of the data. For classification we could attempt'to detect unusual examples and reject them, rather than trusting the classifier's output. We could also compress, interpolate, suppress noise, extend resolution, fuse multiple images, etc. Many image analysis algorithms use probability concepts, but few treat the distribution of images. Zhu, Wu and Mumford [9] do this by computing the maximum entropy distribution given a set of statistics for some features, This seems to work well for textures but it is not clear how well it will model the appearance of more structured objects. There are several algorithms for modeling the distributions of features extracted from the image, instead of the image itself. The Markov Random Field (MRF) models are an example of this line of development; see, e.g., [5, 4]. Unfortunately they tend to be very expensive computationally. In De Bonet and Viola's flexible histogram approach [2, 1], features are extracted at multiple image scales, and the resulting feature vectors are treated as a set of independent Hierarchical Image Probability (HIP) Models Feature Pyramid Figure 1: Pyramids and feature notation. 849 samples drawn from a distribution. They then model this distribution of feature vectors with Parzen windows. This has given good results, but the feature vectors from neighboring pixels are treated as independent when in fact they share exactly the same components from lower-resolutions. To fix this we might want to build a model in which the features at one pixel of one pyramid level condition the features at each of several child pixels at the next higher-resolution pyramid level. The multi scale stochastic process (MSP) methods do exactly that. Luettgen and Willsky [7], for example, applied a scale-space auto-regression (AR) model to texture discrimination. They use a quadtree or quadtree-like organization of the pixels in an image pyramid, and model the features in the pyramid as a stochastic process from coarse-to-fine levels along the tree. The variables in the process are hidden, and the observations are sums of these hidden variables plus noise. The Gaussian distributions are a limitation of MSP models. The result is also a model of the probability of the observations on the tree, not of the image. All of these methods seem well-suited for modeling texture, but it is unclear how we might build the models to capture the appearance of more structured objects. We will argue below that the presence of objects in images can make local conditioning like that of the flexible histogram and MSP approaches inappropriate. In the following we present a model for probability distributions of images, in which we try to move beyond texture modeling. This hierarchical image probability (HIP) model is similar to a hidden Markov model on a tree, and can be learned with the EM algorithm. In preliminary tests of the model on classification tasks the performance was comparable to that of other algorithms. 2 Coarse-to-fine factoring of image distributions Our goal will be to write the image distribution in a form similar to Pr(I) "-' Pr(Fo I Fd Pr(Fl I F 2) ... , where FI is the set of feature images at pyramid levell. We expect that the short-range dependencies can be captured by the model's distribution of individual feature vectors, while the long-range dependencies can be captured somehow at low resolution. The large-scale structures affect finer scales by the conditioning. In fact we can prove that a coarse-to-fine factoring like this is correct. From an image I we build a Gaussian pyramid (repeatedly blur-and-subsample, with a Gaussian filter). Call the l-th level II, e.g., the original image is 10 (Figure 1). From each Gaussian level II we extract some set of feature images Fl. Sub-sample these to get feature images GI. Note that the images in GI have the same dimensions as 11+1 . We denote by G I the set of images containing 11+1 and the images in G I. We further denote the mapping from II to GI by gl. Suppose now that go : 10 t-+ Go is invertible. Then we can think of go as a change of vari850 C. D. Spence and L. Parra abIes. If we have a distribution on a space, its expressions in two different coordinate systems are related by multiplying by the Jacobian. In this case we get Pr( 10 ) = Igo I Pr( Go). Since Go = (Go , II), we can factor Pr(Go) to get Pr(Io) = Igol Pr(Go I h) Pr(h). If gl is invertible for alll E {O, . .. , L - 1} then we can simply repeat this change of variable and factoring procedure to get Pr(I) = [rflgdPr(GIIII+d] Pr(h) 1=0 (1) This is a very general result, valid for all Pr(I), no doubt with some rather mild restrictions to make the change of variables valid. The restriction that gl be invertible is strong, but many such feature sets are known to exist, e.g., most wavelet transforms on images. We know of a few ways that this condition can be relaxed, but further work is needed here. 3 The need for hidden variables For the sake of tractability we want to factor Pr(GI 111+1 ) over positions, something like Pr(I) ,.... ITI ITxEI,+l Pr(gl(x) I fl+! (x)) where gl(x) and fl+! (x) are the feature vectors at position x. The dependence of gl on fi+l expresses the persistence of image structures across scale, e.g., an edge is usually detectable as such in several neighboring pyramid levels. The flexible histogram and MSP methods share this structure. While it may be plausible that fl+1 (x) has a strong influence on gl (x), we argue now that this factorization and conditioning is not enough to capture some properties of real images. Objects in the world cause correlations and non-local dependencies in images. For example, the presence of a particular object might cause a certain kind of texture to be visible at levell. Usually local features fi+l by themselves will not contain enough information to infer the object's presence, but the entire image II+! at that layer might. Thus gl(x) is influenced by more of 11+1 than the local feature vector. Similarly, objects create long-range dependencies. For example, an object class might result in a kind of texture across a large area of the image. If an object of this class is always present, the distribution may factor, but if such objects aren't always present and can't be inferred from lower-resolution information, the presence of the texture at one location affects the probability of its presence elsewhere. We introduce hidden variables to represent the non-local information that is not captured by local features. They should also constrain the variability of features at the next finer scale. Denoting them collectively by A, we assume that conditioning on A allows the distributions over feature vectors to factor. In general, the distribution over images becomes Pr(I) ()( L {IT II Pr(gl(x) I fi+l (x) , A) Pr(A I h)} Pr(h) . (2) A 1=0 xEI,+l As written this is absolutely general, so we need to be more specific. In particular we would like to preserve the conditioning of higher-resolution information on coarser-resolution information, and the ability to factor over positions. Hierarchical Image Probability (HIP) Models 851 Figure 2: Tree structure of the conditional dependency between hidden variables in the HIP model. With subsampling by two, this is sometimes called a quadtree structure. As a first model we have chosen the following structure for our HIP model: 1 Pr(I)<x L IT II [pr(gl(X) \f1+1(x),al(X)) Pr(al(x) \a1+1(X))]. (3) Ao, .. . ,AL_1l=O xE11+l To each position x at each level l we attach a hidden discrete index or label al (x). The resulting label image Al for levell has the same dimensions as the images in G1• Since al(x) codes non-local information we can think of the labels Al as a segmentation or classification at the l-th pyramid level. By conditioning al(x) on al+l (x), we mean that al(x) is conditioned on al+1 at the parent pixel of x. This parent-child relationship follows from the sub-sampling operation. For example, if we sub-sample by two in each direction to get G1 from Fl, we condition the variable al at (x, y) in level l on al+l at location (Lx /2 J , Ly /2 J) in levell + 1 (Figure 2). This gives the dependency graph of the hidden variables a tree structure. Such a probabilistic tree of discrete variables is sometimes referred to as a belief network. By conditioning child labels on their parents information propagates though the layers to other areas of the image while accumulating information along the way. For the sake of simplicity we've chosen Pr(gl I fl+1 ,al) to be normal with mean (kal + Mal fl+ 1 and covariance ~al. We also constrain Mal and ~al to be diagonal. 4 EM algorithm Thanks to the tree structure, the belief network for the hidden variables is relatively easy to train with an EM algorithm. The expectation step (summing over ai's) can be performed directly. If we had chosen a more densely-connected structure with each child having several parents, we would need either an approximate algorithm or Monte Carlo techniques. The expectation is weighted by the probability of a label or a parent-child pair of labels given the image. This can be computed in a fine-to-coarse-to-fine procedure, i.e. working from leaves to the root and then back out to the leaves. The method is based on belief propagation [6]. With some care an efficient algorithm can be worked out, but we omit the details due to space constraints. Once we can compute the expectations, the normal distribution makes the M-step tractable; we simply compute the updated gal' ~al' Mal' and Pr(al I al+d as combinations of various expectation values. I The proportionality factor includes Pr(AL , h) which we model as I1x Pr(gdX) I adx)) Pr(adx)). This is the I = L factor of Equation 3, which should be read as having no quantities fL +1 or aL+l. 852 HP HPNN • 0.4 0.5 0.6 C. D. Spence and L. Parra Plane ROC are8t! .. .. .. . 0.7 Az 0.8 o 000 000 0.9 Figure 3: Examples of aircraft ROls. On the right are Az values from a jack-knife study of detection performance of HIP and HPNN models. Figure 4: SAR images of three types of vehicles to be detected. 5 Experiments We applied HIP to the problem of detecting aircraft in an aerial photograph of Logan airport. A simple template-matching algorithm was used to select forty candidate aircraft, twenty of which were false positives (Figure 3). Ten of the plane examples were used for training one HIP model and ten negative examples were used to train another. Because of thesmall number of examples, we performed a jack-knife study with ten random splits of the data. For features we used filter kernels that were polynomials of up to third order multiplying Gaussians. The HIP pyramid used subsampling by three in each direction. The test set ROC area for HIP had a mean of Az = 0.94, while our HPNN algorithm [8] gave a mean Az of 0.65. The individual values shown in Figure 3. (We compared with the HPNN because it had given Az = 0.86 on a larger set of aircraft images including these with a different set of features and subsampling by two.) We also performed an experiment with the three target classes in the MSTAR public targets data set, to compare with the results of the flexible histogram approach of De Bonet, et al [lJ. We trained three HIP models, one for each of the target vehicles BMP-2, BTR-70 and T-72 (Figure 4). As in [1J we trained each model on ten images of its class, one image for each of ten aspect angles, spaced approximately 36° apart. We trained one model for all ten images of a target, whereas De Bonet et al trained one model per image. We first tried discriminating between vehicles of one class and other objects by thresholding log Pr(I I class), i.e., no model of other objects is used. For the tests, the other objects were taken from the test data for the two other vehicle classes, plus seven other vehicle classes. .. ' " ROC uamg P, ( II t.rget1) 1 Pr( II_rget2) BMP-2 vs T -72: Az = 0.79 BMP-2 ys BTR-70: Az = 0.82 T- 72 ys BTR-70: Az = 0.89 0 1 02 0.3 04 0.5 0.6 07 08 09 f.lII9poallVes b 853 Figure 5: ROC curves for vehicle detection in SAR imagery. (a) ROC curves by thresholding HIP likelihood of desired class. (b) ROC curves for inter-class discrimination using ratios of likelihoods as given by HIP models. There were 1,838 image from these seven other classes, 391 BMP2 test images, 196 BTR70 test images, and 386 Tn test images. The resulting ROC curves are shown in Figure 5a. We then tried discriminating between pairs target classes using HIP model likelihood ratios, i.e., log Pr(I I classl) - log Pr(I I class2). Here we could not use the extra seven vehicle classes. The resulting ROC curves are shown in Figure 5b. The performance is comparable to that of the flexible histogram approach. 6 Conditional distributions of features To further test the HIP model's fit to the image distribution, we computed several distributions of features 9i(X) conditioned on the parent feature Ii+! (x).2 The empirical and computed distributions for a particular parent-child pair of features are shown in Figure 6. The conditional distributions we examined all had similar appearance, and all fit the empirical distributions well. Buccigrossi and Simoncelli [3] have reported such "bow-tie" shape conditional distributions for a variety of features. We want to point out that such conditional distributions are naturally obtained for any mixture of Gaussian distributions with varying scales and zero means. The present HIP model learns such conditionals, in effect describing the features as non-stationary Gaussian variables. 7 Conclusion We have developed a class of image probability models we call hierarchical image probability or HIP models. To justify these, we showed that image distributions can be exactly represented as products over pyramid levels of distributions of sub-sampled feature images conditioned on coarser-scale image information. We argued that hidden variables are needed to capture long-range dependencies while allowing us to further factor the distributions over position. In our current model the hidden variables act as indices of mixture 2This is somewhat involved; Pr(gl I /l+d is not just Pr(gl I /1+1 , al) Pr(al) summed over aI, but Lal Pr(gl, all /l+d = Lal Pr(gl 1/1+1, al) Pr(all /l+t}. 854 C. D. Spence and L. Parra Condllional distirbulion of dala Condilional dislirbulion of HIP model 0 0 W ~ >..!!! ..!!! '" '" !! ~ ~ j c;, c;, f. fealure 9 layer 1 I; fealure 9 layer 1 Figure 6: Empirical and HIP estimates of the distribution of a feature 91 (X) conditioned on its parent feature 11+1 (x). components. The resulting model is somewhat like a hidden Markov model on a tree. Our early results on classification problems showed good performance. Acknowledgements We thank Jeremy De Bonet and John Fisher for kindly answering questions about their work and experiments. Supported by the United States Government. References [1] J. S. De Bonet, P. Viola, and 1. W. Fisher III. Flexible histograms: A multiresolution target discrimination model. In E. G. Zelnio, editor, Proceedings of SPIE, volume 3370,1998. [2] Jeremy S. De Bonet and Paul Viola. Texture recognition using a non-parametric multiscale statistical model. In Conference on Computer Vision and Pattern Recognition. IEEE,1998. [3] Robert W. Buccigrossi and Eero P. Simoncelli. Image compression via joint statistical characterization in the wavelet domain. Technical Report 414, U. Penn. GRASP Laboratory, 1998. Available at ftp:l/ftp.cis.upenn.eduJpub/eero!buccigrossi97.ps.gz. [4] Rama Chellappa and S. Chatterjee. Classification of textures using Gaussian Markov random fields. IEEE Trans. ASSP, 33:959-963, 1985. [5] Stuart Geman and Donald Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. PAMI, PAMI-6(6):194-207, November 1984. [6] Michael 1. Jordan, editor. Learning in Graphical Models, volume 89 of NATO Science Series D: Behavioral and Brain Sciences. Kluwer Academic, 1998. [7] Mark R. Luettgen and Alan S. Will sky. Likelihood calculation for a class of multiscale stochastic models, with application to texture discrimination. IEEE Trans. Image Proc., 4(2):194-207, 1995. [8] Clay D. Spence and Paul Sajda. Applications of multi-resolution neural networks to mammography. In Michael S. Kearns, Sara A. SolI a, and David A. Cohn, editors, NIPS 11, pages 981-988, Cambridge, MA, 1998. MIT Press. [9] Song Chun Zhu, Ying Nian Wu, and David Mumford. Minimax entropy principle and its application to texture modeling. Neural Computation, 9(8): 1627-1660, 1997.
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Reinforcement Learning for Spoken Dialogue Systems Satinder Singh AT&T Labs Michael Keams AT&T Labs Diane Litman AT&T Labs Marilyn Walker AT&T Labs {baveja,mkeams,diane, walker} @research.att.com Abstract Recently, a number of authors have proposed treating dialogue systems as Markov decision processes (MDPs). However, the practical application ofMDP algorithms to dialogue systems faces a number of severe technical challenges. We have built a general software tool (RLDS, for Reinforcement Learning for Dialogue Systems) based on the MDP framework, and have applied it to dialogue corpora gathered from two dialogue systems built at AT&T Labs. Our experiments demonstrate that RLDS holds promise as a tool for "browsing" and understanding correlations in complex, temporally dependent dialogue corpora. 1 Introduction Systems in which human users speak to a computer in order to achieve a goal are called spoken dialogue systems. Such systems are some of the few realized examples of openended, real-time, goal-oriented interaction between humans and computers, and are therefore an important and exciting testbed for AI and machine learning research. Spoken dialogue systems typically integrate many components, such as a speech recognizer, a database backend (since often the goal of the user is to retrieve information), and a dialogue strategy. In this paper we are interested in the challenging problem of automatically inferring a good dialogue strategy from dialogue corpora. Research in dialogue strategy has been perhaps necessarily ad-hoc due to the open-ended nature of dialogue system design. For example, a common and critical design choice is between a system that always prompts the user to select an utterance from fixed menus (system initiative), and one that attempts to determine user intentions from unrestricted utterances (mixed initiative). Typically a system is built that explores a few alternative strategies, this system is tested, and conclusions are drawn regarding which of the tested strategies is best for that domain [4, 7, 2]. This is a time-consuming process, and it is difficult to rigorously compare and evaluate alternative systems in this fashion, much less design improved ones. Recently, a number of authors have proposed treating dialogue design in the formalism of Markov decision processes (MDPs)[ 1, 3, 7]. In this view, the popUlation of users defines the stochastic environment, a dialogue system's actions are its (speech-synthesized) utterances and database queries, and the state is represented by the entire dialogue so far. The goal is to design a dialogue system that takes actions so as to maximize some measure of reward. Viewed in this manner, it becomes possible, at least in principle, to apply the framework and algorithms of reinforcement learning (RL) to find a good dialogue strategy. However, the practical application of RL algorithms to dialogue systems faces a number of severe technical challenges. First, representing the dialogue state by the entire dialogue so Reinforcement Learningfor Spoken Dialogue Systems 957 far is often neither feasible nor conceptually useful, and the so-called belief state approach is not possible, since we do not even know what features are required to represent the belief state. Second, there are many different choices for the reward function, even among systems providing very similar services to users. Previous work [7] has largely dealt with these issues by imposing a priori limitations on the features used to represent approximate state, and then exploring just one of the potential reward measures. In this paper, we further develop the MDP formalism for dialogue systems, in a way that does not solve the difficulties above (indeed, there is no simple "solution" to them), but allows us to attenuate and quantify them by permitting the investigation of different notions of approximate state and reward. Using our expanded formalism, we give one of the first applications of RL algorithms to real data collected from multiple dialogue systems. We have built a general software tool (RLDS, for Reinforcement Learning for Dialogue Systems) based on our framework, and applied it to dialogue corpora gathered from two dialogue systems built at AT&T Labs, the TOOT system for voice retrieval of train schedule information [4] and the ELVIS system for voice retrieval of electronic mail [7]. Our experiments demonstrate that RLDS holds promise not just as a tool for the end-toend automated synthesis of complicated dialogue systems from passive corpora a "holy grail" that we fall far short of here1 but more immediately, as a tool for "browsing" and understanding correlations in complex, temporally dependent dialogue corpora. Such correlations may lead to incremental but important improvements in existing systems. 2 The TOOT and ELVIS Spoken Dialogue Systems The TOOT and ELVIS systems were implemented using a general-purpose platform developed at AT&T, combining a speaker-independent hidden Markov model speech recognizer, a text-to-speech synthesizer, a telephone interface, and modules for specifying data-access functions and dialogue strategies. In TOOT, the data source is the Amtrak train schedule web site, while in ELVIS, it is the electronic mail spool of the user. In a series of controlled experiments with human users, dialogue data was collected from both systems, resulting in 146 dialogues from TOOT and 227 dialogues from ELVIS. The TOOT experiments varied strategies for information presentation, confirmation (whether and how to confirm user utterances) and initiative (system vs. mixed), while the ELVIS experiments varied strategies for information presentation, for summarizing email folders, and initiative. Each resulting dialogue consists of a series of system and user utterances augmented by observations derived from the user utterances and the internal state of the system. The system's utterances (actions) give requested information, ask for clarification, provide greetings or instructions, and so on. The observations derived from the user's utterance include the speech-recognizer output, the corresponding log-likelihood score, the semantic labels assigned to the recognized utterances (such as the desired train departure and arrival cities in TOOT, or whether the user prefers to hear their email ordered by date or sender in ELVIS); indications of user barge-ins on system prompts; and many more. The observations derived from the internal state include the grammar used by the speech recognizer during the tum, and the results obtained from a query to the data source. In addition, each dialogue has an associated survey completed by the user that asks a variety of questions relating to the user's experience. See [4, 7] for details. 3 Spoken Dialogue Systems and MDPs Given the preceding discussion, it is natural to formally view a dialogue as a sequence d d = (a 1, 01, rt), (a2' 02, r2), ... , (at, Ot, rt). -------------1 However, in recent work we have applied the methodology described here to significantly improve the perfonnance of a new dialogue system [5]. 958 S. Singh, M Kearns, D. Litman and M Walker Here ai is the action taken by the system (typically a speech-synthesized utterance, and less frequently, a database query) to start the ith exchange (or tum, as we shall call it), OJ consists of all the observations logged by the system on this turn, as discussed in the last section, and rj is the reward received on this turn. As an example, in roOT a typical turn might indicate that the action aj was a system utterance requesting the departure city, and the 0; might indicate several observations: that the recognized utterance was "New York", that the log-likelihood of this recognition was -2.7, that there was another unrecognized utterance as well, and so on. We will use d[ i] to denote the prefix of d that ends following the ith turn, and d· (a, 0, r) to denote the one-turn extension of dialogue d by the turn (a, 0, r). The scope of the actions aj and observations 0; is determined by the implementation of the systems (e.g. if some quantity was not logged by the system, we will not have access to it in the 0; in the data). Our experimental results will use rewards derived from the user satisfaction surveys gathered for the roOT and ELVIS data We may view any dialogue d as a trajectory in a well-defined true MOP M. The states 2 of M are all possible dialogues, and the actions are all the possible actions available to the spoken dialogue system (utterances and database queries). Now from any state (dialogue) d and action a, the only possible next states (dialogues) are the one-turn extensions d· (a, 0, r). The probability of transition from d to d·(a, 0, r) is exactly the probability, over the stochastic ensemble of users, that 0 and r would be generated following action a in dialogue d. It is in general impractical to work directly on M due to the unlimited size of the state (dialogue) space. Furthermore, M is not known in advance and would have to be estimated from dialogue corpora. We would thus like to permit a flexible notion of approximate states. We define state estimator SE to be a mapping from any dialogue d into some space S. For example, a simple state estimator for roOT might represent the dialogue state with boolean variables indicating whether certain pieces of information had yet been obtained from the user (departure and arrival cities, and so on), and a continuous variable tracking the average log-likelihood of the recognized utterances so far. Then sE(d) would be a vector representing these quantities for the dialogue d. Once we have chosen a state estimator SE, we can transform the dialogue d into an S-trajectory, starting from the initial empty state So E S: So -tal SE(d[l]) -ta2 sE(d[2]) -ta3 . .. -tat SE(d[t]) where the notation -tao SE(d[i]) indicates a transition to SE(d[i]) E S following action aj. Given a set of dialogues d1, .. . , dn , we can construct the empirical MOP MSE • The state space of MSE is S, the actions are the same as in M, and the probability oftransition from s to s' under action a is exactly the empirical probability of such a transition in the S-trajectories obtained from d1 , .•• ,dn . Note that we can build MSE from dialogue corpora, solve for its optimal policy, and analyze the resulting value function. The point is that by choosing SE carefully, we hope that the empirical MOP MSE will be a good approximation of M. By this we mean that MSE renders dialogues (approximately) Markovian: the probability in M of transition from any dialogue d to anyone-turn extension d · (a, 0, r) is (approximately) the probability of transition from sE(d) to sE(d · (a, 0, r)) in MSE • We hope to find state estimators SE which render dialogues approximately Markovian, but for which the amount of data and computation required to find good policies in MSE will be greatly reduced compared to working directly in dialogue space. While conceptually appealing, this approach is subject to at least three important caveats: First, the approach is theoretically justified only to the extent that the chosen state estimator renders dialogues Markovian. In practice, we hope that the approach is robust, in that "small" violations of the Markov property will still produce useful results. Second, while 2These are not to be confused with the internal states of the spoken dialogue system(s) during the dialogue, which in our view merely contribute observations. Reinforcement Learningfor Spoken Dialogue Systems 959 state estimators violating the Markov property may lead to meaningful insights, they cannot be directly compared. For instance, if the optimal value function derived from one state estimator is larger than the optimal value function for another state estimator, we cannot necessarily conclude that the first is better than the second. (This can be demonstrated formally.) Third, even with a Markovian state estimator SE, data that is sparse with respect to SE limits the conclusions we can draw; in a large space S, certain states may be so infrequently visited in the dialogue corpora that we can say nothing about the optimal policy or value function there. 4 The RLDS System We have implemented a software tool (written in C) called RLOS that realizes the above formalism. RLOS users specify an input file of sample dialogues; the dialogues include the rewards received at each turn. Users also specify input files defining S and a state estimator SEe The system has command-line options that specify the discount factor to be used, and a lower bound on the number of times a state s E S must be visited in order for it to be included in the empirical MOP USE (to control overfitting to sparse data). Given these inputs and options, RLOS converts the dialogues into S -trajectories, as discussed above. It then uses these trajectories to compute the empirical MOP USE specified by the data that is, the data is used to compute next-state distributions and average reward in the obvious way. States with too few visits are pruned from USE' RLOS then uses the standard value iteration algorithm to compute the optimal policy and value function [6] for USE, all using the chosen discount factor. 5 Experimental Results The goal of the experiments reported below is twofold: first, to confirm that our RLOS methodology and software produce intuitively sensible policies; and second, to use the value functions computed by the RLOS software to discover and understand correlations between dialogue properties and performance. We have space to present only a few of our many experiments on TOOT and ELVIS data. Each experiment reported below involves choosing a state estimator, running RLOS using either the TOOT or ELVIS data, and then analyzing the resulting policy and value function. For the TOOT experiments, the reward function was obtained from a question in the user satisfaction survey: the last turn in a dialogue receives a reward of +1 if the user indicated that they would use the system again, a reward of 0 if the user answered "maybe", and a reward of -1 if the user indicated that they would not use the system again. All turns other than the last receive reward 0 (Le., a reward is received only at the end of a dialogue). For the ELVIS experiments, we used a summed (over several questions) user-satisfaction score to reward the last turn in each dialogue (this score ranges between 8 and 40). Experiment 1 (A Sensible Policy): In this initial "sanity check" experiment, we created a state estimator for TOOT whose boolean state variables track whether the system knows the value for the following five informational attributes: arrival city (denoted AC), departure city (DC), departure date (~O), departure hour (OH), and whether the hour is AM or PM (AP) 3 . Thus, if the dialogue so far includes a turn in which TOOT prompts the user for their departure city, and the speech recognizer matches the user utterance with "New York", the boolean state variable GotOC? would be assigned a value of 1. Note that this ignores the actual values of the attributes. In addition, there is another boolean variable called ConfirmedAll? that is set to 1 if and only ifthe system took action ConfirmAll (which prompts the user to explicitly verify the attribute values perceived by TOOT) and perceived a "yes" utterance in response. Thus. the state vector js sjmply the binary vector 3Remember that TOOT can only track its perceptions of these attributes, since errors may have occurred in speech recognition. 960 S. Singh, M Kearns, D. Litman and M Walker [ GotAC? , GotAP? , GotDC? , GotDD? , GotDH? , ConfirmedAll? ) Among the actions (the system utterances) available to TOOT are prompts to the user to specify values for these informational attributes; we shall denote these actions with labels AskOC, AskAC, AskOO, AskOH, and AskAP. The system takes several other actions that we shall mention as they arise in our results. The result of running RLOS was the following policy, where we have indicated the action to be taken from each state: [0,0,0,0,0,0): SayGreeting [1,0,0,0,0,0) : AskDC [1,0,1,0,0,0): AskAp [1,0,1,1,0,0): AskDH [0,0,0,1,1,0): AskAP [1,0,0,1,1,0): AskAP [0, 1, 0, 1, 1, 0): AskAll [1, 1 , 0, 1 , 1, 0): AskAll [ 1, 0, 1, 1, 1, 0): AskAP [1,1,1,1,1,0): ConfirmAll [1,1,1,1,1,1): Close Thus, RLOS finds a sensible policy, always asking the user for information which it has not already received, confirming the user's choices when it has all the necessary information, and then presenting the closest matching train schedule and closing the dialogue (action Close). Note that in some cases it chooses to ask the user for values for all the informational attributes even though it has values for some of them. It is important to emphasize that this policy was derived purely through the application of RLOS to the dialogue data, without any knowledge of the "goal" of the system. Furthermore, the TOOT data is such that the empirical MOP built by RLOS for this state estimator does include actions considerably less reasonable than those chosen above from many states. Examples include confirming the values of specific informational attributes such as DC (since we do not represent whether such confirmations were successful, this action would lead to infinite loops of confirmation), and requesting values for informational attributes for which we already have values (such actions appear in the empirical MOP due to speech recognition errors). The mere fact that RLOS was driven to a sensible policy that avoided these available pitfalls indicates a correlation between the chosen reward measure (whether the user would use the system again) and the intuitive system goal of obtaining a completely specified train trip. It is interesting to note that RLOS finds it better to confirm values for all 5 attributes when it has them, as opposed to simply closing the dialogue without confirmation. In a similar experiment on ELVIS, RLOS again found a sensible policy that summarizes the user's inbox at the beginning of the dialogue, goes on to read the relevant e-mail messages until done, and then closes. (a) 0..2<1,---..-----,..-----,----,----, 0.24 I = Number of Information Attributes 0..22 0..2 0.18 !l 01 0..'6 > 0.14 0.12 ' 0.1 II' 0.08 " 1=2 C ·08CL-----''------'2-----'3--~------' Number of Attributes Confirmed (b) 0.35,---.-----,..-----,..-----,----, D = Number of Distress Features 0..3 0.25 !l 0.2 01 r---~ > 0..'5 0..,,---__ 0.05 D=I ---~-_/ D= ~t=~~c===~2~====3==~==.--~ Number of Information Attributes Figure I: a) Role of Confirmation. b) Role of Distress Features (indicators that the dialogue is in trouble). See description of Experiments 2 and 3 respectively in the text for details. Experiment 2 (Role of Confirmation): Here we explore the effect of confirming with the user the values that TOOT perceives for the informational attributes that is, whether the Reinforcement Learningfor Spoken Dialogue Systems 961 trade-off between the increased confidence in the utterance and the potential annoyance to the user balances out in favor of confirmation or not (for the particular reward function we are using). To do so, we created a simple state estimator with just two state variables. The first variable counts the number of the informational attributes (DC, AC, etc.) that roar believes it has obtained, while the second variable counts the number of these that have been confirmed with the user. Figure 1 (a) presents the optimal value as a function of the number of attributes confirmed. Each curve in the plot corresponds to a different setting of the first state variable. For instance, the curve labeled with "1=3" corresponds to the states where the system has obtained 3 informational attributes. We can make two interesting observations from this figure. First, the value function grows roughly linearly with the number of confirmed attributes. Second, and perhaps more startlingly, the value function has only a weak dependence on the first feature the value for states when some number of attributes have been confirmed seems independent of how many attributes (the system believes) have been obtained. This is evident from the lack of separation between the plots for varying values of the state variable I. In other words, our simple (and preliminary) analysis suggests that for our reward measure, confirmed information influences the value function much more strongly than unconfirmed information. We also repeated this experiment replacing attribute confirmation with thresholded speech recognition log-likelihood scores, and obtained qualitatively similar results. Experiment 3 (Role of Distress Features): Dialogues often contain timeouts (user silence when system expected response), resets (user asks for current context of dialogue to be abandoned and the system is reinitialized), user requests for help, and other indicators that the dialogue is potentially in trouble. Do such events correlate with low value? We created a state estimator for roar that, in addition to our variable I counting informational attributes, counted the number of such distress events in the dialogue. Figure l(b) presents the optimal value as a function of the number of attributes obtained. Each curve corresponds to a different number of distress features. This figure confirms that the value of the dialogue is lower for states with a higher number of distress features. (a) 0.7 T = Number of Turns 0.8 0.5 T<4 0°·4 " '" > 0.3 4<=T<8 0.2 0.1 8 <=T< 12 12 <=T< 16 0 0 2 3 Number of 1nfonnation Attributes (b) 35 30 2S !! '" > 20 15 10 0 P=TaskProgrcss =3 2 3 4 5 8 Number of Turns divided by 4 Figure 2: a) Role of Dialogue Length in roar. b) Role of Dialogue Length in ELVIS. See description of Experiment 4 in the text for details. Experiment 4 (Role of the Dialogue Length): All other things being equal (e.g. extent of task completion), do users prefer shorter dialogues? To examine this question, we created a state estimator for TOOT that counts the number of informational attributes obtained (variable I as in Experiment 2), and a state estimator for ELVIS that measures "task progress" (a measure analogous to the variable I for roar; details omitted). In both cases, a second variable tracks the length of the dialogue. 962 S. Singh, M. Kearns, D. Litman and M. Walker Figure 2(a) presents the results for TOOT. It plots the optimal value as a function of the number I of informational values; each curve corresponds to a different range of dialogue lengths. It is immediately apparent that the longer the dialogue, the lower the value, and that within the same length of dialogue it is better to have obtained more attributes 4. Of course, the effect of obtaining more attributes is weak for the longest dialogue length; these are dialogues in which the user is struggling with the system, usually due to multiple speech recognition errors. Figure 2(b) presents the results for ELVIS from a different perspective. The dialogue length is now the x-axis, while each curve corresponds to a different value of P (task progress). It is immediately apparent that the value increases with task progress. More interestingly, unlike TOOT, there seems to be an "optimal" or appropriate dialogue length for each level of task progress, as seen in the inverse U-shaped curves. Experiment 5 (Role of Initiative): One ofthe important questions in dialogue theory is how to choose between system and mixed initiative strategies (cf. Section 1). Using our approach on both TOOT and ELVIS data, we were able to confirm previous results [4, 7] showing that system initiative has a higher value than mixed initiative. Experiment 6 (Role of Reward Functions): To test the robustness of our framework, we repeated Experiments 1-4 for TOOT using a new reward function based on the user's perceived task completion. We found that except for a weaker correlation between number of turns and value function, the results were basically the same across the two reward functions. 6 Conclusion This paper presents a new RL-based framework for spoken dialogue systems. Using our framework, we developed RLDS, a general-purpose software tool, and used it for empirical studies on two sets of real dialogues gathered from the TOOT and ELVIS systems. Our results showed that RLDS was able to find sensible policies, that in ELVIS there was an "optimal" length of dialogue, that in TOOT confirmation of attributes was highly correlated with value, that system initiative led to greater user satisfaction than mixed initiative, and that the results were robust to changes in the reward function. Acknowledgements: We give warm thanks to Esther Levin, David McAllester, Roberto Pieraccini, and Rich Sutton for their many contributions to this work. References [1] A. W. Biennann and P. M. Long. The composition of messages in speech-graphics interactive systems. In Proceedings of the i996international Symposium on Spoken Dialogue. 97-100, 1996. [2] A. L. Gorin, B. A. Parker, R. M. Sachs and J. G. Wilpon. How May I Help You. In Proceedings of International Symposium on Spoken Dialogue. 57-60, 1996. [3] E. Levin, R. Pieraccini and W. Eckert. Learning dialogue strategies within the Markov decision process framework. In Proc. iEEE Workshop on Automatic Speech Recognition and Understanding 1997. [4] D. J. Litman and S. Pan. Empirically Evaluating an Adaptable Spoken Dialogue System. In Proceedings of the 7th International Conference on User Modeling 1999. [5) S. Singh, M. Kearns, D. Litman, and M. Walker. In preparation. [6] R. S. Sutton and A. G. Barto. ReinforcementLearning: An Introduction MIT Press, 1998. [7) M. A. Walker, J. C. Fromer and S. Narayanan. Learning Optimal Dialogue Strategies: A Case Study of a Spoken Dialogue Agent for Email. In Proceedings of the 36th Annual Meeting of the Association of Computational Linguistics, COLINGIACL 98 1345-1352, 1998. 4There is no contradiction with Experiment 2 in this statement, since here we are not separating confirmed and unconfirmed attributes.
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Distributed Synchrony of Spiking Neurons in a Hebbian Cell Assembly David Horn Nir Levy School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel horn~neuron.tau.ac.il nirlevy~post.tau.ac.il Isaac Meilijson Eytan Ruppin School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel isaco~math.tau.ac.il ruppin~math.tau.ac.il Abstract We investigate the behavior of a Hebbian cell assembly of spiking neurons formed via a temporal synaptic learning curve. This learning function is based on recent experimental findings. It includes potentiation for short time delays between pre- and post-synaptic neuronal spiking, and depression for spiking events occuring in the reverse order. The coupling between the dynamics of the synaptic learning and of the neuronal activation leads to interesting results. We find that the cell assembly can fire asynchronously, but may also function in complete synchrony, or in distributed synchrony. The latter implies spontaneous division of the Hebbian cell assembly into groups of cells that fire in a cyclic manner. We invetigate the behavior of distributed synchrony both by simulations and by analytic calculations of the resulting synaptic distributions. 1 Introduction The Hebbian paradigm that serves as the basis for models of associative memory is often conceived as the statement that a group of excitatory neurons (the Hebbian cell assembly) that are coupled synaptically to one another fire together when a subset of the group is being excited by an external input. Yet the details of the temporal spiking patterns of neurons in such an assembly are still ill understood. Theoretically it seems quite obvious that there are two general types of behavior: synchronous neuronal firing, and asynchrony where no temporal order exists in the assembly and the different neurons fire randomly but with the same overall rate. Further subclassifications were recently suggested by [BruneI, 1999]. Experimentally this question is far from being settled because evidence for the associative 130 D. Hom, N. Levy, 1. Meilijson and E. Ruppin memory paradigm is quite scarce. On one hand, one possible realization of associative memories in the brain was demonstrated by [Miyashita, 1988] in the inferotemporal cortex. This area was recently reinvestigated by [Yakovlev et al., 1998] who compared their experimental results with a model of asynchronized spiking neurons. On the other hand there exists experimental evidence [Abeles, 1982] for temporal activity patterns in the frontal cortex that Abeles called synfire-chains. Could they correspond to an alternative type of synchronous realization of a memory attractor? To answer these questions and study the possible realizations of attractors in cortical-like networks we investigate the temporal structure of an attractor assuming the existence of a synaptic learning curve that is continuously applied to the memory system. This learning curve is motivated by the experimental observations of [Markram et al., 1997, Zhang et al., 1998] that synaptic potentiation or depression occurs within a critical time window in which both pre- and post-synaptic neurons have to fire. If the pre-synaptic neuron fires first within 30ms or so, potentiation will take place. Depression is the rule for the reverse order. The regulatory effects of such a synaptic learning curve on the synapses of a single neuron that is subjected to external inputs were investigated by [Abbott and Song, 1999] and by [Kempter et al., 1999]. We investigate here the effect of such a rule within an assembly of neurons that are all excited by the same external input throughout a training period, and are allowed to influence one another through their resulting sustained activity. 2 The Model We study a network composed of N E excitatory and NJ inhibitory integrate-and-fire neurons. Each neuron in the network is described by its subthreshold membrane potential Vi{t) obeying . 1 Vi{t) = - - Vi{t) + R1i(t) Tn (1) where Tn is the neuronal integration time constant. A spike is generated when Vi{t) reaches the threshold Vrest + fJ, upon which a refractory period of TRP is set on and the membrane potential is reset to Vreset where Vrest < Vreset < Vrest + fJ. Ii{t) is the sum of recurrent and external synaptic current inputs. The net synaptic input charging the membrane of excitatory neuron i at time t is R1i(t) = L J~E{t) L 0 (t t~ - Td) - L Ji~J L 0 (t - tj - Td) + r xt (2) j I j m summing over the different synapses of j = 1, ... , NE excitatory neurons and of j = 1, ... ,NJ inhibitory neurons, with postsynaptic efficacies J~E{t) and Ji~J respectively. The sum over 1 (m) represents a sum on different spikes arriving at synapse j, at times t = t; + Td (t = tj + Td), where t~ (tj) is the emission time of the l-th (m-th) spike from the excitatory (inhibitory) neuron j and Td is the synaptic delay. Iext, the external current, is assumed to be random and independent at each neuron and each time step, drawn from a Poisson distribution with mean A ext. Analogously, the synaptic input to the inhibitory neuron i at time tis j j m We assume full connectivity among the excitatory neurons, but only partial connectivity between all other three types of possible connnections, with connection Distributed Synchrony of Spiking Neurons in a Hebbian Cell Assembly 131 probabilities denoted by eEl, e l E and Cl I. In the following we will report simulation results in which the synaptic delays Td were assigned to each synapse, or pair of neurons, randomly, chosen from some finite set of values. Our analytic calculation will be done for one fixed value of this delay parameter. The synaptic efficacies between excitatory neurons are assumed to be potentiated or depressed according to the firing patterns of the pre- and post-synaptic neurons. In addition we allow for a uniform synaptic decay. Thus each excitatory synapse obeys (4) where the synaptic decay constant Ts is assumed to be very large compared to the membrane time constant Tn. J/JE(t) are constrained to vary in the range [0, Jma :~ ]. The change in synaptic efficacy is defined by Fij (t), as Fij(t) = L [6(t - t:)Kp(t; - t:) + 6(t - t;)KD(t; - t:)] (5) k,l where Kp and KD are the potentiation and depression branches of the kernel function K(6) = -cO exp [- (a6 + b)2] (6) plotted in Figure 1. Following [Zhang et al., 1998] we distinguish between the situation where the postsynaptic spike, at t~, appears after or before the presynaptic spike, at t~, using the asymmetric kernel that captures the essence of their experimental observations. ... -o !.'--~-~~--'o-~-~~-----' /I =t'_tk , I Figure 1: The kernel function whose left part, Kp, leads to potentiation of the synapse, and whose right branch, KD, causes synaptic depression. 3 Distributed Synchrony of a Hebbian Assembly We have run our system with synaptic delays chosen randomly to be either 1, 2, or 3ms, and temporal parameters Tn chosen as 40ms for excitatory neurons and 20ms for inhibitory ones. Turning external input currents off after a while we obtained sustained firing activities in the range of 100-150 Hz. We have found, in addition to synchronous and asynchronous realizations of this attractor, a mode of distributed synchrony. A characteristic example of a long cycle is shown in Figure 2: The 100 excitatory neurons split into groups such that each group fires at the same frequency and at a fixed phase difference from any other group. The J/JE synaptic efficacies 132 D. Horn, N Levy, 1. Meilijson and E. Ruppin ':~ : I : I : r I :1 :tI : I : r I : I ] ':f I : r I : I : I :1 ':f r I : 1 : 1 : 1 :1 j I :1 : 1 : 1 : n j : I : I : I : r ~ , " " " " JO I Figure 2: Distributed synchronized firing mode. The firing patterns of six cell assemblies of excitatory neurons are displayed vs time (in ms). These six groups of neurons formed in a self-organized manner for a kernel function with equal potentiation and depression. The delays were chosen randomly from three values, 1 2 or 3ms, and the system is monitored every 0.5ms. are initiated as small random values. The learning process leads to the self-organized synaptic matrix displayed in Figure 3(a). The block form of this matrix represents the ordered couplings that are responsible for the fact that each coherent group of neurons feeds the activity of groups that follow it. The self-organized groups form spontaneously. When the synapses are affected by some external noise, as can come about from Hebbian learning in which these neurons are being coupled with other pools of neurons, the groups will change and regroup, as seen in Figure 3(b) and 3(c). (a) (b) (c) Figure 3: A synaptic matrix for n = 6 distributed synchrony. The synaptic matrix between the 100 excitatory neurons of our system is displayed in a grey-level code with black meaning zero efficacy and white standing for the synaptic upper-bound. (a) The matrix that exists during the distributed synchronous mode of Figure 2. Its basis is ordered such that neurons that fire together are grouped together. (b) Using the same basis as in (a) a new synaptic matrix is shown, one that is formed after stopping the sustained activity of Figure 2, introducing noise in the synaptic matrix, and reinstituting the original memory training. (c) The same matrix as (b) is shown in a new basis that exhibits connections that lead to a new and different realization of distributed synchrony. A stable distributed synchrony cycle can be simply understood for the case of a single synaptic delay setting the basic step, or phase difference, of the cycle. When several delay parameters exist, a situation that probably more accurately represents the a-function character of synaptic transmission in cortical networks, distributed Distributed Synchrony of Spiking Neurons in a Hebbian Cell Assembly 133 synchrony may still be obtained, as is evident from Figure 2. After some time the cycle may destabilize and regrouping may occur by itself, without external interference. The likelihood of this scenario is increased because different synaptic connections that have different delays can interfere with one another. Nonetheless, over time scales of the type shown in Figure 2, grouping is stable. 4 Analysis of a Cycle In this section we analyze the dynamics of the network when it is in a stable state of distributed synchrony. We assume that n groups of neurons are formed and calculate the stationary distribution of JffE(t) . In this state the firing pattern of every two neurons in the network can be characterized by their frequency l/(t) and by their relative phase 8. We assume that 8 is a random normal variable with mean J.Lo and standard deviation 0'0 . Thus, Eq. 4 can be rewritten as the following stochastic differential equation dJi~E(t) = [J.LFij(t) - :s J!jE(t)] dt+O'Fij(t)dW(t) (7) where Fij (t) (Eq. 5) is represented here by a drift term J.LFij (t) and a diffusion term O'Fij (t) which are its mean and standard deviation. W(t) describes a Wiener process. Note that both J.LFij (t) and O'Fij (t) are calculated for a specific distribution of 8 and are functions of J.Lo and 0'0. The stochastic process that satisfies Eq. 7 will satisfy the Fokker-Plank equation for the probability distribution f of JIfE, .J' = ___ .. (t) _ _ JPlE f(JPlE t) Fij tJ' (8) 8f(JPlE t) 8 [( 1) ] 0'2 (t) 82f(JPlE t) 8t 8JPlE J.LF.] T'J 'J' + 2 8JEE2 1J S ij with reflecting boundary conditions imposed by the synaptic bounds, 0 and Jmax . Since we are interested in the stable state of the process we solve the stationary equation. The resulting density function is EE N [1 ( EE 1 EE2) 1 f(Jij ,J.Lt5, 0'15) = O'}ij (t) exp O'}ij (t) 2J.LFij Jij Ts Jij (9) where (10) Eq. 9 enables us to calculate the stationary distribution of the synaptic efficacies between the presynaptic neuron i and the post-synaptic neuron j given their frequency l/ and the parameters J.Lo and 0'15. An example of a solution for a 3-cycle is shown in Figure 4. In this case all neurons fire with frequency l/ = (3Td)-1 and J.Lt5 takes one of the values -Td, 0, Td. Simulation results of a 3-cycle in a network of excitatory and inhibitory integrateand-fire neurons described in Section 2 are given in Figure 5. As can be seen the results obtained from the analysis match those observed in the simulation. 5 Discussion The interesting experimental observations of synaptic learning curves [Markram et al., 1997, Zhang et al., 1998] have led us to study their implications for the firing patterns of a Hebbian cell assembly. We find that, in addition 134 D. Horn, N. Levy, 1. Meilijson and E. Ruppin (a) (b) 70 60 so 40 30 20 01 0.2 0.3 0 4 , 0 ) 0 0 0.' 0.2 0.3 0.4 0.5 JEE " Figure 4: Results of the analysis for n = 3, a6 = 2ms and Td = 2.5ms. (a) The synaptic matrix. Each of the nine blocks symbolizes a group of connections between neurons that have a common phase-lag J..l6 . The mean of Ji~E was calculated for each cell by Eq. 9 and its value is given by the gray scale tone. (b) The distribution of synaptic values between all excitatory neurons. (a) (b) 0. ' 0.2 0.3 0.4 0. 5 5o00,-------------, 4500 4000 3500 3000 2500 2000 1500 500 o 0"---0 .... ' --' ... 0 .2:--~0.3::----::"" 0.4c--'"-:" 0. 5 JEE , Figure 5: Simulation results for a network of N E = 100 and NJ = 50 integrateand-fire neurons, when the network is in a stable n = 3 state. Tn = 10ms for both excitatory and inhibitory neurons. The average frequency of the neurons is 130 Hz. (a) The excitatory synaptic matrix. (b) Histogram of the synaptic efficacies. to the expected synchronous and asynchronous modes, an interesting behavior of distributed synchrony can emerge. This is the phenomenon that we have investigated both by simulations and by analytic evaluation. Distributed synchrony is a mode in which the Hebbian cell assembly breaks into an n-cycle. This cycle is formed by instantaneous symmetry breaking, hence specific classification of neurons into one of the n groups depends on initial conditions, noise, etc. Thus the different groups of a single cycle do not have a semantic invariant meaning of their own. It seems perhaps premature to try and identify these cycles with synfire chains [Abeles, 1982] that show recurrence of firing patterns of groups of neurons with periods of hundreds of ms. Note however, that if we make such an identification, it is a different explanation from the model of [Herrmann et al., 1995J, which realizes the synfire chain by combining sets of preexisting patterns into a cycle. The simulations in Figures 2 and 3 were carried out with a learning curve that possessed equal potentiation and depression branches, i.e. was completely antisymmetric in its argument. In that case no synaptic decay was allowed. Figure 5, on the other hand, had stronger potentiation than depression, and a finite synaptic Distributed Synchrony of Spiking Neurons in a Hebbian Cell Assembly 135 decay time was assumed. Other conditions in these nets were different too, yet both had a window of parameters where distributed synchrony showed up. Using the analytic approach of section 4 we can derive the probability distribution of synaptic values once a definite cyclic pattern of distributed synchrony is formed. An analytic solution of the combined dynamics of both the synapses and the spiking neurons is still an open challenge. Hence we have to rely on the simulations to prove that distributed synchrony is a natural spatiotemporal behavior that follows from combined neuronal dynamics and synaptic learning as outlined in section 2. To the extent that both types of dynamics reflect correctly the dynamics of cortical neural networks, we may expect distributed synchrony to be a mode in which neuronal attractors are being realized. The mode of distrbuted synchrony is of special significance to the field of neural computation since it forms a bridge between the feedback and feed-forward paradigms. Note that whereas the attractor that is formed by the Hebbian cell assembly is of global feedback nature, i.e. one may regard all neurons of the assembly as being connected to other neurons within the same assembly, the emerging structure of distributed synchrony shows that it breaks down into groups. These groups are connected to one another in a self-organized feed-forward manner, thus forming the cyclic behavior we have observed. References [Abbott and Song, 1999] L. F. Abbott and S. Song. Temporally asymmetric hebbian learning, spike timing and neuronal response variability. In M. S. Kearns, S. A. Solla, and D. A. Cohn, editors, Advances in Neural Information Processing Systems 11: Proceedings of the 1998 Conference, pages 69 - 75. MIT Press, 1999. [Abeles, 1982] M. Abeles. Local Cortical Circuits. Springer, Berlin, 1982. [BruneI, 1999] N. BruneI. Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. Journal of Computational Neuroscience, 1999. [Herrmann et al., 1995] M. Herrmann, J . Hertz, and A. Prugel-Bennet. Analysis of synfire chains. Network: Compo in Neural Systems, 6:403 - 414, 1995. [Kempter et al. , 1999] R. Kempter, W. Gerstner, and J. Leo van Hemmen. Spikebased compared to rate-based hebbian learning. In M. S. Kearns, S. A. Solla, and D. A. Cohn, editors, Advances in Neural Information Processing Systems 11: Proceedings of the 1998 Conference, pages 125 - 131. MIT Press, 1999. [Markram et al., 1997] H. Markram, J. Lubke, M. Frotscher, and B. Sakmann. Regulation of synaptic efficacy by coincidence of postsynaptic aps and epsps. Science, 275(5297):213 - 215, 1997. [Miyashita, 1988] Y. Miyashita. Neuronal correlate of visual associative long-term memory in the primate temporal cortex. Nature, 335:817 - 820, 1988. [Yakovlev et al., 1998] V. Yakovlev, S. Fusi, E. Berman, and E. Zohary. Inter-trial neuronal activity in inferior temporal cortex: a putative vehicle to generate longterm visual associations. Nature Neurosc., 1(4):310 - 317, 1998. [Zhang et al., 1998] L. I. Zhang, H. W. Tao, C. E. Holt, W. A. Harris, and M. Poo. A critical window for cooperation and competition among developing retinotectal synapses. Nature, 395:37 - 44, 1998.
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Image representations for facial expression coding Marian Stewart Bartlett* V.C. San Diego marni<Osalk.edu Javier R. Movellan V.C. San Diego movellan<ocogsci.ucsd.edu Paul Ekman V.C. San Francisco ekman<ocompuserve.com Gianluca Donato Di~ital Persona, Redwood City, CA glanlucad<Odigitalpersona.com Joseph C. Hager Network Information Res., SLC, Utah jchager<Oibm.com Terrence J. Sejnowski Howard Hughes Medical Institute The Salk Institute; V.C. San Diego terry<osalk.edu Abstract The Facial Action Coding System (FACS) (9) is an objective method for quantifying facial movement in terms of component actions. This system is widely used in behavioral investigations of emotion, cognitive processes, and social interaction. The coding is presently performed by highly trained human experts. This paper explores and compares techniques for automatically recognizing facial actions in sequences of images. These methods include unsupervised learning techniques for finding basis images such as principal component analysis, independent component analysis and local feature analysis, and supervised learning techniques such as Fisher's linear discriminants. These data-driven bases are compared to Gabor wavelets, in which the basis images are predefined. Best performances were obtained using the Gabor wavelet representation and the independent component representation, both of which achieved 96% accuracy for classifying 12 facial actions. The ICA representation employs 2 orders of magnitude fewer basis images than the Gabor representation and takes 90% less CPU time to compute for new images. The results provide converging support for using local basis images, high spatial frequencies, and statistical independence for classifying facial actions. 1 Introduction Facial expressions provide information not only about affective state, but also about cognitive activity, temperament and personality, truthfulness, and psychopathology. The Facial Action Coding System (FACS) (9) is the leading method for measuring facial movement in behavioral science. FACS is performed manually by highly trained human experts. A FACS coder decomposes a facial expression into component muscle movements (Figure 1). Ekman and Friesen described 46 distinct facial movements, and over 7000 distinct combinations of such movements have * To whom correspondence should be addressed. (UCSD 0523, La Jolla, CA 92093.) This research was supported by NIH Grant No. IF32 MH12417-01. Image Representations for Facial Expression Coding 887 been observed in spontaneous behavior. An automated system would make facial expression measurement more widely accessible as a research tool for behavioral science and medicine. Such a system would also have application in human-computer interaction tools and low bandwidth facial animation coding. A number of systems have appeared in the computer vision literature for classifying facial expressions into a few basic categories of emotion, such as happy, sad, or surprised. While such approaches are important, an objective and detailed measure of facial activity such as FACS is needed for basic research into facial behavior. In a system being developed concurrently for automatic facial action coding, Cohn and colleagues (7) employ feature point tracking of a select set of image points. Techniques employing 2-D image filters have proven to be more effective than featurebased representations for face image analysis [e.g. (6)]. Here we examine image analysis techniques that densely analyze graylevel information in the face image. This work surveys and compares techniques for face image analysis as applied to automated FACS encoding. l The analysis focuses on methods for face image representation in which image graylevels are described as a linear superposition of basis images. The techniques were compared on a common image testbed using common similarity measures and classifiers. We compared four representations in which the basis images were learned from the statistics of the face image ensemble. These include unsupervised learning techniques such as principal component analysis (PCA), and local feature analysis (LFA), which are learned from the second-order dependences among the image pixels, and independent component analysis (ICA) which is learned from the highorder dependencies as well. We also examined a representation obtained through supervised learning on the second-order image statistics, Fisher's linear discriminants (FLD). Classification performances with these data-driven basis images were compared to Gabor wavelets, in which the basis images were pre-defined. We examined properties of optimal basis images, where optimal was defined in terms of classification. Generalization to novel faces was evaluated using leave-one-out cross-validation. Two basic classifiers were employed: nearest neighbor and template matching, where the templates were the mean feature vectors for each class. Two similarity measures were employed for each classifier: Euclidean distance and cosine of the angle between feature vectors. 2 1 a. 4 4 AU 1 Inner brow raiser AU 2 Outer brow raiser AU 4 Brow lowerer h. Figure 1: a. The facial muscles underlying six of the 46 facial actions. b. Cropped face images and 8-images for three facial actions (AU's). 1 A detailed description of this work appears in (8). 888 M S. Bartlett, G. Donato, J. R. Movellan, J. C. Hager, P. Ekman and T. J. Sejnowski 2 Image Database We collected a database of image sequences of subjects performing specified facial actions. The database consisted of image sequences of subjects performing specified facial actions. Each sequence began with a neutral expression and ended with a high magnitude muscle contraction. For this investigation, we used 111 sequences from 20 subjects and attempted to classify 12 actions: 6 upper face actions and 6 lower face actions. Upper and lower-face actions were analyzed separately since facial motions in the lower face do not effect the upper face, and vice versa (9). The face was located in the first frame in each sequence using the centers of the eyes and mouth. These coordinates were obtained manually by a mouse click. The coordinates from Frame 1 were used to register the subsequent frames in the sequence. The aspect ratios of the faces were warped so that the eye and mouth centers coincided across all images. The three coordinates were then used to rotate the eyes to horizontal, scale, and finally crop a window of 60 x 90 pixels containing the upper or lower face. To control for variations in lighting, logistic thresholding and luminance scaling was performed (13). Difference images (b-images) were obtained by subtracting the neutral expression in the first image of each sequence from the subsequent images in the sequence. 3 Unsupervised learning 3.1 Eigenfaces (peA) A number of approaches to face image analysis employ data-driven basis vectors learned from the statistics of the face image ensemble. Techniques such as eigenfaces (17) employ principal component analysis, which is an unsupervised learning method based on the second-order dependencies among the pixels (the pixelwise covariances). PCA has been applied successfully to recognizing facial identity (17), and full facial expressions (14). Here we performed PCA on the dataset of b-images, where each b-image comprised a point in Rn given by the brightness of the n pixels. The PCA basis images were the eigenvectors of the covariance matrix (see Figure 2a), and the first p components comprised the representation. Multiple ranges of components were tested, from p = 10 to P = 200, and performance was also tested excluding the first 1-3 components. Best performance of 79.3% correct was obtained with the first 30 principal components, using the Euclidean distance similarity measure and template matching classifier. Padgett and Cottrell (14) found that a local PCA representation outperformed global PCA for classifying full facial expressions of emotion. Following the methods in (14), a set of local basis images was derived from the principal components of 15x 15 image patches from randomly sampled locations in the b-images (see Figure 2d.) The first p principal components comprised a basis set for all image locations, and the representation was downsampled by a factor of 4. Best performance of 73.4% was obtained with components 2-30, using Euclidean distance and template matching. Unlike the findings in (14), local basis images obtained through PCA were not more effective than global PCA for facial action coding. A second local implementation of PCA, in which the principal components were calculated for fixed 15x 15 image patches also failed to improve over global PCA. 3.2 Local Feature Analysis (LFA) Penev and Atick (15) recently developed a local, topographic representation based on second-order image statistics called local feature analysis (LF A). The kernels are derived from the principal component axes, and consist of a "whitening" step to equalize the variance of the PCA coefficients, followed by a rotation to pixel space. Image Representations for Facial Expression Coding 889 a. h. c. d. Figure 2: a. First 4 PCA basis images. b. Four ICA basis images. The ICA basis images are local, spatially opponent, and adaptive. c. Gabor kernels are local, spatially opponent, and predefined. d. First 4 local PCA basis images. We begin with the matrix P containing the principal component eigenvectors in its columns, and Ai are the corresponding eigenvalues. Each row of the matrix K serves as an element of the LFA image dictionary2 K = pVpT where V = D-! = diag( ~) i = 1, ... ,p (1) V Ai where Ai are the eigenvalues. The rows of K were found to have spatially local properties, and are "topographic" in the sense that they are indexed by spatial location (15). The dimensionality of the LFA representation was reduced by employing an iterative sparsification algorithm based on multiple linear regression described in (15). The LFA representation attained 81.1 % correct classification performance. Best performance was obtained using the first 155 kernels, the cosine similarity measure, and nearest neighbor classifier. Classification performance using LFA was not significantly different from the performance using peA. Although a face recognition algorithm based on the principles of LFA outperformed Eigenfaces in the March 1995 FERET competition, the exact algorithm has not been disclosed. Our results suggest that an aspect of the algorithm other than the LFA representation accounts for the difference in performance. 3.3 Independent Component Analysis (ICA) Representations such as Eigenfaces, LFA, and FLD are based on the second-order dependencies among the pixels, but are insensitive to the high-order dependencies. High-order dependencies are relationships that cannot be described by a linear predictor. Independent component analysis (ICA) learns the high-order dependencies in addition to the second-order dependencies among the pixels. 2 An image dictionary is a set of images that decomposes other images, e.g. through inner product. Here it finds the coefficients for the basis set K- 1 890 M. S. Bartlett, G. Donato, J. R. Movellan, J. C. Hager, P. Ekman and T. J. Sejnowski The ICA representation was obtained by performing Bell & Sejnowski's infomax algorithm (4) (5) on the ensemble of ~-images in the rows of the matrix X. The images in X were assumed to be a linear mixture of an unknown set of independent source images which were recovered through ICA. In contrast to PCA, the ICA source images were local in nature (see Figure 2b). These source images provided a basis set for the expression images. The coefficients of each image with respect to the new basis set were obtained from the estimated mixing matrix A ~ W- 1 , where W is the ICA weight matrix [see (1), (2)]. Unlike PCA, there is no inherent ordering to the independent components of the dataset. We therefore selected as an ordering parameter the class discriminability of each component, defined as the ratio of between-class to within-class variance. Best performance of 95.5% was obtained with the first 75 components selected by class discriminability, using the cosine similarity measure, and nearest neighbor classifier. Independent component analysis gave the best performance among all of the datadriven image kernels. Class discriminability analysis of a PCA representation was previously found to have little effect on classification performance with PCA (2). 4 Supervised learning: Fisher's Linear Discriminants (FLD) A class specific linear projection of a PCA representation of faces was recently shown to improve identity recognition performance (3). The method employs a classic pattern recognition technique, Fisher's linear discriminant (FLD), to project the images into a c - 1 dimensional subspace in which the c classes are maximally separated. Best performance was obtained by choosingp = 30 principal components to first reduce the dimensionality of the data. The data was then projected down to 5 dimensions via the FLD projection matrix, W,ld. The FLD image dictionary was thus Wpca * W,ld. Best performance of 75.7% correct was obtained with the Euclidean distance similarity measure and template matching classifier. FLD provided a much more compact representation than PCA. However, unlike the results obtained by (3) for identity recognition, Fisher's Linear Discriminants did not improve over basic PCA (Eigenfaces) for facial action classification. The difference in performance may be due to the low dimensionality of the final representation here. Class discriminations that are approximately linear in high dimensions may not be linear when projected down to as few as 5 dimensions. 5 Predefined image kernels: Gabor wavelets An alternative to the adaptive bases described above are wavelet decompositions based on predefined families of Gabor kernels. Gabor kernels are 2-D sine waves modulated by a Gaussian (Figure 2c). Representations employing families of Gabor filters at multiple spatial scales, orientations, and spatial locations have proven successful for recognizing facial identity in images (11). Here, the ~-images were convolved with a family of Gabor kernels 'l/Ji, defined as ( .... ) IIkill2 _"kjlI2IzI2 [}'k.i _0'2] 'l/Ji X = --2-e 20'2 e' - e 2 u (2) where ki = ( it c~s'Pl-' ) Iv = 2-~7r, 'PI-' = J.t'!!g. J v sm 'PI-' ' Following (11), the representation consisted of the amplitudes at 5 frequencies (v = 0-4) and 8 orientations (J.t = 1 - 8). Each filter output was downsampled by a factor q and normalized to unit length. We tested the performance of the system using q = 1,4,16 and found that q = 16 yielded the best generalization rate. Best performance was obtained with the cosine similarity measure and nearest neighbor Image Representations for Facial Expression Coding 891 classifier. Classification performance with the Gabor representation was 95.5%. This performance was significantly higher than all of the data-driven approaches in the comparison except independent component analysis, with which it tied. 6 Results and Conclusions PCA Local PCA LFA ICA FLD Gabor 79.3 ±3.9 73.4 ±4.2 81.1 ±3.7 95.5 ±2.0 75.7 ±4.1 95.5 ±2.0 Table 1: Summary of classification performance for 12 facial actions. We have compared a number of different image analysis methods on a difficult classification problem, the classification of facial actions. Best performances were obtained with the Gabor and ICA representations, which both achieved 95.5% correct classification (see Table 1). The performance of these two methods equaled the agreement level of expert human subjects on these images (94%). Image representations derived from the second-order statistics of the dataset (PCA and LFA) performed in the 80% accuracy range. An image representation derived from supervised learning on the second-order statistics (FLD) also did not significantly differ from PCA. We also obtained evidence that high spatial frequencies are important for classifying facial actions. Classification with the three highest frequencies of the Gabor representation (1/ = 0,1,2, cycles/face = 15,18,21 cycles/face) was 93% compared to 84% with the three lowest frequencies (1/ = 2,3,4, cycles/face = 9,12,15). The two representations that significantly outperformed the others, Gabor and Independent Components, employed local basis images, which supports recent findings that local basis images are important for face image analysis (14) (10) (12). The local property alone, however, does not account for the good performance of these two representations, as LFA performed no better than PCA on this classification task, nor did local implementations of PCA. In addition to spatial locality, the ICA representation and the Gabor filter representation share the property of redundancy reduction, and have relationships to representations in the visual cortex. The response properties of primary visual cortical cells are closely modeled by a bank of Gabor kernels. Relationships have been demonstrated between Gabor kernels and independent component analysis. Bell & Sejnowski (5) found using ICA that the kernels that produced independent outputs from natural scenes were spatially local, oriented edge kernels, similar to a bank of Gabor kernels. It has also been shown that Gabor filter outputs of natural images are at least pairwise independent (16). The Gabor wavelets and ICA each provide a way to represent face images as a linear superposition of basis functions. Gabor wavelets employ a set of pre-defined basis images, whereas ICA learns basis images that are adapted to the data ensemble. The Gabor wavelets are not specialized to the particular data ensemble, but would be advantageous when the amount of data is small. The ICA representation has the advantage of employing two orders of magnitude fewer basis images. This can be an advantage for classifiers that involve parameter estimation. In addition, the ICA representation takes 90% less CPU time than the Gabor representation to compute once the ICA weights are learned, which need only be done once. In summary, this comparison provided converging support for using local basis images, high spatial frequencies, and statistical independence for classifying facial actions. Best performances were obtained with Gabor wavelet decomposition and independent component analysis. These two representations employ gray level basis functions that share properties of spatial locality, independence, and have relationships to the response properties of visual cortical neurons. An outstanding issue is whether our findings depend on the simple recognition engines we employed. Would a smarter recognition engine alter the relative per892 M. S. Bartlett. G. Donato. J. R. Movellan. J. C. Hager, P. Ekman and T. J. Sejnowski formances? Our preliminary investigations suggest that is not the case. Hidden Markov models (HMM's) were trained on the PCA, ICA and Gabor representations. The Gabor representation was reduced to 75 dimensions using PCA before training the HMM. The HMM improved classification performance with ICA to 96.3%, and it did not change the overall findings, as it gave similar percent improvements to the PCA and PCA-reduced Gabor representations over their nearest neighbor performances. The dimensionality reduction of the Gabor representation, however, caused its nearest neighbor performance to drop, and the performance with the HMM was 92.7%. The lower dimensionality of the ICA representation was an advantage when employing the HMM. 7 References [1) M.S. Bartlett. Face Image Analysis by Unsupervised Learning and Redundancy Reduction. PhD thesis, University of California, San Diego, 1998. [2) M.S. Bartlett, H.M. Lades, and T.J. Sejnowski. Independent component representations for face recognition. In T. Rogowitz, B. & Pappas, editor, Proceedings of the SPIE Symposium on Electonic Imaging: Science and Technology; Human Vision and Electronic Imaging III, volume 3299, pages 528-539, San Jose, CA, 1998. SPIE Press. [3) P.N. Belhumeur, J .P. Hespanha, and D.J. Kriegman. Eigenfaces vs. fisherfaces: Recognition using class specific linear projection. IEEE Transations on Pattern Analysis and Machine Intelligence, 19(7):711-720, 1997. [4) A.J. Bell and T.J. Sejnowski. An information-maximization approach to blind separation and blind deconvolution. Neural Computation, 7(6):1129-1159, 1995. [5) A.J. Bell and T.J. Sejnowski. The independent components of natural scenes are edge filters. Vision Research, 37(23):3327-3338, 1997. [6) R. Brunelli and T. Poggio. Face recognition: Features versus templates. IEEE transactions on pattern analysis and machine intelligence, 15(10):1042-1052, 1993. [7) J.F. Cohn, A.J. Zlochower, J.J. Lien, Y-T Wu, and T. Kanade. Automated face coding: A computer-vision based method of facial expression analysis. Psychophysiology, 35(1):35-43, 1999. [8) G. Donato, M. Bartlett, J . Hager, P. Ekman, and T. Sejnowski. Classifying facial actions. IEEE Transactions on Pattern Analysis and Machine Intelligence, 21(10):974989, 1999. [9) P. Ekman and W. Friesen. Facial Action Coding System: A Technique for the Measurement of Facial Movement. Consulting Psychologists Press, Palo Alto, CA, 1978. [10) M.S. Gray, J. Movellan, and T.J. Sejnowski. A comparison of local versus global image decomposition for visual speechreading. In Proceedings of the 4th Joint Symposium on Neural Computation, pages 92-98. Institute for Neural Computation, La Jolla, CA, 92093-0523, 1997. [11) M. Lades, J . Vorbruggen, J. Buhmann, J. Lange, W. Konen, C. von der Malsburg, and R. Wurtz. Distortion invariant object recognition in the dynamic link architecture. IEEE Transactions on Computers, 42(3):300-311, 1993. [12) D.D. Lee and S. Seung. Learning the parts of objects by non-negative matrix factorization. Nature, 401:788-791, 1999. [13) J.R. Movellan. Visual speech recognition with stochastic networks. In G. Tesauro, D.S. Touretzky, and T. Leen, editors, Advances in Neural Information Processing Systems, volume 7, pages 851-858. MIT Press, Cambridge, MA, 1995. [14) C. Padgett and G. Cottrell. Representing face images for emotion classification. In M. Mozer, M. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems, volume 9, Cambridge, MA, 1997. MIT Press. [15) P.S. Penev and J .J. Atick. Local feature analysis: a general statistical theory for object representation. Network: Computation in Neural Systems, 7(3):477-500, 1996. [16) E. P. Simoncelli. Statistical models for images: Compression, restoration and synthesis. In 31st Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, November 2-5 1997. [17) M. Turk and A. Pentland. Eigenfaces for recognition. Journal of Cognitive Neuroscience, 3(1):71-86, 1991.
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Algorithms for Independent Components Analysis and Higher Order Statistics Daniel D. Lee Bell Laboratories Lucent Technologies Murray Hill, NJ 07974 Uri Rokni and Haim Sompolinsky Racah Institute of Physics and Center for Neural Computation Hebrew University Jerusalem, 91904, Israel Abstract A latent variable generative model with finite noise is used to describe several different algorithms for Independent Components Analysis (lCA). In particular, the Fixed Point ICA algorithm is shown to be equivalent to the Expectation-Maximization algorithm for maximum likelihood under certain constraints, allowing the conditions for global convergence to be elucidated. The algorithms can also be explained by their generic behavior near a singular point where the size of the optimal generative bases vanishes. An expansion of the likelihood about this singular point indicates the role of higher order correlations in determining the features discovered by ICA. The application and convergence of these algorithms are demonstrated on a simple illustrative example. Introduction Independent Components Analysis (lCA) has generated much recent theoretical and practical interest because of its successes on a number of different signal processing problems. ICA attempts to decompose the observed data into components that are as statistically independent from each other as possible, and can be viewed as a nonlinear generalization of Principal Components Analysis (PCA). Some applications of ICA include blind separation of audio signals, beamforming of radio sources, and discovery of features in biomedical traces [I] . There have also been a number of approaches to deriving algorithms for ICA [2, 3, 4]. Fundamentally, they all consider the problem of recovering independent source signals {s} from observations {x} such that: M Xi = L WijSj , i = l..N (I) j = 1 Here, Wij is a N x M mixing matrix where the number of sources M is not greater than the dimensionality N of the observations. Thus, the columns of W represent the different independent features present in the observed data. Bell and Sejnowski formulated their Infomax algorithm for ICA as maximizing the mutual information between the data and a nonlinearly transformed version of the data [5]. The 492 D. D. Lee. U. Rokni and H. Sompolinsky covariant version of this algorithm uses the natural gradient of the mutual information to iteratively update the estimate for the demixing matrix W- 1 in terms of the estimated componentss = W - 1x [6]: .6.W-1 ex: [1 - (g(s)sT)] W- 1, (2) The nonlinearity g( s) differentiates the features learned by the lnfomax ICA algorithm from those found by conventional PCA. Fortunately, the exact form of the nonlinearity used in Eq. 2 is not crucial for the success of the algorithm, as long as it preserves the sub-Gaussian or super-Gaussian nature of the sources [7] . Another approach to ICA due to Hyvarinen and Oja was derived from maximizing objective functions motivated by projection pursuit [8]. Their Fixed Point ICA algorithm attempts to self-consistently solve for the extremum of a nonlinear objective function. The simplest formulation considers a single source M = 1 so that the mixing matrix is a single vector w, constrained to be unit length Iwl = 1. Assuming the data is first preprocessed and whitened, the Fixed Point ICA algorithm iteratively updates the estimate of w as follows: w t(xg(wT x) ACW w (3) w tIwl' where g(wT x) is a nonlinear function and AC is a constant given by the integral over the Gaussian: (4) The Fixed Point algorithm can be extended to an arbitrary number M ~ N of sources by using Eq. 3 in a serial deflation scheme. Alternatively, the M columns of the mixing matrix W can be updated simultaneously by orthogonalizing the N x M matrix: (5) Under the assumption that the observed data match the underlying ICA model, x = W s, it has been shown that the Fixed Point algorithm converges locally to the correct solution with at least quadratic convergence. However, the global convergence of the generic Fixed Point ICA algorithm is uncertain. This is in contrast to the gradient-based lnfomax algorithm whose convergence is guaranteed as long as a sufficiently small step size is chosen. In this paper, we first review the latent variable generative model framework for Independent Components Analysis. We then consider the generative model in the presence of finite noise, and show how the Fixed Point ICA algorithm can be related to an ExpectationMaximization algorithm for maximum likelihood. This allows us to elucidate the conditions under which the Fixed Point algorithm is guaranteed to globally converge. Assuming that the data are indeed generated from independent components, we derive the optimal parameters for convergence. We also investigate how the optimal size of the ICA mixing matrix varies as a function of the added noise, and demonstrate the presence of a singular point. By expanding the likelihood about this singular point, the behavior of the ICA algorithms can be related to the higher order statistics present in the data. Finally, we illustrate the application and convergence of these ICA algorithms on some artificial data. Generative model A convenient method for interpreting the different ICA algorithms is in terms of the hidden, or latent, variable generative model shown in Fig. 1 [9, 10]. The hidden variables {s j} leA Algorithms and Higher Order Statistics 493 M hidden variables N visible variables Figure 1: Generative model for ICA algorithms. s are the hidden variables, Ij are additive Gaussian noise terms, and x = W s + Ij are the visible variables. correspond to the different independent components and are assumed to have the factorized non-Gaussian prior probability distribution: M P(s) = II e-F(Sj). (6) j=l Once the hidden variables are instantiated, the visible variables {x t } are generated via a linear mapping through the generative weights W: N 1 [1 1 P(xls) = II ~ exp - 2lj2 (Xi - L WijSj)2 , i= l 7T1j j (7) where 1j2 is the variance of the Gaussian noise added to the visible variables. The probability of the data given this model is then calculated by integrating over all possible values of the hidden variables: P(x) = f ds P(s)P(xls) = (27T1j;) N/2 f ds exp [-F(S) 2~2 (x - WS)2] (8) In the limit that the added noise vanishes, 1j2 -T 0, it has previously been shown that maximizing the likelihood of Eq. 8 is equivalent to the Infomax algorithm in Eq. 2 [11]. In the following analysis, we will consider the situation when the variance of the noise is nonzero, 1j2 1= o. Expectation-Maximization We assume that the data has initially been preprocessed and spherized: (XiXj ) = Oij . Unfortunately, for finite noise 1j2 and an arbitrary prior F(sj) , deriving a learning rule for W in closed form is analytically intractable. However, it becomes possible to derive a simple Expectation-Maximization (EM) learning rule under the constraint: W = ~Wo , wlwo = I , (9) which implies that W is orthogonal, and ~ is the length of the individual columns of W . Indeed, for data that obeys the ICA model, x = W s, it can be shown that the optimal W must satisfy this orthogonality condition. By assuming the constraint in Eq. 9 for arbitrary data, the posterior distribution P(slx) becomes conveniently factorized: F(.lx) ()( i! exp [-F(';) + :' I(WT x);,; - ~e';ll· (10) 494 D. D. Lee, U. Rokni and H. Sompolinsky For the E-step, this factorized form allows the expectation function J ds P(slx)s = g(WT x) to be analytically evaluated. This expectation is then used in the M-step to find the new estimate W': (xg(WT x)T) - AsW' = 0, (11 ) where As is a symmetric matrix of Lagrange multipliers that constrain the new W' to be orthogonal. Eq. 11 is easily solved by taking the reduced singular value decomposition of the rectangular matrix: (12) where UTU = VVT = I and D is a diagonal M x M matrix. Then the solution for the EM estimate of the mixing matrix is given by: W' As (13) (14) As a specific example, consider the following prior for binary hidden variables: P( s) = ~[8(s - 1) + 8(s + 1)]. In this case, the expectation J ds P(slx)s = tanh(WT X/(j2) and so the EM update rule is given by onhogonalizing the matrix: W f- (xtanh(:2 WT X)) . (15) Fixed Point leA Besides the presence of the linear term AC Win Eq. 5, the EM update rule looks very much like that of the Fixed Point leA algorithm. It turns out that without this linear term, the convergence of the naive EM algorithm is much slower than that of Eq. 5. Here we show that it is possible to interpret the role of this linear term in the Fixed Point leA algorithm within the framework of this generative model. Suppose that the distribution of the observed data PD (x) is actually a mixture between an isotropic distribution Po(x) and a non-isotropic distribution P1 (x): PD(X) = aPo(x) + (1 - a)P1 (x). (16) Because the isotropic part does not break rotational symmetry, it does not affect the choice of the directions of the learned basis W . Thus, it is more efficient to apply the learning algorithm to only the non-isotropic portion of the distribution, Pt (x) (X PD(X) - aPo(x), rather than to the whole observed distribution PD(X). Applying EM to P1 (x) results in a correction term arising from the subtracted isotropic distribution. With this correction, the EM update becomes: W f- (xg(WT x)) - aAcW (17) which is equivalent to the Fixed Point leA algorithm when a = 1. Unfortunately, it is not clear how to compute an appropriate value for a to use in fitting data. Taking a very small value, a « 1, will result in a learning rule that is very similar to the naive EM update rule. This implies that the algorithm will be guaranteed to monotonically converge, albeit very slowly, to a local maximum of the likelihood. On the other hand, choosing a large value, a » 1, will result in a subtracted probability density P1 (x) that is negative everywhere. In this case, the algorithm will converge slowly to a local minimum of the likelihood. For the Fixed Point algorithm which operates in the intermediate regime, a ~ 1, the algorithm is likely to converge most rapidly. However, it is also in this situation that the subtracted density P1 (x) could have both positive and negative regions, and the algorithm is no longer guaranteed to converge. leA Algorithms and Higher Order Statistics 495 Noise 0 2 Figure 2: Size of the optimal generative bases as a function of the added noise (J2, showing the singular point behavior around (J~ ~ 1. Optimal value of a In order to determine the optimal value of a, we make the assumption that the observed data obeys the ICA model, x = A8. Note that the statistics of the sources in the data need not match the assumed prior distribution of the sources in the generative model Eq. 6. With this assumption, which is not related to the mixture assumption in Eq. 16, it is easy to show that W = A is a fixed point of the algorithm. By analyzing the behavior of the algorithm in the vicinity of this fixed point, a simple expression emerges for the change in deviations from this fixed point, 8W, after a single iteration of Eq. 17: (g'(8)) - aAG 3 8Wij +- ( ()) A 8Wij + O(8W ) (18) 8g 8 - a G where the averaging here is over the true source distribution, assumed for simplicity to be identical for all sources. Thus, the algorithm converges most rapidly if one chooses: (g' (8)) aopt = AG ' (19) so that the local convergence is cubic. From Eq. 18 one can show that the condition for the stability of the fixed point is given by a < ae , where: (8g(8) + g'(8)) ac = 2AG . (20) Thus, for a = 0, the stability criterion in Eq. 18 is equivalent to (8g( 8)) > (g' (8)). For the cubic nonlinearity g( 8) = S3, this implies that the algorithm will find the true independent features only if the source distribution has positive kurtosis. Singular point expansion Let us now consider how the optimal size ~ of the weights W varies as a function of the noise parameter (J2. For very small (J2 « 1, the weights W are approximately described by the Infomax algorithm of Eq. 2, and the lengths of the columns should be unity in order to match the covariance of the data. For large (12 » 1, however, the optimal size of the weights should be very small because the covariance of the noise is already larger than that of the data. In fact, for Factor Analysis which is a special case of the generative model with F(s) = ~s2 in Eq. 6, it can be shown that the weights are exactly zero, W = 0, for (J2 > 1. Thus, the size of the optimal generative weights W varies with (J2 as shown qualitatively in Fig. 2. Above a certain critical noise value (J~ ~ 1, the weights are exactly equal to 496 D. D. Lee, U. Rokni and H. Sompolinsky 0.81 r----~---~--_, a=O.9 0.77 a=1.5 0.76'------'------'-------' o 5 10 15 Iteration Figure 3: Convergence of the modified EM algorithm as a function of a . With 9(S) = tanh(s) as the nonlinearity, the likelihood (In cosh{WT x)) is plotted as a function of the iteration number. The optimal basis W are plotted on the two-dimensional data distribution when the likelihood is maximized (top) and minimized (bottom). zero, W = O. Only below this critical value do the weights become nonzero. We expand the likelihood of the generative model in the vicinity of this singular point. This expansion is well-behaved because the size of the generative weights W acts as a small perturbative parameter in this expansion. The log likelihood of the model around this singular value is then given by: L = -~Tr [WWT - {I _ (j2)J] 2 4 1 + 4! L kurt{sm) (XiXjXkXI)c WimWjmWkmWlm ijklm +0(1 _ (j2)3, (21) where kurt(sm) represents the kurtosis of the prior distribution over the hidden variables. Note that this expansion is valid for any symmetric prior, and differs from other expansions that assume small deviations from a Gaussian prior [12, 13]. Eq. 21 shows the importance of the fourth-order cumulant of the observed data in breaking the rotational degeneracy of the weights W. The generic behavior of ICA is manifest in optimizing the cumulant term in Eq.21, and again depends crucially on the sign of the kurtosis that is used for the prior. Example with artificial data As an illustration of the convergence of the algorithm in Eq. 17, we consider the simple two-dimensional uniform distribution: P(x x) = {1/12, -vls~. Xl, X2 ~ vis (22) I, 2 0, otherWIse With 9(S) = tanh(s) as the nonlinearity, Fig. 3 shows how the overall likelihood converges for different values of the parameter a as the algorithm is iterated. For a ~ 1.0, the algorithm converges to a maximum of the likelihood, with the fastest convergence at aopt = 0.9. However, for a > 1.2, the algorithm converges to a minimum of the likelihood. At an intermediate value, a = 1.1, the likelihood does not converge at all, fluctuating wildly between the maximum and minimum likelihood solutions. The maximum leA Algorithms and Higher Order Statistics 497 likelihood solution shows the basis vectors in W aligned with the sides of the square distribution, whereas the minimum likelihood solution has the basis aligned with the diagonals. These solutions can also be understood as maximizing and minimizing the kurtosis terms in Eq. 21. Discllssion The utility of the latent variable generative model is demonstrated on deriving algorithms for leA. By constraining the generative weights to be orthogonal, an EM algorithm is analytically obtained. By interpreting the data to be fitted as a mixture of isotropic and non-isotropic parts, a simple correction to the EM algorithm is derived. Under certain conditions, this modified algorithm is equivalent to the Fixed Point leA algorithm, and converges much more rapidly than the naive EM algorithm. The optimal parameter for convergence is derived assuming the data is consistent with the leA generative model. There also exists a critical value for the noise parameter in the generative model, about which a controlled expansion of the likelihood is possible. This expansion makes clear the role of higher order statistics in determining the generic behavior of different leA algorithms. We acknowledge the support of Bell Laboratories, Lucent Technologies, the US-Israel Binational Science Foundation, and the Israel Science Foundation. We also thank Hagai Attias, Simon Haykin, Juha Karhunen, Te-Won Lee, Erkki Oja, Sebastian Seung, Boris Shraiman, and Oren Shriki for helpful discussions. References [1] Haykin, S (1999). Neural networks: a comprehensivefoundation. 2nd ed., Prentice-Hall, Upper Saddle River, NJ. [2] Jutten, C & Herault, J (1991). Blind separation of sources, part I: An adaptive algorithm based on neuromimetic architecture. Signal Processing 24, 1-10. [3] Comon, P (1994). Independent component analysis: a new concept? Signal Processing 36, 287-314. [4] Roth, Z & Baram, Y (1996). Multidimensional density shaping by sigmoids. IEEE Trans. Neural Networks 7, 1291-1298. [5] Bell, AJ & Sejnowski, TJ (1995). An information maximization approach to blind separation and blind deconvolution. Neural Computation 7,1129-1159. [6] Amari, S, Cichocki, A & Yang, H (1996). A new learning algorithm for blind signal separation. Advances in Neural Information Processing Systems 8, 757-763. [7] Lee, TW, Girolami, M, & Sejnowski, TJ (1999). Independent component analysis using an extended infomax algorithm for mixed sub-gaussian and super-gaussian sources. Neural Computation 11, 609-633. [8] Hyvarinen, A & Oja, E (1997). A fast fixed-point algorithm for independent component analysis. Neural Computation 9, 1483-1492. [9] Hinton, G & Ghahramani, Z (1997). Generative models for discovering sparse distributed representations. Philosophical Transactions Royal Society B 352, 1177-1190. [10] Attias, H (1998). Independent factor analysis. Neural Computation 11, 803-851. [11] Pearlmutter, B & Parra, L (1996). A context-sensitive generalization of ICA. In ICONIP '96, 151-157. [12] Nadal, JP & Parga, N (1997). Redundancy reduction and independent component analysis: conditions on cumulants and adaptive approaches. Neural Computation 9, 1421-1456. [13] Cardoso, JF (1999). High-order contrasts for independent component analysis. Neural Computation 11,157-192.
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A SNoW-Based Face Detector Ming-Hsuan Yang Dan Roth Narendra Ahuja Department of Computer Science and the Beckman Institute University of Illinois at Urbana-Champaign Urbana, IL 61801 mhyang~vision.ai.uiuc.edu danr~cs.uiuc.edu ahuja~vision.ai.uiuc.edu Abstract A novel learning approach for human face detection using a network of linear units is presented. The SNoW learning architecture is a sparse network of linear functions over a pre-defined or incrementally learned feature space and is specifically tailored for learning in the presence of a very large number of features. A wide range of face images in different poses, with different expressions and under different lighting conditions are used as a training set to capture the variations of human faces. Experimental results on commonly used benchmark data sets of a wide range of face images show that the SNoW-based approach outperforms methods that use neural networks, Bayesian methods, support vector machines and others. Furthermore, learning and evaluation using the SNoW-based method are significantly more efficient than with other methods. 1 Introduction Growing interest in intelligent human computer interactions has motivated a recent surge in research on problems such as face tracking, pose estimation, face expression and gesture recognition. Most methods, however, assume human faces in their input images have been detected and localized. Given a single image or a sequence of images, the goal of face detection is to identify and locate human faces regardless of their positions, scales, orientations, poses and illumination. To support automated solutions for the above applications, this has to be done efficiently and robustly. The challenge in building an efficient and robust system for this problem stems from the fact that human faces are highly non-rigid objects with a high degree of variability in size, shape, color and texture. Numerous intensity-based methods have been proposed recently to detect human faces in a single image or a sequence of images. Sung and Poggio [24J report an example-based learning approach for locating vertical frontal views of human faces. They use a number of Gaussian clusters to model the distributions of face and non-face patterns. A small window is moved over an image to determine whether a face exists using the estimated distributions. In [16], a detection algorithm is proposed that combines template matching and feature-based detection method using hierarchical Markov random fields (MRF) and maximum a posteriori probability (MAP) estimation. Colmenarez and Huang [4) apply Kullback relative information for maximal discrimination between positive and negative examples of faces. They use a family of discrete Markov processes to model faces and background patterns and estimate the density functions. Detection of a face is based on the likelihood A SNoW-Based Face Detector 863 ratio computed during training. Moghaddam and Pentland [12] propose a probabilistic method that is based on density estimation in a high dimensional space using an eigenspace decomposition. In [20], Rowleyet al. use an ensemble of neural networks to learn face and non-face patterns for face detection. Schneiderman et al. describe a probabilistic method based on local appearance and principal component analysis [23]. Their method gives some preliminary results on profile face detection. Finally, hidden Markov models [17], higher order statistics [17], and support vector machines (SVM) [14] have also been applied to face detection and demonstrated some success in detecting upright frontal faces under certain lighting conditions. In this paper, we present a face detection method that uses the SNoW learning architecture [18, 3] to detect faces with different features and expressions, in different poses, and under different lighting conditions. SNoW (Sparse Network of Winnows) is a sparse network of linear functions that utilizes the Winnow update rule [10]. SNoW is specifically tailored for learning in domains in which the potential number of features taking part in decisions is very large, but may be unknown a priori. Some of the characteristics of this learning architecture are its sparsely connected units, the allocation of features and links in a data driven way, the decision mechanism and the utilization of an efficient update rule. SNoW has been used successfully on a variety of large scale learning tasks in the natural language domain [18, 13, 5, 19] and this is its first use in the visual processing domain. In training the SNoW-based face detector, we use a set of 1,681 face images from Olivetti [22], UMIST [6], Harvard [7], Yale [1] and FERET [15] databases to capture the variations in face patterns. In order to compare our approach with other methods, our experiments involve two benchmark data sets [20, 24] that have been used in other works on face detection. The experimental results on these benchmark data sets (which consist of 225 images with 619 faces) show that our method outperforms all other methods evaluated on this problem, including those using neural networks [20], Kullback relative information [4], naive Bayes [23] and support vector machines [14], while being significantly more efficient computationally. Along with these experimental results we describe further experiments that provide insight into some of the theoretical and practical considerations of SNoW-based learning systems. In particular, we study the effect of learning with primitive as well as with multi-scale features, and discuss some of the sources of the success of the approach. 2 The SN oW System The SNoW (Sparse Network of Winnows) learning architecture is a sparse network of linear units over a common pre-defined or incrementally learned feature space. Nodes in the input layer of the network represent simple relations over the input and are being used as the input features. Each linear unit is called a target node and represents relations which are of interest over the input examples; in the current application, only two target nodes are being used, one as a representation for a face pattern and the other for a non-face pattern. Given a set of relations (Le., types of features) that may be of interest in the input image, each input image is mapped into a set of features which are active (present) in it; this representation is presented to the input layer of SNoW and propagates to the target nodes. (Features may take either binary value, just indicating the fact that the feature is active (present) or real values, reflecting its strength; in the current application, all features are binary. See Sec 3.1.) Target nodes are linked via weighted edges to (some of the) input features. Let At = {i1 , ... , im } be the set of features that are active in an example and are linked to the target node t. Then the linear unit is active if and only if 2:iEAt wf > Ot, where wf is the weight on the edge connecting the ith feature to the target node t, and Ot is its threshold. In the current application a single SNoW unit which includes two subnetworks, one 864 M-H Yang, D. Roth and N. Ahuja for each of the targets, is used. A given example is treated autonomously by each target subnetwork; that is, an image labeled as a face is used as a positive example for the face target and as a negative example for the non-face target, and vice-versa. The learning policy is on-line and mistake-driven; several update rules can be used within SNoW. The most successful update rule, and the only one used in this work is a variant of Littlestone's Winnow update rule, a mUltiplicative update rule tailored to the situation in which the set of input features is not known a priori, as in the infinite attribute model [2]. This mechanism is implemented via the sparse architecture of SNoW. That is, (1) input features are allocated in a data driven way - an input node for the feature i is allocated only if the feature i is active in the input image and (2) a link (Le., a non-zero weight) exists between a target node t and a feature i if and only if i has been active in an image labeled t. Thus, the architecture also supports augmenting the feature types at later stages or from external sources in a flexible way, an option we do not use in the current work. The Winnow update rule has, in addition to the threshold fh at the target t, two update parameters: a promotion parameter a > 1 and a demotion parameter 0 < f3 < 1. These are being used to update the current representation of the target t (the set of weights w;) only when a mistake in prediction is made. Let At = {il' ... , im } be the set of active features that are linked to the target node t. If the algorithm predicts 0 (that is, LiEAt w~ ::; fh) and the received label is 1, the active weights in the current example are promoted in a mUltiplicative fashion: 'Vi E At, wf +- a . w~. If the algorithm predicts 1 (LiEAt wf > Ot) and the received label is 0, the active weights in the current example are demoted: 'Vi E At, w~ +- f3. wf. All other weights are unchanged. The key property of the Winnow update rule is that the number of examplesl it requires to learn a linear function grows linearly with the number of relevant features and only logarithmically with the total number of features. This property seems crucial in domains in which the number of potential features is vast, but a relatively small number of them is relevant (this does not mean that only a small number of them will be active, or have non-zero weights). Winnow is known to learn efficiently any linear threshold function and to be robust in the presence of various kinds of noise and in cases where no linear-threshold function can make perfect classification, and still maintain its abovementioned dependence on the number of total and relevant attributes [11, 9]. Once target subnetworks have been learned and the network is being evaluated, a winner-take-all mechanism selects the dominant active target node in the SNoW unit to produce a final prediction. In general, but not in this work, units' output may be cached and processed along with the output of other SNoW units to produce a coherent output. 3 Learning to detect faces For training, we use a set of 1,681 face images (collected from Olivetti [22], UMIST [6], Harvard [7], Yale [1] and FE RET [15] databases) which have wide variations in pose, facial expression and lighting condition. For negative examples we start with 8,422 non-face examples from 400 images of landscapes, trees, buildings, etc. Although it is extremely difficult to collect a representative set of non-face examples, the bootstrap method [24] is used to include more non-face examples during training. For positive examples, each face sample is manually cropped and normalized such that it is aligned vertically and its size is 20 x 20 pixels. To make the detection method less sensitive to scale and rotation variation, 10 face examples are generated from each original sample. The images are produced by randomly rotating the images by up to 15 degrees with scaling between 80% and 120%. This produces 16,810 face samples. Then, histogram equalization is performed that maps the lIn the on-line setting [10] this is usually phrased in terms of a mistake-bound but is known to imply convergence in the PAC sense [25, 8]. A SNoW-Based Face Detector 865 intensity values to expand the range of intensities. The same procedure is applied to input images in detection phase. 3.1 Primitive Features The SNoW-based face detector makes use of Boolean features that encode the positions and intensity values of pixels. Let the pixel at (x, y) of an image with width wand height h have intensity value I(x, y) (O :::; I{x, y) :::; 255). This information is encoded as a feature whose index is 256{y * w + x) + I{x, y). This representation ensures that different points in the {position x intensity} space are mapped to different features. (That is, the feature indexed 256{y * w + x) + I{x, y) is active if and only if the intensity in position (x, y) is I{x, y).) In our experiments, the values for wand hare 20 since each face sample has been normalized to an image of 20 x 20 pixels. Note that although the number of potential features in our representation is 102400 (400 x 256), only 400 of those are active (present) in each example, and it is plausible that many features will never be active. Since the algorithm's complexity depends on the number of active features in an example, rather than the total number of features, the sparseness also ensures efficiency. 3.2 Multi-scale Features Many vision problems have utilized multi-scale features to capture the structures of an object. However, extracting detailed multi-scale features using edge or region information from segmentation is a computationally expensive task. Here we use the SNo W paradigm to extract Boolean features that represent multi-scale information. This is done in a similar way to the {position x intensity} used in Sec. 3.1, only that in this case we encode, in addition to position, the mean and variance of a multi-scale pixel. The hope is that the multi-scale feature will capture information that otherwise requires many pixel-based features to represent, and thus simplify the learning problem. Uninformative multi-scale features will be quickly assigned low weights by the learning algorithm and will not degrade performance. Since each face sample is normalized to be a rectangular image of the same size, it suffices to consider rectangular sub-images with varying size from face samples, and for each generate features in terms of the means and variances of their intensity values. Empirical results show that faces can be described effectively this way. Instead of using the absolute values of the mean and variance when encoding the features, we discretize these values into a predefined number of classes. Since the distribution of the mean values as well as the variance values is normal, the discretization is finer near the means of these distributions. The total number of values was determined empirically to be 100, out of which 80 ended up near the mean. Given that, we use the same scheme as in Sec. 3.1 to map the {position x intensi ty mean x intensity variance} space into the Boolean feature space. This is done separately for four different sub-image scales, of 1 x 1, 2 x 2, 4 x 4 to 10 x 10 pixels. The multi-scale feature vector consists of active features corresponding to all these scales. The number of active features in each example is therefore 400 + 100 + 25 + 4, although the total number of features is much larger. In recent work we have used more sophisticated conjunctive features for this purpose yielding even better results. However, the emphasis here is that with the SNoW approach, even very simplistic features support excellent performance. 4 Empirical Results We tested the SNoW-based approach with both sets of features on the two sets of images collected by Rowley [20], and Sung [24]. Each image is scanned with a rectangular window to determine whether a face exists in the window or not. To detect faces of different scales, each input image is repeatedly subsampled by a factor of 1.2 and scanned through for 10 iterations. Table 1 shows the reported 866 M-H. Yang, D. Roth and N Ahuja experimental results of the SNoW-based face detectors and several face detection systems using the two benchmark data sets (available at http://www.cs.cmu.edu/ -har/ faces.html). The first data set consists of 130 images with 507 frontal faces and the second data set consists of 23 images with 155 frontal faces. There are a few hand drawn faces and cartoon faces in both sets. Since some methods use intensity values as their features, systems 1-4 and 7 discard these such hand drawn and cartoon faces. Therefore, there are 125 images with 483 faces in test set 1 and 20 images with 136 faces in test set 2 respectively. The reported detection rate is computed as the ratio between the number of faces detected in the images by the system and the number of faces identified there by humans. The number of false detections is the number of non-faces detected as faces. It is difficult to evaluate the performance of different methods even though they use the same benchmark data sets because different criteria (e.g. training time, number of training examples involved, execution time, number of scanned windows in detection) can be applied to favor one over another. Also, one can tune the parameters of one's method to increase the detection rates while increasing also the false detections. The methods using neural networks [20], distribution-based [24], Kullback relative information [4] and naive Bayes [23] report several experimental results based on different sets of parameters. Table 1 summarizes the best detection rates and corresponding false detections of these methods. Although the method in [4] has the highest detection rates in one benchmark test, this was done by significantly increasing the number of false detections. Other than that, it is evident that the SNoW-based face detectors outperforms others in terms of the overall performance. These results show the credibility of SNoW for these tasks, as well Table 1: Experimental results on images from test set 1 (125 images with 483 faces) in [20] and test set 2 (20 images with 136 faces) in [24] (see text for details) II Test Set 1 Test Set 2 Method /I Detect Rate False Detects Detect Rate 1 False Detects SNoW w/ priDlitive features 94.2'70 84 93.6'70 3 SNoW wi Dlulti-scale features 94.8% 78 94.1% 3 Mixture of factor analyzers [261 92.3'70 82 89.4'70 3 Fisher linear discriminant [271 93.6'7. 74 91.5'7. 1 Distribution-based. [24 J N~ N/A 81.9% 13 Neural network [20J 92.5J"o 862 90.3% 42 Naive Bayes [23J 93.0% 88 91.2% 12 Kullback relative information [41 98.0'7. 12758 NjA NjA Support vector machine [14J N/A N/A 74.2'7. 20 as exhibit the improvement achieved by increasing the expressiveness of the features. This may indicate that further elaboration of the features, which can be done in a very general and flexible way within SNoW, would yield further improvements. In addition to comparing feature sets, we started to investigate some of the reasons for the success of SNoW in this domain, which we discuss briefly below. Two potential contributions are the Winnow update rule and the architecture. First, we studied the update rule in isolation, independent of the SNoW architecture. The results we got when using the Winnow simply as a discriminator were fairly poor (63.9%/65.3% for Test Set 1, primitive and multi-scale features, respectively, and similar results for the Test Set 2.). The results are not surprising, given that Winnow is used here only as a discriminator and is using only positive weights. Investigating the architecture in isolation reveals that weighting or discarding features based on their contribution to mistakes during training, as is done within SNoW, is crucial. Considering the active features uniformly (separately for faces and non-faces) yields poor results. Specifically, studying the resulting SNoW network shows that the total number of features that were active with non-faces is 102,208, out of 102,400 possible A SNoW-Based Face Detector 867 (primitive) features. The total number of active features in faces was only 82,608, most of which are active only a few times. In retrospect, this is clear given the diverse set of images used as negative examples, relative to the somewhat restricted (by nature) set of images that constitute faces. (Similar phenomenon occurs with the multi-scale features, where the numbers are 121572 and 90528, respectively, out of 135424.) Overall it exhibits that the architecture, the learning regime and the update rule all contribute significantly to the success of the approach. Figure 1 shows some faces detected in our experiments. Note that profile faces and faces under heavy illumination are detected. Experimental results show that profile faces and faces under different illumination are detected very well by our method. Note that although there may exist several detected faces around each face, only one window is drawn to enclose each detected face for clear presentation . . f?,~' "ru . i ... ' Figure 1: Sample experimental results using our method on images from two benchmark data sets. Every detected face is shown with an enclosing window. 5 Discussion and Conclusion Many theoretical and experimental issues are to be addressed before a learning system of this sort can be used to detect faces efficiently and robustly under general conditions. In terms of the face detection problem, the presented method is still not able to detect rotated faces. A recent method [21], addresses this problem by building upon a upright face detector [20] and rotating each test sample to upright position. However, it suffers from degraded detection rates and more false detections. Given our results, we believe that the SNoW approach, if adapted in similar ways, would generalize very well to detect faces under more general conditions. In terms of the SNoW architecture, although the main ingredients of it are understood theoretically, more work is required to better understand its strengths. This is increasingly interesting given that the architecture has been found to perform very well in large-scale problem in the natural language domain as well 868 M-H. Yang, D. Roth and N Ahuja The contributions of this paper can be summarized as follows. We have introduced the SNoW learning architecture to the domain of visual processing and described an approach that detect faces regardless of their poses, facial features and illumination conditions. Experimental results show that this method outperforms other methods in terms of detection rates and false detectionss, while being more efficient both in learning and evaluation. References [1) P. Belhumeur, J. Hespanha, and D. Kriegman. Eigenfaces vs. fisherfaces: Recognition using class specific linear projection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 19(7):711-720, 1997. [2) A. Blum. Learning boolean functions in an infinite attribute space. Machine Learning, 9(4):373386, 1992. [3) A. Carleson, C. Cumby, J. Rosen, and D. Roth. The SNoW learning architecture. Technical Report UIUCDCS-R-99-2101, UIUC Computer Science Department, May 1999. [4) A. J. Colmenarez and T . S. Huang. Face detection with information-based maximum discrimination. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pages 782-787, 1997. [5) A. R. Golding and D. Roth. A winnow based approach to context-sensitive spelling correction. Machine Learning, 34:107-130, 1999. Special Issue on Machine Learning and Natural Language. [6) D. B . Graham and N. M. Allinson. Characterizing virtual eigensignatures for general purpose face recognition. In H. Wechsler, P. J. Phillips, V. Bruce, F. Fogelman-Soulie, and T . S. Huang, editors, Face Recognition: From Theory to Applications, volume 163 of NATO ASI Series F, Computer and Systems Sciences, pages 446-456. Springer, 1998. [7) P. Hallinan. A Deformable Model for Face Recognition Under Arbitrary Lighting Conditions. PhD thesis, Harvard University, 1995. [8) D. Helmbold and M. K. Warmuth. On weak learning. Journal of Computer and System Sciences, 50(3):551-573, June 1995. [9) J . kivinen and M. K. Warmuth. Exponentiated gradient versus gradient descent for linear predictors. In Proceedings oj the Annual ACM Symposium on the Theory of Computing, 1995. [10) [11) [12) [13) [14) [15) [16) [17) [18) [19) [20) [21) [22) [23) [24) [27) N. Littlestone. Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. Machine Learning, 2:285-318, 1988. N. Littlestone. Redundant noisy attributes, attribute errors, and linear threshold learning using winnow. In Proceedings oj the fourth Annual Workshop on Computational Learning Theory, pages 147-156, 1991. B. Moghaddam and A . Pentland. Probabilistic visual learning for object recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 19(7):696-710, 1997. M. Munoz, V. Punyakanok, D. Roth, and D. Zimak. A learning approach to shallow parsing. In EMNLP- VLC'99, the Joint SIGDAT Conference on Empirical Methods in Natural Language Processing and Very Large Corpora, June 1999. E. Osuna, R . Freund, and F. Girosi. Training support vector machines: an application to face detection. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pages 130-136, 1997. P. J. Phillips, H. Moon, S. Rizvi, and P. Rauss. The feret evaluation. In H. Wechsler, P. J. Phillips, V . Bruce, F. Fogelman-Soulie, and T. S. Huang, editors, Face Recognition: From Theory to Applications, volume 163 of NATO ASI Series F, Computer and Systems Sciences, pages 244-261. Springer, 1998. R. J. Qian and T. S. Huang. Object detection using hierarchical mrf and map estimation. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pages 186-192, 1997. A. N. Rajagopalan, K. S. Kumar, J. Karlekar, R. Manivasakan, and M. M. Patil. Finding faces in photographs. In Proceedings of the Sixth International Conference on Computer Vision, pages 640-645, 1998. D. Roth. Learning to resolve natural language ambiguities: A unified approach. In Proceedings of the Fifteenth National Conference on Artificial Intelligence, pages 806-813, 1998. D. Roth and D. Zelenko. Part of speech tagging using a network of linear separators. In COLINGACL 98, The 17th Int. Conference on Computational Linguistics, pages 1136-1142, 1998. H. Rowley, S. Baluja, and T. Kanade. Neural network-based face detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 20(1):23-38, 1998. H. Rowley, S. Baluja, and T . Kanade . Rotation invariant neural network-based face detection. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pages 38-44, 1998. F. S. Samaria. Face Recognition Using Hidden Markov Models. PhD thesis, University of Cambridge, 1994. H. Schneiderman and T . Kanade. Probabilistic modeling of local appearance and spatial relationships for object recognition. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pages 45- 51, 1998. K.-K. Sung and T . Poggio. Example-based learning for view-based human face detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 20(1):39-51, 1998. L. G. Valiant. A theory of the learnable. Commun. ACM, 27(11):1134-1142, Nov. 1984. M.-H. Yang, N. Ahuja, and D. Kriegman. Face detection using a mixture of factor analyzers. In Proce'edings of the IEEE International Conference on Image Processing, 1999. M.-H. Yang, N. Ahuja, and D. Kriegman. Mixtures of linear subspaces for face detection. In Proceedings of the Foruth IEEE International Conference on Automatic Face and Gesture Recognition, 2000.
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The Entropy Regularization Information Criterion Alex J. Smola Dept. of Engineering and RSISE Australian National University Canberra ACT 0200, Australia Alex.Smola@anu.edu.au Bernhard Scholkopf Microsoft Research Limited St. George House, 1 Guildhall Street Cambridge CB2 3NH bsc@microsoft.com John Shawe-Taylor Royal Holloway College University of London Egham, Surrey 1W20 OEX, UK john@dcs.rhbnc.ac.uk Robert C. Williamson Dept. of Engineering Australian National University Canberra ACT 0200, Australia Bob. Williamson @anu.edu.au Abstract Effective methods of capacity control via uniform convergence bounds for function expansions have been largely limited to Support Vector machines, where good bounds are obtainable by the entropy number approach. We extend these methods to systems with expansions in terms of arbitrary (parametrized) basis functions and a wide range of regularization methods covering the whole range of general linear additive models. This is achieved by a data dependent analysis of the eigenvalues of the corresponding design matrix. 1 INTRODUCTION Model selection criteria based on the Vapnik-Chervonenkis (VC) dimension are known to be difficult to obtain, worst case, and often not very tight. Yet they have the theoretical appeal of providing bounds, with few or no assumptions made. Recently new methods [8, 7, 6] have been developed which are able to provide a better characterization of the complexity of function classes than the VC dimension, and moreover, are easily obtainable and take advantage of the data at hand (i.e. they employ the concept of luckiness). These techniques, however, have been limited to linear functions or expansions of functions in terms of kernels as happens to be the case in Support Vector (SV) machines. In this paper we show that the previously mentioned techniques can be extended to expansions in terms of arbitrary basis functions, covering a large range of practical algorithms such as general linear models, weight decay, sparsity regularization [3], and regularization networks [4]. The Entropy Regularization Information Criterion 343 2 SUPPORT VECTOR MACHINES Support Vector machines carry out an effective means of capacity control by minimizing a weighted sum of the training error (1) and a regularization term Q[J] = ~llwI12; i.e. they minimize the regularized risk functional 1 m A Rreg[J] := Remp[f] + AQ[f] = m ~ C(Xi, Yi, f(Xi)) + "2llwI12. (2) t=l Here X := {Xl, ... Xm} C X denotes the training set, Y := {YI, ... Ym} C }j the corresponding labels (target values), X, }j the corresponding domains, A > a a regularization constant, C : X X }j X }j -+ JRt a cost function, and f : X -+ }j is given by f(x) := (x, w), or in the nonlinear case f(x) := (4l(x), w). (3) Here 4l : X -+ l' is a map into a feature space 1'. Finally, dot products in feature space can be written as (4l(x), 4l(X')) = k(x, x') where k is a so-called Mercer kernel. For n E N, ~n denotes the n-dimensional space of vectors x = (Xl, ... , Xn). We define spaces f; as follows: as vector spaces, they are identical to ~n, in addition, they are endowed with p-norms: fora < p < 00 forp = 00 We write fp = fr;:o Furthermore let Ue~ := {x: Ilxlle~ ::; I} be the unitf;-baU. For model selection purposes one wants to obtain bounds on the richness of the map S x Sx : w f-t (f(xd, ... , f(xm)) = ((4l(xd, w), ... , (4l(xm), w)). (4) where w is restricted to an f2 unit ball of some radius A (this is equivalent to choosing an appropriate value of A an increase in A decreases A and vice versa). By the "richness" of S x specificaUy we mean the f: €-covering numbers N( €, S X (AUe;;, ), f1:J of the set Sx(AUlm). In the standard COLT notation, we mean p { There exists a set {Zl, ... zn} C F such that for all } N(€, SX(AUl;;')' f:) := min n Z E Sx(AUem) we have min liz - zililm < € p l::;i::;n 00 See [8] for further details. When carrying out model selection in this case, advanced methods [6] exploit the distribution of X mapped into feature space 1', and thus of the spectral properties of the operator Sx by analyzing the spectrum of the Gram matrix G = [gij]ij, where gij := k(Xi, Xj). All this is possible since k(Xi,Xj) can be seen as a dot product of Xi,Xj mapped into some feature space 1', i.e. k(Xi, Xj) = (4l(Xi), 4l(Xj )) . This property, whilst true for SV machines with Mercer kernels, does not hold in general case where f is expanded in terms of more or less arbitrary basis functions. 344 A. J. Smola. J. Shawe-Taylor, B. Sch61kopf and R. C. Williamson 3 THE BASIC PROBLEMS One basic problem is that when expanding 1 into n (5) i=l with Ii (x) being arbitrary functions, it is not immediately obvious how to regard 1 as a dot product in some feature space. One can show that the VC dimension of a set of n linearly independent functions is n. Hence one would intuitively try to restrict the class of admissible models by controlling the number of basis functions n in terms of which 1 can be expanded. Now consider an extreme case. In addition to the n basis functions Ii defined previously, we are given n further basis functions II, linearly independent of the previous ones, which differ from Ii only on a small domain X', i.e. Iilx\x1 = IIlx\xl. Since this new set of functions is linearly independent, the VC dimension of the joint set is given by 2n. On the other hand, if hardly any data occurs on the domain X', one would not notice the difference between Ii and II. In other words, the joint system of functions would behave as if we only had the initial system of n basis functions. An analogous situation occurs if II = Ii + €gi where € is a small constant and gi was bounded, say, within [0, 1J. Again, in this case, the additional effect of the set offunctions II would be hardly noticable, but still, the joint set of functions would count as one with VC dimension 2n. This already indicates, that simply counting the number of basis functions may not be a good idea after all. .' ''~ Figure 1: From left to right: (a) initial set of functions h, ... , 15 (dots on the x-axis indicate sampling points); (b) additional set of functions IL ... , I~ which differ globally, but only by a small amount; (c) additional set offunctions IL ... , I~ which differ locally, however by a large amount; (d) spectrum of the corresponding design matrices - the bars denote the cases (a)-(c) in the corresponding order. Note that the difference is quite small. On the other hand, the spectra of the corresponding design matrices (see Figure 1) are very similar. This suggests the use of the latter for a model selection criterion. Finally we have the practical problem that capacity control, which in SV machines was carried out by minimizing the length of the "weight vector" w in feature space, cannot be done in an analogous way either. There are several ways to do this. Below we consider three that have appeared in the literature and for which there exist effective algorithms. Example 1 (Weight Decay) Define Q[IJ := ~ L:i ar .. i.e. the coefficients ai of the junction expansion are constrained to an £2 ball. In this case we can consider the following operator S(1)· £n -t £m where X . 2 00' Sr): aM (f(xd, ... , I(xm)) = ((f(Xl), a), . .. , (f(Xm), a)) = Fa (6) Here I(x):= Ul(x) , .. ·In(x)), Fij := Ii(Xj), a'- (al, ... ,an) and a E AUl'2for some A> O. The Entropy Regularization Information Criterion 345 Example 2 (Sparsity Regularization) In this case Q[J] := Li lail, i.e. the coefficients ai of the function expansion are constrained to an £1 ball to enforce sparseness [3]. Thus sC;) : £1 -t £~ with sC;) mapping a as in (6) except a E AUlI. This is similar to expansions encountered in boosting or in linear programming machines. Example 3 (Regularization Networks) Finally one could set Q[J] := ~a T Qa for some positive definite matrix Q. For instance, Qij could be obtainedfrom (Ph, P fj) where P is a regularization operator penalizing non-smooth functions [4J. In this case a lives inside some n-dimensional ellipsoid. By substituting a' := Q% a one can reduce this setting to the case of example 1 with a different set of basis functions (f'(x) = Q-% f(x)) and consider an evaluation operator s~) : £2 -t £: given by s~): a' f-+ (f(xd, . .. , f(xm)) = ((Q-% f(X1), a'), . .. , (Q-t f(xm), a')) = Q-t Fa' (7) where a' E AUl2 for some A> 0 and Fij = fi(xj) as in example 1. Example 4 (Support Vector Machines) An important special case of example 3 are Support Vector Machines where we have Qij = k(Xi,Xj) andfi(x) = k(Xi,X), henceQ = F. Hence the possible values generated by a Support Vector Machine can be written as s~): a' f-+ (f(X1), ... , f(xm)) = ((Q-% f(xd, a'), . .. , (Q-% f(xm), a')) = Ft a' (8) where a' E AUl2 for some A > o. 4 ENTROPY NUMBERS Covering numbers characterize the difficulty of learning elements of a function class. Entropy numbers of operators can be used to compute covering numbers more easily and more tightly than the traditional techniques based on VC-like dimensions such as the fat shattering dimension [1]. Knowing el (S x) = € (see below for the definition) tells one that 10g:N(€ , F,£~) ::; I, where F is the effective class of functions used by the regularised learning machines under consideration. In this section we summarize a few basic definitions and results as presented in [8] and [2]. The lth entropy number €l (F) of a set F with a corresponding metric d is the precision up t~ whicI:! F can _be approximated by 1 elements of F; i.e. for all f E F there exists some fi E {h, ···, fd such that d(f, fi) ::; €l. Hence €1(F) is the functional inverse of the covering number of F. The entropy number of an bounded linear operator T: A -t B between normed linear spaces A and B is defined as €1(T) := €1(T(UA)) with the metric d being induced by II . liB. The dyadic entropy numbers el are defined by el := €2'+1 (the latter quantity is often more convenient to deal with since it corresponds to the log of the covering number). We make use of the following three results on entropy numbers of the identity mapping from £;1 into £;2' diagonal operators, and products of operators. Let id;l ,P2 : £;1 -t £;2 ; id;1 ,P2 : x f-+ x The following result is due to Schlitt; the constants 9.94 and 1.86 were obtained in [9]. Proposition 1 (Entropy numbers for identity operators) Be mEN. Then 1 el(id~,2) ::; 9.94 (t log (1 + T) ) 2 1 & el (id~, (xJ ::; 1.86 (t log (1 + T) ) 2 (9) 346 A. J Smola, J Shawe-Taylor, B. SchOlkopfand R. C. Williamson Proposition 2 (Carl and Stephani [2, p.11]) Let E, F, G be Banach spaces, R : F -+ G, and S: E -+ F. Then,forn, tEN, en+t-l (RS) ~ en(R)et(S), en(RS) ~ en(R)IISII and en(RS) ~ en(S)IIRII. (to) Note that the latter two inequalities follow directly from the fact that €l (R) = IIRIlfor all R: F -+ G by definition of the operator norm IIRII. Proposition 3 Let 0"1 ~ 0"2 ~ . .. ~ O"j ~ . .. ~ 0, 1 ~ p ~ 00 and (11) for x = (Xl, X2, ... , Xj, . .. ) E f!p be the diagonal operator from f!p into itself, generated by the sequence (0" j ) j. Then for all n E N, 5 THE MAIN RESULT We can now state the main theorem which gives bounds on the entropy numbers of S~) for the first three examples of model selection described above (since Support Vector Machines are a special case of example 3 we will not deal with it separately). Proposition 4 Let! be expanded in a linear combination of basis functions as ! .L~=l adi and the coefficients a restricted to one of the convex sets as described in the examples 1 to 3. Moreover denote by Fij := !j(Xi) the design matrix on a particular sample X, and by Q the regularization matrix in the case of example 3. Then the following bound on Sx holds. 1. In the case of weight decay (ex. 1 )(with h + l2 ~ l + 1) el(S~)) ~ 1.96 (llllog(1 +m/h))t eI2(~)' (13) 2. 1n the case of weight sparsity regularization (ex. 2) (with h + l2 + l3 ~ l + 2) el(S~)) ~ 18.48 (lillog (1 + m/h)) t el2 (~) (l3'llog (1 + m/l3)) t. (14) 3. Finally, in the case of regularization networks (ex. 3) (with II + l2 ~ l + 1) el (Sr)) ~ 1.96 (lillog (1 + m/h)) t el2 (~). (15) Here ~ is a diagonal scaling operator (matrix) with (i, i) entries .j(ii and (.j(ii)i are the eigenvalues (sorted in decreasing order) of the matrix FFT in the case of examples 1 and 2, and FQ-l FT in the case of example 3. The entropy number of ~ is readily bounded in terms of (O"i)i by using (3). One can see that the first setting (weight decay) is a special case of the third one, namely when Q = 1, i.e. when Q is just the identity matrix. Proof The proofrelies on a factorization of S~) (i = 1,2,3) in the following way. First we consider the equivalent operator S x mapping from f!~ to f!r and perform a singular value decomposition [5] of the latter into S x = V~W where V, W are operators of norm 1, and ~ contains the singular values of S~), i.e. the singular values of F and FQ- t The Entropy Regularization Information Criterion 347 respectively. The latter, however, are identical to the square root of the eigenvalues of F FT or FQ-l FT. Consequently we can factorize S~) as in the diagram (16) Finally, in order to compute the entropy number of the overall operator one only has to use the factorization of Sx into S~) = id~oo VL:W for i E {1,3} and into S~ ) = id~oo VL:Wid~, 2 for example 2, and apply Proposition 2 several times. We also exploit the fact that for singular value decompositions IIVI\' IIWII s l. • The present theorem allows us to compute the entropy numbers (and thus the complexity) of a class of functions on the current sample X. Going back to the examples of section 3, which led to large bounds on the VC dimension one can see that the new result is much less susceptible to such modifications: the addition of f{. ... f~ to h, ... f n does not change the eigenspectrum L: of the design matrix significantly (possibly only doubling the nominal value of the singular values), if the functions fi differ from fi only slightly. Consequently also the bounds will not change significantly even though the number of basis functions just doubled. Also note that the current error bounds reduce to the results of [6] in the SV case: here Q ij = Fij = k( Xi, X j) (both the design matrix F and the regularization matrix Q are determined by kernels) and therefore FQ-l F = Q. Thus the analysis of the singular values of FQ-l F leads to an analysis of the eigenvalues of the kernel matrix, which is exactly what is done when dealing with SV machines. 6 ERROR BOUNDS To use the above result we need a bound on the expected error of a hypothesis f in terms of the empirical error (training error) and the observed entropy numbers €n(J'). We use [6, Theorem 4.1] with a small modification. Theorem 1 Let:1' be a set of linear junctions as described in the previous examples with en(Sx) as the corresponding bound on the observed entropy numbers of:1' on the dataset X. Moreover suppose thatforafixed threshold b E [?for some f E :1', sgn(f - b) correctly classifies the set X with a margin 'Y := minlSiSm If(Xi) - bl. Finally let U := min{ n E N with en(Sx) s 'Y /8.001} and a(U, <5) := 3.08(1 + bIn t). Then with confidence 1- <5 over X (drawn randomly from pm where P is some probability distribution) the expected error ofsgn(f - b) is boundedfrom above by €(m,U,<5) =! (U(1+a(U,~)log(5t-m)log(17m)) + log (l6r)) . (17) The proof is essentially identical to that of [6, Theorem 4.1] and is omitted. [6] also shows how to compute en (S x) efficiently including an explicit formula for evaluating el (L:). 7 DISCUSSION We showed how improved bounds could be obtained on the entropy numbers of a wide class of popular statistical estimators ranging from weight decay to sparsity regularization 348 A. J Smola. J Shawe-Taylor, B. SchOllropf and R. C. Williamson (with SV machines being a special case thereof). The results are given in a way that is directly useable for practicioners without any tedious calculations of the VC dimension or similar combinatorial quantities. In particular, our method ignores (nearly) linear dependent basis functions automatically. Finally, it takes advantage of favourable distributions of data by using the observed entropy numbers as a base for stating bounds on the true entropy numbers with respect to the function class under consideration. Whilst this leads to significantly improved bounds (we achieved an improvement of approximately two orders of magnitude over previous VC-type bounds involving only the radius of the data R and the weight vector IIwll in the experiments) on the expected risk, the bounds are still not good enough to become predictive. This indicates that possibly rather than using the standard uniform convergence bounds (as used in the previous section) one might want to use other techniques such as a PAC-Bayesian treatment (as recently suggested by Herbrich and Graepel) in combination with the bounds on eigenvalues of the design matrix. Acknowledgements: This work was supported by the Australian Research Council and a grant of the Deutsche Forschungsgemeinschaft SM 62/1-1. References [1] N. Alon, S. Ben-David, N. Cesa-Bianchi, and D. Haussler. Scale-sensitive Dimensions, Uniform Convergence, and Learnability. 1. of the ACM, 44(4):615-631,1997. [2] B. Carl and I. Stephani. Entropy, compactness, and the approximation of operators. Cambridge University Press, Cambridge, UK, 1990. [3] S. Chen, D. Donoho, and M. Saunders. Atomic decomposition by basis pursuit. Technical Report 479, Department of Statistics, Stanford University, 1995. [4] F. Girosi, M. Jones, and T. Poggio. Regularization theory and neural networks architectures. Neural Computation, 7:219-269,1995. [5] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, 1992. [6] B. Scholkopf, J. Shawe-Taylor, A. J. Smola, and R. C. Williamson. Generalization bounds via eigenvalues of the gram matrix. Technical Report NC-TR-99-035, NeuroColt2, University of London, UK, 1999. [7] J. Shawe-Taylor and R. C. Williamson. Generalization performance of classifiers in terms of observed covering numbers. In Proc. EUROCOLT'99, 1999. [8] R. C. Williamson, A. J. Smola, and B. Scholkopf. Generalization performance of regularization networks and support vector machines via entropy numbers of compact operators. NeuroCOLT NC-TR-98-019, Royal Holloway College, 1998. [9] R. C. Williamson, A. J. Smola, and B. SchOlkopf. A Maximum Margin Miscellany. Typescript, 1999.
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A Winner-Take-All Circuit with Controllable Soft Max Property Shih-Chii Lin Institute for Neuroinformatics, ETHjUNIZ Winterthurstrasse 190, CH-8057 Zurich Switzerland shih@ini.phys.ethz.ch Abstract I describe a silicon network consisting of a group of excitatory neurons and a global inhibitory neuron. The output of the inhibitory neuron is normalized with respect to the input strengths. This output models the normalization property of the wide-field directionselective cells in the fly visual system. This normalizing property is also useful in any system where we wish the output signal to code only the strength of the inputs, and not be dependent on the number of inputs. The circuitry in each neuron is equivalent to that in Lazzaro's winner-take-all (WTA) circuit with one additional transistor and a voltage reference. Just as in Lazzaro's circuit, the outputs of the excitatory neurons code the neuron with the largest input. The difference here is that multiple winners can be chosen. By varying the voltage reference of the neuron, the network can transition between a soft-max behavior and a hard WTA behavior. I show results from a fabricated chip of 20 neurons in a 1.2J.Lm CMOS technology. 1 Introduction Lazzaro and colleagues (Lazzaro, 1988) were the first to implement a hardware model of a winner-take-all (WTA) network. This network consists of N excitatory cells that are inhibited by a global signal. Improvements of this network with addition of positive feedback and lateral connections have been described (Morris, 1998; Indiveri, 1998). The dynamics and stability properties of networks of coupled excitatory and inhibitory neurons have been analyzed by many (Amari, 1982; Grossberg, 1988). Grossberg described conditions under which these networks will exhibit WTA behavior. Lazzaro's network computes a single winner as reflected by the outputs of the excitatory cells. Several winners can be chosen by using more localized inhibition. In this work, I describe two variants of a similar architecture where the outputs of the excitatory neurons code the relative input strengths as in a soft-max computation. The relative values of the outputs depend on the number of inputs, their relative strengths and two parameter settings in the network. The global inhibitory 718 8.-c. Liu Figure 1: Network model of recurrent inhibitory network. signal can also be used as an output. This output saturates with increasing number of active inputs, and the saturation level depends on the input strengths and parameter settings. This normalization property is similar to the normalization behavior of the wide-field direction-selective cells in the fly visual system. These cells code the temporal frequency of the visual inputs and are largely independent of the stimulation size. The circuitry in each neuron in the silicon network is equivalent to that in Lazzaro et. al.'s hard WTA network with an additional transistor and a voltage reference. By varying the voltage reference, the network can transition between a soft-max computation and a hard WTA computation. In the two variants, the outputs of the excitatory neurons either code the strength of the inputs or are normalized with respect to a constant bias current. Results from a fabricated network of 20 neurons in a 1.2J.Lm AMI CMOS show the different regimes of operation. 2 Network with Global Inhibition The generic architecture of a recurrent network with excitatory neurons and a single inhibitory neuron is shown in Figure 1. The excitatory neurons receive an external input, and they synapse onto a global inhibitory neuron. The inhibitory neuron, in turn, inhibits the excitatory neurons. The dynamics of the network is described as follows: N dYi ~ dt = -Yi + ei g(~ WjYj) (1) j=l where Wj is the weight of the synapse between the jth excitatory neuron and the inhibitory neuron, and Yj is the state of the jth neuron. Under steady-state conditions, Yi = ei - YT, where YT = g(L:~l WjYj)· Assume a linear relationship between YT and Yj, and letting Wj = W, N "N _ ~ _ W L..Jj=l ej YT - W ~ Yj 1 + wN j=l As N increases, YT = L:;l ej • If all inputs have the same level, e, then YT = e. A Winner-Take-All Circuit with Controllable Soft Max Property 719 Figure 2: First variant of the architecture. Here we show the circuit for two excitatory neurons and the global inhibition neuron, M 4 • The circuit in each excitatory neuron consists of an input current source, h, and transistors, M1 to M3 . The inhibitory transistor is a fixed current source, lb . The inputs to the inhibitory transistor, 101 and I~2 are normalized with respect to lb. 3 First Variant of Network with Fixed Current Source In Sections 3 and 4, I describe two variants of the architecture shown in Figure 1. The two variants differ in the way that the inhibition signal is generated. The first network in Figure 2 shows the circuitry for two excitatory neurons and the inhibition neuron. Each excitatory neuron is a linear threshold unit and consists of an input current, h, and transistors, Ml, M 2 , and M 3 . The state of the neuron is represented by the current, Ir1 . The diode-connected transistor, M2 , introduces a rectifying nonlinearity into the system since Ir1 cannot be negative. The inhibition current, Ir, is sunk by M 1 , and is determined by the gate voltage, VT. The inhibition neuron consists of a current source, Ib, and VT is determined by the corresponding current, Ir1 and the corresponding transistor, M3 in each neuron. Notice that IT cannot be greater than the largest input to the network and the inputs to this network can only be excitatory. The input currents into the transistor, M 4 , are defined as 101 and 102 and are normalized with respect to the current source, h. In the hard WTA condition, the output current of the winning neuron is equal to the bias current, h. This network exhibits either a soft-maximum behavior or a hard WTA behavior depending on the value of an external bias, Va. The inhibition current, IT, is derived as: (2) where N is the number of "active" excitatory neurons (that is, neurons whose Ii > IT), Ii is the same input current to each neuron, and Ia = Ioe",vQ/uT. In deriving the above equation, we assumed that K, = 1. The inhibition current, IT, is a linear combination of the states of the neurons because Ir = 2:f Iri x Ial h· Figure 3(a) shows the response of the common-node voltage, VT, as a function of the number of inputs for different input values measured from a fabricated silicon network of 20 neurons. The input current to each neuron is provided by a pFET transistor that is driven by the gate voltage, Yin. All input currents are equal in this figure. The saturation behavior of the network as a function of the number 720 s.-c. Liu 0.8,---~-~-~-~-~----, Vin=3.9V 0.7 -' . 0.6 Vin=4.3V .II'~''''--.'' ......... •• ' .. ---... ... ..... .. 5 10 15 20 25 30 5 10 15 20 25 30 Number of inputs Number of inputs (a) (b) Figure 3: (a) Common-node voltage, VT, as a function of the number of input stimuli. Va = O.8V. (b) Common-node voltage, VT, as a function of the number of inputs with an input voltage of 4.3V and Vb = O.7V. The curves correspond to different values of Va . of inputs can be seen in the different traces and the saturation level increases as Vin decreases. As seen in Equation 2, the point at which the response saturates is dependent on the ratio, h / I a. In Figure 3(b), I show how the curve saturates at different points for different values of Va and a fixed hand Vin. In Figure 4, I set all inputs to zero except for two inputs, Vin1 and Vin2 that are set to the same value. I measured 101 and 101 as a function of Va as shown in Figure 4(a). The four curves correspond to four values of Vin. Initially both currents 101 and 102 are equal as is expected in the soft-max condition. As Va increases, the network starts exhibiting a WTA behavior. One of the output currents finally goes to zero above a critical value of Va. This critical value increases for higher input currents because of transistor backgate effects. In Figure 4(b), I show how the output currents respond as a function of the differential voltage between the two inputs as shown in Figure 4. Here, I fixed one input at 4.3V and swept the second input differentially around it. The different curves correspond to different values of Va. For a low value of Va, the linear differential input range is about lOOmV. This linear range decreases as Va is increased (corresponding to the WTA condition). 4 Second Variant with Diode-Connected Inhibition Transistor In the second variant shown in Figure 5, the current source, M4 is replaced by a diode-connected transistor and the output currents, 10i' follow the magnitude of the input currents. The inhibition current, Ir, can be expressed as follows: (3) where la is defined in Section 3. We sum Equation 3 over all neurons and assuming equal inputs, we get Ir = J'LJri x la. This equation shows that the feedback signal has a square root dependence on the neuron states. As we will see, this causes the feedback signal to saturate quickly with the number of inputs. A Winner-Take-All Circuit with Controllable Soft Max Property 721 6 r-... ".\'0" 5 ~4 ~ , 2 2.5 Va=O 5~'~\ I /~. " ~ 2 Va=O 6V Al\ ',/' Va=O.4V -: 1.5 Va=O.7V ,i ..s l~ ) ' ~ ,. , <\ ,,~ / .. \ \> < .,.~J \ \ '" 0.5 -0.2 -0.1 0 0.1 0.2 0.3 Vio2-Viol (V) (a) (b) Figure 4: (a) Output currents, 101 and 102 , as a function of Va: for a subthreshold bias current and Yin = 4.0V to 4.3V. (b) Outputs, 101 and 102 , as a function of the differential input voltage, ~ Vin, with Yinl = 4.3V. Figure 5: Second variant of network. The schematic shows two excitatory neurons with diode-connected inhibition transistor. Substituting lri = Ii - IT in Equation 3, we solve for Jr, N IT = -1a:N + (Ia:N )2 + 41a: L Ii (4) From measurements from a fabricated circuit with 20 neurons, I show the dependence of VT (the natural logarithm of Jr) on the number of inputs in Figure 6(a). The output saturates quickly with the number of inputs and the level of saturation increases with increased input strengths. All the inputs have the same value. The network can also act as a WTA by changing Va:. Again, all inputs are set to zero except for two inputs whose gate voltages are both set at 4.2V. As shown in Figure 6(b), the output currents, 101 and 102 , are initially equal, and as Va: increases above 0.6V, the output currents split apart and eventually, 102 = OA. The final value of 101 depends on the maximum input current. This data shows that the network acts as a WTA circuit when Va: > 0.73V. If I set Vin2 = 4.25V instead, the output currents split at a lower value of Va:. 722 0.45.--~-~-~-~-~------, 0.20'---:-2 -~4-~6:--8=---1-:':0:----:'12 Number of inputs (a) 5 4 $ ~3 2 Vinl=4.2V. Vin2=4.25V 0.6 0.7 Va (V) (b) s.-c. Liu 0.8 0.9 Figure 6: (a) Common-node voltage, VT, as a function of the number of inputs for input voltages, 3.9V, 4.06V, and 4.3V for Va = O.4V. (b) Outputs, 101 and 1 02 , as a function of Va for Vinl = 4.2V, Vin2 = 4.25V for the 2 curves with asterisks and for Vinl = Vin2 = 4.2V for the 2 curves with circles. 5 Inhibition The WTA property arises in both variants of this network if the gain parameter, Va, is increased so that the diode-connected transistor, M2 , can be ignored. Both variants then reduce to Lazzaro's network. In the first variant, the feedback current (Ir) is a linear combination ofthe neuron states. However, when the gain parameter is increased so that M2 can be ignored, the feedback current is now a nonlinear combination of the input states so the WTA behavior is exhibited by these reduced networks. Under hard WTA conditions, if Ir is initially smaller than all the input currents, the capacitances C at the nodes Vr1 and Vr2 are charged up by the difference between the individual input current and IT, i.e., d~t = liCIT. Since the inhibition current is a linear combination of Iri and Iri is exponential in Vri , we can see that IT is a sum of the exponentials of the input currents, h Hence the feedback current is nonlinear in the input currents. Another way of viewing this condition in electronic terms is that in the soft WTA condition, the output node of each neuron is a softimpedance node, or a low-gain node. In the hard WTA case, the output node is now a high-impedance node or a high-gain node. Any input differences are immediately amplified in the circuit. 6 Discussion Hahnloser (Hahnloser, 1998) recently implemented a silicon network of linear threshold excitatory neurons that are coupled to a global inhibitory neuron. The inhibitory signal is a linear combination of the output states of the excitatory neurons. This network does not exhibit WTA behavior unless the excitatory neurons include a self-excitatory term. The inhibition current in his network is also generated via a diode-connected transistor. The circuitry in two variants described here is more compact than the circuitry in his network. Recurrent networks with the architecture described in this paper have been proposed by Reichardt and colleagues (Reichardt, 1983) in modelling the aggregation property A Winner-Take-All Circuit with Controllable Soft Max Property 723 of the wide-field direction-selective cells in flies. The synaptic inputs are inhibited by a wide-field cell that pools all the synaptic inputs. Similar networks have also been used to model cortical processing, for example, orientation selectivity (Douglas, 1995). The network implemented here can model the aggregation property of the directionselective cells in the fly. By varying a voltage reference, the network implements either a soft-max computation or a hard WTA computation. This circuitry will be useful in hardware models of cortical processing or motion processing in invertebrates. Acknowledgments I thank Rodney Douglas for supporting this work, and the MOSIS foundation for fabricating this circuit. I also thank Tobias Delbriick for proofreading this document. This work was supported in part by the Swiss National Foundation Research SPP grant and the U.S. Office of Naval Research. References Amari, S., and Arbib, M. A., "Competition and cooperation in neural networks," New York, Springer-Verlag, 1982. Grossberg, W., "Nonlinear neural networks: Principles, mechanisms, and architectures," Neural Networks, 1, 17-61, 1988. Hanhloser, R., "About the piecewise analysis of networks of linear threshold neurons," Neural Networks, 11,691- 697, 1988. Hahnloser, R., "Computation in recurrent networks of linear threshold neurons: Theory, simulation and hardware implementation," Ph.D. Thesis, Swiss Federal Institute of Technology, 1998. Lazzaro, J., Ryckebusch, S. Mahowald, M.A., and Mead. C., "Winner-take-all networks of O(n) complexity," In Tourestzky, D. (ed), Advances in Neural Information Processing Systems 1, San Mateo, CA: Morgan Kaufman Publishers, pp. 703-711, 1988. Morris, T.G., Horiuchi, T. and Deweerth, S.P., "Object-based selection within an analog VLSI visual attention system," IEEE Trans. on Circuits and Systems II, 45:12, 1564-1572, 1998. Indiveri, G., "Winner-take-all networks with lateral excitation," Neuromorphic Systems Engineering, Editor, Lande, TS., 367-380, Kluwer Academic, Norwell, MA, 1998. Reichardt, W., Poggio, T., and Hausen, K., "Figure-ground discrimination by relative movement in the visual system of the fly," BioI. Cybern., 46, 1-30, 1983. Douglas, RJ., Koch, C., Mahowald, M., Martin, KAC., and Suarez, HH., "Recurrent excitation in neocortical circuits," Science, 269:5226,981-985, 1995.
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Bayesian model selection for Support Vector machines, Gaussian processes and other kernel classifiers Matthias Seeger Institute for Adaptive and Neural Computation University of Edinburgh 5 Forrest Hill, Edinburgh EHI 2QL seeger@dai.ed.ac.uk Abstract We present a variational Bayesian method for model selection over families of kernels classifiers like Support Vector machines or Gaussian processes. The algorithm needs no user interaction and is able to adapt a large number of kernel parameters to given data without having to sacrifice training cases for validation. This opens the possibility to use sophisticated families of kernels in situations where the small "standard kernel" classes are clearly inappropriate. We relate the method to other work done on Gaussian processes and clarify the relation between Support Vector machines and certain Gaussian process models. 1 Introduction Bayesian techniques have been widely and successfully used in the neural networks and statistics community and are appealing because of their conceptual simplicity, generality and consistency with which they solve learning problems. In this paper we present a new method for applying the Bayesian methodology to Support Vector machines. We will briefly review Gaussian Process and Support Vector classification in this section and clarify their relationship by pointing out the common roots. Although we focus on classification here, it is straightforward to apply the methods to regression problems as well. In section 2 we introduce our algorithm and show relations to existing methods. Finally, we present experimental results in section 3 and close with a discussion in section 4. Let X be a measure space (e.g. X = ~d) and D = (X,t) = {(Xl,tt), ... , (Xn,tn)), Xi E X, ti E {-l,+l} a noisy LLd. sample from a latent function y : X -+ lR, where P(tly) denotes the noise distribution. Given further points X. we wish to predict t. so as to minimize the error probability P(tlx., D), or (more difficult) to estimate this probability. Generative Bayesian methods attack this problem by placing a stochastic process prior P(y(·)) over the space of latent functions and 604 M Seeger then compute posterior and predictive distributions P(yID), P(y.lx., D) as P(yID) = P(Dly)P(y) P(D) , P(y.ID,x.) = f P(y.ly)P(yID) dy (1) where y = (y(Xi»i, y. = y(x.), the likelihood P(Dly) = TIi P(tiIYi) and P(D) is a normalization constant. P(tlx., D) can then be obtained by averaging P(tly.) over P(y.lx., D). Gaussian process (GP) or spline smoothing models use a Gaussian process prior on y(.) which can be seen as function of X into a set of random variables such that for each finite XI C X the corresponding variables are jointly Gaussian (see [15] for an introduction). A GP is determined by a mean function 1 x 1-4 E[y(x)] and a positive definite covariance kernel K(x,x). Gaussian process classification (GPC) amounts to specifying available prior knowledge by choosing a class of kernels K(x, xIO), 0 E e, where 0 is a vector of hyperparameters, and a hyperprior P(O). Usually, these choices are guided by simple attributes of y(.) such as smoothness, trends, differentiability, but more general approaches to kernel design have also been considered [5]. For 2-class classification the most common noise distribution is the binomial one where P(tly) = (j(ty), (j(u) = (1 + exp( _U»-1 the logistic function, and y is the logit 10g(P( +llx)/ P( -llx» of the target distribution. For this noise model the integral in (1) is not analytically tractable, but a range of approximative techniques based on Laplace approximations [16], Markov chain Monte Carlo [7], variational methods [2] or mean field algorithms [8] are known. We follow [16]. The Laplace approach to GPC is to approximate the posterior P(yID,O) by the Gaussian distribution N(y, 1i-1) where y = argmaxP(yID, 0) is the posterior mode and 1i = \7~\7y(-logP(YID,O», evaluated at y. Then it is easy to show that the predictive distribution is Gaussian with mean k(x.),K-l y and variance k. - k(x.)'K-1k(x.) where K is the covariance matrix (K(Xi,Xj»ij, k(·) = (K(Xi, '»i, k. = K(x., x.) and the prime denotes transposition. The final discriminant is therefore a linear combination of the K (Xi, .). The discriminative approach to the prediction problem is to choose a loss function get, y), being an approximation to the misclassification loss2 I{tY:5o} and then to search for a discriminant y(.) which minimizes E [get, y(x.»] for the points x. of interest (see [14]). Support Vector classification (SVC) uses the c-insensitive loss (SVC loss) get, y) = [1 - ty]+, [u]+ = uI{u~o} which is an upper bound on the misclassification loss, and a reproducing kernel Hilbert space (RKHS) with kernel K(x,xIO) as hypothesis space for y(.). Indeed, Support Vector models and the Laplace method for Gaussian processes are special cases of spline smoothing models in RKHS where the aim is to minimize the functional n ~9(ti'Yi) + AllyOIl~ (2) i=l where II . 11K denotes the norm of the RKHS. It can be shown that the minimizer of (2) can be written as k(·), K-1y where y maximizes n ~9(ti'Yi) - Ay'K-ly. (3) i=l All these facts can be found in [13]. Now (3) is, up to terms not depending on y, the log posterior in the above GP framework if we choose g(t,y) = -logP(tly) and lW.l.O.g. we only consider GPs with mean function 0 in what follows. 2 I A denotes the indicator function of the set A c lR.. Bayesian Model Selection for Support Vector Machines 605 absorb A into O. For the SVC loss, (3) can be transformed into a dual problem via y = Ka, where a is a vector of dual variables, which can be efficiently solved using quadratic programming techniques. [12] is an excellent reference. Note that the SVC loss cannot be written as the negative log of a noise distribution, so we cannot reduce SVC to a special case of a Gaussian process classification model. Although a generative model for SVC is given in [11], it is easier and less problematic to regard SVC as efficient approximation to a proper Gaussian process model. Various such models have been proposed (see [8],[4]). In this work, we simply normalize the SVC loss pointwise, i.e. use a Gaussian process model with the normalized BVe loss g(t, y) = [1 - ty]+ + log Z(y), Z(y) = exp( -[1 - y]+) + exp( -[1 + y]+). Note that g(t, y) is a close approximation of the (unnormalized) SVC loss. The reader might miss the SVM bias parameter which we dropped here for clarity, but it is straightforward to apply this semiparametric extension to GP models to03. 2 A variational method for kernel classification The real Bayesian way to deal with the hyperparameters 0 is to average P(y.lx., D, 0) over the posterior P( OlD) in order to obtain the predictive distribution P(y.lx., D). This can be approximated by Markov chain Monte Carlo methods [7], [16] or simply by P(y.lx.,D,9), 9 = argmaxP(OID). The latter approach, called maximum a-posteriori (MAP), can be justified in the limit of large n and often works well in practice. The basic challenge of MAP is to calculate the evidence P(DI9) = ! P(D,yI9)dy = ! exp (-t.9(ti,Yi») N(yIO,K(9))dy. (4) Our plan is to attack (4) by a variational approach. Let P be a density from a model class r chosen to approximate the posterior P(yID, 0). Then: -logP(DIO) = -JP(Y)lOg (P(D'YIO)~(Y)) dy P(y ID, O)P(y) - ! (P(y)) = F(P, 0) P(y) log P(y ID, 0) dy (5) where we call F(P, 0) = Ep[-log P(D, yIO)] +Ep[logP(y)] the variational free energy. The second term in (5) is the well-known Kullback-Leibler divergence between P and the posterior which is nonnegative and equals zero iff P(y) = P(yID,O) almost everywhere with respect to the distribution P. Thus, F is an upper bound on - log P (D I 0), and changing (P, 0) to decrease F enlarges the evidence or decreases the divergence between the posterior and its approximation, both being favourable. This idea has been introduced in [3] as ensemble learning4 and has been successfully applied to MLPs [1]. The latter work also introduced the model class r we use here, namely the class of Gaussians with mean IL and factor-analyzed covariance ~ = V + L,~1 Cjcj, V diagonal with positive elements5 . Hinton and 3This is the "random effects model with improper prior" of [13], p.19, and works by placing a flat improper prior on the bias parameter. 4We average different discriminants (given by y) over the ensemble P. 5 Although there is no danger of overfitting, the use of full covariances would render the optimization more difficult, time and memory consuming. 606 M Seeger van Camp [3] used diagonal covariances which would be M = 0 in our setting. By choosing a small M, we are able to track the most important correlations between the components in the posterior using O( M n) parameters to represent P. Having agreed on r, the criterion F and its gradients with respect to (J and the parameters of P can easily and efficiently be computed except for the generic term (6) a sum of one-dimensional Gaussian expectations which are, depending on the actual g, either analytically tractable or can be approximated using a quadrature algorithm. For example, the expectation for the normalized SVC loss can be decomposed into expectations over the (unnormalized) SVC loss and over log Z(y) (see end of section 1). While the former can be computed analytically, the latter expectation can be handled by replacing log Z (y) by a piecewise defined tight bound such that the integral can be solved analytically. For the GPC loss (6) cannot be solved analytically and was in our experiments approximated by Gaussian quadrature. We can optimize F using a nested loop algorithm as follows. In the inner loop we run an optimizer to minimize F w.r.t. P for fixed (J. We used a conjugate gradients optimizer since the number of parameters of P is rather large. The outer loop is an optimizer minimizing F w.r.t. (J, and we chose a Quasi-Newton method here since the dimension of e is usually rather small and gradients w.r.t. (J are costly to evaluate. We can use the resulting minimizer (P,O) of F in two different ways. The most natural is to discard P, plug 0 into the original architecture and predict using the mode of P(y ID, 0) as an approximation to the true posterior mode, benefitting from a kernel now adapted to the given data. This is particularly interesting for Support Vector machines due to the sparseness of the final kernel expansion (typically only a small fraction of the components in the weight vector K-1iJ is non-zero, the corresponding datapoints are termed Support Vectors) which allows very efficient predictions for a large number of test points. However, we can also retain P and use it as a Gaussian approximation of the posterior P(yID, 0). Doing so, we can use the variance of the approximative predictive distribution P(y.lx., D) to derive error bars for our predictions, although the interpretation of these figures is somewhat complicated in the case of kernel discriminants like SVM whose loss function does not correspond to a noise distribution. 2.1 Relations to other methods Let us have a look at alternative ways to maximize (4). If the loss get, y) is twice differentiable everywhere, progress can be made by replacing g by its second order Taylor expansion around the mode of the integrand. This is known as Laplace approximation and is used in [16] to maximize (4) approximately. However, this technique cannot be used for nondifferentiable losses of the c-insensitive type6 • Nevertheless, for the SVC loss the evidence (4) can be approximated in a Laplacelike fashion [11], and it will be interesting to compare the results of this work with ours. This approximation can be evaluated very efficiently, but is not continuous 7 6The nondifferentiabilities cannot be ignored since with probability one a nonzero number of the ih sit exactly at these margin locations. 7 Although continuity can be accomplished by a further modification, see [11]. Bayesian Model Selection for Support Vector Machines 607 w.r.t. (J and difficult to optimize if the dimension of e is not small. Opper and Winther [8] use mean field ideas to derive an approximate leave-one-out test error estimator which can be quickly evaluated, but suffers from the typical noisiness of cross-validation scores. Kwok [6] applies the evidence framework to Support Vector machines, but the technique seems to be restricted to kernels with a finite eigenfunction expansion (see [13] for details). It is interesting to compare our variational method to the Laplace method of [16] and the variational technique of [2]. Let g(t, y) be differentiable and suppose that for given (J we restrict ourselves to approximate (6) by replacing g(ti' Yi) by the expansion 8g ( ( ) 182g ( A)( )2 g(ti' JLi) + By ti, JLi) Yi - JLi + 2. 8y2 ti, Yi Yi - JLi , (7) where fj is the posterior mean. This will change the criterion F to Fapproz, say. Then it is easy to show that the Gaussian approximation to the posterior employed by the Laplace method, namely N(fj, (K-1 + W)-l), W = diag(u(Yi)(1-u(Yi»), minimizes Fapproz w.r.t. P if full covariances ~ are used, and plugging this minimizer into Fapproz we end up with the evidence approximation which is maximized by the Laplace method. The latter is not a variational technique since the approximation (7) to the loss function is not an upper bound, and works only for differentiable loss functions. If we upper bound the loss function g(t,y) by a quadratic polynomial and add the variational parameters of this bound to the parameters of P, our method becomes broadly similar to the lower bound algorithm of [2]. Indeed, since for fixed variational parameters of the polynomials we can easily solve for the mean and covariance of P, the former parameters are the only essential ones. However, the quadratic upper bound is poor for functions like the SVC loss, and in these cases our bound is expected to be tighter. 3 Experiments We tested our variational algorithm on a number of datasets from the UCI machine learning repository and the DELVE archive of the University of Toront08 : Leptograpsus crabs, Pima Indian diabetes, Wisconsin Breast Cancer, Ringnorm, Twonorm and Waveform (class 1 against 2). Descriptions may be found on the web. In each case we normalized the whole set to zero mean, unit variance in all input columns, picked a training set at random and used the rest for testing. We chose (for X = JRd) the well-known squared-exponential kernel (see [15]): K(x,xI9) = C (exp ( - 2~ t Wi (Xi - Xi)') + v), 9 = «Wi):'C,V)'. (8) All parameters are constrained to be positive, so we chose the representation ()i = v'f. We did not use a prior on (J (see comment at end of this section). For comparison we trained a Gaussian Process classifier with the Laplace method (also without hyperprior) and a Support Vector machine using lO-fold cross-validation to select the free parameters. In the latter case we constrained the scale parameters Wi to be equal (it is infeasible to adapt d + 2 hyperparameters to the data using crossvalidation) and dropped the v parameter while allowing for a bias parameter. As mentioned above, within the variational method we can use the posterior mode fj 8See http://vvv . cs. utoronto. cal ...... del ve and http://vvv.ics.uci.edu/ ...... mlearn/MLRepository.html . 608 M Seeger Name train test Var.GP GP Var. SVM SVM Lin. size size I y IL Lapl. y IL 10-CV discr. crabs 80 120 3 4 4 4 4 4 3 pima 200 332 66 66 68 64 66 67 67 wdbc 300 269 11 11 8 10 10 9 19 twonorm 300 7100 233 224 297 260 223 163 207 ringnorm 400 7000 119 124 184 129 126 160 1763 waveform 800 2504 206 204 221 211 206 197 220 Table 1: Number of test errors for various methods. as well as the mean IL of P for prediction, and we tested both methods. Error bars were not computed. The baseline method was a linear discriminant trained to minimize the squared error. Table 1 shows the test errors the different methods attained. These results show that the new algorithm performs equally well as the other methods we considered. They have of course to be regarded in combination with how much effort was necessary to produce them. It took us almost a whole day and a lot of user interactions to do the cross-validation model selection. The rule-of-thumb that a lot of Support Vectors at the upper bound indicate too large a parameter C in (8) failed for at least two of these sets, so we had to start with very coarse grids and sweep through several stages of refinement. An effect known as automatic relevance determination (ARD) (see [7]) can be nicely observed on some of the datasets, by monitoring the length scale parameters Wi in (8). Indeed, our variational SVC algorithm almost completely ignored (by driving their length scales to very small values) 3 of the 5 dimensions in "crabs", 2 of 7 in "pima" and 3 of 21 in "waveform". On "wdbc", it detected dimension 24 as particularly important with regard to separation, all this in harmony with the GP Laplace method. Thus, a sensible parameterized kernel family together with a method of the Bayesian kind allows us to gain additional important information from a dataset which might be used to improve the experimental design. Results of experiments with the methods tested above and hyperpriors as well as a more detailed analysis of the experiments can be found in [9]. 4 Discussion We have shown how to perform model selection for Support Vector machines using approximative Bayesian variational techniques. Our method is applicable to a wide range of loss functions and is able to adapt a large number of hyperparameters to given data. This allows for the use of sophisticated kernels and Bayesian techniques like automatic relevance determination (see [7]) which is not possible using other common model selection criteria like cross-validation. Since our method is fully automatic, it is easy for non-experts to use9 , and as the evidence is computed on the training set, no training data has to be sacrificed for validation. We refer to [9] where the topics of this paper are investigated in much greater detail. A pressing issue is the unfortunate scaling of the method with the training set 9 As an aside, this opens the possibility of comparing SVMs against other fullyautomatic methods within the DELVE project (see section 3). Bayesian Model Selection for Support Vector Machines 609 size n which is currently O(n3)1O. We are currently explori~g the applicability of the powerful approximations of [10] which might bring us very much closer to the desired O(n2) scaling (see also [2]). Another interesting issue would be to connect our method with the work of [5] who use generative models to derive kernels in situations where the "standard kernels" are not applicable or not reasonable. Acknowledgments We thank Chris Williams, Amos Storkey, Peter Sollich and Carl Rasmussen for helpful and inspiring discussions. This work was partially funded by a scholarship of the Dr. Erich Muller foundation. We are grateful to the Division of Informatics for supporting our visit in Edinburgh, and to Chris Williams for making it possible. References [1] David Barber and Christopher Bishop. Ensemble learning for multi-layer networks. In Advances in NIPS, number 10, pages 395-401. MIT Press, 1997. [2] Mark N. Gibbs. Bayesian Gaussian Processes for Regression and Classification. PhD thesis, University of Cambridge, 1997. [3] Geoffrey E. Hinton and D. Van Camp. Keeping neural networks simple by minimizing the description length of the weights. In Proceedings of the 6th annual conference on computational learning theory, pages 5- 13, 1993. [4] Tommi Jaakkola, Marina Meila, and Tony Jebara. Maximum entropy discrimination. In Advances in NIPS, number 13. MIT Press, 1999. [5] Tommi S. Jaakkola and David Haussler. Exploiting generative models in discriminative classifiers. In Advances in NIPS, number 11, 1998. [6] James Tin-Tau Kwok. Integrating the evidence framework and the Support Vector machine. Submitted to ESANN 99, 1999. [7] Radford M. Neal. Monte Carlo implementation of Gaussian process models for Bayesian classification and regression. Technical Report 9702, Department of Statistics, University of Toronto, January 1997. [8] Manfred Opper and Ole Winther. GP classification and SVM: Mean field results and leave-one-out estimator. In Advances in Large Margin Classifiers. MIT Press, 1999. [9] Matthias Seeger. Bayesian methods for Support Vector machines and Gaussian processes. Master's thesis, University of Karlsruhe, Germany, 1999. Available at http://vvw.dai.ed.ac.uk/-seeger. [10] John Skilling. Maximum entropy and Bayesian methods. Cambridge University Press, 1988. [11] Peter Sollich. Probabilistic methods for Support Vector machines. In Advances in NIPS, number 13. MIT Press, 1999. [12] Vladimir N. Vapnik. Statistical Learning Theory. Wiley, 1998. [13] Grace Wahba. Spline Models for Observational Data. CBMS-NSF Regional Conference Series. SIAM, 1990. [14] Grace Wahba. Support Vector machines, reproducing kernel Hilbert spaces and the randomized GACV. Technical Report 984, University of Wisconsin, 1997. [15] Christopher K. 1. Williams. Prediction with Gaussian processes: From linear regression to linear prediction and beyond. In M. 1. Jordan, editor, Learning in Graphical Models. Kluwer, 1997. [16] Christopher K.I. Williams and David Barber. Bayesian classification with Gaussian processes. IEEE 7rans. PAMI, 20(12):1342-1351, 1998. laThe running time is essentially the same as that of the Laplace method, thus being comparable to the fastest known Bayesian GP algorithm.
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Kirchoff Law Markov Fields for Analog Circuit Design Richard M. Golden * RMG Consulting Inc. 2000 Fresno Road, Plano, Texas 75074 RMGCONSULT@AOL.COM, www.neural-network.com Abstract Three contributions to developing an algorithm for assisting engineers in designing analog circuits are provided in this paper. First, a method for representing highly nonlinear and non-continuous analog circuits using Kirchoff current law potential functions within the context of a Markov field is described. Second, a relatively efficient algorithm for optimizing the Markov field objective function is briefly described and the convergence proof is briefly sketched. And third, empirical results illustrating the strengths and limitations of the approach are provided within the context of a JFET transistor design problem. The proposed algorithm generated a set of circuit components for the JFET circuit model that accurately generated the desired characteristic curves. 1 Analog circuit design using Markov random fields 1.1 Markov random field models A Markov random field (MRF) is a generalization of the concept of a Markov chain. In a Markov field one begins with a set of random variables and a neighborhood relation which is represented by a graph. Each random variable will be assumed in this paper to be a discrete random variable which takes on one of a finite number of possible values. Each node of the graph indexs a specific random variable. A link from the jth node to the ith node indicates that the conditional probability distribution of the ith random variable in the field is functionally dependent upon the jth random variable. That is, random variable j is a neighbor of random variable i. The only restriction upon the definition of a Markov field (Le., the positivity condition) is that the probability of every realization of the field is strictly positive. The essential idea behind Markov field design is that one specifies a potential (energy) function for every clique in the neighborhood graph such that the subset of random variables associated with that clique obtain their optimal values when that clique'S potential function obtains its minimal value (for reviews see [1]-[2]) . • Associate Professor at University of Texas at Dallas (www.utdallas.eduj-901den) 908 R. M. Golden Markov random field models provide a convenient mechanism for probabilistically representing and optimally combining combinations of local constraints. 1.2 Analog circuit design using SPICE In some mixed signal ASIC (Application Specific Integrated Circuit) design problems, most of the circuit design specifications are well known but the introduction of a single constraint (e.g., an increase in substrate noise) could result in a major redesign of an entire circuit. The industry standard tool for aiding engineers in solving analog circuit design problems is SPICE which is a software environment for simulation of large scale electronic circuits. SPICE does have special optimization options for fitting circuit parameters to desired input-output characteristics but typically such constraints are too weak for SPICE to solve analog circuit design problems with large numbers of free parameters (see [3] for an introduction to SPICE). Another difficulty with using SPICE is that it does not provide a global confidence factor for indicating its confidence in a generated design or local confidence factors for determining the locations of "weak points" in the automatically generated circuit design solution. 1.3 Markov field approaches to analog circuit design In this paper, an approach for solving real-world analog circuit design problems using an appropriately constructed Markov random is proposed which will be referred to as MRFSPICE. Not only are desired input-output characteristics directly incorporated into the construction of the potential functions for the Markov field but additional constraints based upon Kirchoff's current law are directly incorporated into the field. This approach thus differs from the classic SPICE methodology because Kirchoff current law constraints are explicitly incorporated into an objective function which is minimized by the "optimal design". This approach also differs from previous Markov field approaches (Le., the "Harmony" neural network model [4] and the "Brain-State-in-a-Box" neural network model [5]) designed to qualitatively model human understanding of electronic circuit behavior since those approaches used pair-wise correlational (quadratic) potential functions as opposed to the highly nonlinear potential functions that will be used in the approach described in this paper. 1.4 Key contributions This paper thus makes three important contributions to the application of Markov random fields to the analog circuit design problem. First, a method for representing highly nonlinear and non-continuous analog circuits using Kirchoff current law potential functions within the context of a Markov field is described. Second, a relatively efficient algorithm for optimizing the Markov field objective function is briefly described and the convergence proof is briefly sketched. And third, empirical results illustrating the strengths and limitations of the approach is provided within the context of a JFET transistor design problem. 2 Modeling assumptions and algorithms 2.1 Probabilistic modeling assumptions A given circuit circuit design problem consists of a number of design decision variables. Denote those design decision variables by the discrete random variables Kirchoff Law Markov Fields for Analog Circuit Design 909 Xl, ... ,Xd' Let the MRF be denoted by the set x = [Xl"'" Xd] so that a realization of x is the d-dimensional real vector x. A realization of x is referred to as a circuit design solution. Let the joint (global) probability mass function for x be denoted by Po. It is assumed that po(x) > po(y) if and only if the circuit design solution x is preferred to the circuit design solution y. Thus, po(x) specifies a type of probabilistic fuzzy measure [1]. For example, the random variable Xi might refer to a design decision concerning the choice of a particular value for a capacitor C14 . From previous experience, it is expected that the value of CI4 may be usually constrained without serious difficulties to one of ten possible values: O.IJ.LF, 0.2J.LF, 0.3J.LF, O.4J.LF, 0.5J.LF, 0.6J.LF, 0.7J.LF, 0.8J.LF, 0.9J.LF, ar 1J.LF. Thus, ki = 10 in this example. By limiting the choice of CI4 to a small number of finite values, this permits the introduction of design expertise hints directly into the problem formulation without making strong committments to the ultimate choice of the value of capacitor C14 • Other examples of design decision variable values include: resistor values, inductor values, transistor types, diode types, or even fundamentally different circuit topologies. The problem that is now considered will be to assign design preference probabilities in a meaningful way to alternative design solutions. The strategy for doing this will be based upon constructing po with the property that if po(x) > po(y), then circuit design solution x exhibits the requisite operating characteristics with respect to a set of M "test circuits" more effectively than circuit design solution y. An optimal analog circuit design solution x* then may be defined as a global maximum of Po. The specific details of this strategy for constructing Po are now discussed by first carefully defining the concept of a "test circuit". Let V = {O, 1, 2, ... ,m} be a finite set of integers (i.e., the unique "terminals" in the test circuit) which index a set of m complex numbers, Vo, VI, V2, .•. ,Vm which will be referred to as voltages. The magnitude of Vk indicates the voltage magnitude while the angle of Vk indicates the voltage phase shift. By convention the ground voltage, Vo , is always assigned the value of O. Let d E V x V (i.e., an ordered pair of elements in V). A circuit component current source is defined with respect to V by a complex-valued function ia,b whose value is typically functionally dependent upon Va and Vb but may also be functionally dependent upon other voltages and circuit component current sources associated with V. For example, a "resistor" circuit component current source would be modeled by choosing ia,b = (Vb Va) / R where R is the resistance in ohms of some resistor, Vb is the voltage observed on one terminal of the resistor, and Va is the voltage observed on the other terminal of the resistor. The quantity ia,b is the current flowing through the resistor from terminal a to terminal b. Similarly, a "capacitor" circuit component current source would be modeled by choosing ia,b = (Vb - Va) /[27rj f] where j = A and f is the frequency in Hz of the test circuit. A "frequency specific voltage controlled current source" circuit component current source may be modeled by making ia,b functionally dependent upon some subset of voltages in the test circuit. See [6] for additional details regarding the use of complex arithmetic for analog circuit analysis and design. An important design constraint is that Kirchoff's current law should be satisfied at every voltage node. Kirchoff's current law states that the sum of the currents entering a voltage node must be equal to zero [6]. We will now show how this physical law can be directly embodied as a system of nonlinear constraints on the 910 R. M. Golden behavior of the MRF. We say that the kth voltage node in test circuit q is clamped if the voltage Vk is known. For example, node k in circuit q might be directly grounded, node k might be directly connected to a grounded voltage source, or the voltage at node k, Vk, might be a desired known target voltage. If voltage node k in test circuit q is damped, then Kirchoff's current law at voltage node k in circuit q is simply assumed to be satisfied which, in turn, implies that the voltage potential function «lq,k = O. Now suppose that voltage node k in test circuit q is not clamped. This means that the voltage at node k must be estimated. If there are no controlled current sources in the test circuit (Le., only passive devices), then the values of the voltages at the undamped nodes in the circuit can be calculated by solving a system of linear equations where the current choice of circuit component values are treated as constants. In the more general case where controlled current sources exist in the test circuit, then an approximate iterative gradient descent algorithm (such as the algorithm used by SPICE) is used to obtain improved estimates of the voltages of the undamped nodes. The iterative algorithm is always run for a fixed number of iterations. Now the value of «lq,k must be computed. The current entering node k via arc j in test circuit q is denoted by the two-dimensional real vector Ik,i whose first component is the real part of the complex current and whose second component is the imaginary part. The average current entering node k in test circuit q is given by the formula: nit -q ~ q Ik = (link) L- Ik,i' j=l pesign circuit components (e.g., resistors, capacitors, diodes, etc.) which minimize It will satisfy Kirchoff's current law at node k in test circuit q. However, the measure It is an not entirely adequate indicator of the degree to which Kirchoff's current law is satisfied since 1% may be small in magnitude not necessarily because Kirchoff's current law is satisfied but simply because all currents entering node k are small in magnitude. To compensate for this problem, a normalized current signal magnitude to current signal variability ratio is minimized at node k in test circuit q. This ratio decreases in magnitude if 1% has a magnitude which is small relative to the magnitude of individual currents entering node k in test circuit q. The voltage potential function, «l q,k, for voltage node k in test circuit q is now formally defined as follows. Let Let AI, ... , Au be those eigenvalues of Qk,q whose values are strictly greater than some small positive number €. Let ei be the eigenvector associated with eigenvalue Ai. Define u Qk,~ = L(l/Aj)ejeJ. j=l Thus, if Qk,q has all positive eigenvalues, then Qk,q is simply the matrix inverse of Qk,~. Using this notation, the voltage potential function for the undamped voltage Kirchoff Law Markov Fields for Analog Circuit Design 911 node k in test circuit q may be expressed by the formula: ;¥,. [-Iq]TQ-I-Iq 'i!q,1e = Ie Ie' Now define the global probability or "global preference" of a particular design configuration by the formula: PG(x) = (l/Z)exp( -U(x» (1) where U = (liN) Lq Lk <I>q,k and where N is the total number of voltage nodes across all test circuits. The most preferred (Le., "most probable") design are the design circuit components that maximize PG. Note that probabilities have been assigned such that circuit configurations which are less consistent with Kirchoff's current law are considered "less probable" (Le., "less preferred"). Because the normalization constant Z in (I) is computationally intractable to compute, it is helpful to define the easily computable circuit confidence factor, CCF, given by the formula: CCF{x) = exp( -U{x» = ZPG{x). Note that the global probability P is directly proportional to the CCF. Since U is always non-negative and complete satisfaction of Kirchoff's current laws corresponds to the case where U = 0, it follows that CCF(x) has a lower bound of 0 (indicating "no subjective confidence" in the design solution x) and an upper bound of 1 (indicating" absolute subjective confidence" in the design solution x). In addition, local conditional probabilities of the form can be computed using the formula: Such local conditional probabilities are helpful for explicitly computing the probability or "preference" for selecting one design circuit component value given a subset of other design component values have been accepted. Remember that probability (Le., "preference") is essentially a measure of the degree to which the chosen design components and pre-specified operating characteristic voltage versus frequency curves of the circuit satisfy Kirchoff's current laws. 2.2 MRFSPICE algorithm The MRFSPICE algorithm is a combination of the Metropolis and Besag's ICM (Iterated Conditional Modes) algorithms [1]-[2]. The stochastic Metropolis algorithm (with temperature parameter set equal to one) is used to sample from p(x). As each design solution is generated, the CCF for that design solution is computed and the design solution with the best CCF is kept as an initial design solution guess Xo. Next, the deterministic ICM algorithm is then initialized with Xo and the ICM algorithm is applied until an equilibrium point is reached. A simulated annealing method involving decreasing the temperature parameter according to a logarithmic cooling schedule in Step 1 through Step 5 could easily be used to guarantee convergence in distribution to a uniform distribution over the global maxima of PG (Le., convergence to an optimal solution) [1]-[2]. However, for the test problems considered thus far, equally effective results have been obtained by using the above fast heuristic algorithm which is guaranteed to converge to a local maximum as opposed to a global maximum. It is proposed that in situations where the convergence rate is slow or the local maximum generated by MRFSPICE is a 912 R. M. Golden poor design solution with low CCF, that appropriate local conditional probabilites be computed and provided as feedback to a human design engineer. The human design engineer can then make direct alterations to the sample space of PG (Le., the domain of CC F) in order to appropriately simply the search space. Finally, the ICM algorithm can be easily viewed as an artificial neural network algorithm and in fact is a generalization of the classic Hopfield (1982) model as noted in [1]. !!.TEST + "'"""----~ ~OQ1 !GTEST .sQ1 Figure 1: As external input voltage generator EGTEST and external supply voltage EDT EST are varied, current ffiTEST flowing through external resistor RTEST is measured. 3 JFET design problem In this design problem, specific combinations of free parameters for a macroequivalent JFET transistor model were selected on the basis of a given set of characteristic curves specifying how the drain to source current of the JFET varied as a function of the gate voltage and drain voltage at OH z and 1M H z. Specifically, a .JFET transis~ tor model ~;as simulated using the classic Shichman and Hodges (1968) large-signal n-channel .JFET model as described by Vladimirescu [3] (pp. 96-100). The circuit diagram of this transistor model is shown in Figure 1. The only components in the circuit diagram which are not part of the JFET transistor model are the external voltage generators EDTEST and EGTEST, and external resistor RTEST. The specific functions which describe how IDIQGDl, CDIQGD1, RDIQGD1, IDIQGSl, CDIQGS1, RDIQGSl, CGDQ1, and CGSQ1 change as a function of EGTEST and the current IRTEST (which Hows through RTEST) are too long and complex to be Kirchoff Law Markov Fields for Analog Circuit Design 913 presented here (for more details see [3] pp. 96-100). Five design decision variables were defined. The first design decision variable, XDIQGS1, specified a set of parameter values for the large signal gate to source diode model portion of the JFET model. There were 20 possible choices for the value of XDIQGS1. Similarly, the second design decision variable, XDIQGDl, had 20 possible values and specified a set of parameter values for the large signal gate to drain diode model portion of the JFET model. The third design decision variable was XQl which also had 20 possible values were each value specified a set of choices for JFET -type specific parameters. The fourth and fifth design decision variables were the resistors RSQl and RSDI each of which could take on one of 15 possible values. The results of the JFET design problem are shown in Table 1. The phase angle for IRTEST at 1M H z was specified to be approximately 10 degrees, while the observed phase angle for IRTEST ranged from 7 to 9 degrees. The computing time was approximately 2 - 4 hours using unoptimized prototype MATLAB code on a 200 MHZ Pentium Processor. The close agreement between the desired and actual results suggests further research in this area would be highly rewarding. Table 1: Evaluation of MRFSPICE-generated JFET design EGTEST EDTEST IRTEST @ DC (rna) IRTEST @ IMHZ _~ma) (desired/ actual) (desired/ actual) 0 1.5 1.47/1.50 1.19 1.21 0 2.0 1.96/1.99 1.60 1.62 0 3.0 2.94/2.99 2.43 2.43 -0.5 1.5 1.47/1.50 1.07/1.11 -0.5 2.0 1.96/2.00 1.49/1.52 -0.5 3.0 2.95/2.99 2.34/2.35 -1.0 1.5 1.48/1.50 0.96/1.02 -1.0 2.0 1.97/2.00 1.39/1.44 -1.0 3.0 2.9613.00 2.27/2.29 Acknowledgments This research was funded by Texas Instruments Inc. through the direct efforts of Kerry Hanson. Both Kerry Hanson and Ralph Golden provided numerous key insights and knowledge substantially improving this project's quality. References [1] Golden, R. M. (1996) Mathematical methods for neural network analysis and design. Cambridge: MIT Press. [2] Winkler, G. (1995) Image analysis, random fields, and dynamic Monte Carlo methods: A mathematical introduction. New York: Springer-Verlag. [3] Vladimirescu, A. (1994) The SPICE book. New York: Wiley. [4] Smolensky, P. (1986). Information processing in dynamical systems: Foundations of Harmony theory. In D. E. Rumelhart and J. L. McClelland (eds.), Parallel distributed processing. Volume 1: Foundations, pp. 194-281. Cambridge: MIT Press. [5] Anderson, J. A. (1995). An introduction to neural networks. Cambridge: MIT Press. [6] Skilling, H. (1959) Electrical engineering circuits. New York: Wiley.
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Can VI mechanisms account for figure-ground and medial axis effects? Zhaoping Li Gatsby Computational Neuroscience Unit University College London zhaoping~gatsby.ucl.ac.uk Abstract When a visual image consists of a figure against a background, V1 cells are physiologically observed to give higher responses to image regions corresponding to the figure relative to their responses to the background. The medial axis of the figure also induces relatively higher responses compared to responses to other locations in the figure (except for the boundary between the figure and the background). Since the receptive fields of V1 cells are very small compared with the global scale of the figure-ground and medial axis effects, it has been suggested that these effects may be caused by feedback from higher visual areas. I show how these effects can be accounted for by V1 mechanisms when the size of the figure is small or is of a certain scale. They are a manifestation of the processes of pre-attentive segmentation which detect and highlight the boundaries between homogeneous image regions. 1 Introduction Segmenting figure from ground is one of the most important visual tasks. We neither know how to execute it on a computer in general, nor do we know how the brain executes it. Further, the medial axis of a figure has been suggested as providing a convenient skeleton representation of its shape (Blum 1973). It is therefore exciting to find that responses of cells in V1, which is usually considered a low level visual area, differentiate between figure and ground (Lamme 1995, Lamme, Zipser, and Spekreijse 1997, Zipser, Lamme, Schiller 1996) and highlight the medial axis (Lee, Mumford, Romero, and Lamme 1998). This happens even though the receptive fields in V1 are much smaller than the scale of these global and perceptually significant phenomena. A common assumption is that feedback from higher visual areas is mainly responsible for these effects. This is supported by the finding that the figure-ground effects in V1 can be strongly reduced or abolished by anaesthesia or lesions in higher visual areas (Lamme et al 1997). However, in a related experiment (Gallant, van Essen, and Nothdurft 1995), V1 cells were found to give higher responses to global boundaries between two texture regions. Further, this border effect was significant only 10-15 milliseconds after the initial responses of the cells and was present even under anaesthesia. It is thus Can VI Mechanisms Account/or Figure-Ground and Medial Axis Effects? 137 plausible that VI mechanisms is mainly responsible for the border effect. In this paper, I propose that the figure-ground and medial axis effects are manifestations of the border effect, at least for apropriately sized figures. The border effect is significant within a limited and finite distance from the figure border. Let us call the image region within this finite distance from the border the effective border region. When the size of the figure is small enough, all parts of the figure belong to the effective border region and can induce higher responses. This suggests that the figure-ground effect will be reduced or diminished as the size of the figure becomes larger, and the VI responses to regions of the figure far away from the border will not be significantly higher than responses to background. This suggestion is supported by experimental findings (Lamme et al 1997). Furthermore, the border effect can create secondary ripples as the effect decays with distance from the border. Let us call the distance from the border to the ripple the ripple wavelength. When the size of a figure is roughly twice the ripple wavelength, the ripples from the two opposite borders of the figure can reinforce each other at the center of the figure to create the medial axis effect, which, indeed, is observed to occur only for figures of appropriate sizes (Lee et al 1998). I validate this proposal using a biologically based model of VI with intra-cortical interactions between cells with nearby but not necessarily overlapping receptive fields. Intra-cortical interactions cause the responses of a cell be modulated by nearby stimuli outside its classical receptive fields the contextual influences that are observed physiologically (Knierim and van Essen 1992, Kapadia et al 1995). Contextual influences make VI cells sensitive to global image features, despite their local receptive fields, as manifested in the border and other effects. 2 The VI model We have previously constructed a VI model and shown it to be able to highlight smooth contours against a noisy background (Li 1998, 1999, 1999b) and also the boundaries between texture regions in images the border effect. Its behavior agrees with physiological observations (Knierim and van Essen 1992, Kapadia et al 1995) that the neural response to a bar is suppressed strongly by contextual bars of similar orientatons iso-orientation suppression; that the response is less suppressed by orthogonally or randomly oriented contextual bars; and that it is enhanced by contextual bars that are aligned to form a smooth contour in which the bar is within the receptive field contour enhancement. Without loss of generality, the model ignores color, motion, and stereo dimensions, includes mainly layer 2-3 orientation selective cells, and ignores the intra-hypercolumnar mechanism by which their receptive fields are formed. Inputs to the model are images filtered by the edge- or bar-like local receptive fields (RFs) of VI cells. l Cells influence each other contextually via horizontal intra-cortical connections (Rockland and Lund 1983, Gilbert, 1992), transforming patterns of inputs to patterns of cell responses. Fig. 1 shows the elements of the model and their interactions. At each location i there is a model VI hypercolumn composed of K neuron pairs. Each pair (i, fJ) has RF center i and preferred orientation fJ = k1r / K for k = 1, 2, ... K , and is called (the neural representation of) an edge segment. Based on experimental data (White, 1989), each edge segment consists of an excitatory and an inhibitory neuron that are interconnected, and each model cell represents a collection of local cells of similar types. The excitatory cell receives the visual input; its output is used as a measure of the response or salience of the edge segment and projects to higher visual areas. The inhibitory cells are treated as interneurons. Based on observations IThe terms 'edge' and 'bar' will be used interchangeably. 138 A Visual space, edge detectors, and their interactions B Neural connection pattern. Model Neural Elements Solid: J, Dashed: W ~ 0'·':. ,-, .• : ,-, .• :: ~ ~ ~ 0'·': 0'·': ~ ~ ~~~-~~~ ~ ~ >< ,-, .• :; ~ ~ ~ ,-, .. :: :, .. :: :, .. :: ~ Edge outputs to higher visual areas Inputs Ie to inhibitory cells ~~-r--+-~~--r--+~--~ An interconnected --~ neuron pair for - : edge segment i e '-0-I Visual inputs, filtered through the receptive fields, to the excitatory cells. Inhibitory interneurons Excitatory neurons Z. Li Figure 1: A: Visual inputs are sampled in a discrete grid of edge/bar detectors. Each grid point i has K neuron pairs (see C), one per bar segment, tuned to different orientations () spanning 1800 • Two segments at different grid points can interact with each other via monosynaptic excitation J (the solid arrow from one thick bar to anothe r) or disynaptic inhibition W (the dashed arrow to a thick dashed bar). See also C. B: A schematic of the neural connection pattern from the center (thick solid) bar to neighboring bars within a few sampling unit distances. J's contacts are shown by thin solid bars. W's are shown by thin dashed bars. The connection pattern is translation and rotation invariant. C: An input bar segment is directly processed by an interconnected pair of excitatory and inhibitory cells, each cell models abstractly a local group of cells of the same type. The excitatory cell receives visual input and sends output 9x (Xii}) to higher centers. The inhibitory cell is an interneuron. Visual space is taken as having periodic boundary conditions. by Gilbert, Lund and their colleagues (Rockland and Lund, 1983, Gilbert 1992) horizontal connections JiO,jO' (respectively WiO,jO') mediate contextual influences via monosynaptic excitation (respectively disynaptic inhibition) from j(}' to i(} which have nearby but different RF centers, i -::j:. j, and similar orientation preferences, () '" ()'. The membrane potentials follow the equations: XiO -axXiO - 2: 'l/J(f),,(})9y(Yi,9+flO) + J 09x(XiO) + 2: JiO ,jO' 9x (XjOl ) + fio + fo flO j#.i,O' YiO -ayYiO+9x(XiO)+ 2: W iO,jOl9x(Xjol)+fc j#i,O' Can VI Mechanisms Account/or Figure-Ground and Medial Axis Effects? 139 where O:zXie and O:yYie model the decay to resting potential, 9z(X) and 9y(Y) are sigmoid-like functions modeling cells' firing rates in response to membrane potentials x and y, respectively, 1/J(6.8) is the spread of inhibition within a hypercolumn, J09z(Xie) is self excitation, Ie and 10 are background inputs, including noise and inputs modeling the general and local normalization of activities (see Li (1998) for more details). Visual input lie persists after onset, and initializes the activity levels 9z(Xie). The activities are then modified by the contextual influences. Depending on the visual input, the system often settles into an oscillatory state (Gray and Singer, 1989, see the details in Li 1998). Temporal averages of 9z(Xie) over several oscillation cycles are used as the model's output. The nature of the computation performed by the model is determined largely by the horizontal connections J and W, which are local (spanning only a few hypercolumns), and translation and rotation invariant (Fig. IB). A: Input image (li8) to model B: Model output -------------111111111 II III -------------1.1 I III II I I I I I -------------11111 I I I I I I I I I -------------111 I I I I II I I I I I -------------1 I I I I I I I I II I I I -------------111 I I I 1 II I I II I -------------1 I I I I I I I I I I III -------------111 I 1 I 1 II I I I I I -------------1 I I III I I I I I I I I -------------1 I I 1 I I I I I 1 I III -------------111 I III II I I I I I -------------111 I I I 1 II II I I I -------------1 I I I I I I I II I I I I -------------1 I I I 1 I I I I I I I I I -------------111 I I I I II II I I I -------------111 I I I I II II I I I -------------1 I II I I II I I I I I I -------------111 II I I II II II I -------------1111111 I III III -------------1.1 I I II II II I I I -------------1 1 I 1 I I I I I I I I I I -------------111 I I I I II II I I I Figure 2: An example of the performance of the model. A: Input li9 consists of two regions; each visible bar has the same input strength. B: Model output for A, showing non-uniform output strengths (temporal averages of 9" (Xi9)) for the edges. The input and output strengths are proportional to the bar widths. Because of the noise in the system, the saliencies of the bars in the same column are not exactly the same, this is also the case in other figures. The model was applied to some texture border and figure-ground stimuli, as shown in examples in the figures. The input values fir} are the same for all visible bars in each example. The differences in the outputs are caused by intracortical interactions. They become significant about one membrane time constant after the initial neural response (Li, 1998). The widths of the bars in the figures are proportional to input and output strengths. The plotted region in each picture is often a small region of an extended image. The same model parameters (e.9. the dependence of the synaptic weights on distances and orientations, the thresholds and gains in the functions 9z0 and 9yO, and the level of input noise in 10 ) are used for all the simulation examples. Fig. 2 demonstrates that the model indeed gives higher responses to the boundaries between texture regions. This border effect is highly significant within a distance of about 2 texture element spacings from the border. Thus the effective border region is about 2 in texture element spacings in this example. Furthermore, at about 9 texture element spacings to the right of the texture border there is a much smaller but Significant (visible on the figure) secondary peak in the response amplitude. Thus the ripple wavelength is about 9 texture element spacings here. The border effect is mainly caused by the fact that the texture elements at the border experience less iso-orientation suppression (which reduces the response levels to other texture bars in the middle of a homogeneous (texture) region) the texture elements at the border have fewer neighboring texture bars of a similar orientation than the texture elements in the centers of the regions. The stronger responses to the effective border region cause extra iso-orientation suppression to texture bars near but right outside the effective border region. Let us call this region of stronger 140 Model Input --111 11111111111111111111111 111-==11111111111111111111111111111== ==111 11111111111111111111111 111== --11111111111111111111111111111---111 11111111111111111111111111-==11111111111111111111111111111== ==111 11111111111111111111111111== --11111111111111111111111111111---11111111111111111111111111111--------1 I I I I I I -------1 II I I II -------1 I I I I I I ------- I I I I I I I ------- I I I I I I I -------1111 1 11 ------- I I I I I I I -------1 I I I I I I -------1 I I I I I I ------- I I I I I I I :::::= IIII III ------1 I I I I I I ---------1 I ----------1 I --------- 1 I ----------1 I ---------- I I ---------1 I --------- I I ---------- 1 I ---------1 I ----------1 I ---------- I I ----------1 I --------11 III I II I I II I III I I I II I II I I I I I I II I II I II I III III III III III III III III III III III III 111------111------I I 1------I I 1------111------I I 1------11 1-----111-----I I 1------111------111------111------111------I 1--------I 1--------I I --------I 1--------I 1--------1-------11-------I 1--------I 1--------I 1--------I 1--------I 1--------11---------------------1 I 1------------- - ----------- 1 I 1------------------------- 1 I 1-------------------------1 I 1-------------------------1 I 1-------------------------1 I 1-------------------------1 I 1-------------------------1 I 1------------- - -----------1 I 1-------------------------1 I 1-------------------------1 I 1-------------------------1 I 1-------------------------1 I 1------------Model Output --•• 1111111111111111111111111 •• -==1111111 11111111111"111 11111== --•• 1111111111111111 11111 I 111.-=::= III III : 11I11 111I11 : I I III 1.:== - .... 1"11"1111111111"111111 •• -==11 11"11 I111 111111 111== _'1 I I II I" II " I II I I I I I I I 1'.-- •• 1111 11111111111111111 •• -=:= 11I111111111111111111 1111 =:== ------•• I I I I -------••• I I I -------. I I I I I ------. I' I I I -------••• I I I -------. '1 1 I I ::=:=11.111 ------1.1 1 I I ------I I I I ------. III I I -------11 I I I ------• I I I ----------•• 1 ---------1·· --------- .1 ----------.. , ----------·11 ----------. ::::::::::111 ----------1. 1 ---------- ., ----------1·1 ----------. ---------... II II I I II II I I II II I I I I II I I .. ------.. ------.. ------.. ------.. ------.. ------I ------.. ------.. ------.. ------.. ------.. -----.. ------I I I 11.--------I I I I 1-------I I I I. --------I I I I •• --------I II I •• --------I 111·--------II -------I I I I 1--------I II 1 •• --------I I 1 •• --------I I I 1 •• --------I I I ••• --------I II ••• --------------------··1-------------------------.. -------------------------1 •• -----------------------•• 1------------------------•• 1-------------------------1.1------------------------... -------------------------1.1-------------------------... -------------------------1 •• ------------------------... -------------------------•• 1-------------------------1.1------------Z. Li Figure 3: Dependence on the size of the figure. The figure-ground effect is most evident only for small figures, and the medial axis effect is most evident only for figures of finite and appropriate sizes. suppression from the border the border suppression region, which is significant and visible in Fig. (2B). This region can reach no further than the longest length of the horizontal connenctions (mediating the sup presion) from the effective border region. Consequently, texture bars right outside the border suppression region not only escape the stronger suppression from the border, but also experience weaker iso-orientation suppression from the weakened texture bars in the nearby border suppression region. As a result, a second saliency peak appears the ripple effect, and we can hence conclude that the ripple wavelength is of the same order of magnitude as the longest connection length of the cortical lateral connections mediating intra-cortical interactions. Fig. 3 shows that for very small figures, the whole figure belongs to the effective border region and is highlighted in the Vl responses. As the figure size increases, the responses in the inside of the figure become smaller than the responses in the border region. However, when the size of the figure is appropriate, namely about twice the ripple wavelength, the center of the figure induces a secondary response highlight. In this case, the ripples or the secondary saliency peaks from both borders superpose onto each other at the same spatial location at the center of the figure. This reinforces the saliency peak at this medial axis since it has two border suppression regions (from two opposite borders), one on each side of it, as its contextual stimuli. For even larger figures, the medial axis effect diminishes because the ripples from Can VI Mechanisms Account for Figure-Ground and Medial Axis Effects? Model Input I I 11111111111111111 11111111111111111 11111111111111111 11111111111111111 1---------------1 1---------------1 1---------------1 1---------------1 1---------------1 1---------------1 1---------------1 1---------------1 1---------------1 1---------------1 1---------------1 1---------------1 1---------------1 1---------------1 111111111IIII11I1 11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111 -------1 II I I II I II I I I 11------:::::::1 I I I I II I II I I I I I::::::: :::::::1 II I I II I I I I I I I I::::::: :::::::1 II I I I I I II I I II I::::::: -------1 II I I I I I I I I I I 11-------------1 II I I I I I I I I I 111-------------1 II I I I I I I I I I I 11------:::::::1 II I I I I I II I I I I I::::::: -------1 II I I I I I II I I II 1-------::-:-:-:-=-1 I 11~1?71711 I 1--:-:-:-:-::=:-:-: -_-_-==_-_-_-:..-_-(j' ,',',I,',',',"""'if ___ -_-=_-===-_-___ ---------111111111111111------------~-------1111111111111111111111111111111Ir:----=:-:-::===-----:-,,',',1,',1,',',',',',',1,',',',','--=---==..-:------===------,',',',',',',',',',',',',',',',',',',------=.---: :--~-----11111111111111111111111111111111111_-:---==----- -------1111 IIIIIIIIII 1111-------------11 II IIIII t III 1111---------------1111111111111111---------------111 III1IIIII '11----------------1 I I I I I I I I I I I I 1----------_-_-===-_-_-_-_-_-_=.! ,',1,',',',',1,'.', !.=.,..-_-_-_-___ -===--_-_-_ -----=-----_-~-..=..'_I_'-'_I~-::_-:-~----"""'///////////////""'" """'////////'/'////'"",, """'/'/','/'//'////'"",, """,'//'/'/"/'////'""" """"'/'//','////'/'"",, """'///'//////'////'"",, """'///'/'////'////'"",, """'///'//////'////'"",, """,'//'/"/'/'//'"""" """,'//'//////'///'""", """'///'///////////'"",, """'///////////////""'" """'///////////////""'" 1I1I1 I II II II II I I'''' 1111 11111 II II I I1III II II I II II I II II I 11---------------1 I I I II I 11---------------1 I I I II I 11---------------1 I I I II I 11---------------1 I I I II I 11---------------1 I I I II I 11---------------1 I I I II I 11:::::::::::::::1 I I I II I 11---------------1 I I I II I 11---------------1 I I I II I 11---------------11 I I II I 11---------------11 I I I I I 11---------------11 I I II I Model Output 1111111111111111111 "1"11111111111111 II 111'111"1111'11 III I "1" 'I' 111111 I III 1111 I ................. I ••••••••••••••••• 11:::::::::::::::11 11:::::::::::::::1' 11---------------11 .=:~~~~~~~=I .1---------------11 •1---------------1 1---------------1 .. ---------------.. I••••• .... ••••••••• ..... " .......... 1 1111111.11111111111 1111111111111111111 '""11"'1"1'1111 1111111111,11,1111, '111111111111111111 1111111111111111111 -------.111,,1 •• , I 111.-----------•• 11 ".1" 1111.-------------.11 I I 11I1 I I 111.------:::::::11: I I 1IIII I III:::::::: -------··1 I I 'II1 I I I '11------:::::=U I I. I I II I I ::::::: -------•• 1 I I ,.1" 111.1-------------.11 I I "." I I , •• ------:::::::111 II 1III1 I III:::::::: -------•• , I 111I11 I I 1 •• ------"""""""""""""", """""""""""""", """""""""""""", """""""""""""", """""""""""""", """""""""""""", """""""""""""", """""""""""""", """""""""""""", """""""""""""", """""""""""""", """""""""""""", """""""""""""", I I' I ,.1------------111 I I I I I II I 111:::::::::::::::1.11 II I I II I , •• -------------•• , I II I 1111111===--=====111, : II I II I '.1--------------11' I II I I I I I " ------------I I I I I I II 1 111:::::::::::::::.11 I II 1 I II I 11::::::::::::::::111 I II I I II I , •• --------------•• , I II I Figure 4: Dependence on the shape and texture feature of the figures. the two opposite borders of the figure no longer reinforce each other. 141 Fig. 4 demonstrates that the border effect and its consequences for the medial axis also depend on the shape of the figures and the nature of the texture they contain (eg the orientations of the elements). Bars in the texture parallel to the border induce stronger highlights, and as a consequence, cause stronger ripple effects and medial axis highlights. This comes from the stronger co-linear, contour enhancing, inputs these bars receive than bars not parallel to the border. 142 Z. Li 3 Summary and Discussion The model of V1 was originally proposed to account for pre-attentive contour enhancement and visual segmentation (Li 1998, 1999, 1999b). The contextual influences mediated by intracortical interactions enable each V1 neuron to process inputs from a local image area substantially larger than its classical receptive field. This enables cortical neurons to detect image locations where translation invariance in the input image breaks down, and highlight these image locations with higher neural activities, making them conspicuous. These highlights mark candidate locations for image region (or object surface) boundaries, smooth contours and small figures against backgrounds, serving the purpose of pre-attentive segmentation. This paper has shown that the figure-ground and medial axis effects observed in the recent experiments can be accounted for using a purely V1 mechanism for border highlighting, provided that the sizes of the figures are small enough or of finite and appropriate scale. This has been the case in the existing experiments. We therefore suggest that feedbacks from higher visual areas are not necessary to explain the experimental observations, although we cannot, of course, exclude the possibilities that they also contribute. References [1] Lamme V.A. (1995) Journal of Neuroscience 15(2), 1605-15. [2] Lee T.S, Mumford D, Romero R. and Lamme V. A.F. (1998) Vis. Res. 38: 2429-2454. [3] Zipser K., Lamme V. A., and Schiller P. H. (1996) J. Neurosci. 16 (22), 737689. [4] Lamme V. A. F., Zipser K. and Spekreijse H. Soc. Neuroscience Abstract 603.1, 1997. [5] Blum H. (1973) Biological shape and visual science J. Theor. Bioi. 38: 205-87. [6] Gallant J.L., van Essen D.C., and Nothdurft H.C. (1995) In Early vision and beyond eds. T. Papathomas, Chubb C, Gorea A., and Kowler E. (MIT press), pp 89-98. [7] C. D. Gilbert (1992) Neuron. 9(1), 1-13. [8] C. M. Gray and W. Singer (1989) Proc. Natl. Acad. Sci. USA 86, 1698-1702. [9] M. K. Kapadia, M. Ito, C. D. Gilbert, and G. Westheimer (1995) Neuron. 15(4), 843-56. [10] J. J. Knierim and D. C. van Essen (1992) J. Neurophysiol. 67, 961-980. [11] Z. Li (1998) Neural Computation 10(4) p 903-940. [12] Z. Li (1999) Network: computations in neural systems 10(2). p. 187-212. [13] Z. Li (1999b) Spatial Vision 13(1) p. 25-50. [14] K.S. Rockland and J. S. Lund (1983) J. Compo Neurol. 216, 303-318 [15] E. L. White (1989) Cortical circuits (Birkhauser).
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Policy Gradient Methods for Reinforcement Learning with Function Approximation Richard S. Sutton, David McAllester, Satinder Singh, Yishay Mansour AT&T Labs - Research, 180 Park Avenue, Florham Park, NJ 07932 Abstract Function approximation is essential to reinforcement learning, but the standard approach of approximating a value function and determining a policy from it has so far proven theoretically intractable. In this paper we explore an alternative approach in which the policy is explicitly represented by its own function approximator, independent of the value function, and is updated according to the gradient of expected reward with respect to the policy parameters. Williams's REINFORCE method and actor-critic methods are examples of this approach. Our main new result is to show that the gradient can be written in a form suitable for estimation from experience aided by an approximate action-value or advantage function. Using this result, we prove for the first time that a version of policy iteration with arbitrary differentiable function approximation is convergent to a locally optimal policy. Large applications of reinforcement learning (RL) require the use of generalizing function approximators such neural networks, decision-trees, or instance-based methods. The dominant approach for the last decade has been the value-function approach, in which all function approximation effort goes into estimating a value function, with the action-selection policy represented implicitly as the "greedy" policy with respect to the estimated values (e.g., as the policy that selects in each state the action with highest estimated value). The value-function approach has worked well in many applications, but has several limitations. First, it is oriented toward finding deterministic policies, whereas the optimal policy is often stochastic, selecting different actions with specific probabilities (e.g., see Singh, Jaakkola, and Jordan, 1994). Second, an arbitrarily small change in the estimated value of an action can cause it to be, or not be, selected. Such discontinuous changes have been identified as a key obstacle to establishing convergence assurances for algorithms following the value-function approach (Bertsekas and Tsitsiklis, 1996). For example, Q-Iearning, Sarsa, and dynamic programming methods have all been shown unable to converge to any policy for simple MDPs and simple function approximators (Gordon, 1995, 1996; Baird, 1995; Tsitsiklis and van Roy, 1996; Bertsekas and Tsitsiklis, 1996). This can occur even if the best approximation is found at each step before changing the policy, and whether the notion of "best" is in the mean-squared-error sense or the slightly different senses of residual-gradient, temporal-difference, and dynamic-programming methods. In this paper we explore an alternative approach to function approximation in RL. 1058 R. S. Sutton, D. McAl/ester. S. Singh and Y. Mansour Rather than approximating a value function and using that to compute a deterministic policy, we approximate a stochastic policy directly using an independent function approximator with its own parameters. For example, the policy might be represented by a neural network whose input is a representation of the state, whose output is action selection probabilities, and whose weights are the policy parameters. Let 0 denote the vector of policy parameters and p the performance of the corresponding policy (e.g., the average reward per step). Then, in the policy gradient approach, the policy parameters are updated approximately proportional to the gradient: ap ~O~CtaO' (1) where Ct is a positive-definite step size. If the above can be achieved, then 0 can usually be assured to converge to a locally optimal policy in the performance measure p. Unlike the value-function approach, here small changes in 0 can cause only small changes in the policy and in the state-visitation distribution. In this paper we prove that an unbiased estimate of the gradient (1) can be obtained from experience using an approximate value function satisfying certain properties. Williams's (1988, 1992) REINFORCE algorithm also finds an unbiased estimate of the gradient, but without the assistance of a learned value function. REINFORCE learns much more slowly than RL methods using value functions and has received relatively little attention. Learning a value function and using it to reduce the variance of the gradient estimate appears to be ess~ntial for rapid learning. Jaakkola, Singh and Jordan (1995) proved a result very similar to ours for the special case of function approximation corresponding to tabular POMDPs. Our result strengthens theirs and generalizes it to arbitrary differentiable function approximators. Konda and Tsitsiklis (in prep.) independently developed a very simialr result to ours. See also Baxter and Bartlett (in prep.) and Marbach and Tsitsiklis (1998). Our result also suggests a way of proving the convergence of a wide variety of algorithms based on "actor-critic" or policy-iteration architectures (e.g., Barto, Sutton, and Anderson, 1983; Sutton, 1984; Kimura and Kobayashi, 1998). In this paper we take the first step in this direction by proving for the first time that a version of policy iteration with general differentiable function approximation is convergent to a locally optimal policy. Baird and Moore (1999) obtained a weaker but superficially similar result for their VAPS family of methods. Like policy-gradient methods, VAPS includes separately parameterized policy and value functions updated by gradient methods. However, VAPS methods do not climb the gradient of performance (expected long-term reward), but of a measure combining performance and valuefunction accuracy. As a result, VAPS does not converge to a locally optimal policy, except in the case that no weight is put upon value-function accuracy, in which case VAPS degenerates to REINFORCE. Similarly, Gordon's (1995) fitted value iteration is also convergent and value-based, but does not find a locally optimal policy. 1 Policy Gradient Theorem We consider the standard reinforcement learning framework (see, e.g., Sutton and Barto, 1998), in which a learning agent interacts with a Markov decision process (MDP). The state, action, and reward at each time t E {O, 1, 2, . . . } are denoted St E S, at E A, and rt E R respectively. The environment's dynamics are characterized by state transition probabilities, P:SI = Pr { St+ 1 = Sf I St = s, at = a}, and expected rewards 'R~ = E {rt+l 1st = s, at = a}, 'r/s, Sf E S, a E A. The agent's decision making procedure at each time is characterized by a policy, 1l'(s, a, 0) = Pr {at = alst = s, O}, 'r/s E S,a E A, where 0 E ~, for l « lSI, is a parameter vector. We assume that 1l' is diffentiable with respect to its parameter, i.e., that a1f~~a) exists. We also usually write just 1l'(s, a) for 1l'(s, a, 0). Policy Gradient Methods for RL with Function Approximation 1059 With function approximation, two ways of formulating the agent's objective are useful. One is the average reward formulation, in which policies are ranked according to their long-term expected reward per step, p(rr): p(1I") = lim .!.E{rl +r2 + ... +rn 11I"} = '" £ff(s) "'1I"(s,a)'R.:, n-+oon ~ ~ II Q where cP (s) = limt-+oo Pr {St = slso, 11"} is the stationary distribution of states under 11", which we assume exists and is independent of So for all policies. In the average reward formulation, the value of a state-action pair given a policy is defined as 00 Q1r(s,a) = LE {rt - p(1I") I So = s,ao = a,1I"}, Vs E S,a E A. t=l The second formulation we cover is that in which there is a designated start state So, and we care only about the long-term reward obtained from it. We will give our results only once, but they will apply to this formulation as well under the definitions p(1I") = E{t. "(t-lrt I 80 ,1I"} and Q1r(s,a) = E{t. "(k-lrt+k 1St = s,at = a, 11" }. where,,( E [0,1] is a discount rate ("( = 1 is allowed only in episodic tasks). In this formulation, we define d1r (8) as a discounted weighting of states encountered starting at So and then following 11": cP(s) = E:o"(tpr{st = slso,1I"}. Our first result concerns the gradient of the performance metric with respect to the policy parameter: Theorem 1 (Policy Gradient). For any MDP, in either the average-reward or start-state formulations, ap = "'.ftr( )'" a1l"(s,a)Q1r( ) ao ~ u s ~ ao s, a . II Q Proof: See the appendix. (2) This way of expressing the gradient was first rtiscussed for the average-reward formulation by Marbach and Tsitsiklis (1998), based on a related expression in terms of the state-value function due to Jaakkola, Singh, and Jordan (1995) and Coo and Chen (1997). We extend their results to the start-state formulation and provide simpler and more direct proofs. Williams's (1988, 1992) theory of REINFORCE algorithms can also be viewed as implying (2). In any event, the key aspect of both expressions for the gradient is that their are no terms of the form adiJII): the effect of policy changes on the distribution of states does not appear. This is convenient for approximating the gradient by sampling. For example, if 8 was sampled from the distribution obtained by following 11", then Ea a1r~~,a) Q1r (s, a) would be an unbiased estimate of ~. Of course, Q1r(s, a) is also not normally known and must be estimated. One approach is to use the actual returns, Rt = E~l rt+k - p(1I") (or Rt = E~l "(k-lrt+k in the start-state formulation) as an approximation for each Q1r (St, at). This leads to Williams's episodic REINFORCE algorithm, t::..Ot oc a1r~~,at2 Rt (1 ) (the ~a 7r St,at 7r\St,Ut) corrects for the oversampling of actions preferred by 11"), which is known to follow ~ in expected value (Williams, 1988, 1992). 2 Policy Gradient with Approximation Now consider the case in which Q1r is approximated by a learned function approximator. If the approximation is sufficiently good, we might hope to use it in place of Q1r 1060 R. S. Sutton, D. MeAl/ester, S. Singh and Y. Mansour in (2) and still point roughly in the direction of the gradient. For example, Jaakkola, Singh, and Jordan (1995) proved that for the special case of function approximation arising in a tabular POMDP one could assure positive inner product with the gradient, which is sufficient to ensure improvement for moving in that direction. Here we extend their result to general function approximation and prove equality with the gradient. Let fw : S x A ~ be our approximation to Q7f, with parameter w. It is natural to learn f w by following 1r and updating w by a rule such as AWt oc I,u [Q7f (St, at) fw(st,at)]2 oc [Q7f(st,at) fw(st,at)]alw~~,ad, where Q7f(st,at) is some unbiased estimator of Q7f(st, at), perhaps Rt. When such a process has converged to a local optimum, then LcF(s):E 1r(s,a)[Q7f(s,a) - fw(s,a)] 8f~~,a) = o. (3) /I a Theorem 2 (Policy Gradient with Function Approximation). If fw satisfies (3) and is compatible with the policy parameterization in the sense thatl 8fw(s, a) 81r(s, a) 1 = 8w 80 1r(s, a) , (4) then 8p ~ ~ 81r(s, a) ao = ~cF(s) ~ ao fw(s,a). (5) II a Proof: Combining (3) and (4) gives Ld7f(s) L 87r1~a) [Q7f(s,a) - fw(s,a)] = 0 (6) II a which tells us that the error in fw(s, a) is orthogonal to the gradient of the policy parameterization. Because the expression above is zero, we can subtract it from the policy gradient theorem (2) to yield ap ao = L cF(s) L a1r1~ a) Q7f(s, a) - :E cF(s) :E a1r1~ a) [Q7f (s, a) - fw(s, a)] II 11 II a ~ ~ a1r(s,a) ~cF(s)~ ao [Q7f(s,a)-Q7f(s,a)+fw(s,a)] /I a ~ ~ a1r(s,a) = ~ cF(s) ~ ao fw(s, a). Q.E.D. II a 3 Application to Deriving Algorithms and Advantages Given a policy parameterization, Theorem 2 can be used to derive an appropriate form for the value-function parameterization. For example, consider a policy that is a Gibbs distribution in a linear combination of features: 'is E S,s E A, ITsitsiklis (personal communication) points out that /w being linear in the features given on the righthand side may be the only way to satisfy this condition. Policy Gradient Methods for RL with Function Approximation 1061 where each <Psa is an i-dimensional feature vector characterizing state-action pair s, a. Meeting the compatibility condition (4) requires that ofw(s,a) _ o1r(s,a) 1 _ A _ L ( b)A O 00 ( ) - 'l'sa 1r S, 'l'sb, W 7rS,a b so that the natural parameterization of fw is fw(s,a) ~ wT ["',. - ~"(S,b)""bl · In other words, fw must be linear in the same features as the policy, except normalized to be mean zero for each state. Other algorithms can easily be derived for a variety of nonlinear policy parameterizations, such as multi-layer backpropagation networks. The careful reader will have noticed that the form given above for f w requires that it have zero mean for each state: l:a 1r(s, a)fw(s, a) = 0, Vs E S . In this sense it is better to think of f w as an approximation of the advantage function, A7r(s,a) = Q7r(s,a) - V7r(s) (much as in Baird, 1993), rather than of Q7r . Our convergence requirement (3) is really that fw get the relative value of the actions correct in each state, not the absolute value, nor the variation from state to state. Our results can be viewed as a justification for the special status of advantages as the target for value function approximation in RL. In fact, our (2), (3), and (5), can all be generalized to include an arbitrary function of state added to the value function or its approximation. For example, (5) can be generalized to ~ = l:s d7r(s) l:a 87r~~,a) [fw(s, a) + v(s)] ,where v : S ---+ R is an arbitrary function. (This follows immediately because l:a 87r~~a) = 0, Vs E S.) The choice of v does not affect any of our theorems, but can substantially affect the variance of the gradient estimators. The issues here are entirely analogous to those in the use of reinforcement baselines in earlier work (e.g., Williams, 1992; Dayan, 1991; Sutton, 1984). In practice, v should presumably be set to the best available approximation of V7r. Our results establish that that approximation process can proceed without affecting the expected evolution of fw and 1r. 4 Convergence of Policy Iteration with Function Approximation Given Theorem 2, we can prove for the first time that a form of policy iteration with function approximation is convergent to a locally optimal policy. Theorem 3 (Policy Iteration with Function Approximation). Let 1r and fw be any differentiable function approximators for the policy and value function respectively that satisfy the compatibility condition (4) and for which maxe,s,a,i,j 18;~~9;) I < B < 00. Let {Ok}~o be any step-size sequence such that limk-+oo Ok = 0 and l:k Ok = 00. Then, for any MDP with bounded rewards, the sequence {p(1rk)}r=o, defined by any 00, 1rk = 1r(.,., Ok), and '"' '"' [ ]ofw(s,a) Wk w such that ~crk(S) ~ 1rk(s,a) Q7rk(s,a) - fw(s,a) ow = ° s a '"' '"' 01rk(S, a) Ok+l = Ok+Ok~crk(S)~ 00 fWk(s,a), s a converges such that limk-+oo 8P~;k) = o. Proof: Our Theorem 2 assures that the Ok update is in the direction of the gradient. 8 2 7r(s a) ....£..£....The bounds on 89;89j and on the MDP's rewards together assure us that 89i89j 1062 R. S. Sutton, D. MeAl/ester. S. Singh and Y. Mansour is also bounded. These, together with the step-size requirements, are the necessary conditions to apply Proposition 3.5 from page 96 of Bertsekas and Tsitsiklis (1996), which assures convergence to a local optimum. Q.E.D. Acknowledgements The authors wish to thank Martha Steenstrup and Doina Precup for comments, and Michael Kearns for insights into the notion of optimal policy under function approximation. References Baird, L. C. (1993). Advantage Updating. Wright Lab. Technical Report WL-TR-93-1l46. Baird, L. C. (1995). Residual algorithms: Reinforcement learning with function approximation. Proc. of the Twelfth Int. Co,:4. on Machine Learning, pp. 30-37. Morgan Kaufmann. Baird, L. C., Moore, A. W. (1999) . Gradient descent for general reinforcement learning. NIPS 11. MIT Press. Barto, A. G., Sutton, R. S., Anderson, C. W. (1983). Neuronlike elements that can solve difficult learning control problems. IEEE 1rans. on Systems, Man, and Cybernetics 19:835. Baxter, J., Bartlett, P. (in prep.) Direct gradient-based reinforcement learning: I. Gradient estimation algorithms. Bertsekas, D. P., Tsitsiklis, J. N. (1996). Neuro-Dynamic Programming. Athena Scientific. Cao, X.-R., Chen, H.-F. (1997). Perturbation realization, potentials, and sensitivity analysis of Markov Processes, IEEE 1hlns. on Automatic Control 42{1O):1382-1393. Dayan, P. (1991). Reinforcement comparison. In D. S. Touretzky, J. L. Elman, T. J. Sejnowski, and G. E. Hinton (eds.), Connectionist Models: Proceedings of the 1990 Summer School, pp. 45-51. Morgan Kaufmann. Gordon, G. J. (1995). Stable function approximation in dynamic programming. Proceedings of the Twelfth Int. Conf. on Machine Learning, pp. 261-268. Morgan Kaufmann. Gordon, G. J. (1996). Chattering in SARSA(A). CMU Learning Lab Technical Report. Jaakkola, T., Singh, S. P., Jordan, M. I. (1995) Reinforcement learning algorithms for partially observable Markov decision problems, NIPS 7, pp. 345-352. Morgan Kaufman. Kimura, H., Kobayashi, S. (1998). An analysis of actor/critic algorithms using eligibility traces: Reinforcement learning with imperfect value functions. Proc. ICML-98, pp. 278-286. Konda, V. R., Tsitsiklis, J. N. (in prep.) Actor-critic algorithms. Marbach, P., Tsitsiklis, J. N. (1998) Simulation-based optimization of Markov reward processes, technical report LIDS-P-2411, Massachusetts Institute of Technology. Singh, S. P., Jaakkola, T., Jordan, M. I. (1994). Learning without state-estimation in partially observable Markovian decision problems. Proc. ICML-94, pp. 284-292. Sutton, R. S. (1984). Temporal Credit Assignment in Reinforcement Learning. Ph.D. thesis, University of Massachusetts, Amherst. Sutton, R. S., Barto, A. G. (1998). Reinforcement Learning: An Introduction. MIT Press. Tsitsiklis, J. N. Van Roy, B. (1996). Feature-based methods for large scale dynamic programming. Machine Learning 22:59-94. Williams, R. J. (1988). Toward a theory of reinforcement-learning connectionist systems. Technical Report NU-CCS-88-3, Northeastern University, College of Computer Science. Williams, R. J. (1992). Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning 8:229-256. Appendix: Proof of Theorem 1 We prove the theorem first for the average-reward formulation and then for the startstate formulation. 8V1I'(s) 8fJ def = a ~ [87r(S, a) 11'() 8 11'( ] ~ 80 Q s,a +7r(s,a)80Q s,a) a Policy Gradient Methods for RL with Function Approximation 1063 Therefore, ap = '""" [a1r(S,a)Q1T( ) ( ),"""pa aV1T(S')]_ aV1T(s) ae Lao s, a + 1r s, a Lss' ao ae a ~ Summing both sides over the stationary distribution d1T, = '""" 1T ( ) '""" a1r( s, a) Q1T ( ) '""" ..nr ( ) '""" ( ) '""" a aV1T (s') L- d s Lae s, a + L- U s L- 7r s, a L- Pss' ae s a s a s' _ L~(s)av;O(s), s but since ~ is stationary, _ Ld1T(s) av;o(s) s :: = Ld1T(s) L a1r~~a) Q1T(s,a). Q.E.D. s a For the start-state formulation: aV1T ( s ) def a '""" 1T ae = ae L- 1r(s, a)Q (s, a) 'risE S a = L [a1r~~ a) Q1T(S, a) + 1r(s, a) :eQ1T(s, a)] a ~ ~ [inr~~ a) Q'(s,a) +--(s,a) :0 ['R~ + ~ ~P:., V'(S')]] ~ ~ [inr~~ a) Q'(s,a) +--(s,a) ~ ~P:.,! V'(S')] (7) '"""~ It ( ),"""a1r(x,a)Q1T() = L- L- 'Y Pr s -+ x, k, 1r Lae x, a , x It=o a after several steps of unrolling (7), where Pr(s -+ x, k, 1r) is the probability of going from state s to state x in k steps under policy 1r. It is then immediate that ap a {~t-l I } a 71' ae = aoE ti 'Y rt So,1r = ae v (so) '"""~ It ( ,"",,87r(s,a)Q7I'( ) = L- L- 'Y Pr So -+ s, k, 1r) Lae s, a s k=O a = L ~(s) L a1r~~ a) Q7I'(s, a). Q.E.D. s a
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Lower Bounds on the Complexity of Approximating Continuous Functions by Sigmoidal Neural Networks Michael Schmitt Lehrstuhl Mathematik und Informatik FakuWit ftir Mathematik Ruhr-Universitat Bochum D-44780 Bochum, Germany mschmitt@lmi.ruhr-uni-bochum.de Abstract We calculate lower bounds on the size of sigmoidal neural networks that approximate continuous functions. In particular, we show that for the approximation of polynomials the network size has to grow as O((logk)1/4) where k is the degree of the polynomials. This bound is valid for any input dimension, i.e. independently of the number of variables. The result is obtained by introducing a new method employing upper bounds on the Vapnik-Chervonenkis dimension for proving lower bounds on the size of networks that approximate continuous functions. 1 Introduction Sigmoidal neural networks are known to be universal approximators. This is one of the theoretical results most frequently cited to justify the use of sigmoidal neural networks in applications. By this statement one refers to the fact that sigmoidal neural networks have been shown to be able to approximate any continuous function arbitrarily well. Numerous results in the literature have established variants of this universal approximation property by considering distinct function classes to be approximated by network architectures using different types of neural activation functions with respect to various approximation criteria, see for instance [1, 2, 3, 5, 6, 11, 12, 14, 15]. (See in particular Scarselli and Tsoi [15] for a recent survey and further references.) All these results and many others not referenced here, some of them being constructive, some being merely existence proofs, provide upper bounds for the network size asserting that good approximation is possible if there are sufficiently many network nodes available. This, however, is only a partial answer to the question that mainly arises in practical applications: "Given some function, how many network nodes are needed to approximate it?" Not much attention has been focused on establishing lower bounds on the network size and, in particular, for the approximation of functions over the reals. As far as the computation of binary-valued Complexity of Approximating Continuous Functions by Neural Networks 329 functions by sigmoidal networks is concerned (where the output value of a network is thresholded to yield 0 or 1) there are a few results in this direction. For a specific Boolean function Koiran [9] showed that networks using the standard sigmoid u(y) = 1/(1 + e-Y ) as activation function must have size O(nl/4) where n is the number of inputs. (When measuring network size we do not count the input nodes here and in what follows.) Maass [13] established a larger lower bound by constructing a binary-valued function over IRn and showing that standard sigmoidal networks require O(n) many network nodes for computing this function. The first work on the complexity of sigmoidal networks for approximating continuous functions is due to DasGupta and Schnitger [4]. They showed that the standard sigmoid in network nodes can be replaced by other types of activation functions without increasing the size of the network by more than a polynomial. This yields indirect lower bounds for the size of sigmoidal networks in terms of other network types. DasGupta and Schnitger [4] also claimed the size bound AO(I/d) for sigmoidal networks with d layers approximating the function sin(Ax). In this paper we consider the problem of using the standard sigmoid u(y) = 1/(1 + e-Y) in neural networks for the approximation of polynomials. We show that at least O«logk)1/4) network nodes are required to approximate polynomials of degree k with small error in the loo norm. This bound is valid for arbitrary input dimension, i.e., it does not depend on the number of variables. (Lower bounds can also be obtained from the results on binary-valued functions mentioned above by interpolating the corresponding functions by polynomials. This, however, requires growing input dimension and does not yield a lower bound in terms of the degree.) Further, the bound established here holds for networks of any number of layers. As far as we know this is the first lower bound result for the approximation of polynomials. From the computational point of view this is a very simple class of functions; they can be computed using the basic operations addition and multiplication only. Polynomials also play an important role in approximation theory since they are dense in the class of continuous functions and some approximation results for neural networks rely on the approximability of polynomials by sigmoidal networks (see, e.g., [2, 15]). We obtain the result by introducing a new method that employs upper bounds on the Vapnik-Chervonenkis dimension of neural networks to establish lower bounds on the network size. The first use of the Vapnik-Chervonenkis dimension to obtain a lower bound is due to Koiran [9] who calculated the above-mentioned bound on the size of sigmoidal networks for a Boolean function. Koiran's method was further developed and extended by Maass [13] using a similar argument but another combinatorial dimension. Both papers derived lower bounds for the computation of binary-valued functions (Koiran [9] for inputs from {O, 1 }n, Maass [13] for inputs from IRn). Here, we present a new technique to show that and how lower bounds can be obtained for networks that approximate continuous functions. It rests on two fundamental results about the Vapnik-Chervonenkis dimension of neural networks. On the one hand, we use constructions provided by Koiran and Sontag [10] to build networks that have large Vapnik-Chervonenkis dimension and consist of gates that compute certain arithmetic functions. On the other hand, we follow the lines of reasoning of Karpinski and Macintyre [7] to derive an upper bound for the VapnikChervonenkis dimension of these networks from the estimates of Khovanskil [8] and a result due to Warren [16]. In the following section we give the definitions of sigmoidal networks and the VapnikChervonenkis dimension. Then we present the lower bound result for function approximation. Finally, we conclude with some discussion and open questions. 330 M Schmitt 2 Sigmoidal Neural Networks and VC Dimension We briefly recall the definitions of a sigmoidal neural network and the VapnikChervonenkis dimension (see, e.g., [7, 10]). We consider /eed/orward neural networks which have a certain number of input nodes and one output node. The nodes which are not input nodes are called computation nodes and associated with each of them is a real number t, the threshold. Further, each edge is labelled with a real number W called weight. Computation in the network takes place as follows: The input values are assigned to the input nodes. Each computation node applies the standard sigmoid u(y) = 1/(1 + e- V) to the sum W1Xl + ... + WrXr - t where Xl, .•. ,Xr are the values computed by the node's predecessors, WI, ••• ,Wr are the weights of the corresponding edges, and t is the threshold. The output value of the network is defined to be the value computed by the output node. As it is common for approximation results by means of neural networks, we assume that the output node is a linear gate, i.e., it just outputs the sum WIXI + ... + WrXr - t. (Clearly, for computing functions on finite sets with output range [0, 1] the output node may apply the standard sigmoid as well.) Since u is the only sigmoidal function that we consider here we will refer to such networks as sigmoidal neural networks. (Sigmoidal functions in general need to satisfy much weaker assumptions than u does.) The definition naturally generalizes to networks employing other types of gates that we will make use of (e.g. linear, multiplication, and division gates). The Vapnik-Chervonenkis dimension is a combinatorial dimension of a function class and is defined as follows: A dichotomy of a set S ~ IRn is a partition of S into two disjoint subsets (So, Sl) such that So U SI = S. Given a set F offunctions mapping IRn to {O, I} and a dichotomy (So, Sd of S, we say that F induces the dichotomy (So, Sd on S if there is some f E F such that /(So) ~ {O} and f(Sd ~ {I}. We say further that F shatters S if F induces all dichotomies on S. The VapnikChervonenkis (VC) dimension of F, denoted VCdim(F), is defined as the largest number m such that there is a set of m elements that is shattered by F. We refer to the VC dimension of a neural network, which is given in terms of a "feedforward architecture", i.e. a directed acyclic graph, as the VC dimension of the class of functions obtained by assigning real numbers to all its programmable parameters, which are in general the weights and thresholds of the network or a subset thereof. Further, we assume that the output value of the network is thresholded at 1/2 to obtain binary values. 3 Lower Bounds on Network Size Before we present the lower bound on the size of sigmoidal networks required for the approximation of polynomials we first give a brief outline of the proof idea. We will define a sequence of univariate polynomials (Pn)n>l by means of which we show how to construct neural architectures Nn consistmg of various types of gates such as linear, multiplication, and division gates, and, in particular, gates that compute some of the polynomials. Further, this architecture has a single weight as programmable parameter (all other weights and thresholds are fixed). We then demonstrate that, assuming the gates computing the polynomials can be approximated by sigmoidal neural networks sufficiently well, the architecture Nn can shatter a certain set by assigning suitable values to its programmable weight. The final step is to reason along the lines of Karpinski and Macintyre [7] to obtain via Khovanskil's estimates [8] and Warren's result [16] an upper bound on the VC dimension of Nn in terms of the number of its computation nodes. (Note that we cannot directly apply Theorem 7 of [7] since it does not deal with division gates.) Comparing this bound with the cardinality of the shattered set we will then be able Complexity of Approximating Continuous Functions by Neural Networks (3) W 1 (2) W 1 (1) W 1 W (3) (2) 1 P3 n Wi P 2 n Wj P1 n (3) Wn (2) Wn (1) Wn j --------------------------------~ k--------------------------------------------------~ 331 (1) Wk Figure 1: The network Nn with values k, j, i, 1 assigned to the input nodes Xl, X2, X3, X4 respectively. The weight W is the only programmable parameter of the network. to conclude with a lower bound on the number of computation nodes in Nn and thus in the networks that approximate the polynomials. Let the sequence (Pn)n2: l of polynomials over IR be inductively defined by { 4x(1 - x) n = 1 , Pn(X) = P(Pn-dx)) n 2:: 2 . Clearly, this uniquely defines Pn for every n 2:: 1 and it can readily be seen that Pn has degree 2n. The main lower bound result is made precise in the following statement. Theorem 1 Sigmoidal neural networks that approximate the polynomials (Pn)n>l on the interval [0,1] with error at most O(2-n ) in the 100 norm must have at least n(nl/4) computation nodes. Proof. For each n a neural architecture Nn can be constructed as follows: The network has four input nodes Xl, X2, X3, X4. Figure 1 shows the network with input values assigned to the input nodes in the order X4 = 1, X3 = i, X2 = j, Xl = k. There is one weight which we consider as the (only) programmable parameter of Nn . It is associated with the edge outgoing from input node X4 and is denoted by w. The computation nodes are partitioned into six levels as indicated by the boxes in Figure 1. Each level is itself a network. Let us first assume, for the sake of simplicity, that all computations over real numbers are exact. There are three levels labeled with II, having n + 1 input nodes and one output node each, that compute so-called projections 7r : IRnH -+ IR where 7r(YI,"" Yn, a) = Ya for a E {I, ... , n}. The levels labeled P3 , P2 , PI have one input node and n output nodes each. Level P3 receives the constant 1 as input and thus the value W which is the parameter of the network. We define the output values of level PA for>. = 3,2, 1 by (A) ( wb = Pbon"'-l v) , b= 1, ... ,n where v denotes the input value to level PA. This value is equal to w for>. = 3 and ( (A+l) ()..+l) ) h . OUT h (A) bid f 7r WI , .•. , Wn ,XA+l ot erWlse. vve observe t at wb+l can e calcu ate rom 332 M Schmitt w~A) as Pn>'_l(W~A»). Therefore, the computations of level PA can be implemented using n gates each of them computing the function Pn>.-l. We show now that Nn can shatter a set of cardinality n 3 • Let S = {I, ... ,n p. It has been shown in Lemma 2 of [10] that for each (/31 , ... , /3r) E {O, 1 Y there exists some W E [0,1] such that for q = 1, ... ,T pq(w) E [0,1/2) if /3q = 0, and pq(w) E (1/2,1] if /3q = 1. This implies that, for each dichotomy (So, Sd of S there is some W E [0,1] such that for every (i, j, k) E S Pk (pj.n (Pi.n2 (w))) < 1/2 Pk(Pj.n(Pi.n2 (w))) > 1/2 if ( i, j, k) E So , if (i,j,k)ES1' Note that Pk(Pj.n(Pi.n2 (w))) is the value computed by Nn given input values k, j, i, 1. Therefore, choosing a suitable value for w, which is the parameter of Nn , the network can induce any dichotomy on S. In other words, S is shattered by Nn . It has been shown in Lemma 1 of [10] that there is an architecture An such that for each E > ° weights can be chosen for An such that the function in,€ computed by this network satisfies lim€~o in,€(Yl, ... ,Yn, a) = Ya. Moreover, this architecture consists of O(n) computation nodes, which are linear, multiplication, and division gates. (Note that the size of An does not depend on E.) Therefore, choosing E sufficiently small, we can implement the projections 1r in Nn by networks of O(n) computation nodes such that the resulting network N~ still shatters S. Now in N~ we have O(n) computation nodes for implementing the three levels labeled II and we have in each level PA a number of O(n) computation nodes for computing Pn>.-l, respectively. Assume now that the computation nodes for Pn>.-l can be replaced by sigmoidal networks such that on inputs from S and with the parameter values defined above the resulting network N:: computes the same functions as N~. (Note that the computation nodes for Pn>.-l have no programmable parameters.) We estimate the size of N::. According to Theorem 7 of Karpinski and Macintyre [7] a sigmoidal neural network with I programmable parameters and m computation nodes has VC dimension O((ml)2). We have to generalize this result slightly before being able to apply it. It can readily be seen from the proof of Theorem 7 in [7] that the result also holds if the network additionally contains linear and multiplication gates. For division gates we can derive the same bound taking into account that for a gate computing division, say x/y, we can introduce a defining equality x = z . Y where z is a new variable. (See [7] for how to proceed.) Thus, we have that a network with I programmable parameters and m computation nodes, which are linear, multiplication, division, and sigmoidal gates, has VC dimension O((ml)2). In particular, if m is the number of computation nodes of N::, the VC dimension is O(m2 ). On the other hand, as we have shown above, N:: can shatter a set of cardinality n3 • Since there are O(n) sigmoidal networks in N:: computing the functions Pn>.-l, and since the number of linear, multiplication, and division gates is bounded by O(n), for some value of A a single network computing Pn>.-l must have size at least O(fo). This yields a lower bound of O(nl/4) for the size of a sigmoidal network computing Pn. Thus far, we have assumed that the polynomials Pn are computed exactly. Since polynomials are continuous functions and since we require them to be calculated only on a finite set of input values (those resulting from S and from the parameter values chosen for w to shatter S) an approximation of these polynomials is sufficient. A straightforward analysis, based on the fact that the output value of the network has a "tolerance" close to 1/2, shows that if Pn is approximated with error O(2-n) Complexity of Approximating Continuous Functions by Neural Networks 333 in the loo norm, the resulting network still shatters the set S. This completes the proof of the theorem. D The statement of the previous theorem is restricted to the approximation of polynomials on the input domain [0,1]. However, the result immediately generalizes to any arbitrary interval in llt Moreover, it remains valid for multivariate polynomials of arbitrary input dimension. Corollary 2 The approximation of polynomials of degree k by sigmoidal neural networks with approximation error O(ljk) in the 100 norm requires networks of size O((log k)1/4). This holds for polynomials over any number of variables. 4 Conclusions and Open Questions We have established lower bounds on the size of sigmoidal networks for the approximation of continuous functions. In particular, for a concrete class of polynomials we have calculated a lower bound in terms of the degree of the polynomials. The main result already holds for the approximation of univariate polynomials. Intuitively, approximation of multivariate polynomials seems to become harder when the dimension increases. Therefore, it would be interesting to have lower bounds both in terms of the degree and the input dimension. Further, in our result the approximation error and the degree are coupled. Naturally, one would expect that the number of nodes has to grow for each fixed function when the error decreases. At present we do not know of any such lower bound. We have not aimed at calculating the constants in the bounds. For practical applications such values are indispensable. Refining our method and using tighter results it should be straightforward to obtain such numbers. Further, we expect that better lower bounds can be obtained by considering networks of restricted depth. To establish the result we have introduced a new method for deriving lower bounds on network sizes. One of the main arguments is to use the functions to be approximated to construct networks with large VC dimension. The method seems suitable to obtain bounds also for the approximation of other types of functions as long as they are computationally powerful enough. Moreover, the method could be adapted to obtain lower bounds also for networks using other activation functions (e.g. more general sigmoidal functions, ridge functions, radial basis functions). This may lead to new separation results for the approximation capabilities of different types of neural networks. In order for this to be accomplished, however, an essential requirement is that small upper bounds can be calculated for the VC dimension of such networks. Acknowledgments I thank Hans U. Simon for helpful discussions. This work was supported in part by the ESPRIT Working Group in Neural and Computational Learning II, NeuroCOLT2, No. 27150. References [1] A. Barron. Universal approximation bounds for superposition of a sigmoidal function. IEEE Transactions on Information Theory, 39:930--945, 1993. 334 M Schmitt [2J C. K. Chui and X. Li. Approximation by ridge functions and neural networks with one hidden layer. Journal of Approximation Theory, 70:131-141,1992. [3J G. Cybenko. Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals, and Systems, 2:303-314, 1989. [4J B. DasGupta and G. Schnitger. The power of approximating: A comparison of activation functions. In C. L. Giles, S. J. Hanson, and J. D. Cowan, editors, Advances in Neural Information Processing Systems 5, pages 615-622, Morgan Kaufmann, San Mateo, CA, 1993. [5] K. Hornik. Approximation capabilities of multilayer feedforward networks. Neural Networks, 4:251-257, 1991. [6] K. Hornik, M. Stinchcombe, and H. White. Multilayer feedforward networks are universal approximators. Neural Networks, 2:359-366, 1989. [7] M. Karpinski and A. Macintyre. Polynomial bounds for VC dimension of sigmoidal and general Pfaffian neural networks. Journal of Computer and System Sciences, 54:169-176, 1997. [8] A. G. Khovanskil. Fewnomials, volume 88 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1991. [9] P. Koiran. VC dimension in circuit complexity. In Proceedings of the 11th Annual IEEE Conference on Computational Complexity CCC'96, pages 81-85, IEEE Computer Society Press, Los Alamitos, CA, 1996. [10] P. Koiran and E. D. Sontag. Neural networks with quadratic VC dimension. Journal of Computer and System Sciences, 54:190-198, 1997. [11] V. Y. Kreinovich. Arbitrary nonlinearity is sufficient to represent all functions by neural networks: A theorem. Neural Networks, 4:381-383, 1991. [12] M. Leshno, V. Y. Lin, A. Pinkus, and S. Schocken. Multilayer feedforward networks with a nonpolynomial activation function can approximate any function. Neural Networks, 6:861-867, 1993. [13] W. Maass. Noisy spiking neurons with temporal coding have more computational power than sigmoidal neurons. In M. Mozer, M. 1. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems 9, pages 211-217. MIT Press, Cambridge, MA, 1997. [14] H. Mhaskar. Neural networks for optimal approximation of smooth and analytic functions. Neural Computation, 8:164-177, 1996. [15J F. Scarselli and A. C. Tsoi. Universal approximation using feedforward neural networks: A survey of some existing methods and some new results. Neural Networks, 11:15-37, 1998. [16] H. E. Warren. Lower bounds for approximation by nonlinear manifolds. Transactions of the American Mathematical Society, 133:167-178, 1968.
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Evolv .......... JIiIIIIIIo. Bradley Tookes Dept of Comp. Sci. and Elec. Engineering University of Queensland Queensland, 4072 Australia btonkes@csee.uq. edu.au Alan Blair Department of Computer Science University of Melbourne Parkville, Victoria, 3052 Australia blair@cs.mu.oz.au Janet Wiles Dept of Comp. Sci. and Elec. Engineering , School of Psychology , University of Queensland Queensland, 4072 Australia janetw@csee.uq.edu.au Abstract Recent theories suggest that language acquisition is assisted by the evolution of languages towards forms that are easily learnable. In this paper, we evolve combinatorial languages which can be learned by a recurrent neural network quickly and from relatively few examples. Additionally, we evolve languages for generalization in different "worlds", and for generalization from specific examples. We find that languages can be evolved to facilitate different forms of impressive generalization for a minimally biased, general purpose learner. The results provide empirical support for the theory that the language itself, as well as the language environment of a learner, plays a substantial role in learning: that there is far more to language acquisition than the language acquisition device. 1 Introduction: Factors in language learnability In exploring issues oflanguage learnability, the special abilities of humans to learn complex languages have been much emphasized, with one dominant theory based on innate, domain-specific learning mechanisms specifically tuned to learning human languages. It has been argued that without strong constraints on the learning mechanism, the complex syntax of language could .not be learned from the sparse data that a 'child observes [1]. More recent theories challenge this claim and emphasize the interaction between learner and environment [~]. In addition to these two theories is the proposal that rather than "language-savvy infants", languages themselves adapt to human learners, and the ones that survive are "infant-friendly languages" [3-5]. To date, relatively few empirical studies have explored how such adaptation of language facilitates learning. Hare and Elman [6] demonstrated that Evolving Learnable Lan~ages 67 classes of past tense forms could evolve over simulated generations in response to changes in the frequency of verbs, using neural networks. Kirby [7] showed, using a symbolic system, how compositional languages are more likely to emerge when learning is constrained to a limited set of examples. Batali [8] has evolved recurrent networks that communicate simple structured,concepts. Our argument is not that humans are general purpose learners. Rather, current research questions require exploring the nature and extent of biases that learners bring to language learning, and the ways in which languages exploit those biases [2]. Previous theories suggesting that many aspects of language were unlearnable without strong biases are graduallybrealdng down as new aspects of language are shown to be learnable with much weaker biases. Studies include the investigation of how languages may exploit biases as subtle as attention ~d memory limitations in children [9]. A complementary study has shown that general purpose learners can evolve biases in the form of initial starting weights that facilitate the learning of a family of recursive languages [10] .. In this paper we present an empirical paradigm for continuing the exploration of factors that contribute to language learnability. The paradigm we propose necessitates the evolution of languages comprising recursive sentences over symbolic strings languages whose sentences cannot be. conveyed without combinatorial composition of symbols drawn from a finite alphabet. The paradigm is not based on any specific natural language, but rather, it is the simplest task we could find to illustrate the point that languages with compositional structure can be evolved to be learnable from few sentences.. The simplicity of the communication task allows us to analyze the language and its generalizability, and highlight the nature of the generalization properties. We start with the evolution of a recursive language that can be learned easily from five sentences by a minimally biased learner. We then address issues of robust learning of evolved languages, showing that different languages support generalization in different ways. We also address a factor to which scant regard has been paid, namely that languages may evolve not just to their learners, but also to be easily generalizable from a specific set of concepts. It seems almost axiomatic that learning paradigms should sample randomly from the training domain. It may be that human languages are not learnable from random sentences, but are easily generalizable from just those examples that a child is likely to be exposed to in its environment. In the third series of simulations, we test whether a language can adapt to be learnable from a core·set of concepts. 2 A paradig:m for exploring language learnability We consider a simple language task in which two recurrent neural networks try to communicate a "concept" represented by a point in the unit interval, [0, 1] over a symbolic· channeL An encoder network sends a sequence of symbols (thresholded outputs) for each concept, which a decoder network receives and processes back into a concept (the framework is described in greater detail in [11]). For communication to be successful, the decoder's output should approximate the encoder's input for all concepts. The architecture for the encoder is a recurrent network with one input unit and five output units, and with recurrent connections from both the output and hidden units back to the hidden units. The encoder produces a sequence of up to five symbols (states of the output units) taken from ~ = {A, ....., J}, followed by the $ symbol, for each concept taken from .[0, 1]. To encode a value x E [0,1], the network 68 B. Tonkes, A. Blair and J. Wiles ~~-----I S I ~ ,,---A-i ~ "~~Lf tm e A E Wl e A B IE e E B ABC E B A E /'" ~ Il\ I f\ ~ I I I I 1/'\ /\ I I II A\ I B C [BI B A ECAEB B Q A E E I3l E E 1B1 E AB C Be E E A C BC A BB ~ ~/\I\II\IIIIII\II/\IIl\/\/\1l\1/\111l\11/\/\IIl\II/\II/\ iiiiiiffliiiiiiiiffliiiiil liiiii fiiiiiiiiiiiiiiiiiiiiiiiff SSSSSssOOSSSS$$$SSsOOSS$$$SSmSSS$SS mSSSSSSSSSS$SSSSSSSSSSSssm Figure 1: Hierarchical decomposition of the language produced by an encoder, with the first symbols produced appearing near the root of the tree. The ordering of leaves in the tree represent the input space, smaller inputs being encoded by those sentences on the left. The examples used to train the best decoder found during . evolution are highlighted. The decoder must generalize to all other branches. LTI order to learn the task, the decoder must generalize systematically to novel states in the tree, including generalizing to symbols in different positions in the sequence. (Figure 2 shows the sequence of states of a successful decoder.) is presented with a sequence of inputs (x, 0, 0, ..).At each step, the output units of the network assume one of eleven states: all zero if no output is greater than 0.5 (denoted by $); or the saturation of the two highest activations at 1.0 and the remainder at 0.0 (denoted by A = [1,1,0,0,0] through J = [0,0,0,1, 1]). If the zero output is produced, propagation is halted. Otherwise propagation continues for up to five steps, after which the output units assume the zero ($) state. The decoder is a recurrent network with 5 input units and a single output, and a recurrent hidden layer. Former work [11] has shown that due to conflicting constraints of the encoder and decoder, it is easier for the decoder to process strings which are in the reverse order to those produced by the encoder. Consequently, the input to the decoder is taken to be the reverse of the output from the decoder, except for $, which remains the last symbol. (For clarity, strings are written in the order produced by the encoder.) Each input pattern presented to the decoder matches the output of the encoder either two units are active, or none are. The network is trained with backpropagation through time to produce the desired value, x, on presentation of the final symbol. in the sequence ($). A simple hill-climbing evolutionary strategy with a two-stage evaluation function is used to evolve an initially random encoder into one which produces a language which a random decoder can learn easily from few examples. The evaluation of an encoder, mutated from the current "champion" by the addition of Gaussian noise to the weights, is performed against two criteria. (1) The mutated network must produce a greater variety of sequences over the range of inputs; and (2) a decoder with initially small random weights, trained on the mutated encoder's output, must yield lower sum-squared error across the entire range of inputs than the champion. Each mutant encoder is paired with a single decoder with initially random weights. If the mutant encoder-decoder pair is more successful than the champion, the mutant becomes champion and the ptocess is repeated. Since the encoder's input space is continuous and impossible to examine in its entirety, the input range is approximated with 100 uniformly distributed examples from 0.00 to 0.99. The final output from the hill-climber is the language gen~rated by the best encoder found. Evolving Learnable Languages . 2.1 Evolving an easily learnable language 69 Humans learn from sparse data. In the first series of simulations we test whether a compositional language can be evolved that learners can reliably and effectively learn from only five examples. From just five training examples, it seems unreasonable to expect that any decoder would learn the task. The task is intentionally hard in that a language is restricted to sequences of discrete symbols with which it must describe a continuous space. Note that simple linear interpolation is not possible due to the symbolic alphabet of the languages. Recursive solutions are possible but are unable to be learned by an unbiased learner. The decoder is a minimally-biased learner and as the simulations showed, performed much better than arguments based on learnability theory would predict. Ten languages were evolved with the hill-climbing algorithm (outlined above) for 10000 generations.1 For each language, 100 new random decoders were trained under the same conditions as during evolution (five examples, 400 epochs). All ten runs used encoders and decoders with five hidden units. All of the evolved languages were learnable by some decoders (minimum 20, maximum 72, mean 48). A learner is said to have effectively learned the language if its sum-squared-error across the 100 points in the space is less than 1.0.2 Encoders employed on average 36 sentences (minimum 21, maximum 60) to communicate the 100 points. The 5 training examples for each decoder were sampled randomly from [0, 1] and hence some decoders faced very difficult generalization tasks. The difficulty of the task is demonstrated by the language analyzed in Figures 1 and 2. The evolved languages all contained' similar compositional structure to that of the language described in Figures 1 and 2. The inherent biases of the decoder, although minimal, are clearly sufficient for learning the compositional structure. 3 Evolving languages for particular generalization The first series of simulations demonstrate that we can find languages for which a minimally biased learner can generalize from few examples. In the next simulations we consider whether languages can be evolved to facilitate specific forms of generalization in their users. Section 2.1 considered the case· where the decoder's required output was the same as the encoder's input. This setup yields the approximation to the line y == x in Figure 2. The compositional structure of the evolved languages allows the decoder to generalize to unseen regions of the space. In the following series of simulations we consider the relationship between the structure of a language and the way in which the decoder is required to generalize. This association is studied by altering the desired relationship between the encoder'~ input (x) and the decoder's output (y). Two sets of ten languages were evolved, one set requiring y = x (identity, as in section 2.1), the other using a function resembling a series of five steps at random heights: y == r(L5xJ); r = (0.3746, 0.5753,0.8102,0.7272,0.4527) (random step)3. All conditions were as for section 2, with the exception that 10 training examples were used and the hill-climber ran for 1000 generations. On completion of evolution, 100 decoders were trained on the 20 final languages under both conditions above as lOne generation represents the creation of a more variable, mutated e~coder and the subsequent training of a decoder. 2A language is said to be reliably learnable when at least 50% of random decoders are able to effectively learn it within 400 epochs. 3 L5xJ provides an index into the array r, based on the mag~tudeof x. 70 (t;~O.;2 0.3 0.4 0.5 0.8 07 01 0.9 t (a) Tonkes) A. Blair and J. Wiles (b) (c) ·:~-q!JL~;l J~,~"",.".,.~~ oQ 0, ~:l ·o:i~,4cio6 G.:; ,{;;I o. (d) 1,~ '~-' ....--.....---..---..---...---r-_~ Figure 2: Decoder output after seeing the first n symbols in the message, for n == 1 (a) to n == 6 (f) (from the language in Figure 1). The X-axis is the encoder's input, the Y-axis is the decoder's output at that point in the sequence. The five points that the decoder was trained on are shown as crosses in each graph. After the first symbol (A, B, G, E or $), the decoder outputs one of five values (a); after the second symbol, more outputs are possible (b). Subsequent symbols in each string specify finer gradations in the output. Note that the output is not constructed monotonically, with each symbol providing a closer approximation to the target function, but rather recursively, only approximating the linear target at the final position in each sequence. Structure inherent in the sequences allows the system to generalize to parts of the space it has never seen. Note that the generalization is not based on interpolation between symbol values, but rather on their compositional structure. well as two others, a sine function and a cubic function. The results show that languages can be evolved to enhance generalization preferentially for one "world" over another. On average, the languages performed far better when tested in the world in which they were evolved than in other worlds. Languages evolved for the identity mapping were on average learned by 64% of decoders trained on the identity task compared with just 5% in the random step case. Languages evolved for the random step task were learned by 60% of decoders trained on the random step task but only 24% when trained on the identity task. Decoders generally performed poorly on the cubic function, and no decoder learned the sine task from either set of evolved languages. The second series of simulations show that the manner in which the decoder generalizes is not restricted to the task of section 2.1. Rather, the languages evolve to facilitate generalization by the decoder in different ways, aided by its minimal biases. Evolving Learnable LaliZJ!UGlJ!es 71 4 core concepts' In the former simulations, randomly selected concepts were used to train decoders. In some cases a pathological distribution of points made learning extremely difficult. In contrast, it seems likely that human children learn language based on a common set of semantically-constrained core concepts ("Mom", "I want milk", "no", etc). For the third series of simulations, we tested whether selecting a fortuitous set of training concepts could have a positive affect on the success of an evolved language. The simulations with alternative generalization functions (section 3) indicated that decoders had difficulty generalizing to the sine function. Even when encoders were evolved specifically on the sine task, in the best of 10 systems only 13 of 100 random decoders successfully learned. We evolved a new language on a specifically chosen set of 10 points for generalization to the sine function. One hundred decoders were then trained on the resulting language ush"1.g either the same set of 10 points, or a random set. Of the networks trained on the fixed set, 92 learned the tasked, compared with 5 networks trained on the random sets. That a language evolves to communicate a restricted set of concepts is not particularly unusual. But what this simulation shows is the more surprising result that a language can evolve to generalize from specific core concepts to a whole recursive langUage in a particular way (in this case, a sine function). 5 Discussion The first series of simulations show that a compositional language can be learned from five strings by an recurrent network. Generalization performance included correct decoding of novel branches and symbols in novel positions (Figure 1). The second series of simulations highlight how a language can be evolved to facilitate different forms of generalization in the decoder. The final simulation demonstrates that languages can also be tailored to generalize from a specific set of examples. The three series of simulations modify the language environment of the decoder in three different ways: (1) the relationship between utterances and meaning; (2) the type of generalization required from the decoder; and (3) the particular utterances and meanings to which a learner is exposed. In each case, the language environment of the learner was sculpted to exploit the minimal biases present in the learner. While taking an approach similar to [10] of giving the learner' an additional bias in the form of initial weights was also likely to have been effective, the purpose of the simulations was to investigate how strongly external factors could assist in simplifying learning. 6 Conclusions "The key to understanding language learnability does not lie in the richly social context of language training, nor _in the incredibly prescient guesses of young language learners; rather, it lies in a process that seems otherwise far remote from the microcosm of toddlers and caretakers language change. Although the rate of social evolutionary change in learning structure appears unchang~ ing compared to the time it takes a child to develop language abilities, this process is crucial to understanding how the child can learn a language that on the surface appears impossibly complex and poorly taught." [3, p115]. 72 B. Tonkes, A. Blair and J. Wiles In this paper we studied ways in which languages can adapt to their learners. running simulations of a language evolution process, 'We contribute additional components to the list of aspects of language that can be learned by minimally-biased, general-purpose learners, namely that recursive structure can be learned from few examples, that languages can evolve to facilitate generalization in a particular way, and that they can evolve to be easily learnable from common sentences. In all the simulations in this paper, enhancement of language learnability is achieved through changes to the learner's environment without resorting to adding biases in the language acquisition device. This work was supported by an APA to Bradley Tonkes, a UQ PostdoCtoral Fellowship to Alan Blair and an ARC grant to Janet Wiles. References [1] N. Chomsky. Language and Mind. Harcourt, Brace, New York, 1968. [2] J. L. Elman, E. A. Bates, M. H. Johnson, A. Karmiloff-Smith, D. Parisi, and K. Plunkett. Rethinking Innateness: A Connectionist Perspective on Development. MIT Press, Boston, 1996. [3] T. W. Deacon. The Symbolic Species: The Co-Evolution of Language and~the Brain. W. W. Norton and Company, New York, 1997. [4] S. Kirby. Fitness and the selective adaptation of language. In J. Hurford, C. Knight, and M. Studdert-Kennedy, editors, Approaches to the Evolution of Language. Cambridge University Press, Cambridge, 1998. [5] M. H. Christiansen. Language as an organism implications for the evolution and acquisition of language. Unpublished manuscript, February 1995. [6] M. Hare and J. L. Elman. Learning and morphological change. Cognition, 56:61-98, 1995. [7] S. Kirby. Syntax without natural selection: How compositionality emerges from vocabulary in a population of learners. In C. Knight, J. Hurford, and M.. StuddertKennedy, editors, The Evolutionary Emergence of Language: Social function and the origins of linguistic form. Cambridge University Press, Cambridge, 1999. [8] J. Batali. Computational simulations of the emergence of grammar. In J. Hurford, C. Knight, and M. Studdert-Kennedy, editors, Approaches to the Evolution of Language, pages 405-426. Cambridge University Press, Cambridge, 1998.. [9] J. L. Elman. Learning and development in neural networks: The importance of starting smalL Cognition, 48:71-99, 1993. [10] J. Batali. Innate biases and critical periods: Combining evolution and learning in the acquisition of syntax. In R. Brooks and P. Maes, editors, Proceedings of the Fourth Artificial Life Workshop, pages 160-171. MIT Press, 1994. [11] B. Tonkes, A. Blair, and J. Wiles. A paradox of neural encoders and decoders, Of, why don't we talk backwards? In B. McKay, X. Yao, C. S. Newton, J. -H. Kim, and T. Furuhashi, editors, Simulated Evolution and Learning, volume 1585 of Lecture Notes in Artificial Intelligence. Springer, 1999.
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Large Margin DAGs for Multiclass Classification John C. Platt Microsoft Research 1 Microsoft Way Redmond, WA 98052 jpiatt@microsojt.com Nello Cristianini Dept. of Engineering Mathematics University of Bristol Bristol, BS8 1 TR - UK nello.cristianini@bristol.ac.uk John Shawe-Taylor Department of Computer Science Royal Holloway College - University of London EGHAM, Surrey, TW20 OEX - UK j.shawe-taylor@dcs.rhbnc.ac.uk Abstract We present a new learning architecture: the Decision Directed Acyclic Graph (DDAG), which is used to combine many two-class classifiers into a multiclass classifier. For an N -class problem, the DDAG contains N(N - 1)/2 classifiers, one for each pair of classes. We present a VC analysis of the case when the node classifiers are hyperplanes; the resulting bound on the test error depends on N and on the margin achieved at the nodes, but not on the dimension of the space. This motivates an algorithm, DAGSVM, which operates in a kernel-induced feature space and uses two-class maximal margin hyperplanes at each decision-node of the DDAG. The DAGSVM is substantially faster to train and evaluate than either the standard algorithm or Max Wins, while maintaining comparable accuracy to both of these algorithms. 1 Introduction The problem of multiclass classificatIon, especially for systems like SVMs, doesn't present an easy solution. It is generally simpler to construct classifier theory and algorithms for two mutually-exclusive classes than for N mutually-exclusive classes. We believe constructing N -class SVMs is still an unsolved research p~oblem. The standard method for N -class SVMs [10] is to construct N SVMs. The ith SVM will be trained with all of the examples in the ith class with positive labels, and all other examples with negative labels. We refer to SVMs trained in this way as J -v-r SVMs (short for oneversus-rest). The final output of the N l-v-r SVMs is the class that corresponds to the SVM with the highest output value. Unfortunately, there is no bound on the generalization error for the l-v-r SVM, and the training time of the standard method scales linearly with N. Another method for constructing N -class classifiers from SVMs is derived from previous research into combining two-class classifiers. Knerr [5] suggested constructing all possible two-class classifiers from a training set of N classes, each classifier being trained on only 548 J C. Platt, N. Cristianini and J Shawe-Taylor two out of N classes. There would thus be K = N (N - 1) /2 classifiers. When applied to SVMs, we refer to this as J -v-J SVMs (short for one-versus-one). Knerr suggested combining these two-class classifiers with an "AND" gate [5]. Friedman [4] suggested a Max Wins algorithm: each I-v-l classifier casts one vote for its preferred class, and the final result is the class with the most votes. Friedman shows circumstances in which this algorithm is Bayes optimal. KreBel [6] applies the Max Wins algorithm to Support Vector Machines with excellent results. A significant disadvantage of the I-v-l approach, however, is that, unless the individual classifiers are carefully regularized (as in SVMs), the overall N -class classifier system will tend to overfit. The "AND" combination method and the Max Wins combination method do not have bounds on the generalization error. Finally, the size of the I-v-l classifier may grow superlinearly with N, and hence, may be slow to evaluate on large problems. In Section 2, we introduce a new multiclass learning architecture, called the Decision Directed Acyclic Graph (DDAG). The DDAG contains N(N - 1)/2 nodes, each with an associated I-v-l classifier. In Section 3, we present a VC analysis of DDAGs whose classifiers are hyperplanes, showing that the margins achieved at the decision nodes and the size of the graph both affect their performance, while the dimensionality of the input space does not. The VC analysis indicates that building large margin DAGs in high-dimensional feature spaces can yield good generalization performance. Using such bound as a guide, in Section 4, we introduce a novel algorithm for multiclass classification based on placing l-v-l SVMs into nodes of a DDAG. This algorithm, called DAGSVM, is efficient to train and evaluate. Empirical evidence of this efficiency is shown in Section 5. 2 Decision DAGs A Directed Acyclic Graph (DAG) is a graph whose edges have an orientation and no cycles. A Rooted DAG has a unique node such that it is the only node which has no arcs pointing into it. A Rooted Binary DAG has nodes which have either 0 or 2 arcs leaving them. We will use Rooted Binary DAGs in order to define a class of functions to be used in classification tasks. The class of functions computed by Rooted Binary DAGs is formally defined as follows. Definition 1 Decision DAGs (DDAGs). Given a space X and a set of boolean functions F = {f : X -t {a, I}}, the class DDAG(F) of Decision DAGs on N classes over Fare functions which can be implemented using a rooted binary DAG with N leaves labeled by the classes where each of the K = N(N - 1)/2 internal nodes is labeled with an element of F. The nodes are arranged in a triangle with the single root node at the top, two nodes in the second layer and so on until the jinallayer of N leaves. The i-th node in layer j < N is connected to the i-th and (i + 1)-st node in the (j + 1)-st layer. To evaluate a particular DDAG G on input x EX, starting at the root node, the binary function at a node is evaluated. The node is then exited via the left edge, if the binary function is zero; or the right edge, if the binary function is one. The next node's binary function is then evaluated. The value of the decision function D (x) is the value associated with the final leaf node (see Figure l(a». The path taken through the DDAG is known as the evaluation path. The input x reaches a node of the graph, if that node is on the evaluation path for x. We refer to the decision node distinguishing classes i and j as the ij-node. Assuming that the number of a leaf is "its class, this node is the i-th node in the (N - j + i)-th layer provided i < j. Similarly the j-nodes are those nodes involving class j, that is, the internal nodes on the two diagonals containing the leaf labeled by j. The DDAG is equivalent to operating on a list, where each node eliminates one class from the list. The list is initialized with a list of all classes. A test point is evaluated against the decision node that corresponds to the first and last elements of the list. If the node prefers Large Margin DAGs for Multiclass Classification test points on this SIde of hyperplane cannot be in class 1 549 3 4 1 1 vs4 SVM 4 3 2 (a) 1 1 1 1 1 1 1 (b) test pOInts on this Side of hyperplane cannot be In class 4 Figure 1: (a) The decision DAG for finding the best class out of four classes. The equivalent list state for each node is shown next to that node. (b) A diagram of the input space of a four-class problem. A I-v-l SVM can only exclude one class from consideration. one of the two classes, the other class is eliminated from the list, and the DDAG proceeds to test the first and last elements of the new list. The DDAG terminates when only one class remains in the list. Thus, for a problem with N classes, N - 1 decision nodes will be evaluated in order to derive an answer. The current state of the list is the total state of the system. Therefore, since a list state is reachable in more than one possible path through the system, the decision graph the algorithm traverses is a DAG, not simply a tree. Decision DAGs naturally generalize the class of Decision Trees, allowing for a more efficient representation of redundancies and repetitions that can occur in different branches of the tree, by allowing the merging of different decision paths. The class of functions implemented is the same as that of Generalized Decision Trees [1], but this particular representation presents both computational and learning-theoretical advantages. 3 Analysis of Generalization In this paper we study DDAGs where the node-classifiers are hyperplanes. We define a Perceptron DDAG to be a DDAG with a perceptron at every node. Let w be the (unit) weight vector correctly splitting the i and j classes at the ij-node with threshold O. We define the margin of the ij-node to be I = minc(x)==i,j {I(w, x) - Ol}, where c(x) is the class associated to training example x. Note that, in this definition, we only take into account examples with class labels equal to i or j . Theorem 1 Suppose we are able to classify a random m sampLe of LabeLed examples using a Perceptron DDAG on N classes containing K decision nodes with margins Ii at node i, then we can bound the generalization error with probability greater than 1 - 6 to be less than 130R2 ( 2(2m)K) --:;;;:D' log ( 4em) log( 4m) + log 6 ' where D' = L~l ~, and R is the radius of a ball containing the distribution's support. Proof: see Appendix 0 550 J. C. Platt, N. Cristianini and J. Shawe-Taylor Theorem 1 implies that we can control the capacity of DDAGs by enlarging their margin. Note that, in some situations, this bound may be pessimistic: the DDAG partitions the input space into poly topic regions, each of which is mapped to a leaf node and assigned to a specific class. Intuitively, the only margins that should matter are the ones relative to the boundaries of the cell where a given training point is assigned, whereas the bound in Theorem 1 depends on all the margins in the graph. By the above observations, we would expect that a DDAG whose j-node margins are large would be accurate at identifying class j , even when other nodes do not have large margins. Theorem 2 substantiates this by showing that the appropriate bound depends only on the j-node margins, but first we introduce the notation, Ej(G) = P{x : (x in class j and x is misclassified by G) or x is misclassified as class j by G}. Theorem 2 Suppose we are able to correctly distinguish class j from the other classes in a random m-sample with a DDAG Gover N classes containing K decision nodes with margins 'Yi at node i, then with probability 1 - J, 130R2 ( 2(2m)N-l) Ej(G) ~ ----;:;;D'log(4em) log(4m) + log J ' where D' = ~ .. d ~,and R is the radius of a ball containing the support of the L-tErno es "Y; distribution. Proof: follows exactly Lemma 4 and Theorem I, but is omitted.O 4 The DAGSVM algorithm Based on the previous analysis, we propose a new algorithm, called the Directed Acyclic Graph SVM (DAGSVM) algorithm, which combines the results of I-v-I SVMs. We will show that this combination method is efficient to train and evaluate. The analysis of Section 3 indicates that maximizing the margin of all of the nodes in a DDAG will minimize a bound on the generalization error. This bound is also independent of input dimensionality. Therefore, we will create a DDAG whose nodes are maximum margin classifiers over a kernel-induced feature space. Such a DDAG is obtained by training each ij-node only on the subset of training points labeled by i or j. The final class decision is derived by using the DDAG architecture, described in Section 2. The DAGSVM separates the individual classes with large margin. It is safe to discard the losing class at each I-v-l decision because, for the hard margin case, all of the examples of the losing class are far away from the decision surface (see Figure 1 (b)). For the DAGSVM, the choice of the class order in the list (or DDAG) is arbitrary. The experiments in Section 5 simply use a list of classes in the natural numerical (or alphabetical) order. Limited experimentation with re-ordering the list did not yield significant changes in accuracy performance. The DAGSVM algorithm is superior to other multiclass SVM algorithms in both training and evaluation time. Empirically, SVM training is observed to scale super-linearly with the training set size m [7], according to a power law: T = crn"Y, where 'Y ~ 2 for algorithms based on the decomposition method, with some proportionality constant c. For the standard I-v-r multiclass SVM training algorithm, the entire training set is used to create all N classifiers. Hence the training time for I-v-r is T1- v - r = cNm"Y . (1) Assuming that the classes have the same number of examples, training each l-v-I SVM only requires 2m/ N training examples. Thus, training K l-v-I SVMs would require T N(N -1) (2m) "Y '" "Y-1 N2- "Y "Y I-v-l c 2 N '" 2 c m . (2) Large Margin DAGs for Multiclass Classification 551 For a typical case, where 'Y = 2, the amount of time required to train all of the 1-v-1 SVMs is independent of N , and is only twice that of training a single 1-v-r SVM. Vsing 1-v-1 SVMs with a combination algorithm is thus preferred for training time. 5 Empirical Comparisons and Conclusions The DAGSVM algorithm was evaluated on three different test sets: the VSPS handwritten digit data set [10], the VCI Letter data set [2], and the VCI Covertype data set [2]. The USPS digit data consists of 10 classes (0-9), whose inputs are pixels of a scaled input image. There are 7291 training examples and 2007 test examples. The UCI Letter data consists of 26 classes (A-Z), whose inputs are measured statistics of printed font glyphs. We used the first 16000 examples for training, and the last 4000 for testing. All inputs of the VCI Letter data set were scaled to lie in [-1,1]. The VCI Covertype data consists of 7 classes of trees, where the inputs are terrain features. There are 11340 training examples and 565893 test examples. All of the continuous inputs for Covertype were scaled to have zero mean and unit variance. Discrete inputs were represented as a 1-of-n code. On each data set, we trained N 1-v-r SVMs and K 1-v-1 SVMs, using SMO [7], with soft margins. We combined the 1-v-1 SVMs both with the Max Wins algorithm and with DAGSVM. The choice of kernel and of the regularizing parameter C was determined via perfonnance on a validation set. The validation performance was measured by training on 70% of the training set and testing the combination algorithm on 30% of the training set (except for Covertype, where the UCI validation set was used). The best kernel was selected from a set of polynomial kernels (from degree 1 through 6), both homogeneous and inhomogeneous; and Gaussian kernels, with various a. The Gaussian kernel was always found to be best. (1 C Error Kernel Training CPU Classifier Size Rate (%) Evaluations Time (sec) (Kparameters) USPS l-v-r 3.58 100 4.7 2936 3532 760 Max Wins 5.06 100 4.5 1877 307 487 DAGSVM 5.06 100 4.4 819 307 487 Neural Net [10] 5.9 UCI Letter 1-v-r 0.447 100 2.2 8183 1764 148 Max Wins 0.632 100 2.4 7357 441 160 DAGSVM 0.447 10 2.2 3834 792 223 Neural Net 4.3 UCI Covertype l-v-r 1 10 30.2 7366 4210 105 Max Wins 1 10 29.0 7238 1305 107 DAGSVM 1 10 29.2 4390 1305 107 Neural Net [2] 30 Table 1: Experimental Results Table 1 shows the results of the experiments. The optimal parameters for all three multiclass SVM algorithms are very similar for both data sets. Also, the error rates are similar for all three algorithms for both data sets. Neither 1-v-r nor Max Wins is statistically significantly better than DAGSVM using McNemar's test [3] at a 0.05 significance level for USPS or UCI Letter. For VCI Covertype, Max Wins is slightly better than either of the other SVM-based algorithms. The results for a neural network trained on the same data sets are shown for a baseline accuracy comparison. The three algorithms distinguish themselves in training time, evaluation time, and classifier size. The number of kernel evaluations is a good indication of evaluation time. For J-v552 J C. Platt, N. Cristianini and J Shawe-Taylor r and Max Wins, the number of kernel evaluations is the total number of unique support vectors for all SVMs. For the DAGSVM, the number of kernel evaluations is the number of unique support vectors averaged over the evaluation paths through the DDAG taken by the test set. As can be seen in Table 1, Max Wins is faster than I-v-r SVMs, due to shared support vectors between the I-v-1 classifiers. The DAGSVM has the fastest evaluation. The DAGSVM is between a factor of 1.6 and 2.3 times faster to evaluate than Max Wins. The DAGSVM algorithm is also substantially faster to train than the standard I-v-r SVM algorithm: a factor of 2.2 and 11.5 times faster for these two data sets. The Max Wins algorithm shares a similar training speed advantage. Because the SVM basis functions are drawn from a limited set, they can be shared across classifiers for a great savings in classifier size. The number of parameters for DAGSVM (and Max Wins) is comparable to the number of parameters for I-v-r SVM, even though there are N (N - 1) /2 classifiers, rather than N. In summary, we have created a Decision DAG architecture, which is amenable to a VCstyle bound of generalization error. Using this bound, we created the DAGSVM algorithm, which places a two-class SVM at every node of the DDAG. The DAGSVM algorithm was tested versus the standard 1-v-r multiclass SVM algorithm, and Friedman's Max Wins combination algorithm. The DAGSVM algorithm yields comparable accuracy and memory usage to the other two algorithms, but yields substantial improvements in both training and evaluation time. 6 Appendix: Proof of Main Theorem Definition 2 Let F be a set of reaL vaLued functions. We say that a set of points X is ,shattered by F relative to r = (rx)xEx, if there are reaL numbers rx, indexed by x E X, such that for all binary vectors b indexed by X, there is a function fb E F satisfying (2bx - l)fdx) ~ (2bx - l)rx +,. The fat shattering dimension, fatF, of the set F is a function from the positive reaL numbers to the integers which maps a vaLue, to the size of the largest ,-shattered set, if the set is finite, or maps to infinity otherwise. As a relevant example, consider the class Flin = {x -+ (w, x) - (J : Ilwll = I}. We quote the following result from [1]. Theorem 3 Let Flin be restricted to points in a ball ofn dimensions of radius R about the origin. Then We wiIl bound generalization with a technique that closely resembles the technique used in [1] to study Perceptron Decision Trees. We will now give a lemma and a theorem: the lemma bounds the probability over a double sample that the first half has zero error and the second error greater than an appropriate E. We assume that the DDAG on N classes has K = N(N - 1)/2 nodes and we denote fat}'"l· b) by fatb). III Lemma 4 Let G be a DDAG on N classes with K = N(N - 1)/2 decision nodes with margins ,1,,2, ... "K at the decision nodes satisfying ki = fat ( ,i/8), where fat is continuous from the right. Then the following bound hoLds, p2m{xy:::I a graph G : G which separates classes i and j at the ij-node for all x in x, a fraction of points misclassified in y > E(m, K, 6).} < 6 where E(m, K, 6) = ! (D log (8m) + log 2;) and D = L:~1 ki log(4em/ki ). Proof The proof of Lemma 4 is omitted for space reasons, but is formally analogous to the proof of Lemma 4.4 in [8], and can easily be reconstructed from it. 0 Large Margin DAGs for Muldclass Classification 553 Lemma 4 applies to a particular DDAG with a specified margin Ii at each node. In practice, we observe these quantities after generating the DDAG. Hence, to obtain a bound that can be applied in practice, we must bound the probabilities uniformly over all of the possible margins that can arise. We can now give the proof for Theorem 1. Proof of Main Theorem: We must bound the probabilities over different margins. We first use a standard result due to Vapnik [9, page 168] to bound the probability of error in terms of the probability of the discrepancy between the performance on two halves of a double sample. Then we combine this result with Lemma 4. We must consider all possible patterns of ki's over the decision nodes. The largest allowed value of ki is m, and so, for fixed K, we can bound the number of possibilities by m K . Hence, there are m K of applications of Lemma 4 for a fixed N. Since K = N(N - 1)/2, we can let 15k = 8/mK, so thatthe sum L:~l 15k = 8. Choosing € (m ,K, 8;) = 6~2 (D'IOg(4em) log(4m) + log 2(2;)K) (3) in the applications of Lemma 4 ensures that the probability of any of the statements failing to hold is less than 8/2. Note that we have replaced the constant 82 = 64 by 65 in order to ensure the continuity from the right required for the application of Lemma 4 and have upper bounded log(4em/ki ) by log(4em). Applying Vapnik's Lemma [9, page 168] in each case, the probability that the statement of the theorem fails to hold is less than 8. 0 More details on this style of proof, omitted in this paper for space constraints, can be found in [1]. References [1] K. Bennett, N. Cristianini, J. Shawe-Taylor, and D. Wu. Enlarging the margin in perceptron decision trees. Machine Learning (submitted). http://lara.enm.bris.ac.ukJcig/pubsIML-PDT.ps. [2] C. Blake, E. Keogh, and C. Merz. UCI repository of machine leaming databases. Dept. of information and computer sciences, University of Califomia, Irvine, 1998. http://www.ics.uci.edul,,,mleamIMLRepository.html. [3] T. G. Dietterich. Approximate statistical tests for comparing supervised classification leaming algorithms. Neural Computation, 10: 1895-1924, 1998. [4] J. H. Friedman. Another approach to polychotomous classification. Technical report, Stanford Department of Statistics, 1996. htlp:llwww-stat.stanford.edulreports/friedmanlpoly.ps.Z. [5] S. Knerr, L. Personnaz, and G. Dreyfus. Single-layer leaming revisited: A stepwise procedure for building and training a neural network. In Fogelman-Soulie and Herault, editors, Neurocomputing: Algorithms, Architectures and Applications, NATO ASI. Springer, 1990. [6] U. KreGel. Pairwise classification and support vector machines. In B. SchOlkopf, C. J. C. Burges, and A. J. Smola, editors, Advances in Kernel Methods: Support Vector Learning, pages 255-268. MIT Press, Cambridge, MA, 1999. [7] J. Platt. Fast training of support vector machines using sequential minimal optimization. In B. Scholkopf, C. J. C. Burges, and A. J. Smola, editors, Advances in Kernel Methods Support Vector Learning, pages 185-208. MIT Press, Cambridge, MA, 1999. [8] J. Shawe-Taylor and N. Cristianini. Data dependent structural risk minimization for perceptron decision trees. In M. Jordan, M. Keams, and S. SoJla, editors, Advances in Neural Information Processing Systems, volume 10, pages 336-342. MIT Press, 1999. [9] V. Vapnik. Estimation of Dependences Based on Empirical Data [in Russian). Nauka, Moscow, 1979. (English translation: Springer Verlag, New York, 1982). [10] V. Vapnik. Statistical Learning Theory. Wiley, New York, 1998.
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Approximate Planning in Large POMDPs via Reusable Trajectories Michael Kearns AT&T Labs mkearns@research.att.com Yishay Mansour Tel Aviv University mansour@math.tau.ac.il Abstract AndrewY. Ng UC Berkeley ang@cs.berkeley.edu We consider the problem of reliably choosing a near-best strategy from a restricted class of strategies TI in a partially observable Markov decision process (POMDP). We assume we are given the ability to simulate the POMDP, and study what might be called the sample complexity that is, the amount of data one must generate in the POMDP in order to choose a good strategy. We prove upper bounds on the sample complexity showing that, even for infinitely large and arbitrarily complex POMDPs, the amount of data needed can be finite, and depends only linearly on the complexity of the restricted strategy class TI, and exponentially on the horizon time. This latter dependence can be eased in a variety of ways, including the application of gradient and local search algorithms. Our measure of complexity generalizes the classical supervised learning notion of VC dimension to the settings of reinforcement learning and planning. 1 Introduction Much recent attention has been focused on partially observable Markov decision processes (POMDPs) which have exponentially or even infinitely large state spaces. For such domains, a number of interesting basic issues arise. As the state space becomes large, the classical way of specifying a POMDP by tables of transition probabilities clearly becomes infeasible. To intelligently discuss the problem of planning that is, computing a good strategy 1 in a given POMDP compact or implicit representations of both POMDPs, and of strategies in POMDPs, must be developed. Examples include factored next-state distributions [2, 3, 7], and strategies derived from function approximation schemes [8]. The trend towards such compact representations, as well as algorithms for planning and learning using them, is reminiscent of supervised learning, where researchers have long emphasized parametric models (such as decision trees and neural networks) that can capture only limited structure, but which enjoy a number of computational and information-theoretic benefits. Motivated by these issues, we consider a setting were we are given a generative model, or lThroughout, we use the word strategy to mean any mapping from observable histories to actions, which generalizes the notion of policy in a fully observable MDP. 1002 M Kearns. Y. Mansour and A. Y. Ng simulator, for a POMDP, and wish to find a good strategy 7r from some restricted class of strategies II. A generative model is a "black box" that allows us to generate experience (trajectories) from different states of our choosing. Generative models are an abstract notion of compact POMDP representations, in the sense that the compact representations typically considered (such as factored next-state distributions) already provide efficient generative models. Here we are imagining that the strategy class II is given by some compact representation or by some natural limitation on strategies (such as bounded memory). Thus, the view we are adopting is that even though the world (POMDP) may be extremely complex, we assume that we can at least simulate or sample experience in the world (via the generative model), and we try to use this experience to choose a strategy from some "simple" class II. We study the following question: How many calls to a generative model are needed to have enough data to choose a near-best strategy in the given class? This is analogous to the question of sample complexity in supervised learning but harder. The added difficulty lies in the reuse of data. In supervised learning, every sample (x, f(x)) provides feedback about every hypothesis function h(x) (namely, how close h(x) is to f(x)). If h is restricted to lie in some hypothesis class 1i, this reuse permits sample complexity bounds that are far smaller than the size of 1i. For instance, only O(log(I1il)) samples are needed to choose a near-best model from a finite class 1i. If 1i is infinite, then sample sizes are obtained that depend only on some measure of the complexity of1i (such as VC dimension [9]), but which have no dependence on the complexity of the target function or the size of the input domain. In the POMDP setting, we would like analogous sample complexity bounds in terms of the "complexity" of the strategy class II bounds that have no dependence on the size or complexity of the POMDP. But unlike the supervised learning setting, experience "reuse" is not immediate in POMDPs. To see this, consider the "straw man" algorithm that, starting with some 7r E II, uses the generative model to generate many trajectories under 7r, and thus forms a Monte Carlo estimate of V 7r (so). It is not clear that these trajectories under 7r are of much use in evaluating a different 7r' E II, since 7r and 7r' may quickly disagree on which actions to take. The naive Monte Carlo method thus gives 0(1111) bounds on the "sample complexity," rather than O(log(IIII)), for the finite case. In this paper, we shall describe the trajectory tree method of generating "reusable" trajectories, which requires generating only a (relatively) small number of trajectories a number that is independent of the state-space size of the POMDP, depends only linearly on a general measure of the complexity of the strategy class II, and depends exponentially on the horizon time. This latter dependence can be eased via gradient algorithms such as Williams' REINFORCE [10] and Baird and Moore's more recent YAPS [1], and by local search techniques. Our measure of strategy class complexity generalizes the notion of VC dimension in supervised learning to the settings of reinforcement learning and planning, and we give bounds that recover for these settings the most powerful analogous results in supervised learning bounds for arbitrary, infinite strategy classes that depend only on the dimension of the class rather than the size of the state space. 2 Preliminaries We begin with some standard definitions. A Markov decision process (MDP) is a tuple (S, So, A, {P (,1 s, a)}, R), where: S is a (possibly infinite) state set; So E S is a start state; A = {al' . .. ,ad are actions; PC Is, a) gives the next-state distribution upon taking action a from state s; and the reward function R(s, a) gives the corresponding rewards. We assume for simplicity that rewards are deterministic, and further that they are bounded Approximate Planning in Large POMDPs via Reusable Trajectories 1003 in absolute value by Rmax. A partially observable Markov decision process (POMDP) consists of an underlying MOP and observation distributions Q(ols) for each state s, where 0 is the random observation made at s. We have adopted the common assumption of a fixed start state,2 because once we limit the class of strategies we entertain, there may not be a single "best" strategy in the classdifferent start states may have different best strategies in II. We also assume that we are given a POMOP M in the form of a generative model for M that, when given as input any state-action pair (s, a), will output a state S' drawn according to P(·ls, a), an observation o drawn according to Q(·ls), and the reward R(s, a). This gives us the ability to sample the POMOP M in a random-access way. This definition may initially seem unreasonably generous: the generative model is giving us a fully observable simulation of a partially observable process. However, the key point is that we must still find a strategy that performs well in the partially observable setting. As a concrete example, in designing an elevator control system, we may have access to a simulator that generates random rider arrival times, and keeps track of the waiting time of each rider, the number of riders waiting at every floor at every time of day, and so on. However helpful this information might be in designing the controller, this controller must only use information about which floors currently have had their call button pushed (the observables). In any case, readers uncomfortable with the power provided by our generative models are referred to Section 5, where we briefly describe results requiring only an extremely weak form of partially observable simulation. At any time t, the agent will have seen some sequence of observations, 00,··., Ot, and will have chosen actions and received rewards for each of the t time steps prior to the current one. We write its observable history as h (( 00, ao, TO), ... , (Ot-l , at-I, Tt-l ), (Ot, _, _)). Such observable histories, also called trajectories, are the inputs to strategies. More formally, a strategy 7r is any (stochastic) mapping from observable histories to actions. (For example, this includes approaches which use the observable history to track the belief state [5].) A strategy class II is any set of strategies. We will restrict our attention to the case of discounted return,3 and we let, E [0,1) be the discount factor. We define the t::-horizon time to be HE = 10gl'(t::(1 - ,)/2Rmax). Note that returns beyond the first HE-steps can contribute at most t::/2 to the total discounted return. Also, let Vmax = Rmax/(l - ,) bound the value function. Finally, for a POMDP M and a strategy class II, we define opt(M, II) = SUP7rEII V7r (so) to be the best expected return achievable from So using II. Our problem is thus the following: Given a generative model for a POMOP M and a strategy class II, how many calls to the generative model must we make, in order to have enough data to choose a 7r E II whose performance V7r(so) approaches opt(M, II)? Also, which calls should we make to the generative model to achieve this? 3 The Trajectory Tree Method We now describe how we can use a generative model to create "reusable" trajectories. For ease of exposition, we assume there are only two actions al and a2, but our results generalize easily to any finite number of actions. (See the full paper [6].) 2 An equivalent definition is to assume a fixed distribution D over start states, since So can be a "dummy" state whose next-state distribution under any action is D. 3The results in this paper can be extended without difficulty to the undiscounted finite-horizon setting [6]. 1004 M. Keams, Y. Mansour and A. Y. Ng A trajectory tree is a binary tree in which each node is labeled by a state and observation pair, and has a child for each of the two actions. Additionally, each link to a child is labeled by a reward, and the tree's depth will be H~, so it will have about 2H e nodes. (In Section 4, we will discuss settings where this exponential dependence on H~ can be eased.) Each trajectory tree is built as follows: The root is labeled by So and the observation there, 00 ' Its two children are then created by calling the generative model on (so, ad and (so, a2), which gives us the two next-states reached (say s~ and s~ respectively), the two observations made (say o~ and o~), and the two rewards received (r~ = R(so, ad and r~ = R(so, a2). Then (s~ , aD and (s~, o~) label the root's aI-child and a2-child, and the links to these children are labeled r~ and r~. Recursively, we generate two children and rewards this way for each node down to depth H~ . Now for any deterministic strategy tr and any trajectory tree T, tr defines a path through T: tr starts at the root, and inductively, if tr is at some internal node in T, then we feed to tr the observable history along the path from the root to that node, and tr selects and moves to a child of the current node. This continues until a leaf node is reached, and we define R( tr, T) to be the discounted sum of returns along the path taken. In the case that tr is stochastic, tr defines a distribution on paths in T, and R(tr, T) is the expected return according to this distribution. (We will later also describe another method for treating stochastic strategies.) Hence, given m trajectory trees T1 , ... , T m, a natural estimate for V7r(so) is V7r(so) = ,; 2:::1 R(tr, Ti). Note that each tree can be used to evaluate any strategy, much the way a single labeled example (x, f(x)) can be used to evaluate any hypothesis h(x) in supervised learning. Thus in this sense, trajectory trees are reusable. Our goal now is to establish uniform convergence results that bound the error of the estimates V7r (so) as a function of the "sample size" (number of trees) m. Section 3.1 first treats the easier case of deterministic classes II; Section 3.2 extends the result to stochastic classes. 3.1 The Case of Deterministic II Let us begin by stating a result for the special case of finite classes of deterministic strategies, which will serve to demonstrate the kind of bound we seek. Theorem 3.1 Let II be any finite class of deterministic strategies for an arbitrary twoaction POMDP M. Let m trajectory trees be created using a generative modelfor M, and V7r(so) be the resulting estimates. lfm = 0 ((Vrnax /t)2(log(IIII) + log(1/8))), then with probability 1 - 8, I V7r (so) - V7r (so) I :s t holds simultaneously for alltr E II. Due to space limitations, detailed proofs of the results of this section are left to the full paper [6], but we will try to convey the intuition behind the ideas. Observe that for any fixed deterministic tr, the estimates R( tr, Ti) that are generated by the m different trajectory trees Ti are independent. Moreover, each R(tr, Ti ) is an unbiased estimate of the expected discounted H~ -step return of tr, which is in turn t/2-close to V7r(so). These observations, combined with a simple Chernoff and union bound argument, are sufficient to establish Theorem 3.1. Rather than developing this argument here, we instead move straight on to the harder case of infinite II. When addressing sample complexity in supervised learning, perhaps the most important insight is that even though a class 1i may be infinite, the number of possible behaviors of 1i on a finite set of points is often not exhaustive. More precisely, for boolean functions, we say that the set Xl, ... , Xd is shattered by 1i if every of the 2d possible labelings of Approximate Planning in Large POMDPs via Reusable Trajectories 1005 these points is realized by some h E 1i. The VC dimension of 1i is then defined as the size of the largest shattered set [9]. It is known that if the VC dimension of 1i is d, then the number <P d (m) of possible labelings induced by 1i on a set of m points is at most (em J d)d, which is much less than 2m for d « m. This fact provides the key leverage exploited by the classical VC dimension results, and we will concentrate on replicating this leverage in our setting. If II is a (possibly infinite) set of deterministic strategies, then each strategy tr E II is simply a deterministic function mapping from the set of observable histories to the set {al' a2}, and is thus a boolean function on observable histories. We can therefore write VC(II) to denote the familiar VC dimension of the set of binary functions II. For example, if II is the set of all thresholded linear functions of the current vector of observations (a particular type of memoryless strategy), then VC(II) simply equals the number of parameters. We now show intuitively why a class II of bounded VC dimension d cannot induce exhaustive behavior on a set Tl , ... ,T m of trajectory trees for m » d. Note that if trl, tr2 E II are such that their "reward labelings" (R(trl' Tl ), ... ,R(trl' T m)) and (R( tr2, Tt), ... , R( tr2, T m)) differ, then R( trl, Ti) =f. R(tr2' Ti) for some 1 ::; i ::; m. But if trl and tr2 give different returns on Ti , then they must choose different actions at some node in Ti . In other words, every different reward labeling of the set of m trees yields a different (binary) labeling of the set of m . 2H • observable histories in the trees. So, the number of different tree reward labelings can be at most <Pd(m· 2H<) ::; (em· 2H<Jd)d. By developing this argument carefully and applying classical uniform convergence techniques, we obtain the following theorem. (Full proof in [6].) Theorem 3.2 Let II be any class of deterministic strategies for an arbitrary two-action POMDP M, and let VC(II) denote its VC dimension. Let m trajectory trees be created using a generative model for M, and "\I7r (so) be the resulting estimates. If (1) then with probability 1 - 6, I V 7r (so) - "\I7r (so) I ::; € holds simultaneously for alltr E II. 3.2 The Case of Stochastic II We now address the case of stochastic strategy classes. We describe an approach where we transform stochastic strategies into "equivalent" deterministic ones and operate on the deterministic versions, reducing the problem to the one handled in the previous section. The transformation is as follows: Given a class of stochastic strategies II, each with domain X (where X is the set of all observable histories), we first extend the domain to be X x [0,1]. Now for each stochastic strategy tr E II, define a corresponding deterministic transformed strategy tr' with domain X x [0,1], given by: tr'(h, r) = al if r ::; Pr[tr(h) = ad, and 7r'(h,r) = a2 otherwise (for any hEX, r E [0,1]). Let II' be the collection of these transformed deterministic strategies tr'. Since II' is just a set of deterministic boolean functions, its VC dimension is well-defined. We then define the pseudo-dimension of the original set of stochastic strategies II to be p VC(II) = VC(II').4 Having transformed the strategy class, we also need to transform the POMDP, by augmenting the state space S to be S x [0,1]. Informally, the transitions and rewards remain the same, except that after each state transition, we draw a new random variable r uniformly in [0,1], and independently of all previous events. States are now of the form (s, r), and we let r be an observed variable. Whenever in the original POMDP a stochastic strategy tr would 4This is equivalent to the conventional definition of the pseudo-dimension of IT [4], when it is viewed as a set of maps into real-valued action-probabilities. 1006 M Kearns, Y. Mansour and A. Y. Ng have been given a history h, in the transformed POMDP the corresponding deterministic transformed strategy 7r' is given (h, r), where r is the [0, l]-random variable at the current state. By the definition of 7r', it is easy to see that 7r' and 7r have exactly the same chance of choosing each action at any node (randomization over r). We are now back in the deterministic case, so Theorem 3.2 applies, with VC(II) replaced by pVC (II) = VC(II'), and we again have the desired uniform convergence result. 4 Algorithms for Approximate Planning Given a generative model for a POMDP, the preceding section's results immediately suggest a class of approximate planning algorithms: generate m trajectory trees T1 , ... , T m, and search for a 7r E II that maximizes V7r (so) = (1/ m) L R( 7r, Ti). The following corollary to the uniform convergence results establishes the soundness of this approach. Corollary 4.1 Let II be a class of strategies in a POMDP M, and let the number m of trajectory trees be as given in Theorem 3.2. Let it = argmax7rErr{V7r(so)} be the policy in II with the highest empirical return on the m trees. Then with probability 1 - 0, it is near-optimal within II: V7T(SO) ~ opt(M, II) - 2€. (2) If the suggested maximization is computationally infeasible, one can search for a local maximum 7r instead, and uniform convergence again assures us that V7r (so) is a trusted estimate of our true performance. Of course, even finding a local maximum can be expensive, since each trajectory tree is of size exponential in H{. However, in practice it may be possible to significantly reduce the cost of the search. Suppose we are using a class of (possibly transformed) deterministic strategies, and we perform a greedy local search over II to optimize V7r (so). Then at any time in the search, to evaluate the policy we are currently considering, we really need to look at only a single path of length Hf in each tree, corresponding to the path taken by the strategy being considered. Thus, we should build the trajectory trees lazily that is, incrementally build each node of each tree only as it is needed to evaluate R( 7r, Ti) for the current strategy 7r. If there are parts of a tree that are reached only by poor policies, then a good search algorithm may never even build these parts of the tree. In any case, for a fixed number of trees, each step of the local search now takes time only linear in H f •5 There is a different approach that works directly on stochastic strategies (that is, without requiring the transformation to deterministic strategies). In this case each stochastic strategy 7r defines a distribution over all the paths in a trajectory tree, and thus calculating R( 7r, T) may in general require examining complete trees. However, we can view each trajectory tree as a small, deterministic POMDP by itself, with the children of each node in the tree being its successor nodes. So if II = {7re : e E IRd} is a smoothly parameterized family of stochastic strategies, then algorithms such as William's REINFORCE [10] can be used to find an unbiased estimate of the gradient (d/ de) V7r 9 (so), which in turn can be used to 5 See also (Ng and Jordan, in preparation) which, by assuming a much stronger model of a POMDP (a deterministic function 1 such that I(s, a, r) is distributed according to P('ls, a) when r is distributed Uniform[O,l]), gives an algorithm that enjoys uniform convergence bounds similar to those presented here, but with only a polynomial rather than exponential dependence on H,. The algorithm samples a number of vectors r(i) E [0, IjH., each of which, with I, defines an H,-step Monte Carlo evaluation trial for any policy 7r. The bound is on the number of such random vectors needed (rather than on the total number of calls to f). Approximate Planning in Large POMDPs via Reusable Trajectories 1007 perform stochastic gradient ascent to maximize V7r8 (so). Moreover, for a fixed number of trees, these algorithms need only O(H€) time per gradient estimate; so combined with lazy tree construction, we again have a practical algorithm whose per-step complexity is only linear in the horizon time. This line of thought is further developed in the long version of the paper.6 5 The Random Trajectory Method Using a fully observable generative model of a POMDP, we have shown that the trajectory tree method gives uniformly good value estimates, with an amount of experience linear in VC(II), and exponential in H€. It turns out we can significantly weaken the generative model, yet still obtain essentially the same theoretical results. In this harder case, we assume a generative model that provides only partially observable histories generated by a truly random strategy (which takes each action with equal probability at every step, regardless of the history so far). Furthermore, these trajectories always begin at the designated start state, so there is no ability provided to "reset" the POMDP to any state other than so. (Indeed, underlying states may never be observed.) Our method for this harder case is called the Random Trajectory method. It seems to lead less readily to practical algorithms than the trajectory tree method, and its formal description and analysis, which is more difficult than for trajectory trees, are given in the long version of this paper [6]. As in Theorem 3.2, we prove that the amount of data needed is linear in VC(II), and exponential in the horizon time that is, by averaging appropriately over the resulting ensemble of trajectories generated, this amount of data is sufficient to yield uniformly good estimates of the values for all strategies in II. References [1] L. Baird and A. W. Moore. Gradient descent for general Reinforcement Learning. In Advances in Neural Information Processing Systems 11, 1999. [2] C. Boutilier, T. Dean, and S. Hanks. Decision theoretic planning: Structural assumptions and computational leverage. Journal of Artificial Intelligence Research, 1999. [3] X. Boyen and D. Koller. Tractable inference for complex stochastic processes. In Proc. UAI, pages 33-42, 1998. [4] David Haussler. Decision theoretic generalizations of the PAC model for neural net and oter learning applications. Information and Computation, 100:78-150, 1992. [5] L. P. Kaelbling, M. L. Littman, and A. R. Cassandra. Planning and acting in partially observable stochastic domains. ArtifiCial Intelligence, 101, 1998. [6] M. Kearns, Y. Mansour, and A. Y. Ng. Approximate planning in large POMDPs via reusable trajectories. (long version), 1999. [7] 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, 1999. [8] R. S. Sutton and A. G. Barto. Reinforcement Learning. MIT Press, 1998. [9] Y.N. Vapnik. Estimation of Dependences Based on Empirical Data. Springer-Verlag, 1982. [10] R. J. Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 8:229-256, 1992. 6In the full paper, we also show how these algorithms can be extended to find in expected O( He) time an unbiased estimate of the gradient of the true value V 7T8 (so) for discounted infinite horizon problems (whereas most current algorithms either only converge asymptotically to an unbiased estimate of this gradient, or need an absorbing state and "proper" strategies).
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Maximum entropy discrimination Tommi Jaakkola MIT AI Lab 545 Technology Sq. Cambridge, MA 02139 tommi@ai.mit.edu Marina Meila MIT AI Lab 545 Technology Sq. Cambridge, MA 02139 mmp@ai. mit. edu Abstract Tony Jebara MIT Media Lab 20 Ames St. Cambridge, MA 02139 jebara@media. mit. edu We present a general framework for discriminative estimation based on the maximum entropy principle and its extensions. All calculations involve distributions over structures and/or parameters rather than specific settings and reduce to relative entropy projections. This holds even when the data is not separable within the chosen parametric class, in the context of anomaly detection rather than classification, or when the labels in the training set are uncertain or incomplete. Support vector machines are naturally subsumed under this class and we provide several extensions. We are also able to estimate exactly and efficiently discriminative distributions over tree structures of class-conditional models within this framework. Preliminary experimental results are indicative of the potential in these techniques. 1 Introduction Effective discrimination is essential in many application areas. Employing generative probability models such as mixture models in this context is attractive but the criterion (e.g., maximum likelihood) used for parameter/structure estimation is suboptimal. Support vector machines (SVMs) are, for example, more robust techniques as they are specifically designed for discrimination [9]. Our approach towards general discriminative training is based on the well known maximum entropy principle (e.g., [3]). This enables an appropriate training of both ordinary and structural parameters of the model (cf. [5, 7]). The approach is not limited to probability models and extends, e.g., SVMs. 2 Maximum entropy classification Consider a two-class classification problem1 where labels y E {-I, I} are assigned IThe extension to a multi-class is straightforward[4]. The formulation also admits an easy extension to regression problems, analogously to SVMs. Maximum Entropy Discrimination 471 to examples X E X. Given two generative probability distributions P(XIOy ) with parameters Oy, one for each class, the corresponding decision rule follows the sign of the discriminant function: P(XIOl) C(XI8) = log P(XIO-l) + b (1) where 8 = {Ol,O-l,b} and b is a bias term, usually expressed as a log-ratio b = log p/(l - p). The class-conditional distributions may come from different families of distributions or the parametric discriminant function could be specified directly without any reference to models. The parameters Oy may also include the model structure (see later sections). The parameters 8 = {01, 0-1, b} should be chosen to maximize classification accuracy. We consider here the more general problem of finding a distribution P(8) over parameters and using a convex combination of discriminant functions, i.e., J P(8)C(XI8)d8 in the decision rule. The search for the optimal P(8) can be formalized as a maximum entropy (ME) estimation problem. Given a set of training examples {Xl, ... , X T} and corresponding labels {Yl, ... ,YT} we find a distribution P(8) that maximizes the entropy H(P) subject to the classification constraints J P(8) [Yt C(Xt I8)] d8 2: , for all t. Here, > 0 specifies a desired classification margin. The solution is unique (if it exists) since H(P) is concave and the linear constraints specify a convex region. Note that the preference towards high entropy distributions (fewer assumptions) applies only within the admissible set of distributions P'"Y consistent with the constraints. See [2] for related work. We will extend this basic idea in a number of ways. The ME formulation assumes, for example, that the training examples can be separated with the specified margin. We may also have a reason to prefer some parameter values over others and would therefore like to incorporate a prior distribution Po (8). Other extensions and generalizations will be discussed later in the paper. A more complete formulation is based on the following minimum relative entropy principle: Definition 1 Let {Xt, yd be the training examples and labels, C(XI8) a parametric discriminant function, and, = [,1, ... "tl a set of margin variables. Assuming a prior distribution Po(8,,), we find the discriminative minimum relative entropy (MRE) distribution P(8,,) by minimizing D(PIIPo) subject to (2) for all t. Here fj = sign ( J P(8) C(XI8) d8) specifies the decision rule for any new example X. The margin constraints and the preference towards large margin solutions are encoded in the prior Po('). Allowing negative margin values with non-zero probabilities also guarantees that the admissible set P consisting of distributions P(8,,) consistent with the constraints, is never empty. Even when the examples cannot be separated by any discriminant function in the parametric class (e.g., linear), we get a valid solution. The miss-classification penalties follow from Pob) as well. 472 T. Jaakkola, M. Meila and T. Jebara b) -. -. .••... ..... c) -~:C--~-;----;-----.---:. Figure 1: a) Minimum relative entropy (MRE) projection from the prior distribution to the admissible set. b) The margin prior Po(Tt). c) The potential terms in the MRE formulation (solid line) and in SVMs (dashed line). c = 5 in this case. Suppose po(e, ,) = po(e)Po(T) and poe,) = Dt Po (Tt) , where Po(Tt) = ee-c(I-"Yt) for ,t ~ 1, (3) This is shown in Figure lb. The penalty for margins smaller than I-lie (the prior mean of,t) is given by the relative entropy distance between P(T) and Po(T). This is similar but not identical to the use of slack variables in support vector machines. Other choices of the prior are discussed in [4]. The MRE solution can be viewed as a relative entropy projection from the prior distribution po(e,,) to the admissible set P . Figure la illustrates this view. From the point of view of regularization theory, the prior probability Po specifies the entropic regularization used in this approach. Theorem 1 The solution to the MRE problem has the following general form [1] pee,,) = ztA)Po(e,,) el:t At[Yt,C(xtle)-"Y,] (4) where Z (A) is the normalization constant (partition function) and A = {AI, ... , AT} defines a set of non-negative Lagrange multipliers, one for each classification constraint. A are set by finding the unique maximum of the following jointly concave objective function: J(A) = -logZ(A) The solution is sparse, Le., only a few Lagrange mUltipliers will be non-zero. This arises because many of the classification constraints become irrelevant once the constraints are enforced for a small subset of examples. Sparsity leads to immediate but weak generalization guarantees expressed in terms of the number of non-zero Lagrange multipliers [4]. Practicalleave-one-out cross-validation estimates can be also derived. 2.1 Practical realization of the MRE solution We now turn to finding the MRE solution. To begin with, we note that any disjoint factorization of the prior Po (e, ,), where the corresponding parameters appear in distinct additive components in YtC(Xt, e) - ,t, leads to a disjoint factorization of the MRE solution pee, ,). For example, {e \ b, b, ,} provides such a factorization. As a result of this factorization, the bias term could be eliminated by imposing additional constraints on the Lagrange multipliers [4]. This is analogous to the handling of the bias term in support vector machines [9]. We consider now a few specific realizations such as support vector machines and a class of graphical models. Maximum Entropy Discrimination 473 2.1.1 Support vector machines It is well known that the log-likelihood ratio of two Gaussian distributions with equal covariance matrices yields a linear decision rule. With a few additional assumptions, the MRE formulation gives support vector machines: Theorem 2 Assuming C(X, e) = OT X - band po(e, ,) = Po(O)Po(b)Po(,) where Po (0) is N (0,1), Po (b) approaches a non-informative prior, and Po (J) is given by eq. (3) then the Lagrange multipliers A are obtained by maximizing J(A) subject to ° ::; At ::; c and 2:t AtYt = 0, where J(A) = :~:) At + log(l - At/C)]- ~ 2:, AtAt'YtydX [ Xt,) t t,t' (5) The only difference between our J(A) and the (dual) optimization problem for SVMs is the additional potential term log(l - At/C). This highlights the effect of the different miss-classification penalties, which in our case come from the MRE projection. Figure Ib shows, however, that the additional potential term does not always carry a huge effect (for c = 5). Moreover, in the separable case, letting c -+ 00, the two methods coincide. The decision rules are formally identical. We now consider the case where the discriminant function C(X, e) corresponds to the log-likelihood ratio of two Gaussians with different (and adjustable) covariance matrices. The parameters e in this case are both the means and the covariances. The prior paCe) must be the conjugate Normal-Wishart to obtain closed form integrals2 for the partition function, Z. Here, p(e l , e- l ) is P(m1' VdP(m-1, V-d, a density over means and covariances. The prior distribution has the form po(ed = N(m1; mo, Vdk) IW(V1; kVo, k) with parameters (k, mo, Vo) that can be specified manually or one may let k -+ 0 to get a non-informative prior. Integrating over the parameters and the margin, we get Z = Z"( X Zl X Z-l, where (6) .:l -.:l w 6 T -T . N1 = 2:t Wt, Xl = 2:t ~Xt, 3 1 = 2:t WtXtXt - N1X I X 1 . Here, Wt IS a scalar weight given by Wt = u(Yt)+YtAt. For Z-l, the weights are set to Wt = u( -Yt)-YtAt; u(·) is the step function. Given Z, updating A is done by maximizing J(A). The resulting marginal MRE distribution over the parameters (normalized by Zl x Z-d is a Normal-Wishart distribution itself, p(e1) = N(m1; Xl, VdNd IW(V1; 3 1 , N1) with the final A values. Predicting the label for a new example X involves taking expectations of the discriminant function under a Normal-Wishart. This is We thus obtain discriminative quadratic decision boundaries. These extend the linear boundaries without (explicitly) resorting to kernels. More generally, the covariance estimation in this framework adaptively modifies the kernel. 2This can be done more generally for conjugate priors in the exponential family. 474 T. Jaakkola, M Meila and T. Jebara 2.1.2 Graphical models We consider here graphical models with no hidden variables. The ME (or MRE) distribution is in this case a distribution over both structures and parameters. Finding the distribution over parameters can be done in closed form for conjugate priors when the observations are complete. The distribution over structures is, in general, intractable. A notable exception is a tree model that we discuss in the forthcoming. A tree graphical model is a graphical model for which the structure is a tree. This model has the property that its log-likelihood can be expressed as a sum of local terms [8] logP(X,EIO) = 2: hu(X, 0) + 2: wuv(X,O) (8) u uvEE The discriminant function consisting of the log-likelihood ratio of a pair of tree models (depending on the edge sets E1, E_l, and parameters 01, 0_1) can be also expressed in this form. We consider here the ME distribution over tree structures for fixed parameters3 . The treatment of the general case (i.e. including the parameters) is a direct extension of this result. The ME distribution over the edge sets E1 and E-1 factorizes with components P(E±l) = _1_e±2:,)."Yt[2:uvEE±1 w;!'v1(X"O±I)+2:u hU(X"O±I») = h±1 IT W~1 (9) Z±1 Z±1 EE uv ±1 where Z±1, h±l, W±1 are functions of the same Lagrange multipliers..\. To completely define the distribution we need to find ..\ that optimize J(..\) in Theorem 1; for classification we also need to compute averages with respect to P(E±d. For these, it suffices to obtain an expression of the partition function( s) Z±1. P is a discrete distribution over all possible tree structures for n variables (there are nn-2 trees). However, a remarkable graph theory result, called the Matrix Tree Theorem [10], enables us to perform all necessary summations in closed form in polynomial time. On the basis of this result, we find Theorem 3 The normalization constant Z of a distribution of the form (9) is Z h.2: IT Wuv = h 'IQ(W)I, where (10) E uvEE { -Wuv 2:~'=l WV'v Quv(W) u=f:.v u=v (11) This shows that summing over the distribution of all trees, when this distribution factors according to the trees' edges, can be done in closed form by computing the value of a determinant in time O(n3 ). Since we obtain a closed form expression, optimization of the Lagrange multipliers and evaluating the resulting classification rule are also tractable. Figure 2a provides a comparison of the discriminative tree approach and a maximum likelihood tree estimation method on a DNA splice junction problem. 3Each tree relies on a different set of n -1 pairwise node marginals. In our experiments the class-conditional pairwise marginals were obtained directly from data. Maximum Entropy Discrimination a) t··········· ~:: ,//._ .. : 8:: : CD 2" -.. .. "false pOsitives" .. . b) °0 02 04 04 04 -c) o. 000-----=0.-=. ----=-... :---::": 0.C---::": 0 .• ---' .... 475 Figure 2: ROC curves based on independent test sets. a) Tree estimation: discriminative (solid) and ML (dashed) trees. b) Anomaly detection: MRE (solid) and Bayes (dashed). c) Partially labeled case: 100% labeled (solid), 10% labeled + 90% unlabeled (dashed), and 10% labeled + 0% unlabeled training examples (dotted). 3 Extensions Anomaly detection: In anomaly detection we are given a set of training examples representing only one class, the "typical" examples. We attempt to capture regularities among the examples to be able to recognize unlikely members of this class. Estimating a probability distribution P(XIO) on the basis of the training set {Xl, " . , X T} via the ML (or analogous) criterion is not appropriate; there is no reason to further increase the probability of those examples that are already well captured by the model. A more relevant measure involves the level sets X)' = {X EX: log P(X 10) 2:: ,} which are used in deciding the class membership in any case. We estimate the parameters 0 to optimize an appropriate level set. Definition 2 Given a probability model P(XIO), 0 E e, a set of training examples {X 1, ... , X T }, a set of margin variables , = bl, ... , ,T], and a prior distribution Po(O, ,) we find the MRE distribution P(O, ,) such that minimizes D(PIIPo) subject to the constraints J P(O, ,) [log P(XtIO) - ,t] dOd, 2:: 0 for all t = 1, ... ,T. Note that this again a MRE projection whose solution can be obtained as before. The choice of Pob) in Po(O, ,) = Po (O)Po b) is not as straightforward as before since each margin ,t needs to be close to achievable log-probabilities. We can nevertheless find a reasonable choice by relating the prior mean of ,t to some a-percentile of the training set log-probabilities generated through ML or other estimation criterion. Denote the resulting value by la and define the prior Pobt) as Pobt) = ee -c (l ,, -),.) for,t ::; lao In this case the prior mean of,t is la - lie. Figure 2b shows in the context of a simple product distribution that this choice of prior together with the MRE framework leads to a real improvement over standard (Bayesian) approach. We believe, however, that the effect will be more striking for sophisticated models such as HMMs that may otherwise easily capture spurious regularities in the data. An extension of this formalism to latent variable models is provided in [4]. Uncertain or incompletely labeled examples: Examples with uncertain labels are hard to deal with in any (probabilistic or not) discriminative classification method. Uncertain labels can be, however, handled within the maximum entropy formalism: let Y = {Yl ,' .. , YT} be a set of binary variables corresponding to the labels for the training examples. We can define a prior uncertainty over the labels by specifying Po(Y) ; for simplicity, we can take this to be a product distribution 476 T. Jaakkola, M Meila and T. Jebara Po{Y) = TIt Pt,o(Yt) where a different level of uncertainty can be assigned to each example. Consequently, we find the minimum relative entropy projection from the prior distribution po(e", y) = po{e)Po([)Po(Y) to the admissible set of distributions (no longer a function of the labels) that are consistent with the constraints: E y fe ,,"( p(e", y) [YtC(Xt, e) -,tl de d, ~ 0 for all t = 1, ... , T. The MRE principle differs from transduction [9], provides a soft rather than hard assignment of unlabeled examples, and is fundamentally driven by large margin classification. The MRE solution is not, however, often feasible to obtain in practice. We can nevertheless formulate an efficient mean field approach in this context [4]. Figure 2c demonstrates that even the approximate method is able to reap most of the benefit from unlabeled examples (compare, e.g., [6]). The results are for a DNA splice junction classification problem. For more details see [4]. 4 Discussion We have presented a general approach to discriminative training of model parameters, structures, or parametric discriminant functions. The formalism is based on the minimum relative entropy principle reducing all calculations to relative entropy projections. The idea naturally extends beyond standard classification and covers anomaly detection, classification with partially labeled examples, and feature selection. References [1] Cover and Thomas (1991). Elements of information theory. John Wiley & Sons. [2] Kivinen J. and Warmuth M. (1999). Boosting as Entropy Projection. Proceedings of the 12th Annual Conference on Computational Learning Theory. [3] Levin and Tribus (eds.) (1978). The maximum entropy formalism. Proceedings of the Maximum entropy formalism conference, MIT. [4] Jaakkola T., Meila M. and Jebara T. (1999). Maximum entropy discrimination. MIT AITR-1668, http://www.ai .mit. edu;-tommi/papers .html. [5] Jaakkola T. and Haussler D. (1998). Exploiting generative models in discriminative classifiers. NIPS 11. [6] Joachims, T. (1999). Transductive inference for text classification using support vector machines. International conference on Machine Learning. [7] Jebara T. and Pentland A. (1998). Maximum conditional likelihood via bound maximization and the CEM algorithm. NIPS 11. [8] Meila M. and Jordan M. (1998). Estimating dependency structure as a hidden variable. NIPS 11. [9] Vapnik V. (1998). Statistical learning theory. John Wiley & Sons. [10] West D. (1996). Introduction to graph theory. Prentice Hall.
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U nmixing Hyperspectral Data Lucas Parra, Clay Spence, Paul Sajda Sarnoff Corporation, CN-5300, Princeton, NJ 08543, USA {lparra, cspence,psajda} @sarnoff.com Andreas Ziehe, Klaus-Robert Miiller GMD FIRST.lDA, Kekulestr. 7, 12489 Berlin, Germany {ziehe,klaus}@first.gmd.de Abstract In hyperspectral imagery one pixel typically consists of a mixture of the reflectance spectra of several materials, where the mixture coefficients correspond to the abundances of the constituting materials. We assume linear combinations of reflectance spectra with some additive normal sensor noise and derive a probabilistic MAP framework for analyzing hyperspectral data. As the material reflectance characteristics are not know a priori, we face the problem of unsupervised linear unmixing. The incorporation of different prior information (e.g. positivity and normalization of the abundances) naturally leads to a family of interesting algorithms, for example in the noise-free case yielding an algorithm that can be understood as constrained independent component analysis (ICA). Simulations underline the usefulness of our theory. 1 Introduction Current hyperspectral remote sensing technology can form images of ground surface reflectance at a few hundred wavelengths simultaneously, with wavelengths ranging from 0.4 to 2.5 J.Lm and spatial resolutions of 10-30 m. The applications of this technology include environmental monitoring and mineral exploration and mining. The benefit of hyperspectral imagery is that many different objects and terrain types can be characterized by their spectral signature. The first step in most hyperspectral image analysis systems is to perform a spectral unmixing to determine the original spectral signals of some set of prime materials. The basic difficulty is that for a given image pixel the spectral reflectance patterns of the surface materials is in general not known a priori. However there are general physical and statistical priors which can be exploited to potentially improve spectral unmixing. In this paper we address the problem of unmixing hyperspectral imagery through incorporation of physical and statistical priors within an unsupervised Bayesian framework. We begin by first presenting the linear superposition model for the reflectances measured. We then discuss the advantages of unsupervised over supervised systems. Unmixing Hyperspectral Data 943 We derive a general maximum a posteriori (MAP) framework to find the material spectra and infer the abundances. Interestingly, depending on how the priors are incorporated, the zero noise case yields (i) a simplex approach or (ii) a constrained leA algorithm. Assuming non-zero noise our MAP estimate utilizes a constrained least squares algorithm. The two latter approaches are new algorithms whereas the simplex algorithm has been previously suggested for the analysis of hyperspectral data. Linear Modeling To a first approximation the intensities X (Xi>.) measured in each spectral band A = 1, ... , L for a given pixel i = 1, ... , N are linear combinations of the reflectance characteristics S (8m >.) of the materials m = 1, ... , M present in that area. Possible errors of this approximation and sensor noise are taken into account by adding a noise term N (ni>'). In matrix form this can be summarized as X = AS + N, subject to: AIM = lL, A ~ 0, (1) where matrix A (aim) represents the abundance of material m in the area corresponding to pixel i, with positivity and normalization constraints. Note that ground inclination or a changing viewing angle may cause an overall scale factor for all bands that varies with the pixels. This can be incorporated in the model by simply replacing the constraint AIM = lL with AIM ~ lL which does does not affect the discussion in the remainder of the paper. This is clearly a simplified model of the physical phenomena. For example, with spatially fine grained mixtures, called intimate mixtures, multiple reflectance may causes departures from this first order model. Additionally there are a number of inherent spatial variations in real data, such as inhomogeneous vapor and dust particles in the atmosphere, that will cause a departure from the linear model in equation (1). Nevertheless, in practical applications a linear model has produced reasonable results for areal mixtures. Supervised vs. Unsupervised techniques Supervised spectral un mixing relies on the prior knowledge about the reflectance patterns S of candidate surface materials, sometimes called endmembers, or expert knowledge and a series of semiautomatic steps to find the constituting materials in a particular scene. Once the user identifies a pixel i containing a single material, i.e. aim = 1 for a given m and i, the corresponding spectral characteristics of that material can be taken directly from the observations, i.e., 8 m >. = Xi>. [4]. Given knowledge about the endmembers one can simply find the abundances by solving a constrained least squares problem. The problem with such supervised techniques is that finding the correct S may require substantial user interaction and the result may be error prone, as a pixel that actually contains a mixture can be misinterpreted as a pure endmember. Another approach obtains endmembers directly from a database. This is also problematic because the actual surface material on the ground may not match the database entries, due to atmospheric absorption or other noise sources. Finding close matches is an ambiguous process as some endmembers have very similar reflectance characteristics and may match several entries in the database. Unsupervised unmixing, in contrast, tries to identify the endmembers and mixtures directly from the observed data X without any user interaction. There are a variety of such approaches. In one approach a simplex is fit to the data distribution [7, 6, 2]. The resulting vertex points of the simplex represent the desired endmembers, but this technique is very sensitive to noise as a few boundary points can potentially change the location of the simplex vertex points considerably. Another approach by Szu [9] tries to find abundances that have the highest entropy subject to constraints that the amount of materials is as evenly distributed as possible - an assumption 944 L. Parra, C. D. Spence, P Sajda, A. Ziehe and K.-R. Muller which is clearly not valid in many actual surface material distributions. A relatively new approach considers modeling the statistical information across wavelength as statistically independent AR processes [1]. This leads directly to the contextual linear leA algorithm [5]. However, the approach in [1] does not take into account constraints on the abundances, noise, or prior information. Most importantly, the method [1] can only integrate information from a small number of pixels at a time (same as the number of endmembers). Typically however we will have only a few endmembers but many thousand pixels. 2 The Maximum A Posterior Framework 2.1 A probabilistic model of unsupervised spectral unmixing Our model has observations or data X and hidden variables A, S, and N that are explained by the noisy linear model (1). We estimate the values of the hidden variables by using MAP (A SIX) = p(XIA, S)p(A, S) = Pn(XIA, S)Pa(A)ps(S) p , p(X) p(X) (2) with Pa(A), Ps(S), Pn(N) as the a priori assumptions of the distributions. With MAP we estimate the most probable values for given priors after observing the data, A MAP, SMAP = argmaxp(A, SIX) (3) A,S Note that for maximization the constant factor p(X) can be ignored. Our first assumption, which is indicated in equation (2) is that the abundances are independent of the reflectance spectra as their origins are completely unrelated: (AO) A and S are independent. The MAP algorithm is entirely defined by the choices of priors that are guided by the problem of hyperspectral unmixing: (AI) A represent probabilities for each pixel i. (A2) S are independent for different material m. (A3) N are normal i.i.d. for all i, A. In summary, our MAP framework includes the assumptions AO-A3. 2.2 Including Priors Priors on the abundances Positivity and normalization of the abundances can be represented as, (4) where 60 represent the Kronecker delta function and eo the step function. With this choice a point not satisfying the constraint will have zero a posteriori probability. This prior introduces no particular bias of the solutions other then abundance constraints. It does however assume the abundances of different pixels to be independent. Prior on spectra Usually we find systematic trends in the spectra that cause significant correlation. However such an overall trend can be subtracted and/or filtered from the data leaving only independent signals that encode the variation from that overall trend. For example one can capture the conditional dependency structure with a linear auto-regressive (AR) model and analyze the resulting "innovations" or prediction errors [3]. In our model we assume that the spectra represent independent instances of an AR process having a white innovation process em.>. distributed according to Pe(e). With a Toeplitz matrix T of the AR coefficients we Unmixing Hyperspectral Data 945 can write, em = Sm T. The AR coefficients can be found in a preprocessing step on the observations X. If S now represents the innovation process itself, our prior can be represented as, M L L Pe (S) <X Pe(ST) = II II Pe( L sm>.d>.>.,) , (5) m=1 >.=1 >.'=1 Additionally Pe (e) is parameterized by a mean and scale parameter and potentially parameters determining the higher moments of the distributions. For brevity we ignore the details of the parameterization in this paper. Prior on the noise As outlined in the introduction there are a number of problems that can cause the linear model X = AS to be inaccurate (e.g. multiple reflections, inhomogeneous atmospheric absorption, and detector noise.) As it is hard to treat all these phenomena explicitly, we suggest to pool them into one noise variable that we assume for simplicity to be normal distributed with a wavelength dependent noise variance a>., L p(XIA, S) = Pn(N) = N(X - AS,~) = II N(x>. - As>., a>.l) , (6) >.=1 where N (', .) represents a zero mean Gaussian distribution, and 1 the identity matrix indicating the independent noise at each pixel. 2.3 MAP Solution for Zero Noise Case Let us consider the noise-free case. Although this simplification may be inaccurate it will allow us to greatly reduce the number of free hidden variables - from N M + M L to M2 . In the noise-free case the variables A, S are then deterministically dependent on each other through a N L-dimensional 8-distribution, Pn(XIAS) = 8(X - AS). We can remove one of these variables from our discussion by integrating (2). It is instructive to first consider removing A p(SIX) <X I dA 8(X - AS)Pa(A)ps(S) = IS-1IPa(XS- 1 )Ps(S). (7) We omit tedious details and assume L = M and invertible S so that we can perform the variable substitution that introduces the Jacobian determinant IS-II . Let us consider the influence of the different terms. The Jacobian determinant measures the volume spanned by the endmembers S. Maximizing its inverse will therefore try to shrink the simplex spanned by S. The term Pa(XS- 1 ) should guarantee that all data points map into the inside of the simplex, since the term should contribute zero or low probability for points that violate the constraint. Note that these two terms, in principle, define the same objective as the simplex envelope fitting algorithms previously mentioned [2]. In the present work we are more interested in the algorithm that results from removing S and finding the MAP estimate of A. We obtain (d. Eq.(7)) p(AIX) oc I dS 8(X - AS)Pa(A)ps(S) = IA -llps(A- 1X)Pa(A). (8) For now we assumed N = M. 1 If Ps (S) factors over m, i.e. endmembers are independent, maximizing the first two terms represents the leA algorithm. However, lIn practice more frequently we have N > M. In that case the observations X can be mapped into a M dimensional subspace using the singular value decomposition (SVD), X = UDVT , The discussion applies then to the reduced observations X = u1x with U M being the first M columns of U . 946 L. Parra. C. D. Spence. P Sajda. A. Ziehe and K.-R. Muller the prior on A will restrict the solutions to satisfy the abundance constraints and bias the result depending on the detailed choice of Pa(A), so we are led to constrained ICA. In summary, depending on which variable we integrate out we obtain two methods for solving the spectral unmixing problem: the known technique of simplex fitting and a new constrained ICA algorithm. 2.4 MAP Solution for the Noisy Case Combining the choices for the priors made in section 2.2 (Eqs.(4), (5) and (6)) with (2) and (3) we obtain (9) AMAP, SMAP = "''i~ax ft {g N(x", - a,s" a,) ll. P,(t. 'm,d",) } , subject to AIM = lL, A 2: O. The logarithm of the cost function in (9) is denoted by L = L(A, S). Its gradient with respect to the hidden variables is 88L = _AT nm diag(O')-l - fs(sm) (10) Sm where N = X - AS, nm are the M column vectors of N, fs(s) = - olnc;(s). In (10) fs is applied to each element of Sm. The optimization with respect to A for given S can be implemented as a standard weighted least squares (L8) problem with a linear constraint and positivity bounds. Since the constraints apply for every pixel independently one can solve N separate constrained LS problems of M unknowns each. We alternate between gradient steps for S and explicit solutions for A until convergence. Any additional parameters of Pe(e) such as scale and mean may be obtained in a maximum likelihood (ML) sense by maximizing L. Note that the nonlinear optimization is not subject to constraints; the constraints apply only in the quadratic optimization. 3 Experiments 3.1 Zero Noise Case: Artificial Mixtures In our first experiment we use mineral data from the United States Geological Survey (USGS)2 to build artificial mixtures for evaluating our unsupervised unmixing framework. Three target endmembers where chosen (Almandine WS479, Montmorillonite+Illi CM42 and Dickite NMNH106242). A spectral scene of 100 samples was constructed by creating a random mixture of the three minerals. Of the 100 samples, there were no pure samples (Le. no mineral had more than a 80% abundance in any sample). Figure 1A is the spectra of the endmembers recovered by the constrained ICA technique of section 2.3, where the constraints were implemented with penalty terms added to the conventional maximum likelihood ICA algorithm. These are nearly identical to the spectra of the true endmembers, shown in figure 1B, which were used for mixing. Interesting to note is the scatter-plot of the 100 samples across two bands. The open circles are the absorption values at these two bands for endmembers found by the MAP technique. Given that each mixed sample consists of no more than 80% of any endmember, the endmember points on the scatter-plot are quite distant from the cluster. A simplex fitting technique would have significant difficulty recovering the endmembers from this clustering. 2see http://speclab.cr . usgs.gov /spectral.lib.456.descript/ decript04.html Unmixing Hyperspectral Data found endmembers O~------' 50 100 150 200 wavelength A target endmembers O~------' 50 100 150 200 wavelength B 947 observed X and found S g 0.8 o ~ ~0.6 ., ~ ~ 0.4 o 0.2'---~------' 0.4 0.6 0.8 wavelength=30 C Figure 1: Results for noise-free artificial mixture. A recovered endmembers using MAP technique. B "true" target endmembers. C scatter plot of samples across 2 bands showing the absorption of the three endmembers computed by MAP (open circles). 3.2 Noisy Case: Real Mixtures To validate the noise model MAP framework of section 2.4 we conducted an experiment using ground truthed USGS data representing real mixtures. We selected lOxl0 blocks of pixels from three different regions3 in the AVIRIS data of the Cuprite, Nevada mining district. We separate these 300 mixed spectra assuming two endmembers and an AR detrending with 5 AR coefficients and the MAP techniques of section 2.4. Overall brightness was accounted for as explain in the linear modeling of section 1. The endmembers are shown in figure 2A and B in comparison to laboratory spectra from the USGS spectral library for these minerals [8J. Figure 2C shows the corresponding abundances, which match the ground truth; region (III) mainly consists of Muscovite while regions (1)+(I1) contain (areal) mixtures of Kaolinite and Muscovite. 4 Discussion Hyperspectral unmixing is a challenging practical problem for unsupervised learning. Our probabilistic approach leads to several interesting algorithms: (1) simplex fitting, (2) constrained ICA and (3) constrained least squares that can efficiently use multi-channel information. An important element of our approach is the explicit use of prior information. Our simulation examples show that we can recover the endmembers, even in the presence of noise and model uncertainty. The approach described in this paper does not yet exploit local correlations between neighboring pixels that are well known to exist. Future work will therefore exploit not only spectral but also spatial prior information for detecting objects and materials. Acknowledgments We would like to thank Gregg Swayze at the USGS for assistance in obtaining the data. 3The regions were from the image plate2.cuprite95.alpha.2um.image.wlocals.gif in ftp:/ /speclab.cr.usgs.gov /pub/cuprite/gregg.thesis.images/, at the coordinates (265,710) and (275,697), which contained Kaolinite and Muscovite 2, and (143,661), which only contained Muscovite 2. 948 0.65 0.6 0.55 0.5 Muscovite 'c .•.• ", "'0 .. ' ., 0.45 0.4,--~--:-:-:-"-~----:--:--~ 160 190 200 210 220 waveleng1h A L. Parra, C. D, Spence, P Sajda, A. Ziehe and K-R. Muller Kaolinite 0.8 0.7 0.6 0.4 0.3 180 190 200 210 220 wavelength B C Figure 2: A Spectra of computed endmember (solid line) vs Muscovite sample spectra from the USGS data base library. Note we show only part of the spectrum since the discriminating features are located only between band 172 and 220. B Computed endmember (solid line) vs Kaolinite sample spectra from the USGS data base library. C Abundances for Kaolinite and Muscovite for three regions (lighter pixels represent higher abundance). Region 1 and region 2 have similar abundances for Kaolinite and Muscovite, while region 3 contains more Muscovite. References [1] J. Bayliss, J. A. Gualtieri, and R. Cromp. Analyzing hyperspectral data with independent component analysis. In J. M. Selander, editor, Proc. SPIE Applied Image and Pattern Recognition Workshop, volume 9, P.O. Box 10, Bellingham WA 98227-0010, 1997. SPIE. [2] J.W. Boardman and F.A. Kruse. Automated spectral analysis: a geologic example using AVIRIS data, north Grapevine Mountains, Nevada. In Tenth Thematic Conference on Geologic Remote Sensing, pages 407-418, Ann arbor, MI, 1994. Environmental Research Institute of Michigan. [3] S. Haykin. Adaptive Filter Theory. Prentice Hall, 1991. [4] F. Maselli, , M. Pieri, and C. Conese. Automatic identification of end-members for the spectral decomposition of remotely sensed scenes. Remote Sensing for Geography, Geology, Land Planning, and Cultural Heritage (SPIE), 2960:104109,1996. [5] B. Pearlmutter and L. Parra. Maximum likelihood blind source separation: A context-sensitive generalization ofICA. In M. Mozer, M. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems 9, pages 613-619, Cambridge MA, 1997. MIT Press. [6] J.J. Settle. Linear mixing and the estimation of ground cover proportions. International Journal of Remote Sensing, 14:1159-1177,1993. [7] M.O. Smith, J .B. Adams, and A.R. Gillespie. Reference endmembers for spectral mixture analysis. In Fifth Australian remote sensing conference, volume 1, pages 331-340, 1990. [8] U.S. Geological Survey. USGS digital spectral library. Open File Report 93-592, 1993. [9] H. Szu and C. Hsu. Landsat spectral demixing a la superresolution of blind matrix inversion by constraint MaxEnt neural nets. In Wavelet Applications IV, volume 3078, pages 147-160. SPIE, 1997. Invariant Feature Extraction and Classification in Kernel Spaces Sebastian Mikal , Gunnar Ratschl , Jason Weston2 , Bernhard Sch8lkopf3, Alex Smola4 , and Klaus-Robert Mullerl 1 GMD FIRST, Kekulestr. 7,12489 Berlin, Germany 2 Barnhill BioInformatics, 6709 Waters Av., Savannah, GR 31406, USA 3 Microsoft Research Ltd., 1 Guildhall Street, Cambridge CB2 3NH, UK 4 Australian National University, Canberra, 0200 ACT, Australia {mika, raetsch, klaus }@first.gmd.de, jasonw@dcs.rhbnc.ac.uk bsc@microsoft.com, Alex.Smola.anu.edu.au Abstract We incorporate prior knowledge to construct nonlinear algorithms for invariant feature extraction and discrimination. Employing a unified framework in terms of a nonlinear variant of the Rayleigh coefficient, we propose non-linear generalizations of Fisher's discriminant and oriented PCA using Support Vector kernel functions. Extensive simulations show the utility of our approach. 1 Introduction It is common practice to preprocess data by extracting linear or nonlinear features. The most well-known feature extraction technique is principal component analysis PCA (e.g. [3]). It aims to find an orthonormal, ordered basis such that the i-th direction describes as much variance as possible while maintaining orthogonality to all other directions. However, since PCA is a linear technique, it is too limited to capture interesting nonlinear structure in a data set and nonlinear generalizations have been proposed, among them Kernel PCA [14], which computes the principal components of the data set mapped nonlinearly into some high dimensional feature space F. Often one has prior information, for instance, we might know that the sample is corrupted by noise or that there are invariances under which a classification should not change. For feature extraction, the concepts of known noise or transformation invariance are to a certain degree equivalent, i.e. they can both be interpreted as causing a change in the feature which ought to be minimized. Clearly, invariance alone is not a sufficient condition for a good feature, as we could simply take the constant function. What one would like to obtain is a feature which is as invariant as possible while still covering as much of the information necessary for describing the particular data. Considering only one (linear) feature vector wand restricting to first and second order statistics of the data one arrives at a maximization of the so called Rayleigh coefficient (1) Invariant Feature Extraction and Classification in Kernel Spaces 527 where w is the feature vector and Sf, SN are matrices describing the desired and undesired properties of the feature, respectively (e.g. information and noise). If S/ is the data covariance and SN the noise covariance, we obtain oriented PCA [3J. If we leave the field of data description to perform supervised classification, it is common to choose S / as the separability of class centers (between class variance) and SN to be the within class variance. In that case, we recover the well known Fisher Discriminant [7J. The ratio in (1) is maximized when we cover much of the information coded by S/ while avoiding the one coded by SN . The problem is known to be solved, in analogy to PCA, by a generalized symmetric eigenproblem S/w = >"SNW [3], where>.. E ~ is the corresponding (biggest) eigenvalue. In this paper we generalize this setting to a nonlinear one. In analogy to [8, 14J we first map the data via some nonlinear mapping <l> to some high-dimensional feature space F and then optimize (1) in F . To avoid working with the mapped data explicitly (which might be impossible if F is infinite dimensional) we introduce support vector kernel functions [11], the well-known kernel trick. These kernel functions k(x, y) compute a dot product in some feature space F , i.e. k(x, y) = (<l>(x)· <l>(y)) . Formulating the algorithms in Fusing <l> only in dot products, we can replace any occurrence of a dot product by the kernel function k. Possible choices for k which have proven useful e.g. in Support Vector Machines [2] or Kernel PCA [14J are Gaussian RBF, k(x, y) = exp( -llx - yI12/c), or polynomial kernels, k(x, y) = (x· y)d , for some positive constants c E ~ and dEN, respectively. The remainder of this paper is organized as follows: The next section shows how to formulate the optimization problem induced by (1) in feature space. Section 3 considers various ways to find Fisher's Discriminant in F; we conclude with extensive experiments in section 4 and a discussion of our findings. 2 Kernelizing the Rayleigh Coefficient To optimize (1) in some kernel feature space F we need to find a formulation which uses only dot products of <l>-images. As numerator and denominator are both scalars this can be done independently. Furthermore, the matrices S/ and SN are basically covariances and thus the sum over outer products of <l>-images. Therefore, and due to the linear nature of (1) every solution W E F can be written as an expansion in terms of mapped training datal, i.e. l W = L Cti<l>(Xi). (2) i=l To define some common choices in F let X = {Xl , .. . ,xe} be our training sample and, where appropriate, Xl U X2 = X , Xl n X2 = 0, two subclasses (with I Xi I = £i). We get the full covariance of X by 1 1 C = f L (<l>(x) - m)(<l>(x) - m)T with m = f L <l>(x) , (3) ~EX ~EX I SB and Sw are operators on a (finite-dimensional) subspace spanned by the CP(Xi) (in a possibly infinite space). Let w = VI + V2, where VI E Span(CP(Xi) : i = 1, .. . ,f) and V2 1. Span(CP(xi) : i = 1, ... ,f) . Then for S = Sw or S = SB (which are both symmetric) (w , Sw) ((VI + V2) , S(VI + V2)) ((VI + V2)S, VI) (VI ,SVI) As VI lies in the span of the cp(Xi) and S only operates on this subspace there exist an expansion of w which maximizes J(w) . 528 S. Mika. G. Riitsch. J. Weston. B. Scholkopj, A. J. Smola and K.-R. Muller which could be used as Sf in oriented Kernel PCA. For SN we could use an estimate of the noise covariance, analogous to the definition of C but over mapped patterns sampled from the assumed noise distribution. The standard formulation of the Fisher discriminant in F, yielding the Kernel Fisher Discriminant (KFD) [8] is given by Sw = L L (cJ>(x) - mi)(cJ>(x) - mdT and SB = (m2 - mt}(m2 - ml)T, i=I,2 xEX; the within-class scatter Sw (as S N), and the between class scatter S B ( as Sf). Here mi is the sample mean for patterns from class i. To incorporate a known invariance e.g. in oriented Kernel PCA, one could use the tangent covariance matrix [12], 1 T = ft2 L (cJ>(x) - cJ>(£tx))(cJ>(x) - cJ>(£tx))T for some small t> O. (4) :IlEX Here £t is a local I-parameter transformation. T is a finite difference approximation t of the covariance of the tangent of £t at point cJ>(x) (details e.g. in [12]). Using Sf = C and SN = T in oriented Kernel PCA, we impose invariance under the local transformation £t. Crucially, this matrix is not only constructed from the training patterns X. Therefore, the argument used to find the expansion (2) is slightly incorrect. Neverthless, we can assume that (2) is a reasonable approximation for describing the variance induced by T. Multiplying either of these matrices from the left and right with the expansion (2), we can find a formulation which uses only dot products. For the sake of brevity, we only give the explicit formulation of (1) in F for KFD (cf. [8] for details). Defining (I-'i)j = t L:IlEXi k(xj,x) we can write (1) for KFD as J(a) = (aTI-') 2 aTMa aTNa aTNa' (5) where N = KKT - Li=1,2fil-'iI-'T, I-' = 1-'2 - 1-'1 ' M = I-'I-'T, and Kij = k(xi,xj). The results for other choices of Sf and S N in F as for the cases of oriented kernel PCA or transformation invariance can be obtained along the same lines. Note that we still have to maximize a Rayleigh coefficient. However, now it is a quotient in terms of expansion coefficients a, and not in terms of w E F which is a potentially infinite-dimensional space. Furthermore, it is well known that the solution for this special eigenproblem is in the direction of N-1 (1-'2 - 1-'1) [7), which can be solved using e.g. a Cholesky factorization of N. The projection of a new pattern x onto w in F can then be computed by l (w· cJ>(x)) = LQik(xi'x). (6) i=1 3 Algorithms Estimating a covariance matrix with rank up to f from f samples is ill-posed. Furthermore, by performing an explicit centering in F each covariance matrix loses one more dimension, i.e. it has only rank f - 1 (even worse, for KFD the matrix N has rank f - 2). Thus the ratio in (1) is not well defined anymore, as the denominator might become zero. In the following we will propose several ways to deal with this problem in KFD. Furthermore we will tackle the question how to solve the optimization problem of KFD more efficiently. So far, we have an eigenproblem of size .e x .e. If .e becomes large this is numerically demanding. Reformulations of the original problem allow to overcome some of these limitations. Finally, we describe the connection between KFD and RBF networks. Invariant Feature Extraction and Classification in Kernel Spaces 529 3.1 Regularization and Solution on a Subspace As noted before, the matrix N has only rank £ - 2. Besides numerical problems which can cause the matrix N to be not even positive, we could think of imposing some regularization to control capacity in F. To this end, we simply add a mUltiple of the identity matrix to N, Le. replace N by NJ1. where NJ1. := N + /-LI. (7) This can be viewed in different ways: (i) for /-L > 0 it makes the problem feasible and numerically more stable as NJ1. becomes positive; (ii) it can be seen as decreasing the bias in sample based estimation of eigenvalues (cf. [6)); (iii) it imposes a regularization on 110112, favoring solutions with small expansion coefficients. furthermore, one could use other regularization type additives to N, e.g. penalizing IIwl12 in analogy to SVM (by adding the kernel matrix Kij = k(xi' Xj)). To optimize (5) we need to solve an £ x £ eigenproblem, which might be intractable for large £. As the solutions are not sparse one can not directly use efficient algorithms like chunking for Support Vector Machines (cf. [13]). To this end, we might restrict the solution to lie in a subspace, Le. instead of expanding w by (2) we write (8) i=l with m < l. The patterns Zi could either be a subset of the training patterns X or e.g. be estimated by some clustering algorithm. The derivation of (5) does not change, only K is now m x £ and we end up with m x m matrices N and M. Another advantage is, that it increases the rank of N (relative to its size) although there still might be some need for regularization. 3.2 Quadratic optimization and Sparsification Even if N has full rank, maximizing (5) is underdetermined: if 0 is optimal, then so is any multiple thereof. Since 0 T M 0 = (0 T J..L)2, M has rank one. Thus we can seek for a vector 0, such that oTNo is minimal for fixed OTJ..L (e.g. to 1). The solution is unique and we can find the optimal 0 by solving the quadratic optimization problem: (9) Although the quadratic optimization problem is not easier to solve than the eigenproblem, it has an appealing interpretation. The constraint 0 T J..L = 1 ensures, that the average class distance, projected onto the direction of discrimination, is constant, while the intra class variance is minimized, i.e. we maximize the average margin. Contrarily, the SVM approach [2] optimizes for a large minimal margin. Considering (9) we are able to overcome another shortcoming of KFD. The solutions 0 are not sparse and thus evaluating (6) is expensive. To solve this we can add an h-regularizer >'110111 to the objective function, where>. is a regularization parameter allowing us to adjust the degree of sparseness. 3.3 Connection to RBF Networks Interestingly, there exists a close connection between RBF networks (e.g. [9, 1)) and KFD. If we add no regularization and expand in all training patterns, we find that an optimal 0 is given by 0 = K- 1y, where K is the symmetric, positive matrix of all kernel elements k(xi' Xj) and y the ±1 label vector2. A RBF-network with the 2To see this, note that N can be written as N = KDK where D = I -YIyT -Y2Y; has rank e - 2, while Yi is the vector of l/Vli's for patterns from class i and zero otherwise. 530 S. Mika, G. Ratsch, J. Weston, B. SchOlkopf, A. J. Smola and K.-R. Muller Banana B.Cancer Diabetes German Heart Image Ringnorm F.Sonar Splice Thyroid Titanic Twonorm Waveform RBF AB ABR SVM KFD 10.8±O.06 12.3±O.07 10.9±0.04 1l.5±O.07 10.8±O.05 27.6±0.47 30.4±0.47 26.5±0.45 26.o±O.4725.8±0.46 24.3±O.19 26.5±O.23 23.8±O.18 23.5±0.17 23.2±O.16 24.7±O.24 27.5±O.25 24.3±O.21 23.6±O.21 23.1±0.22 17.6±O.33 20.3±O.34 16.5±O.35 16.0±O.33 16.1±0.34 3.3±O.06 2.1±O.01 2.1±O.06 3.o±O.06 4.8±O.06 1.7±O.02 1.9±O.03 1.6±0.01 1.7±O.01 1.5±O.01 34.4±O.20 35.7±O.18 34.2±O.22 32.4±O.18 33.2±0.11 10.o±O.10 10.1±O.05 9.5±O.01 10.9±O.07 10.5±O.06 4.5±O.21 4.4±O.22 4.6±O.22 4.8±O.22 4.2±O.21 23.3±O.13 22.6±O.12 22.6±O.12 22.4±O.10 23.2±O.20 2.9±O.03 3.0±O.03 2.1±O.02 3.0±O.02 2.6±O.02 10.7±O.1l 10.8±O.06 9.8±O.08 9. 9± o. 04 9.9±O.04 Table 1: Comparison between KFD, single RBF classifier, AdaBoost (AB), regul. AdaBoost (ABR) and SVMs (see text). Best result in bold face, second best in italics. same kernel at each sample and fixed kernel width gives the same solution, if the mean squared error between labels and output is minimized. Also for the case of restricted expansions (8) there exists a connection to RBF networks with a smaller number of centers (cf. [4]). 4 Experiments Kernel Fisher Discriminant Figure 1 shows an illustrative comparison of the features found by KFD, and Kernel PCA. The KFD feature discriminates the two classes, the first Kernel PCA feature picks up the important nonlinear structure. To evaluate the performance of the KFD on real data sets we performed an extensive comparison to other state-of-the-art classifiers, whose details are reported in [8j.3 We compared the Kernel Fisher Discriminant and Support Vector Machines, both with Gaussian kernel, to AdaBoost [5], and regularized AdaBoost [10] (cf. table 1). For KFD we used the regularized within-class scatter (7) and computed projections onto the optimal direction w E :F by means of (6). To use w for classification we have to estimate a threshold. This can be done by e.g. trying all thresholds between two outputs on the training set and selecting the median of those with the smallest empirical error, or (as we did here) by computing the threshold which maximizes the margin on the outputs in analogy to a Support Vector Machine, where we deal with errors on the trainig set by using the SVM soft margin approach. A disadvantage of this is, however, that we have to control the regularization constant for the slack variables. The results in table 1 show the average test error and the standard If K has full rank, the null space of D , which is spanned by Yl and Y2' is the null space of N . For a = K-1 Y we get aT N a = 0 and aT J.£ =I O. As we are free to fix the constraint aT J.£ to any positive constant (not just 1), a is also feasible. 3The breast cancer domain was obtained from the University Medical Center, Inst. of Oncology, Ljubljana, Yugoslavia. Thanks to M. Zwitter and M. Soklic for the data. All data sets used in the experiments can be obtained via http://www.first.gmd.de/-raetsch/. Figure 1: Comparison of feature found by KFD (left) and first Kernel PCA feature (right). Depicted are two classes (information only used by KFD) as dots and crosses and levels of same feature value. Both with polynomial kernel of degree two, KFD with the regularized within class scatter (7) (/1 = 10-3 ) . Invariant Feature Extraction and Classification in Kernel Spaces 531 deviation of the averages' estimation, over 100 runs with different realizations of the datasets. To estimate the necessary parameters, we ran 5-fold cross validation on the first five realizations of the training sets and took the model parameters to be the median over the five estimates (see [10] for details of the experimental setup). Using prior knowledge. A toy example (figure 2) shows a comparison of Kernel PCA and oriented Kernel PCA, which used S[ as the full covariance (3) and as noise matrix SN the tangent covariance (4) of (i) rotated patterns and (ii) along the x-axis translated patterns. The toy example shows how imposing the desired invariance yields meaningful invariant features. In another experiment we incorporated prior knowledge in KFD. We used the USPS database of handwritten digits, which consists of 7291 training and 2007 test patterns, ~ach 2?6 .dimensional gray scale ima~es of the digits 0 ... 9: We use? the regulanzed withm-class scatter (7) (p, = 10- ) as SN and added to It an multiple A of the tangent covariance (4), i.e. SN = NJj + AT. As invariance transformations we have chosen horizontal and vertical translation, rotation, and thickening (cf. [12]), where we simply averaged the matrices corresponding to each transformation. The feature was extracted by using the restricted expansion (8), where the patterns Zi were the first 3000 training samples. As kernel we have chosen a Gaussian of width 0.3·256, which is optimal for SVMs [12]. For each class we trained one KFD which classified this class against the rest and computed the 10-class error by the winnertakes-all scheme. The threshold was estimated by minimizing the empirical risk on the normalized outputs of KFD. Without invariances, i.e. A = 0, we achieved a test error of 3.7%, slightly better than a plain SVM with the same kernel (4.2%) [12]. For A = 10-3, using the tangent covariance matrix led to a very slight improvement to 3.6%. That the result was not significantly better than the corresponding one for KFD (3.7%) can be attributed to the fact that we used the same expansion coefficients in both cases. The tangent covariance matrix, however, lives in a slightly different subspace. And indeed, a subsequent experiment where we used vectors which were obtained by clustering a larger dataset, including virtual examples generated by the appropriate invariance transformation, led to 3.1 %, comparable to an SVM using prior knowledge (e.g. [12]; best SVM result 2.9% with local kernel and virtual support vectors). 5 Conclusion In the task of learning from data it is equivalent to have prior knowledge about e.g. invariances or about specific sources of noise. In the case of feature extraction, we seek features which are sufficiently (noise-) invariant while still describing interesting structure. Oriented PCA and, closely related, Fisher's Discriminant, use particularly simple features, since they only consider first and second order statistics for maximizing the Rayleigh coefficient (1). Since linear methods can be too restricted in many real-world applications, we used Support Vector Kernel functions to obtain nonlinear versions of these algorithms, namely oriented Kernel PCA and Kernel Fisher Discriminant analysis. Our experiments show that the Kernel Fisher Discriminant is competitive or in Figure 2: Comparison of first features found by Kernel PCA and oriented Kernel PCA (see text); from left to right: KPCA, OKPCA with rotation and translation invariance; all with Gaussian kernel. 532 S. Mika, G. Riitsch, J. Weston, B. SchOlkopf, A. J. Smola and K.-R. Muller some cases even superior to the other state-of-the-art algorithms tested. Interestingly, both SVM and KFD construct a hyperplane in :F which is in some sense optimal. In many cases, the one given by the solution w of KFD is superior to the one of SVMs. Encouraged by the preliminary results for digit recognition, we believe that the reported results can be improved, by incorporating different invariances and using e.g. local kernels [12]. Future research will focus on further improvements on the algorithmic complexity of our new algorithms, which is so far larger than the one of the SVM algorithm, and on the connection between KFD and Support Vector Machines (cf. [16, 15]). Acknowledgments This work was partially supported by grants of the DFG (JA 379/5-2,7-1,9-1) and the EC STORM project number 25387 and carried out while BS and AS were with GMD First. References [1) C.M. Bishop. Neural Networks for Pattern Recognition. Oxford Univ. Press, 1995. [2] B. Boser, 1. Guyon, and V.N. Vapnik. A training algorithm for optimal margin classifiers. In D. Haussler, editor, Proc. COLT, pages 144- 152. ACM Press, 1992. [3) K.I. Diamantaras and S.Y. Kung. Principal Component Neural Networks. Wiley, New York,1996. [4] B.Q. Fang and A.P. Dawid. Comparison of full bayes and bayes-least squares criteria for normal discrimination. Chinese Journal of Applied Probability and Statistics, 12:401- 410, 1996. [5] Y. Freund and R.E. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. In EuroCOLT 94. LNCS, 1994. [6] J.H. Friedman. Regularized discriminant analysis. Journal of the American Statistical Association, 84(405):165- 175, 1989. [7] K Fukunaga. Introduction to Statistical Pattern Recognition. Academic Press, San Diego, 2nd edition, 1990. [8) S. Mika, G. Ratsch, J. Weston, B. Scholkopf, and K-R. Muller. Fisher discriminant analysis with kernels. In Y.-H. Hu, J . Larsen, E. Wilson, and S. Douglas, editors, Neural Networks for Signal Processing IX, pages 41-48. IEEE, 1999. [9] J. Moody and C. Darken. Fast learning in networks of locally-tuned processing units. Neural Computation, 1(2):281-294, 1989. [10] G. Ratsch, T. Onoda, and K-R. Muller. Soft margins for adaboost. Technical Report NC-TR-1998-021, Royal Holloway College, University of London, UK, 1998. [11] S. Saitoh. Theory of Reproducing Kernels and its Applications. Longman Scientific & Technical, Harlow, England, 1988. [12] B. Scholkopf. Support vector learning. Oldenbourg Verlag, 1997. [13) B. Scholkopf, C.J.C. Burges, and A.J. Smola, editors. Advances in Kernel Methods Support Vector Learning. MIT Press, 1999. [14] B. Scholkopf, A.J. Smola, and K-R. Muller. Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10:1299-1319, 1998. [15] A. Shashua. On the relationship between the support vector machine for classification and sparsified fisher's linear discriminant. Neural Processing Letters, 9(2):129- 139, April 1999. [16) S. Tong and D. Koller. Bayes optimal hyperplanes --+ maximal margin hyperplanes. Submitted to IJCA1'99 WorkshOp on Support Vector Machines (robotics. stanford. edurkoller/), 1999.
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The Relaxed Online Maximum Margin Algorithm Yi Li and Philip M. Long Department of Computer Science National University of Singapore Singapore 119260, Republic of Singapore {liyi,p/ong}@comp.nus.edu.sg Abstract We describe a new incremental algorithm for training linear threshold functions: the Relaxed Online Maximum Margin Algorithm, or ROMMA. ROMMA can be viewed as an approximation to the algorithm that repeatedly chooses the hyperplane that classifies previously seen examples correctly with the maximum margin. It is known that such a maximum-margin hypothesis can be computed by minimizing the length of the weight vector subject to a number of linear constraints. ROMMA works by maintaining a relatively simple relaxation of these constraints that can be efficiently updated. We prove a mistake bound for ROMMA that is the same as that proved for the perceptron algorithm. Our analysis implies that the more computationally intensive maximum-margin algorithm also satisfies this mistake bound; this is the first worst-case performance guarantee for this algorithm. We describe some experiments using ROMMA and a variant that updates its hypothesis more aggressively as batch algorithms to recognize handwritten digits. The computational complexity and simplicity of these algorithms is similar to that of perceptron algorithm, but their generalization is much better. We describe a sense in which the performance of ROMMA converges to that of SVM in the limit if bias isn't considered. 1 Introduction The perceptron algorithm [10, 11] is well-known for its simplicity and effectiveness in the case of linearly separable data. Vapnik's support vector machines (SVM) [13] use quadratic programming to find the weight vector that classifies all the training data correctly and maximizes the margin, i.e. the minimal distance between the separating hyperplane and the instances. This algorithm is slower than the perceptron algorithm, but generalizes better. On the other hand, as an incremental algorithm, the perceptron algorithm is better suited for online learning, where the algorithm repeatedly must classify patterns one at a time, then finds out the correct classification, and then updates its hypothesis before making the next prediction. In this paper, we design and analyze a new simple online algorithm called ROMMA (the Relaxed Online Maximum Margin Algorithm) for classification using a linear threshold The Relaxed Online Maximum Margin Algorithm 499 function. ROMMA has similar time complexity to the perceptron algorithm, but its generalization performance in our experiments is much better on average. Moreover, ROMMA can be applied with kernel functions. We conducted experiments similar to those performed by Cortes and Vapnik [2] and Freund and Schapire [3] on the problem of handwritten digit recognition. We tested the standard perceptron algorithm, the voted perceptron algorithm (for details, see [3]) and our new algorithm, using the polynomial kernel function with d = 4 (the choice that was best in [3]). We found that our new algorithm performed better than the standard perceptron algorithm, had slightly better performance than the voted perceptron. For some other research with aims similar to ours, we refer the reader to [9,4,5,6]. The paper is organized as foIlows. In Section 2, we describe ROMMA in enough detail to determine its predictions, and prove a mistake bound for it. In Section 3, we describe ROMMA in more detail. In Section 4, we compare the experimental results of ROMMA and an aggressive variant of ROMMA with the perceptron and the voted perceptron algorithms. 2 A mistake-bound analysis 2.1 The online algorithms For concreteness, our analysis will concern the case in which instances (also called patterns) and weight vectors are in R n . Fix n EN. In the standard online learning model [7], learning proceeds in trials. In the tth trial, the algorithm is first presented with an instance it ERn. Next, the algorithm outputs a prediction Yt of the classification of it. Finally, the algorithm finds out the correct classification Yt E {-1, 1}. If Yt =I=- Yt, then we say that the algorithm makes a mistake. It is worth emphasizing that in this model, when making its prediction for the tth trial, the algorithm only has access to instance-classification pairs for previous trials. All of the online algorithms that we will consider work by maintaining a weight vector WI which is updated between trials, and predicting Yt = sign( Wt . it), where sign( z) is 1 if z is positive, -1 if z is negative, and 0 otherwise.! The perceptron algorithm. The perceptron algorithm, due to Rosenblatt [10, 11], starts off with Wi = O. When its prediction differs from the label Yt, it updates its weight vector by Wt+i = Wt + Ytit. If the prediction is correct then the weight vector is not changed. The next three algorithms that we will consider assume that all of the data seen by the online algorithm is collectively linearly separable, i.e. that there is a weight vector u such that for all each trial t, Yt = sign( u . xd. When kernel functions are used, this is often the case in practice. The ideal online maximum margin algorithm. On each trial t, this algorithm chooses a weight vector Wt for which for all previous trials s ::; t, sign( Wt . is) = Ys, and which maximizes the minimum distance of any is to the separating hyperplane. It is known [1, 14] that this can be implemented by choosing Wt to minimize Ilwdl subject to the constraints that Ys (Wt . xs) ;::: 1 for all s ::; t. These constraints define a convex polyhedron in weight space which we will refer to as Pt. The relaxed online maximum margin algorithm. This is our new algorithm. The first difference is that trials in which mistakes are not made are ignored. The second difference 'The prediction of 0, which ensures a mistake, is to make the proofs simpler. The usual mistake bound proof for the perceptron algorithm goes through with this change. 500 Y. Li and P. M Long is in how the algorithm responds to mistakes. The relaxed algorithm starts off like the ideal algorithm. Before the second trial, it sets W2 to be the shortest weight vector such that Yl (W2 . i l ) 2:: 1. If there is a mistake on the second trial, it chooses W3 as would the ideal algorithm, to be the smallest element of (1) However, if the third trial is a mistake, then it behaves differently. Instead of choosing W4 to be the smallest element of {w: yI(w· i l ) 2:: I} n {w: Y2(W. i 2) 2:: I} n {w: Y3(W· i3) 2:: I} , it lets W4 be the smallest element of {w: W3 . W 2:: JJw3112} n {w: Y3(W. i3) 2:: I}. This can be thought of as, before the third trial, replacing the polyhedron defined by (1) with the halfspace {w : W3 · W 2:: JJW3JJ2} (see Figure 1). Figure 1: In ROMMA, a convex polyhedron in weight space is replaced with the halfspace with the same smallest element. Note that this halfspace contains the polyhedron of (1); in fact, it contains any convex set whose smallest element is W3. Thus, it can be thought of as the least restrictive convex constraint for which the smallest satisfying weight vector is W3. Let us call this halfspace H3 . The algorithm continues in this manner. If the tth trial is a mistake, then Wt+l is chosen to be the smallest element of Ht n {w : Yt(w· it) 2:: I}, and Ht+l is set to be {w : Wt+l . W 2:: IIwt+lJJ2}. If the tth trial is not a mistake, then Wt+l = Wt and Ht+l = Ht. We will call Ht the old constraint, and {w : Yt (w . it) 2: I} the new constraint. Note that after each mistake, this algorithm needs only to solve a quadratic programming problem with two linear constraints. In fact, there is a simple closed-form expression for Wt+l as a function of Wt, it and Yt that enables it to be computed incrementally using time similar to that of the perceptron algorithm. This is described in Section 3. The relaxed online maximum margin algorithm with aggressive updating. The algorithm is the same as the previous algorithm, except that an update is made after any trial in which yt{Wt . it} < 1, not just after mistakes. 2.2 Upper bound on the number of mistakes made Now we prove a bound on the number of mistakes made by ROMMA. As in previous mistake bound proofs (e.g. [8]), we will show that mistakes result in an increase in a "measure of progress", and then appeal to a bound on the total possible progress. Our proof will use the squared length of Wt as its measure of progress. First we will need the following lemmas. Lemma 1 On any run of ROMMA on linearly separable data, if trial t was a mistake, then the new constraint is binding at the new weight vector; i.e. Yt (Wt+l . it) = 1. Proof: For the purpose of contradiction suppose the new constraint is not binding at the new weight vector Wt+l. Since Wt fails to satisfy this constraint, the line connecting Wt+l and Wt intersects with the border hyperplane of the new constraint, and we denote the intersecting point wq. Then Wq can be represented as Wq = aWt + (l-a)Wt+l, 0 < a < 1. The Relaxed Online Maximum Margin Algorithm 501 Since the square of Euclidean length II . 1\2 is a convex function, the following holds: IIwql\2 ~ allwtll2 + (1 - a) IIwt+d2 Since Wt is the unique smallest member of Ht and Wt+1 i= Wt, we have IIwtl12 < IIwt+11l2, which implies (2) Since Wt and Wt+1 are both in Ht, Wq is too, and hence (2) contradicts the definition of Wt+1· 0 Lemma 2 On any run of ROMMA on linearly separable data, if trial t was a mistake, and not the first one, then the old constraint is binding at the new weight vector, i.e. Wt + 1 . Wt = IIwtV Proof: Let At be the plane of weight vectors that make the new constraint tight, i.e. At = {tV : Yt(w· xd = I}. By Lemma 1, Wt+1 E At . Let at = Ytxtlllxtll2 be the element of At that is perpendicular to it. Then each wE At satisfies IIwII2 = lIatll2 + IIw - at 112, and therefore the length of a vector W in At is minimized when W = at and is monotone in the distance from W to at. Thus, if the old constraint is not binding, then Wt+1 = at. since otherwise the solution could be improved by moving Wt+1 a little bit toward at. But the old constraint requires that (Wt . Wt+d 2: IIwtll2, and if Wt+1 = at = Ytxtlllxtll2, this means that Wt . (YtxtlllxtIl2) 2: Ilwtll2. Rearranging, we get Yt(Wt . xd 2: IIxtll211wtlI2 > 0 (IIXtll > 0 follows from the fact that the data is linearly separable, and IIwt!\ > 0 follows from the fact that there was at least one previous mistake). But since trial t was a mistake, Yt (Wt . Xt) ~ 0, a contradiction. 0 Now we're ready to prove the mistake bound. Theorem 3 Choose mEN, and a sequence (Xl, Yd,···, (xm , Ym) of patternclassijicationpairsinRn x {-1,+1}. LetR = maxtl\xtli. Ifthereisaweightvector ii such that Yt (ii . Xt) 2: 1 for all 1 ~ t ~ m, then the number of mistakes made by ROMMA on (Xl, yd, .. . , (xm, Ym) is at most R211ii1l2. Proof: First, we claim that for all t, ii E Ht. This is easily seen since ii satisfies all the constraints that are ever imposed on a weight vector, and therefore all relaxations of such constraints. Since Wt is the smallest element of Ht. we have IIwtll ~ lliill. We have W2 = Ylxdllid 2, and therefore IIw211 = 1/lIx1\\ 2:: 1/ R which implies IIw2112 2: 1/ R2. We claim that if any trial t > 1 is a mistake, then IIWt+1112 2: IIwtlI 2 + 1/ R2. This will imply by induction that after M mistakes, the squared length of the algorithm's weight vector is at least M / R2, which, since all of the algorithm's weight vectors are no longer than ii, will complete the proof. B Figure 2: At, Bt , and Pt Choose an index t > 1 of a trial in which a mistake is made. Let At = {tV : Yt (w . it) = I} and Bt = {w : (w . Wt) = IIwtIl2}. By Lemmas 1 and 2, Wt+1 EAt n Bt. The distance from Wt to At (call· it pe) satisfies IYt(xt . we) -11 1 1 Pt = IIxtll 2: lIitll 2: R ' (3) since the fact that there was a mistake in trial t implies Yt(Xt . Wt) ~ O. Also, since Wt+1 E At. (4) 502 Y. Li and P. M. Long Because Wt is the normal vector of Bt and Wt+1 E Bt, we have IIWt+1112 = IIWtll2 + IIWt+1 - Wt1l 2. Thus, applying (3) and (4), we have IIWt+d2 IIWt 112 = IIWt+! - welI2 2: p; 2: 1/ R2, which, as discussed above, completes the proof. 0 Using the fact, easily proved using induction, that for all t, Pt ~ Ht , we can easily prove the following, which complements analyses of the maximum margin algorithm using independence assumptions [1, 14, 12]. Details are omitted due to space constraints. Theorem 4 Choose mEN, and a sequence (x\, yd,"', (im , Ym) of patternclassification pairs in R n x {-I, +1}. Let R = maXt lIitli. If there is a weight vector ii such that Yt (ii . it) 2: 1 for all 1 ::; t ::; m, then the number of mistakes made by the ideal online maximum margin algorithm on (Xl, yd, .. " (xm, Ym) is at most R211ii1l2. In the proof of Theorem 3, if an update is made and Yt (Wt . id < 1 - 0 instead of Yt (Wt . it) ::; 0, then the progress made can be seen to be at least 02/ R2. This can be applied to prove the following. Theorem 5 Choose 0 > 0, mEN, and a sequence (Xl, Y1) , ... , (X m, Ym) of patternclassification pairs in R n x {-I, + I}. Let R = maXt lIiell. If there is a weight vector ii such that Yt (ii . Xt) 2: 1 for aliI::; t ::; m, then if (i1' yI), ... , (im, Ym) are presented on line the number of trials in which aggressive ROMMA has Yt (Wt . it) < 1 - 0 is at most R2I1iiIl2/02. Theorem 5 implies that, in a sense, repeatedly cycling through a dataset using aggressive ROMMA will eventually converge to SVM; note however that bias is not considered. 3 An efficient implementation When the prediction of ROMMA differs from the expected label, the algorithm chooses Wt+! to minimize IIWt+!1I subject to AWt+! = b, where A = (~f) and b = ( 11~:"2 ) . Simple calculation shows that Wt+! AT (AAT)-lb ( IIxtII211Wtll2 - Yt(Wt . it)) ~ ("wtIl2(Yt - (Wt · it)) ) ~ lIitll211Wtll2 - (Wt . ie)2 Wt + IIxtll211Wtll2 _ (Wt . XtP Xt· (5) If on trials t in which a mistake is made, Ct = Since the computations required by ROMMA involve inner products together with a few operations on scalars, we can apply the kernel method to our algorithm, efficiently solving the original problem in a very high dimensional space. Computationally, we only need to modify the algorithm by replacing each inner product computation (ii . Xj) with a kernel function computation IC (ii, Xj). To make a prediction for the tth trial, the algorithm must compute the inner product between Xt and prediction vector Wt. In order to apply the kernel function, as in [1, 3], we store each prediction vector Wt in an implicit manner, as the weighted sum of examples on which The Relaxed Online Maximum Margin Algorithm 503 mistakes occur during the training. In particular. each Wt is represented as (t-l) t-l (t_l ) Wt = IT Cj WI + L n Cn djxj J=1 J=1 n=J+l The above formula may seem daunting; however, making use of the recurrence (Wt+l ·x) = Ct (Wt . x) + dt (Xt . x). it is obvious that the complexity of our new algorithm is similar to that of perceptron algorithm. This was born out by our experiments. The implementation for aggressive ROMMA is similar to the above. 4 Experiments We did some experiments using the ROMMA and aggressive ROMMA as batch algorithms on the MNIST OCR database. 2 We obtained a batch algorithm from our online algorithm in. the usual way, making a number of passes over the dataset and using the final weight vector to classify the test data. Every example in this database has two parts, the first is a 28 x 28 matrix which represents the image of the corresponding digit. Each entry in the matrix takes value from {O, . . . , 255}. The second part is a label taking a value from {O,· .. , g} . The dataset consists of 60, 000 training examples and 10,000 test examples. We adopt the following polynomial kernel: K(Xi, Xj) = (1 + (Xi· Xj))d. This corresponds to using an expanded collection of features including all products of at most d components of the original feature vector (see [14]). Let us refer to the mapping from the original feature vector to the expanded feature vector as <1>. Note that one component of <I> (x) is always 1, and therefore the component of the weight vector corresponding to that component can be viewed as a bias. In our experiments, we set WI = <1>(6') rather than (5 to speed up the learning of the coefficient corresponding to the bias. We chose d = 4 since in experiments on the same problem conducted in [3, 2], the best results occur with this value. To cope with multiclass data, we trained ROMMA and aggressive ROMMA once for each of the 10 labels. Classification of an unknown pattern is done according to the maximum output of these ten classifiers. As every entry in the image matrix takes value from {O, · .. , 255}, the order of magnitude of K(x, x) is at least 1026 , which might cause round-off error in the computation of Ci and di . We scale the data by dividing each entry with 1100 when training with ROMMA. Table 1: Experimental results on MNIST data T=l T=2 T=3 T =4 Err MisNo Err MisNo Err MisNo Err MisNo percep 2.84 7970 2.27 10539 1.99 11945 1.85 12800 voted-percep 2.26 7970 1.88 10539 1.76 11945 1.69 12800 ROMMA 2.48 7963 1.96 9995 1.79 10971 1.77 11547 agg-ROMMA 2.14 6077 1.82 7391 1.71 7901 1.67 8139 agg-ROMMACNC) 2.05 5909 1.76 6979 1.67 7339 1.63 7484 Since the performance of online learning is affected by the order of sample sequence, all the results shown in Table 1 average over 10 random permutations. The columns marked 2National Institute for Standards and Technology, special database 3. See http://www.research.att.com/ ... yanniocr for information on obtaining this dataset. 504 Y Li and P M. Long "MisNo" in Table 1 show the total number of mistakes made during the training for the 10 labels. Although online learning would involve only one epoch, we present results for a batch setting until four epochs (T in Table 1 represents the number of epochs). To deal with data which are linearly inseparable in the feature space, and also to improve generalization, Friess et al [4] suggested the use of quadratic penalty in the cost function, which can be implemented using a slightly different kernel function [4, 5]: iC(Xk ' Xj) = K(Xk, Xj) + c5kj ).., where c5kj is the Kronecker delta function. The last row in Table 1 is the result of aggressive ROMMA using this method to control noise ().. = 30 for 10 classifiers). We conducted three groups of experiments, one for the perceptron algorithm (denoted "percep"), the second for the voted perceptron (denoted "voted-percep") whose description is in [3], the third for ROMMA, aggressive ROMMA (denoted "agg-ROMMA"), and aggressive ROMMA with noise control (denoted "agg-ROMMA(NC)"). Data in the third group are scaled. All three groups set 'lih = <1>(0). The results in Table 1 demonstrate that ROMMA has better performance than the standard perceptron, aggressive ROMMA has slightly better performance than the voted perceptron. Aggressive ROMMA with noise control should not be compared with perceptrons without noise control. Its presentation is used to show what performance our new online algorithm could achieve (of course it's not the best, since all 10 classifiers use the same )"). A remarkable phenomenon is that our new algorithm behaves very well at the first two epochs. References [1] B. E. Boser, I. M. Guyon, and V. N. Vapnik. A training algorithm for optimal margin classifiers. Proceedings of the 1992 Workshop on Computational Learning Theory, pages 144-152, 1992. [2] C. Cortes and V. Vapnik. Support-vector networks. Machine Learning, 20(3):273-297,1995. [3] y. Freund and R. E. Schapire. Large margin classification using the perceptron algorithm. Proceedings of the 1998 Conference on Computational Learning Theory, 1998. [4] T. T. Friess, N. Cristianini, and C. Campbell. The kernel adatron algorithm: a fast and simple learning procedure for support vector machines. In Proc. 15th Int. Con! on Machine Learning. Morgan Kaufman Publishers, 1998. [5] S. S. Keerthi, S. K. Shevade, C. Bhattacharyya, and K. R. K. Murthy. A fast iterative nearest point algorithm for support vector machine c1assiifer design. Technical report, Indian Institute of Science, 99. TR-ISL-99-03. [6] Adam Kowalczyk. Maximal margin perceptron. In Smola, Bartlett, Scholkopf, and Schuurmans, editors, Advances in Large Margin Classifiers, 1999. MIT-Press. [7] N. Littlestone. Learning quickly when irrelevant attributes abound: a new linear-threshold algorithm. Machine Learning, 2:285-318, 1988. [8] N. Littlestone. Mistake Bounds and Logarithmic Linear-threshold Learning Algorithms. PhD thesis, UC Santa Cruz, 1989. [9] John C. Platt. Fast training of support vector machines using sequential minimal optimization. In B. Scholkopf, C. Burges, A. Smola, editors, Advances in Kernel Methods: Support Vector Machines, 1998. MIT Press. [10] F. Rosenblatt. The perceptron: A probabilistic model for information storage and organization in the brain. Psychological Review, 65:386-407, 1958. [11] F. Rosenblatt. Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms. Spartan Books, Washington, D. C., 1962. [12] J. Shawe-Taylor, P. Bartlett, R. Williamson, and M. Anthony. A framework for structural risk minimization. In Proc. of the 1996 Conference on Computational Learning Theory, 1996. [13] V. N. Vapnik. Estimation of Dependencies based on Empirical Data. Springer Verlag, 1982. [14] V. N. Vapnik. The Nature of Statistical Learning Theory. Springer, 1995.
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Bayesian modelling of tMRI time series Pedro A. d. F. R. H~jen-S~rensen, Lars K. Hansen and Carl Edward Rasmussen Department of Mathematical Modelling, Building 321 Technical University of Denmark DK-2800 Lyngby, Denmark phs,lkhansen,carl@imrn.dtu.dk Abstract We present a Hidden Markov Model (HMM) for inferring the hidden psychological state (or neural activity) during single trial tMRI activation experiments with blocked task paradigms. Inference is based on Bayesian methodology, using a combination of analytical and a variety of Markov Chain Monte Carlo (MCMC) sampling techniques. The advantage of this method is that detection of short time learning effects between repeated trials is possible since inference is based only on single trial experiments. 1 Introduction Functional magnetic resonance imaging (tMRI) is a non-invasive technique that enables indirect measures of neuronal activity in the working human brain. The most common tMRI technique is based on an image contrast induced by temporal shifts in the relative concentration of oxyhemoglobin and deoxyhemoglobin (BOLD contrast). Since neuronal activation leads to an increased blood flow, the so-called hemodynamic response, the measured tMRI signal reflects neuronal activity. Hence, when analyzing the BOLD signal there are two unknown factors to consider; the task dependent neuronal activation and the hemodynamic response. Bandettini et al. [1993] analyzed the correlation between a binary reference function (representing the stimulus/task sequence) and the BOLD signal. In the following we will also make reference to the binary representation of the task as the paradigm. Lange and Zeger [] 997] discuss a parameterized hemodynamic response adapted by a least squares procedure. Mu]tivariate strategies have been pursued in [Worsley et al. 1997, Hansen et al. ] 999]. Several explorative strategies have been proposed for finding spatio-tempora] activation patterns without explicit reference to the activation paradigm. McKeown et al. [1998] used independent component analysis and found several types of activations including components with "transient task related" response, i.e., responses that could not simply be accounted for by the paradigm. The model presented in this paper draws on the experimental observation that the basic coupling between the net neural activity and hemodynamic response is roughly linear while the relation between neuronal response and stimulus/task parameters is often nonlinear [Dale ]997]. We will represent the neuronal activity (integrated over the voxel and sampling time interval) by a binary signal while we will represent the hemodynamic response as a linear filter of unknown form and temporal extent. Bayesian Modelling ofjMRJ Time Series 755 2 A Bayesian model of fMRI time series Let S = {St : t = 0, ... , T - 1} be a hidden sequence of binary state variables St E {O, 1}, representing the state of a single voxel over time; the time variable, t, indexes the sequence of tMRI scans. Hence, St is a binary representation of the neural state. The hidden sequence is governed by a symmetric first order Hidden Markov Model (HMM) with transition probability a = P(St+1 = jiSt = j). We expect the activation to mimic the blocked structure of the experimental paradigm so for this reason we restrict a to be larger than one half. The predicted signal (noiseless signal) is given by Yt = h*s+()o +()I t, where * denotes the linear convolution and h is the impulse response of a linear system of order M f . The dc off-set and linear trend which are typically seen in tMRI time series are given by ()o and ()b respectively. Finally, it is assumed that the observable is given by Zt = Yt + Ct, where Ct is iid. Gaussian noise with variance £7;. The generative model considered is therefore given by: p(StISt-l ,a) a88t ,8t_1 + (1- a)(l- 88t ,8t_1)' p(zls,£7n ,(),Mf ) '" N(y,£7~I), where Y = {yt} = H8()' andz = {zt}. Furthermore, 88t ,8t - 1 is the usual Kronecker delta and H8 = [1,~, 'Yos, 'YIS, ... , 'YM/ -ISJ, where 1 = (1, ... 1)', ~=(l, .. . ,T)'IT and 'Yi is a i-step shift operator, that is 'YiS = (O, .. . ,O,SO,SI, ... ,ST-I-i)'. The linear parameters are collected in () = (()o'(h , h)'. ~ i So SI S2 S3 ST_I ... ~ Z, ': \:,1 The graphical model. The hidden states X t = (St- l, St-2," ., St-(M/-l)) have been introduced to make the model first or.. \0 der. Xo XI x 2 X3 XT_I 3 Analytic integration and Monte Carlo sampling In this section we introduce priors over the model parameters and show how inference may be performed. The filter coefficients and noise parameters may be handled analytically, whereas the remaining parameters are treated using sampling procedures (a combination of Gibbs and Metropolis sampling). Like in the previous section explicit reference to the filter order l'vlf may be omitted to ease the notation. The dc off-set ()o and the linear trend ()1 are given (improper) uniform priors. The filter coefficients are given priors that are uniform on an interval of length (3 independently for each coefficient: for Ihil < ~, otherwise Assuming that all the values of () for which the associated likelihood has non-vanishing contributions lie inside the box where the prior for () has support, we may integrate out the filter coefficients via a Gaussian integral: 756 P. A. d. F R. Hejen-Serensen, L. K. Hansen and C. E. Rasmussen We have here defined the mean fiLter, 8s = (H;Hs)-l Hsz and mean predicted signaL, Ys = Hils, for given state and filter length. We set the interval-length, (3 to be 4 times the standard deviation of the observed signal z. This is done, since the response from the filter should be able to model the signal, for which it is thought to need an interval of plus/minus two standard deviations. We now proceed to integrate over the noise parameter; using the (improper) noninformative Jeffreys prior, P(O'n) ex 0'~1, we get a Gamma integral: M[-T ! 1 T - M f (ll'(ZIZ Y~Ys)) 2 +1 p(zls, Mf) = p(zIO'n, S, Mf )P(O'n)dO'n = -2r ( 2 - 1) / (3M[ V IH~Hsl The remaining variables cannot be handled analytically, and will be treated using various forms of sampling as described in the following sections. 3.1 Gibbs and Metropolis updates of the state sequence We use a flat prior on the states, p(St = 0) = p(St = 1), together with the first order Markov property for the hidden states and Bayes' rule to get the conditional posterior for the individual states: p(St = jls\st, a, Mf) ex p(St = jISt-1, a)p(st+llst = j , a)p(zls, Mf)· These probabilities may (in normalized form) be used to implement Gibbs updates for the hidden state variables, updating one variable at a time and sweeping through all variables. However, it turns out that there are significant correlations between states which makes it difficult for the Markov Chain to move around in the hidden state-space using only Gibbs sampling (where a single state is updated at a time). To improve the situation we also perform global state updates, consisting of proposing to move the entire state sequence one step forward or backward (the direction being chosen at random) and accepting the proposed state using the Metropolis acceptance procedure. The proposed movements are made using periodic boundary conditions. The Gibbs sweep is computationally involved, since it requires computation of several matrix expressions for every state-variable. 3.2 Adaptive Rejection Sampling for the transition probability The likelihood for the transition probability a is derived from the Hidden Markov Model: T-1 1 p(sla) = p(so) II p(stlst-t,a) = 2'aE(s)(l- af- 1-E(s) , t=l where E(s) = L,;=~1 6S , , 8'_1 is the number of neighboring states in S with identical values. The prior on the transition probabilities is uniform, but restricted to be larger than one half, since we expect the activation to mimic the blocked structure of the experimental paradigm. It is readily seen that p(als) ex p(sla), a E [~, 1] is log-concave. Hence, we may use the Adaptive Rejection Sampling algorithm [Gilks and Wild, t 992] to sample from the distribution for the transition probability. 3.3 Metropolis updates for the filter length In practical applications using real tMRI data, we do typically not know the necessary length of the filter. The problem of finding the "right" model order is difficult and has received a lot of attention. Here, we let the Markov Chain sample over different filter lengths, effectively integrating out the filter-length rather than trying to optimize it. Although the Bayesian Modelling offMRl Time Series 757 value of Mf determines the dimensionality of the parameter space, we do not need to use specialized sampling methodology (such as Reversible Jump MCMC [Green, 1995]), since those parameters are handled analytically in our model. We put a flat (improper) prior on M f and propose new filter lengths using a Gaussian proposal centered on the current value, with a standard deviation of 3 (non-positive proposed orders are rejected). This choice of the standard deviation only effects the mixing rate of the Markov chain and does not have any influence on the stationary distribution. The proposed values are accepted using the Metropolis algorithm, usingp(Mfls, y) ex: p(yls, Mf). 3.4 The posterior mean and uncertainty of the predicted signal Since () has a flat prior the conditional probability for the filter coefficients is proportional to the likelihood p(zl(),') and by (*) we get: p(()lz, s, an, Mf) '" N(Dsz, a;DsD~), Ds = (H~Hs)-l H~ . The posterior mean of the predicted signal, y, is then readily computed as: Y = (y(),un,s,M,)() ~ s M = (Ys)s Mj = (HiJs)s,M, = (Fs)s M,z, 'Vn" f ' , where Fs = HsDs. Here, the average over () and an is done analytically, and the average over the state and filter length is done using Monte Carlo. The uncertainty in the posterior, can also be estimated partly by analytical averaging, and partly Monte Carlo: ~ - «( - ')( - ')') y Y() ,un,s,M, Y Y(),un,s,M, Y () ~ s M ,Vn, , J 1 ((' " , )F F') (F 'F') " , = T _ M _ 2 z z - YsYs s s + sZz s - YY . f 8,M, s,M, 4 Example: synthetic data In order to test the model, we first present some results on a synthetic data set. A signal z of length 100 is generated using a Mf = 10 order filter, and a hidden state sequence s consisting of two activation bursts (indicated by dotted bars in figure I top left). In this example, the hidden sequence is actually not generated from the generative model (*); however, it still exhibits the kind of block structure that we wish to be able to recover. The model is run for 10000 iterations, which is sufficient to generate 500 approximately independent samples from the posterior; figure 2 (right) shows the auto-covariance for Mf as a function of the iteration lag. It is thought that changes in Mf are indicative of correlation time of the overall system. The correlation plot for the hidden states (figure 2, left) shows that the state activation onset correlates strongly with the second onset and negatively with the end of the activation (and vice versa). This indicates that the Metropolis updates described in section 3.] may indeed be effective. Notice also that the very strong correlation among state variables does not strongly carryover to the predicted signal (figure 1, bottom right). To verify that the model can reasonably recover the parameters used to generate the data, posterior samples from some of the model variables are shown in figure 3. For all these parameters the posterior density is large around the correct values. Notice, that there in the original model (*) is an indeterminacy in the simultaneous inference of the state sequence and the filter parameters (but no indeterminacy in the predicted signal); for example, the same signal is predicted by shifting the state sequence backward in time and introducing leading zero filter coefficients. However, the Bayesian methodology breaks this symmetry by penalizing complex models. 758 P. A. d. F. R. Hojen-Sorensen, L. K. Hansen and C. E. Rasmussen N CI:S 4 c: 0> '(j) -0 2 Q.) "-:J (/) ~ 0 ~ _2L---~--~----~--~--~ o 20 40 60 80 100 Scan number, t 6~--~_~ , --~-------------~ : "" , .. ~'.-" ....... . ....... _2L---~--~----~--~--~ o 20 40 60 80 100 Scan number, t o -2~--~~~--~~~--~ 20 40 60 80 100 Scan number, t 0.8 ..... -20 .: Q.) 0.6 -0 40 E 0.4 :J c: 60 c: 0.2 CI:S (.) en 80 o 100 -0.2 20 40 60 80 100 Scan number, t J Figure 1: Experiments with synthetic data. Top left, the measured response from a voxel is plotted for 100 consecutive scans. In the bottom left, the underlying signal is seen in thin, together with the posterior mean, fj (thick), and two std. dev. error-bars in dotted. Top right, the posterior probabilities are shown as a grey level, for each scan. The true activated instances are indicated by the dotted bars and the pseudo MAP estimate of the activation sequence is given by the crossed bars. Bottom right, shows the posterior uncertainty ~Y' The posterior mean and the two standard deviations are plotted in figure 1 bottom left. Notice, however, that the distribution of y is not Gaussian, but rather a mixture of Gaussians, and is not necessarily well characterized by mean and variance alone. In figure] (top left), the distribution of Yt is visualized using grey-scale to represent density. 5 Simulations on real fMRI data and discussion In figure 4 the model has been applied to two measurements in the same voxel in visual cortex. The fMRI scans were acquired every 330 ms. The experimental paradigm consisted of 30 scans of rest followed by 30 scans of activation and 40 rest. Visual activation consisted of a flashing (8 Hz) annular checkerboard pattern. The model readily identifies the activation burst of somewhat longer duration than the visual stimulus and delayed around 2 seconds. The delay is in part caused by the delay in the hemodynamic response. These results show that the integration procedure works in spite of the very limited data at hand. In figure 4 (top) the posterior model size suggests that (at least) two competing models can explain the signal from this trial. One of these models explains the measured signal as a simple square wave function which seems reasonable by considering the signal. Conversely, figure 4 (bottom), suggests that the signal from the second trial can not be explained by a simple model. This too, seems reasonable because of the long signal raise interval suggested in the signal. Bayesian Modelling o!iMRl Time Series '" <Ii (!) :0 '" -~ > * liS c '" u u J: Hidden state variables. s 759 OS" lag Figure 2: The covariance of the hidden states based on a long run of the model is shown to the left. Notice, that the states around the front (back) of the activity "bumps" are highly (anti-) correlated. Right The auto-covariance for the filter length M f as a function of the lag time in iterations. The correlation length is about 20, computed as the sum of autocovariance coefficients from lag -400 to 400. Since the posterior distribution of the filter length is very broad it is questionable whether an optimization based procedure such as maximum likelihood estimation would be able to make useful inference in this case were data is very limited. Also, it is not obvious how one may use cross-validation in this setting. One might expect such optimization based strategies to get trapped in suboptimal solutions. This, of course, remains to be investigated. 6 Conclusion We have presented a model for voxel based explorative data analysis of single trial fMRI signals during blocked task activation studies. The model is founded on the experimental observation that the basic coupling between the net neural activity and hemodynamic response is roughly linear. The preliminary investigation reported here are encouraging in that the model reliably detects reasonable hidden states from the very noisy fMRI data. One drawback of this method is that the Gibbs sampling step is computational expensive. To improve on this step one could make use of the large class of variational/mean field methods known from the graphical models literature. Finally, current work is in progress for generalizing the model to multiple voxels, including spatial correlation due to e.g. spillover effects. 0.15 0.15 0.15 0.3 0.1 0.1 0.1 0.2 0.05 0.05 0.05 A 0.1 rL a a a 0.8 1.2 1.5 2 2.5 -2 -1 a 5 10 cr DC off-set Trend Mf Figure 3: Posterior distributions of various model parameters. The parameters used to generate the data are: a = 1.0, DC off-set = 2, trend = -1 and filter order M f = 10. 760 P. A. d. F R. HfJjen-SfJrensen, L. K. Hansen and C. E. Rasmussen 320~--------------------~---. 180~~----~~~~~~~~--~ o 20 40 60 eo 100 Scan number, t 280.---------__ ----.---____ ---, 260 180 180~------==~~~~--------~ o 20 40 eo 80 Scan number. t ::;r ~nnnen~~~~~nn_ I 12 14 16 18 20 cr ::~r _~cOmmm[[n~~ 1 10 12 14 16 cr Figure 4: Analysis of two experimental trials of the same voxel in visual cortex. The left hand plot shows the posterior inferred signal distribution superimposed by the measured signal. The dotted bar indicates the experimental paradigm and the crossed bar indicates the pseudo MAP estimate of the neural activity. To the right the posterior noise level and inferred filter length are displayed. Acknowledgments Thanks to Egill Rostrup for providing the fMRI data. This work is funded by the Danish Research Councils through the Computational Neural Network Center (CONNECT) and the THOR Center for Neuroinformatics. References Bandettini, P. A. (1993). Processing strategies for time-course data sets in functional MRI of the human brain Magnetic Resonance in Medicine 30, 161-173. Dale, A. M. and R. L. Buckner (1997). Selective Averaging of Individual Trials Using fMRI. Neuro/mage 5, Abstract S47. Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82, 711-732. Gilks, W. R. and P. Wild (1992). Adaptive rejection sampling for Gibbs sampling. Applied Statistics 41, 337-348. Hansen, L. K. et al. (1999). Generalizable Patterns in Neuroimaging: How Many Principal Components? Neuro/mage, to appear. Lange, N. and S. L. Zeger (1997). Non-linear Fourier time series analysis for human brain mapping by functional magnetic resonance imaging. Journal of the Royal Statistical Society - Series C Applied Statistics 46, 1-30. McKeown, M. J. et al. (1998). Spatially independent activity patterns in functional magnetic resonance imaging data during the stroop color-naming task. Proc. Nat!. Acad. Sci. USA. 95,803-810. Worsley, K. J. et al. (1997). Characterizing the Response of PET and fMRI Data Using Multivariate Linear Models (MLM). Neuro/mage 6, 305-319.
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Bayesian averaging is well-temperated Lars Kai Hansen Department of Mathematical Modelling Technical University of Denmark B321 DK-2800 Lyngby, Denmark lkhansen@imm.dtu.dk Abstract Bayesian predictions are stochastic just like predictions of any other inference scheme that generalize from a finite sample. While a simple variational argument shows that Bayes averaging is generalization optimal given that the prior matches the teacher parameter distribution the situation is less clear if the teacher distribution is unknown. I define a class of averaging procedures, the temperated likelihoods, including both Bayes averaging with a uniform prior and maximum likelihood estimation as special cases. I show that Bayes is generalization optimal in this family for any teacher distribution for two learning problems that are analytically tractable: learning the mean of a Gaussian and asymptotics of smooth learners. 1 Introduction Learning is the stochastic process of generalizing from a random finite sample of data. Often a learning problem has natural quantitative measure of generalization. If a loss function is defined the natural measure is the generalization error, i.e., the expected loss on a random sample independent of the training set. Generalizability is a key topic of learning theory and much progress has been reported. Analytic results for a broad class of machines can be found in the litterature [8, 12, 9, 10] describing the asymptotic generalization ability of supervised algorithms that are continuously parameterized. Asymptotic bounds on generalization for general machines have been advocated by Vapnik [11]. Generalization results valid for finite training sets can only be obtained for specific learning machines, see e.g. [5]. A very rich framework for analysis of generalization for Bayesian averaging and other schemes is defined in [6]. A veraging has become popular as a tool for improving generalizability of learning machines. In the context of (time series) forecasting averaging has been investigated intensely for decades [3]. Neural network ensembles were shown to improve generalization by simple voting in [4] and later work has generalized these results to other types of averaging. Boosting, Bagging, Stacking, and Arcing are recent examples of averaging procedures based on data resampling that have shown useful see [2] for a recent review with references. However, Bayesian averaging in particular is attaining a kind of cult status. Bayesian averaging is indeed provably optimal in a 266 L. K. Hansen number various ways (admissibility, the likelihood principle etc) [1]. While it follows by construction that Bayes is generalization optimal if given the correct prior information, i.e., the teacher parameter distribution, the situation is less clear if the teacher distribution is unknown. Hence, the pragmatic Bayesians downplay the role of the prior. Instead the averaging aspect is emphasized and "vague" priors are invoked. It is important to note that whatever prior is used Bayesian predictions are stochastic just like predictions of any other inference scheme that generalize from a finite sample. In this contribution I analyse two scenarios where averaging can improve generalizability and I show that the vague Bayes average is in fact optimal among the averaging schemes investigated. Averaging is shown to reduce variance at the cost of introducing bias, and Bayes happens to implement the optimal bias-variance trade-off. 2 Bayes and generalization Consider a model that is smoothly parametrized and whose predictions can be described in terms of a density function1 . Predictions in the model are based on a given training set: a finite sample D = {Xa}~=l of the stochastic vector x whose density - the teacher - is denoted p(xIOo). In other words the true density is assumed to be defined by a fixed, but unknown, teacher parameter vector 00 . The model, denoted H, involves the parameter vector ° and the predictive density is given by p(xID, H) = ! p(xIO, H)p(OID, H)dO (1) p(OID, H) is the parameter distribution produced in training process. In a maximum likelihood scenario this distribution is a delta function centered on the most likely parameters under the model for the given data set. In ensemble averaging approaches, like boosting bagging or stacking, the distribution is obtained by training on resampled traning sets. In a Bayesian scenario, the parameter distribution is the posterior distribution, p(DIO, H)p(OIH) p(OID, H) = f p(DIO', H)p(O'IH)dO' (2) where p(OIH) is the prior distribution (probability density of parameters if D is empty). In the sequel we will only consider one model hence we suppress the model conditioning label H. The generalization error is the average negative log density (also known as simply the "log loss" - in some applied statistics works known as the "deviance") r(DIOo) = ! -logp(xID)p(xIOo)dx, (3) The expected value of the generalization error for training sets produced by the given teacher is given by f(Oo) = ! ! -logp(xID)p(xIOo)dxp(DIOo)dD. (4) lThis does not limit us to conventional density estimation; pattern recognition and many functional approximations problems can be formulated as density estimation problems as well. Bayesian Averaging is Well-Temperated 267 Playing the game of "guessing a probability distribution" [6] we not only face a random training set, we also face a teacher drawn from the teacher distribution p( Bo) . The teacher averaged generalization must then be defined as r = J f(Bo)p(Bo)dBo . (5) This is the typical generalization error for a random training set from the randomly chosen teacher - produced by the model H. The generalization error is minimized by Bayes averaging if the teacher distribution is used as prior. To see this, form the Lagrangian functional £[q(xID)] = J J J -logq(xID)p(xIBo)dxp(DIBo)dDp(Bo)dBo+A J q(xID)dx (6) defined on positive functions q(xID). The second term is used to ensure that q(xID) is a normalized density in x . Now compute the variational derivative to obtain 6£ 1 J 6q(xID) = - q(xID) p(xIBo)p(DIBo)p(Bo)dBo + A. (7) Equating this derivative to zero we recover the predictive distribution of Bayesian averagmg, J p(DIB)p(B) q(xID) = p(xIB) J p(DIB')p(B')dB' dB, (8) where we used that A = J p(DIB)p(B)dB is the appropriate normalization constant. It is easily verified that this is indeed the global minimum of the averaged generalization error. We also note that if the Bayes average is performed with another prior than the teacher distribution p( Bo), we can expect a higher generalization error. The important question from a Bayesian point of view is then: Are there cases where averaging with generic priors (e.g. vague or uniform priors) can be shown to be optimal? 3 Temperated likelihoods To come closer to a quantative statement about when and why vague Bayes is the better procedure we will analyse two problems for which some analytical progress is possible. We will consider a one-parameter family of learning procedures including both a Bayes and the maximum likelihood procedure, v(DIB) p(BI!3,D,H) = Jpf3(DIB')dB" (9) where !3 is a positive parameter (plying the role of an inverse temperature). The family of procedures are all averaging procedures, and !3 controls the width of the average. Vague Bayes (here used synonymously with Bayes with a uniform prior) is recoved for !3 = 1, while the maximum posterior procedure is obtained by cooling to zero width !3 --+ 00 . In this context the generalization design question can be frased as follows: is there an optimal temperature in the family of the temperated likelihoods? 3.1 Example: ID normal variates Let the teacher distribution be given by p(xIBo) = ~exp (-~(X - Bo)2) 211"<72 2<7 (10) 268 L.K. Hansen The model density is of the same form with (J unknown and u2 assumed to be known. For N examples the posterior (with a uniform prior) is, p(OID) = J 2:U2 exp (-::2 (x - (J)2) , (11) with x = 1/ N Eo: Xo:. The temperated likelihood is obtained by raising to the ,8'th power and normalizing, f7iN (,8N ) p((JID,,8) = V ~ exp - 2u2 (x - (J)2 . (12) The predictive distribution is found by integrating w.r.t. (J, p(xID,,8) = !P(ZIB)P(BID,~)dB; ~exp (--212 (x- X)2) , (13) 21!'u$ u f3 with u~ = u2(1+1/,8N). We note that this distribution is wider for all the averaging procedures than it is for maximum likelihood (,8 -T (0), i.e., less variant. For very small ,8 the predictive distribution is almost independent of the data set, hence highly biased. It is straightforward to compute the generalization error of the predictive distribution for general,8. First we compute the generalization error for the specific training set D, r(D,,8, (Jo) = ! -logp(xID, ,8)p(xl(Jo)dx = log )21!'u$ + 2~~ ((x - (JO)2 + ( 2) , (14) The average generalization error is then found by averaging w.r.t the sampling distribution using x"" N((Jo, u2/N)., r(,8) = ! r(D, ,8)dDp(DI(Jo) = log )21!'u$ + 2:$ (~ + 1) , (15) We first note that the generalization error is independent of the teacher (Jo parameter, this happened because (J is a "location" parameter. The ,8-dependency of the averaged generalization error is depicted in Figure 1. Solving 8r(,8) /8,8 = 0 we find that the optimal ,8 solves u$=U2(,8~+I)=U2(~+I) :=} ,8=1 (16) Note that this result holds for any N and is independent of the teacher parameter. The Bayes averaging at unit temperature is optimal for any given value of (Jo, hence, for any teacher distribution. We may say that the vague Bayes scheme is robust to the teacher distribution in this case. Clearly this is a much stronger optimality than the more general result proven above. 3.2 Bias-variance tradeoff It is interesting to decompose the generalization error in Eq. 15 in bias and variance components. We follow Heskes [7] and define the bias error as the generalization error of the geometric average distribution, B(,8) = ! -logp(x)p(xl(Jo)dx, (17) Bayesian Averaging is Wel/-Temperated 269 0.7 0.& GENERALIZATION 0.5 A 04 1£ § , v 03 02 0.1 0 0 0.5 15 2 25 3 35 45 TEMPERATURE Figure 1: Bias-variance trade-off as function of the width of the temperated likelihood ensemble (temperature = 1/ (3) for N = 1. The bias is computed as the generalization error of the predictive distribution obtained from the geometric average distribution w.r.t. training set fluctuations as proposed by Heskes. The predictive distribution produced by Bayesian averaging corresponds to unit temperature (vertical line) and it achieves the minimal generalization error. Maximum-likelihood estimation for reference is recovered as the zero width/temperature limit. with p(x) = Z-l exp (/ 10g(P(X 1D)]P(D I80)dD) . Inserting from Eq. (13), we find p(z) = ~exp (-~(X -80)2) . 27r0'~ 0' f3 Integrating over the teacher distribution we find, 1 0'2 B(f3) = -2 log 27r0'~ + -2 20'f3 The variance error is given by V(f3) = r(f3) - B(f3) , 0'2 V (f3) = 2N O'~ (18) (19) (20) (21) We can now quantify the statements above. By averaging a bias is introduced -the predictive distribution becomes wider- which decrease the variance contribution initially so that the generalization error being the sum of the two decreases. At still higher temperatures the bias becomes too strong and the generalization error start to increase. The Bayes average at unit temperature is the optimal trade-off within the given family of procedures. 270 L. K. Hansen 3.3 Asymptotics for smoothly parameterized models We now go on to show that a similar result also holds for general learning problems in limit of large data sets. We consider a system parameterized by a finite dimensional parameter vector O. For a given large training set and for a smooth likelihood function, the temperated likelihood is approximately Gaussian centered at the maximum posterior parameters[13]' hence the normalized temperated posterior reads P(OI(3D,H) = I(3NA(~OML) lexp (_(3; 60'A(D,OML)60) (22) where 60 = O-OML, with OML = OML(D) denoting the maximum likelihood solution for the given training sample. The second derivative or Hessian matrix is given by 1 N A(D,O) N LA(xa,O) (23) a=l A(x,O) = {)2 ()O{)O' - log p( x 10) (24) The predictive distribution is given by p(xl(3, D) = ! p(xIO)p(OI(3, D)dO (25) we write p(xIO) = exp(-f(xIO)) and expand f(xIO) around OML to second order, we find p(xIO) ~ p(XIOML) exp (-a(xIOML)'60 ~60' A(xIOML)60) . (26) We are then in position to perform the integration over the posterior to find the normalized predictive distribution, p(xl(3, D) = p(XIOML) I(3N A(D)I 1 , I(3NA(D) + A(x)1 exp ( 2'a(xIOML) A(xIOML)a(xIOML)). (27) Proceeding as above, we compute the generalization error f((3, ( 0 ) = ! ! -logp(xl(3, D)p(xIOo)dxp(DIOo)dD (28) For sufficiently smooth likelihoods, fluctuations in the maximum likelihood parameters will be asymptotic normal, see e.g. [8], and furthermore fluctuations in A(D) can be neglected, this means that we can approximate, A(x) + A(D) ~ (~ + l)Ao, Ao = ! A(xIOo)p(xIOo)dx (29) where Ao is the averaged Fisher information matrix. With these approximations (valid as N --+ (0) the generalization error can be found, d ( 1) d 1+ ~ f((3, ( 0 ) ~ f(oo) + 2 log 1 + (3N - 21 + (3N' (30) with d = dim(O) denoting the dimension of the parameter vector. Like in the ID example (Eq. (15)) we find the generalization error is asymptotically independent of the teacher parameters. It is minimized for (3 = 1 and we conclude that Bayes is well-temperated in the asymptotics and that this holds for any teacher distribution. In the Bayes literature this is refered to as the prior is overwhelmed by data [1]. Decomposing the errors in bias and variance contributions we find similar results as for in ID example, Bayes introduces the optimal bias by averaging at unit temperature. Bayesian Averaging is Well-Temperated 271 4 Discussion We have seen two examples of Bayes averaging being optimal, in particular improving on maximum likelihood estimation. We found that averaging introduces a bias and reduces variance so that the generalization error (being the sum of bias and variance) initially decrease. Bayesian averaging at unit temperature is the optimal width of the averaging distribution. For larger temperatures (widths) the bias is too strong and the generalization error increases. Both examples were special in the sense that they lead to generalization errors that are independent of the random teacher parameter. This is not generic, of course, rather the generic case is that a mis-specified prior can lead to arbitrary large learning catastrophes. Acknowledgments I thank the organizers of the 1999 Max Planck Institute Workshop on Statistical Physics of Neural Networks Michael Biehl, Wolfgang Kinzel and Ido Kanter, where this work was initiated. I thank Carl Edward Rasmussen, Jan Larsen, and Manfred Opper for stimulating discussions on_Bayesian averaging. This work was funded by the Danish Research Councils through the Computational Neural Network Center CONNECT and the THOR Center for Neuroinformatics. References [1] C.P. Robert: The Bayesian Choice - A Decision- Theoretic Motivation. Springer Texts in Statistics, Springer Verlag, New York (1994). A. Ohagan: Bayesian Inference. Kendall's Advanced Theory of Statistics. Vol 2B. The University Press, Cambridge (1994). [2] L. Breiman: Using adaptive bagging to debias regressions. Technical Report 547, Statistics Dept. U.C. Berkeley, (1999). [3] R.T. Clemen Combining forecast: A review and annotated bibliography. Journal of Forecasting 5, 559 (1989). [4] L.K. Hansen and P. Salamon: Neural Network Ensembles. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12, 993-1001 (1990). [5] L.K. Hansen: Stochastic Linear Learning: Exact Test and Training Error Averages. Neural Networks 6, 393-396, (1993) [6] D. Haussler and M. Opper: Mutual Information, Metric Entropy, and Cumulative Relative Entropy Risk Annals of Statistics 25 2451-2492 (1997) [7] T . Heskes: Bias/Variance Decomposition for Likelihood-Based Estimators. Neural Computation 10, pp 1425-1433, (1998). [8] L. Ljung: System Identification: Theory for the User. Englewood Cliffs, New Jersey: Prentice-Hall, (1987). [9] J . Moody: "Note on Generalization, Regularization, and Architecture Selection in Nonlinear Learning Systems," in B.H. Juang, S.Y. Kung & C.A. Kamm (eds.) Proceedings of the first IEEE Workshop on Neural Networks for Signal Processing, Piscataway, New Jersey: IEEE, 1-10, (1991). [10] N. Murata, S. Yoshizawa & S. Amari: Network Information Criterion Determining the Number of Hidden Units for an Artificial Neural Network Model. IEEE Transactions on Neural Networks, vol. 5, no. 6, pp. 865-872, 1994. [11] V. Vapnik: Estimation of Dependences Based on Empirical Data. Springer-Verlag New York (1982). [12] H. White, "Consequences and Detection of Misspecified Nonlinear Regression Models," Journal of the American Statistical Association, 76(374), 419-433, (1981). [13] D.J .C MacKay: Bayesian Interpolation, Neural Computation 4, 415-447, (1992) .
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Policy Search via Density Estimation AndrewY. Ng Computer Science Division u.c. Berkeley Berkeley, CA 94720 ang@cs.berkeley.edu Ronald Parr Computer Science Dept. Stanford University Stanford, CA 94305 parr@cs.stanjord.edu Abstract Daphne Koller Computer Science Dept. Stanford University Stanford, CA 94305 kolle r@cs.stanjord.edu We propose a new approach to the problem of searching a space of stochastic controllers for a Markov decision process (MDP) or a partially observable Markov decision process (POMDP). Following several other authors, our approach is based on searching in parameterized families of policies (for example, via gradient descent) to optimize solution quality. However, rather than trying to estimate the values and derivatives of a policy directly, we do so indirectly using estimates for the probability densities that the policy induces on states at the different points in time. This enables our algorithms to exploit the many techniques for efficient and robust approximate density propagation in stochastic systems. We show how our techniques can be applied both to deterministic propagation schemes (where the MDP's dynamics are given explicitly in compact form,) and to stochastic propagation schemes (where we have access only to a generative model, or simulator, of the MDP). We present empirical results for both of these variants on complex problems. 1 Introduction In recent years, there has been growing interest in algorithms for approximate planning in (exponentially or even infinitely) large Markov decision processes (MDPs) and partially observable MDPs (POMDPs). For such large domains, the value and Q-functions are sometimes complicated and difficult to approximate, even though there may be simple, compactly representable policies which perform very well. This observation has led to particular interest in direct policy search methods (e.g., [9, 8, 1]), which attempt to choose a good policy from some restricted class IT of policies. In our setting, IT = {1ro : (J E ~m} is a class of policies smoothly parameterized by (J E ~m. If the value of 1ro is differentiable in (J, then gradient ascent methods may be used to find a locally optimal 1ro. However, estimating values of 1ro (and the associated gradient) is often far from trivial. One simple method for estimating 1ro's value involves executing one or more Monte Carlo trajectories using 1ro, and then taking the average empirical return; cleverer algorithms executing single trajectories also allow gradient estimates [9, 1]. These methods have become a standard approach to policy search, and sometimes work fairly well. In this paper, we propose a somewhat different approach to this value/gradient estimation problem. Rather than estimating these quantities directly, we estimate the probability density over the states of the system induced by 1ro at different points in time. These time slice Policy Search via Density Estimation 1023 densities completely determine the value of the policy 1re. While density estimation is not an easy problem, we can utilize existing approaches to density propagation [3, 5], which allow users to specify prior knowledge about the densities, and which have also been shown, both theoretically and empirically, to provide robust estimates for time slice densities. We show how direct policy search can be implemented using this approach in two very different settings of the planning problem: In the first, we have access to an explicit model of the system dynamics, allowing us to provide an explicit algebraic operator that implements the approximate density propagation process. In the second, we have access only to a generative model of the dynamics (which allows us only to sample from, but does not provide an explicit representation of, next-state distributions). We show how both of our techniques can be combined with gradient ascent in order to perform policy search, a somewhat subtle argument in the case of the sampling-based approach. We also present empirical results for both variants in complex domains. 2 Problem description A Markov Decision Process (MDP) is a tuple (S, So, A, R, P) where:! S is a (possibly infinite) set of states; So E S is a start state; A is a finite set of actions; R is a reward function R : S f-t [0, Rmax]; P is a transition model P : S x A f-t ils, such that P(s' I s, a) gives the probability oflanding in state s' upon taking action a in state s. A stochastic policy is a map 1r : S f-t ilA, where 1r( a Is) is the probability of taking action a in state s. There are many ways of defining a policy 1r'S "quality" or value. For a horizon T and discount factor 1', the finite horizon discounted value function VT,"Y[1r] is defined by VO,"Y[1r](s) = R(s) ; vt+1,"Y[1r](s) = R(s) + l' L:a 1r(a I s) L:sl P(s' Is, a)vt'''Y[1r](s'). For an infinite state space (here and below), the summation is replaced by an integral. We can now define several optimality criteria. The finite horizon total reward with horizon T is VT[1r] = VT,d1r](so). The infinite horizon discounted reward with discount l' < 1 is V"Y[1r] = limT-HXl VT,"Y[1r](So). The infinite horizon average reward is Vavg [1r] = limT-HXl ~ VT,1 [1r](so), where we assume that the limit exists. Fix an optimality criterion V. Our goal is to find a policy that has a high value. As discussed, we assume we have a restricted set II of policies, and wish to select a good 1r E II. We assume that II = {1re I ° E ffim} is a set of policies parameterized by 0 E ffi.m, and that 1re(a I s) is continuously differentiable in 0 for each s, a. As a very simple example, we may have a one-dimensional state, two-action MDP with "sigmoidal" 1re, such that the probability of choosing action ao at state x is 1re(ao I x) = 1/(1 + exp( -81 - 82x)). Note that this framework also encompasses cases where our family II consists of policies that depend only on certain aspects of the state. In particular, in POMDPs, we can restrict attention to policies that depend only on the observables. This restriction results in a subclass of stochastic memory-free policies. By introducing artificial "memory bits" into the process state, we can also define stochastic limited-memory policies. [6] Each 0 has a value V[O] = V[1re], as specified above. To find the best policy in II, we can search for the 0 that maximizes V[O]. If we can compute or approximate V[O], there are many algorithms that can be used to find a local maximum. Some, such as Nelder-Mead simplex search (not to be confused with the simplex algorithm for linear programs), require only the ability to evaluate the function being optimized at any point. If we can compute or estimate V[O]'s gradient with respect to 0, we can also use a variety of (deterministic or stochastic) gradient ascent methods. IWe write rewards as R(s) rather than R(s, a), and assume a single start state rather than an initial-state distribution, only to simplify exposition; these and several other minor extensions are trivial. 1024 A. Y Ng, R. Parr and D. Koller 3 Densities and value functions Most optimization algorithms require some method for computing V[O] for any 0 (and sometimes also its gradient). In many real-life MOPs, however, doing so exactly is completely infeasible, due to the large or even infinite number of states. Here, we will consider an approach to estimating these quantities, based on a density-based reformulation of the value function expression. A policy 71" induces a probability distribution over the states at each time t. Letting ¢(O) be the initial distribution (giving probability 1 to so), we define the time slice distributions via the recurrence: (1) s a It is easy to verify that the standard notions of value defined earlier can reformulated in terms of ¢(t); e.g., VT,1'[7I"](So) = Ei'=o ,,/(¢(t) . R), where· is the dot-product operation (equivalently, the expectation of R with respect to ¢(t). Somewhat more subtly, for the case of infinite horizon average reward, we have that Vavg [71"] = ¢(oo) . R, where ¢(oo) is the limiting distribution of (1), if one exists. This reformulation gives us an alternative approach to evaluating the value of a policy 71"0: we first compute the time slice densities ¢(t) (or ¢(oo), and then use them to compute the value. Unfortunately, that modification, by itself, does not resolve the difficulty. Representing and computing probability densities over large or infinite spaces is often no easier than representing and computing value functions. However, several results [3, 5] indicate that representing and computing high-quality approximate densities may often be quite feasible. The general approach is an approximate density propagation algorithm, using time-slice distributions in some restricted family 3. For example, in continuous spaces, 3 might be the set of multivariate Gaussians. The approximate propagation algorithm modifies equation (1) to maintain the time-slice densities in 3. More precisely, for a policy 71"0, we can view (1) as defining an operator cf>[0] that takes one distribution in !:1s and returns another. For our current policy 71"0 0 , we can rewrite (1) as: ¢(t+1) = cf>[Oo](¢(t)) . In most cases,=: will not be closed under cf>; approximate density propagation algorithms use some alternative operator 4>, with the properties that, for ¢ E 3: (a) 4>( ¢) is also in 3, and (b) 4>( ¢) is (hopefully) close to cf>(¢). We use 4>[0] to denote the approximation to cf>[0], and ¢(t) to denote (4) [0]) (t) (¢(O)). If 4> is selected carefully, it is often the case that ¢(t) is close to ¢(t). Indeed, a standard contraction analysis for stochastic processes can be used to show: Proposition 1 Assume thatJor all t, 11cf>(¢(t)) - 4>(¢(t))lll ~ c. Then there exists some constant>. such thatJor all t, 1I¢(t) - ¢(t) lit ~ c/ >.. In some cases, >. might be arbitrarily small, in which case the proposition is meaningless. However, there are many systems where>. is reasonable (and independent of c) [3]. Furthermore, empirical results also show that approximate density propagation can often track the exact time slice distributions quite accurately. Approximate tracking can now be applied to our planning task. Given an optimality criterion V expressed with ¢(t) s, we define an approximation V to it by replacing each ¢(t) with ¢(t), e.g., VT,1'[7I"](so) = Ei'=o ,t¢(t) . R. Accuracy guarantees on approximate tracking induce comparable guarantees on the value approximation; from this, guarantees on the performance of a policy 7I"iJ found by optimizing V are also possible: Proposition 2 Assume that,for all t, we have that 11¢(t) - ¢(t) lit ~ 6. ThenJor each fixed T, ,: IVT,1'[7I"](So) - VT,1'[7I"](so)I = 0(6). Policy Search via Density Estimation Proposition 3 Let 0* = argmaxo V[O] and 0 V[O]I ::; €, then V[O*] - V[O] ::; 2€. 4 Differentiating approximate densities 1025 argmaxo V[O]. If maxo!V[O] In this section we discuss two very different techniques for maintaining an approximate density ¢ (t) using an approximate propagation operator <1>, and show when and how they can be combined with gradient ascent to perform policy search. In general, we will assume that :=: is a family of distributions parameterized by e E ffi.l. For example, if :=: is the set of d-dimensional multivariate Gaussians with diagonal covariance matrices, e would be a 2d-dimensional vector, specifying the mean vector and the covariance matrix's diagonal. Now, consider the task of doing gradient ascent over the space of policies, using some optimality criterion V, say VT,.,,[O]. Differentiating it relative to 0, we get '\7oVT,.,, [O] = '£'['=0 ,t d s~t ) . R. To avoid introducing new notation, we also use ¢ (t) to denote the associated vector of parameters e E ffi.l . These parameters are a function of O. Hence, the internal gradient term is represented by an £ x m Jacobian matrix, with entries representing the derivative of a parameter ~i relative to a parameter OJ. This gradient can be computed using a simple recurrence, based on the chain rule for derivatives: The first summand (an £ x m Jacobian) is the derivative of the transition operator <1> relative to the policy parameters O. The second is a product of two terms: the derivative of <1> relative to the distribution parameters, and the result of the previous step in the recurrence. 4.1 Deterministic density propagation Consider a transition operator q, (for simplicity, we omit the dependence on 0). The idea in this approach is to try to get <1>( ¢) to be as close as possible to q,(¢), subject to the constraint that <1>( ¢) E 3. Specifically, we define a projection operator r that takes a distribution 'ljJ not in 3, and returns a distribution in 3 which is closest (in some sense) to 'ljJ. We then define <1>(¢) = r(q,(¢)). In order to ensure that gradient descent applies in this setting, we need only ensure that rand q, are differentiable functions. Clearly, there are many instantiations of this idea for which this assumption holds. We provide two examples. Consider a continuous-state process with nonlinear dynamics, where q, is a mixture of conditional linear Gaussians. We can define 3 to be the set of multivariate Gaussians. The operator r takes a distribution (a mixture of gaussians) 'ljJ and computes its mean and covariance matrix. This can be easily computed from 'ljJ's parameters using simple differentiable algebraic operations. A very different example is the algorithm of [3] for approximate density propagation in dynamic Bayesian networks (DBNs). A DBN is a structured representation of a stochastic process, that exploits conditional independence properties of the distribution to allow compact representation. In a DBN, the state space is defined as a set of possible assignments x to a set of random variables Xl , ' .. ,Xn . The transition model P(x' I x) is described using a Bayesian network fragment over the nodes {Xl, ' " ,Xn , X{, .. . ,X~}. A node X i represents xft) and X: represents xft+1). The nodes X i in the network are forced to be roots (i.e., have no parents), and are not associated with conditional probability distributions. Each node X: is associated with a conditional probability distribution (CPO), which specifies P(X: I Parents(XD). The transition probability P(X' I X) is defined as 1026 A. Y. Ng, R. Parr and D. Koller 11 P(X: I Parents(Xf)). OBNs support a compact representation of complex transition models in MOPs [2]. We can extend the OBN to encode the behavior of an MOP with a stochastic policy 7l' by introducing a new random variable A representing the action taken at the current time. The parents of A will be those variables in the state on which the action is allowed to depend. The CPO of A (which may be compactly represented with function approximation) is the distribution over actions defined by 7l' for the different contexts. In discrete OBNs, the number of states grows exponentially with the number of state variables, making an explicit representation of a joint distribution impractical. The algorithm of [3] defines:::: to be a set of distributions defined compactly as a set of marginals over smaller clusters of variables. In the simplest example, :::: is the set of distributions where XI, ... ,X n are independent. The parameters ~ defining a distribution in :::: are the parameters of n multinomials. The projection operator r simply marginalizes distributions onto the individual variables, and is differentiable. One useful corollary of [3]'s analysis is that the decay rate of a structured ~ over:::: can often be much higher than the decay rate of ~, so that multiple applications of ~ can converge very rapidly to a stationary distribution; this property is very useful when approximating ¢(oo) to optimize relative to Vavg . 4.2 Stochastic density propagation In many settings, the assumption that we have direct access to ~ is too strong. A weaker assumption is that we have access to a generative model a black box from which we can generate samples with the appropriate distribution; i.e., for any s, a, we can generate samples s' from P(s' I s, a). In this case, we use a different approximation scheme, based on [5]. The operator ~ is a stochastic operator. It takes the distribution ¢, and generates some number of random state samples Si from it. Then, for each Si and each action a, we generate a sample s~ from the transition distribution P(· I Si, a). This sample (Si' ai, sD is then assigned a weight Wi = 7l'8(ai I Si), to compensate for the fact that not all actions would have been selected by 7l'e with equal probability. The resulting set of N samples s~ weighted by the WiS is given as input to a statistical density estimator, which uses it to estimate a new density ¢'. We assume that the density estimation procedure is a differentiable function of the weights, often a reasonable assumption. Clearly, this <1> can be used to compute ¢(t) for any t, and thereby approximate 7l'e'S value. However, the gradient computation for ~ is far from trivial. In particular, to compute the derivative 8<1> /8¢, we must consider <1>'s behavior for some perturbed ¢It) other than the one (say, ¢~t) to which it was applied originally. In this case, an entirely different set of samples would probably have been generated, possibly leading to a very different density. It is hard to see how one could differentiate the result of this perturbation. We propose an alternative solution based on importance sampling. Rather than change the samples, we modify their weights to reflect the change in the probability that they would be generated. Specifically, when fitting ¢it+1) , we now define a sample (Si' ai, sD's weight to be ~ (t) . (J.(t) 0) _ ¢1 (Si)7l'e (ai lSi) W t '1'1' ~(t) ¢o (Si) (3) We can now compute <1>'s derivatives at (0o, ¢~t)) with respect to any of its parameters, as required in (2). Let ( be the vector of parameters (0, e). Using the chain rule, we have 8<1> [O](¢) 8<1> [O](¢) 8w 8( = 8w . 8[' The first term is the derivative of the estimated density relative to the sample weights (an £ x N matrix). The second is the derivative of the weights relative to the parameter vector (an N x (m + £) Jacobian), which can easily be computed from (3). Policy Search via Density Estimation ~ 818 ~ (a) 0.042 o. ~ 038 0.36 , ..... 0.34 (J) , 0 () 0.32 0.3 I , , , , , , , , 1027 O~ ~ O~r o 2.0'----:.';O:------:-:'~:---,:7:~:----:200=· --:2~~--::300::---~3~::--::400:----:-!..SO #Function evaluations (b) Figure 1: Driving task: (a) DBN model; (b) policy-search/optimization results (with 1 s.e.) 5 Experimental results We tested our approach in two very different domains. The first is an average-reward DBN-MDP problem (shown in Figure l(a)), where the task is to find a policy for changing lanes when driving on a moderately busy two-lane highway with a slow lane and a fast lane. The model is based on the BAT DBN of [4], the result of a separate effort to build a good model of driver behavior. For simplicity, we assume that the car's speed is controlled automatically, so we are concerned only with choosing the LateraL Action - change Lane or drive straight. The observables are shown in the figure: LCLr and RClr are the clearance to the next car in each lane (close, medium or far). The agent pays a cost of 1 for each step it is "blocked" by (meaning driving close to) the car to its front; it pays a penalty of 0.2 per step for staying in the fast lane. Policies are specified by action probabilities for the 18 possible observation combinations. Since this is a reasonably small number of parameters, we used the simplex search algorithm described earlier to optimize V[O]. The process mixed quite quickly, so ¢(20) was a fairly good approximation to ¢(= ). Bused a fully factored representation of the joint distribution except for a single cluster over the three observables. Evaluations are averages of 300 Monte Carlo trials of 400 steps each. Figure 1 (b) shows the estimated and actual average rewards, as the policy parameters are evolved over time. The algorithm improved quickly, converging to a very natural policy with the car generally staying in the slow lane, and switching to the fast lane only when necessary to overtake. In our second experiment, we used the bicycle simulator of [7]. There are 9 actions corresponding to leaning left/center/right and applying negative/zero/positive torque to the handlebar; the six-dimensional state used in [7] includes variables for the bicycle'S tilt angle and orientation, and the handlebar's angle. If the bicycle tilt exceeds 7r /15, it falls over and enters an absorbing state. We used policy search over the following space: we selected twelve (simple, manually chosen but not fine-tuned) features of each state; actions were chosen with a softmax the probability of taking action ai is exp(x,wi)/ Ej exp(x,wj). As the problem only comes with a generative model of the complicated, nonlinear, noisy bicycle dynamics, we used the stochastic density propagation version of our algorithm, with (stochastic) gradient ascent. Each distribution in B was a mixture of a singleton point consisting of the absorbing-state, and of a 6-D multivariate Gaussian. 1028 A. Y. Ng, R. Pa" and D. Koller The first task in this domain was to balance reliably on the bicycle. Using a horizon of T = 200, discount 'Y = 0.995, and 600 Si samples per density propagation step, this was quickly achieved. Next, trying to learn to ride to a goal2 10m in radius and 1000m away, it also succeeded in finding policies that do so reliably. Formal evaluation is difficult, but this is a sufficiently hard problem that even finding a solution can be considered a success. There was also some slight parameter sensitivity (and the best results were obtained only with ~(O) picked/fit with some care, using in part data from earlier and less successful trials, to be "representative" of a fairly good rider's state distribution,) but using this algorithm, we were able to obtain solutions with median riding distances under 1.1 km to the goal. This is significantly better than the results of [7] (obtained in the learning rather than planning setting, and using a value-function approximation solution), which reported much larger riding distances to the goal of about 7km, and a single "best-ever" trial of about 1.7km. 6 Conclusions We have presented two new variants of algorithms for performing direct policy search in the deterministic and stochastic density propagation settings. Our empirical results have also shown these methods working well on two large problems. Acknowledgements. We warmly thank Kevin Murphy for use of and help with his Bayes Net Toolbox, and Jette Randl~v and Preben Alstr~m for use of their bicycle simulator. A. Ng is supported by a Berkeley Fellowship. The work of D. Koller and R. Parr is supported by the ARO-MURI program "Integrated Approach to Intelligent Systems", DARPA contract DACA 76-93-C-0025 under subcontract to lET, Inc., ONR contract N6600 1-97 -C8554 under DARPA's HPKB program, the Sloan Foundation, and the Powell Foundation. References [1] L. Baird and A.W. Moore. Gradient descent for general Reinforcement Learning. In NIPS II, 1999. [2] C. Boutilier, T. Dean, and S. Hanks. Decision theoretic planning: Structural assumptions and computational leverage. 1. Artijiciallntelligence Research, 1999. [3] X. Boyen and D. Koller. Tractable inference for complex stochastic processes. In Proc. VAl, pages 33-42, 1998. [4] J. Forbes, T. Huang, K. Kanazawa, and S.J. Russell. The BATmobile: Towards a Bayesian automated taxi. In Proc. IlCAI, 1995. [5] D. Koller and R. Fratkina. Using learning for approximation in stochastic processes. In Proc. ICML, pages 287-295, 1998. [6] N. Meuleau, L. Peshkin, K-E. Kim, and L.P. Kaelbling. Learning finite-state controllers for partially observable environments. In Proc. VAIlS, 1999. [7] 1. Randl0v and P. Alstr0m. Learning to drive a bicycle using reinforcement learning and shaping. In Proc. ICML, 1998. [8] J.K. Williams and S. Singh. Experiments with an algorithm which learns stochastic memoryless policies for POMDPs. In NIPS 11, 1999. [9] R.J. Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 8:229-256, 1992. 2For these experiments, we found learning could be accomplished faster with the simulator'S integration delta-time constant tripled for training. This and "shaping" reinforcements (chosen to reward progress made towards the goal) were both used, and training was with the bike "infinitely distant" from the goal. For this and the balancing experiments, sampling from the fallen/absorbingstate portion of the distributions J>(t) is obviously inefficient use of samples, so all samples were drawn from the non-absorbing state portion (i.e. the Gaussian, also with its tails corresponding to tilt angles greater than 7r /15 truncated), and weighted accordingly relative to the absorbing-state portion.
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Low Power Wireless Communication via Reinforcement Learning Timothy X Brown Electrical and Computer Engineering University of Colorado Boulder, CO 80309-0530 tirnxb@colorado.edu Abstract This paper examines the application of reinforcement learning to a wireless communication problem. The problem requires that channel utility be maximized while simultaneously minimizing battery usage. We present a solution to this multi-criteria problem that is able to significantly reduce power consumption. The solution uses a variable discount factor to capture the effects of battery usage. 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 [1,2, 8, 10]. These have demonstrated reinforcement learning can find good policies that significantly increase the application reward within the dynamics of the telecommunication problems. However, a key issue is how to treat the commonly occurring multiple reward and constraint criteria in a consistent way. This paper will focus on power management for wireless packet communication channels. These channels are unlike wireline channels in that channel quality is poor and varies over time, and often one side of the wireless link is a battery operated device such as a laptop computer. In this environment, power management decides when to transmit and receive so as to simultaneously maximize channel utility and battery life. A number of power management strategies have been developed for different aspects of battery operated computer systems such as the hard disk and CPU [4, 5]. Managing the channel is different in that some control actions such as shutting off the wireless transmitter make the state of the channel and the other side of the communication unobservable. In this paper, we consider the problem of finding a power management policy that simultaneously maximizes the radio communication's earned revenue while minimizing battery usage. The problem is recast as a stochastic shortest path problem which in turn is mapped to a discounted infinite horizon with a variable discount factor. Results show significant reductions in power usage. 894 T. X Brown Figure 1: The five components of the radio communication system. 2 Problem Description The problem is comprised of five components as shown in Figure 1: mobile application, mobile radio, wireless channel, base station radio. and base station application. The applications on each end generate packets that are sent via a radio across the channel to the radio and then application on the other side. The application also defines the utility of a given end-to-end performance. The radios implement a simple acknowledgment/retransmit protocol for reliable transmission. The base station is fixed and has a reliable power supply and therefore is not power constrained. The mobile power is limited by a battery and it can choose to turn its radio off for periods of time to reduce power usage. Note that even with the radio off, the mobile system continues to draw power for other uses. The channel adds errors to the packets. The rate of errors depends on many factors such as location of mobile and base station, intervening distance. and levels of interference. The problem requires models for each of these components. To be concrete. the specific models used in this paper are described in the following sections. It should be emphasized that in order to focus on the machine learning issues, simple models have been chosen. More sophisticated models can readily be included. 2.1 The Channel The channel carries fixed-size packets in synchronous time slots. All packet rates are normalized by the channel rate so that the channel carries one packet per unit time in each direction. The forward and reverse channels are orthogonal and do not interfere. Wireless data channels typically have low error rates. Occasionally. due to interference or signal fading, the channel introduces many errors. This variation is possible even when the mobile and base station are stationary. The channel is modeled by a two state Gilbert-Elliot model [3]. In this model, the channel is in either a "good" or a "bad" state with a packet error probabilities Pg and Pb where Pg < Pb· The channel is symmetric with the same loss rate in both directions. The channel stays in each state with a geometrically distributed holding time with mean holding times hg and hb time slots. 2.2 Mobile and Base Station Application The traffic generated by the source is a bursty ON/OFF model that alternates between generating no packets and generating packets at rate TON. The holding times are geometrically distributed with mean holding times hON and hOFF. The traffic in each direction is independent and identically distributed. 2.3 The Radios The radios can transmit data from the application and send it on the channel and simultaneously receive data from the other radio and pass it on to its application. The radios implement a simple packet protocol to ensure reliability. Packets from the sources are queued in the radio and sent one by one. Packets consist of a header and data. The header carries acknowledgements (ACK's) with the most recent packet received without error. The header contains a checksum so that errors in the payload can be detected. Errored packets Low Power Wireless Communication via Reinforcement Learning 895 Parameter Name Symbol Value Channel Error Rate, Good pg 0.01 Channel Error Rate, Bad Pb 0.20 Channel Holding Time, Good hg 100 Channel Holding Time, Bad hb 10 Source On Rate TON 1.0 Source Holding Time, On hON 1 Source Holding Time, Off hOFF 10 Power, Radio Off POFF 7W Power, Radio On PON 8.5W Power, Radio Transmitting PTX lOW Real Time Max Delay dmax 3 Web Browsing Time Scale do 3 Table 1: Application parameters. cause the receiving radio to send a packet with a negative acknowledgment (NACK) to the other radio instructing it to retransmit the packet sequence starting from the errored packet. The NACK is sent immediately even if no data is waiting and the radio must send an empty packet. Only unerrored packets are sent on to the application. The header is assumed to always be received without errorl. Since the mobile is constrained by power, the mobile is considered the master and the base station the slave. The base station is always on and ready to transmit or receive. The mobile can turn its radio off to conserve power. Every ON-OFF and OFF-ON transition generates a packet with a message in the header indicating the change of state to the base station. These message packets carry no data. The mobile expends power at three levels-PoFF, Po N , and Ptx--corresponding to the radio off, receiver on but no packet transmitted, and receiver on packet transmitted. 2.4 Reward Criteria Reward is earned for packets passed in each direction. The amount depends on the application. In this paper we consider three types of applications, an e-mail application, a real-time application, and a web browsing application. In the e-mail application, a unit reward is given for every packet received by the application. In the real time application a unit reward is given for every packet received by the application with delay less than dmax · The reward is zero otherwise. In the web browsing application, time is important but not critical. The value of a packet with delay d is (1 - l/do)d, where do is the desired time scale of the arrivals. The specific parameters used in this experiment are given in Table 1. These were gathered as typical values from [7, 9]. It should be emphasized that this model is the simplest model that captures the essential characteristics of the problem. More realistic channels, protocols, applications, and rewards can readily be incorporated but for this paper are left out for clarity. 1 A packet error rate of 20% implies a bit error rate of less than 1 %. Error correcting codes in the header can easily reduce this error rate to a low value. The main intent is to simplify the protocol for this paper so that time-outs and other mechanisms do not need to be considered. 896 T. X Brown Component States Channel {good,ba~} Application {ON,OFF} Mobile { ON,OFF} Mobile {List of waiting and unacknowledged packets and their current delay} Base Station {List of waiting and unacknowledged packets and their current delay} Table 2: Components to System State. 3 Markov Decision Processes At any given time slot, t, the system is in a particular configuration, x, defined by the state of each of the components in Table 2. The system state is s = (x, t) where we include the time in order to facilitate accounting for the battery. The mobile can choose to toggle its radio between the ON and OFF state and rewards are generated by successfully received packets. The task of the learner is to determine a radio ON/OFF policy that maximizes the total reward for packets received before batteries run out. The battery life is not a fixed time. First, it depends on usage. Second, for a given drain, the capacity depends on how long the battery was charged, how long it has sat since being charged, the age of the battery, etc. In short, the battery runs out at a random time. The system can be modeled as a stochastic shortest path problem whereby there exists a terminal state, So, that corresponds to the battery empty in which no more reward is possible and the system remains permanently at no cost. 3.1 Multi-criteria Objective Formally, the goal is to learn a policy for each possible system state so as to maximize J'(8)=E{t.C(t) 8,,,}, where E{ 'Is, 'Jr} is the expectation over possible trajectories starting from state s using policy 'Jr, c(t) is the reward for packets received at time t, and T is the last time step before the batteries run out. Typically, T is very large and this inhibits fast learning. So, in order to promote faster learning we convert this problem to a discounted problem that removes the variance caused by the random stopping times. At time t, given action a(t), while in state s(t) the terminal state is reached with probability Ps(t) (a(t)). Setting the value of the terminal state to 0, we can convert our new criterion to maximize: r (8) = E { t. c(t) g (1 - p>(T)(a(T))) S,,,}, where the product is the probability of reaching time t. In words, future rewards are discounted by 1 Ps (a), and the discounting is larger for actions that drain the batteries faster. Thus a more power efficient strategy will have a discount factor closer to one which correctly extends the effective horizon over which reward is captured. 3.2 Q-Iearning RL methods solve MDP problems by learning good approximations to the optimal value function, J*, given by the solution to the Bellman optimality equation which takes the Low Power Wireless Communication via Reinforcement Learning following form: J*(s) max [Esf{c(s,a,s') + (l-ps(a))J*(s')}] aEA(s) 897 (1) where A(s) is the set of actions available in the current state s, c(s, a, s') is the effective immediate payoff, and Esf {.} is the expectation over possible next states s'. We learn an appr<;>ximation to J* using Watkin's Q-learning algorithm. Bellman's equation can be rewritten in Q-factor as J*(s) max Q*(s,a) aEA(s) (2) In every time step the following decision is made. The Q-value of turning on in the next state is compared to the Q-value of turning off in the next state. If turning on has higher value the mobile turns on. Else, the mobile turns off. Whatever our decision, we update our value function as follows: on a transition from state s to s' on action a, Q(s, a) (1 - 1')Q(s, a) + l' (C(S, a, s') + (1- ps(a)) max Q(s', b)) (3) bEA(Sf) where l' is the learning rate. In order for Q-Iearning to perform well, all potentially important state-action pairs (s, a) must be explored. At each state, with probability 0.1 we apply a random action instead of the action recommended by the Q-value. However, we still use (3) to update Q-values using the action b recommended by the Q-values. 3.3 Structural Limits to the State Space For theoretical reasons it is desirable to use a table lookup representation. In practice, since the mobile radio decides using information available to it, this is impossible for the following reasons. The state of the channel is never known directly. The receiver only observes errored packets. It is possible to infer the state, but, only when packets are actually received and channel state changes introduce inference errors. Traditional packet applications rarely communicate state information to the transport layer. This state information could also be inferred. But, given the quickly changing application dynamics, the application state is often ignored. For the particular parameters in Table 1, (i.e. rON = 1.0) the application is on if and only if it generates a packet so its state is completely specified by the packet arrivals and does not need to be inferred. The most serious deficiency to a complete state space representation is that when the mobile radio turns OFF, it has no knowledge of state changes in the base station. Even when it is ON, the protocol does not have provisions for transferring directly the state information. Again, this implies that state information must be inferred. One approach to these structural limits is to use a POMDP approach [6] which we leave to future work. In this paper, we simply learn deterministic policies on features that estimate the state. 3.4 Simplifying Assumptions Beyond the structural problems of the previous section we must treat the usual problem that the state space is huge. For instance, assuming even moderate maximum queue sizes and maximum wait times yields 1020 states. If one considers e-mail like applications where 898 TX Brown Component Feature Mobile Radio is radio ON or OFF Mobile Radio number of packets waiting at the mobile Mobile Radio wait time of first packet waiting at the mobile Channel number of errors received in last 4 time slots Base Radio number of time slots since mobile was last ON Table 3: Decision Features Measured by Mobile Radio wait times of minutes (1000's of time slot wait times) with many packets waiting possible, the state space exceeds 10100 states. Thus we seek a representation to reduce the size and complexity of the state space. This reduction is taken in two parts. The first is a feature representation that is possible given the structural limits of the previous section, the second is a function approximation based on these feature vectors. The feature vectors are listed in Table 3. These are chosen since they are measurable at the mobile radio. For function approximation, we use state aggregation since it provably converges. 4 Simulation Results This section describes simulation-based experiments on the mobile radio control problem. For this initial study, we simplified the problem by setting Pg = Pb = 0 (i.e. no channel errors). State aggregation was used with 4800 aggregate states. The battery termination probability, ps(a) was simply PIlOOO where P is the power appropriate for the state and action chosen from Table 1. This was chosen to have an expected battery life much longer than the time scale of the traffic and channel processes. Three policies were learned, one for each application reward criteria. The resulting policies are tested by simulating for 106 time slots. In each test run, an upper and lower bound on the energy usage is computed. The upper bound is the case of the mobile radio always on2 . The lower bound is a policy that ignores the reward criteria but still delivers all the packets. In this policy, the radio is off and packets are accumulated until the latter portion of the test run when they are sent in one large group. Policies are compared using the normalized power savings. This is a measure of how close the policy is to the lower bound with 0% and 100% being the upper and lower bound. The results are given in Table 4. The table also lists the average reward per packet received by the application. For the e-mail application, which has no constraints on the packets, the average reward is identically one. 5 Conclusion This paper showed that reinforcement learning was able to learn a policy that significantly reduced the power consumption of a mobile radio while maintaining a high application utility. It used a novel variable discount factor that captured the impact of different actions on battery life. This was able to gain 50% to 80% of the possible power savings. 2There exist policies that exceed this power, e.g. if they toggle oNand oFFoften and generate many notification packets. But, the always on policy is the baseline that we are trying to improve upon. Low Power Wireless Communication via Reinforcement Learning 899 Application Normalized Average Power Savings Reward E-mail 81% 1 Real Time 49% 1.00 Web Browsing 48% 0.46 Table 4: Simulation Results. In the application the paper used a simple model of the radio, channel, battery, etc. It also used simple state aggregation and ignored the partially observable aspects of the problem. Future work will address more accurate models, function approximation, and POMDP approaches. Acknowledgment This work was supported by CAREER Award: NCR-9624791 and NSF Grant NCR9725778. References [1] Boyan, J.A., Littman, M.L., "Packet routing in dynamically changing networks: a reinforcement learning approach," in Cowan, J.D., et aI., ed. Advances in NIPS 6, Morgan Kauffman, SF, 1994. pp. 671-678. [2] Brown, TX, Tong, H., Singh, S., "Optimizing admission control while ensuring quality of service in multimedia networks via reinforcement learning," in Advances in Neural Information Processing Systems 12, ed. M. Kearns, et aI., MIT Press, 1999, pp. 982-988. [3] Goldsmith, AJ., Varaiya, P.P., "Capacity, mutual information, and coding for finite state Markov channels," IEEE T. on Info. Thy., v. 42, pp. 868-886, May 1996. [4] Govil, K., Chan, E., Wasserman, H., "Comparing algorithms for dynamic speedsetting of a low-power cpu," Proceedings of the First ACM Int. Can! on Mobile Computing and Networking (MOBICOM), 1995. [5] Helmbold, D., Long, D.D.E., Sherrod, B., "A dynamic disk spin-down technique for mobile computing. Proceedings of the Second ACM Int. Can! on Mobile Computing and Networking (MOBICOM), 1996. [6] Jaakola, T., Singh, S., Jordan, M.I., "Reinforcement Learning Algorithm for Partially Observable Markov Decision Problems," in Advances in Neural Information Processing Systems 7, ed. G. Tesauro, et aI., MIT Press, 1995, pp. 345-352. [7] Kravits, R., Krishnan, P., "Application-Driven Power Management for Mobile Communication," Wireless Networks, 1999. [8] Marbach, P., Mihatsch, 0., Schulte, M., Tsitsiklis, J.N., "Reinforcement learning for call admission control and routing in integrated service networks," in Jordan, M., et aI., ed. Advances in NIPS 10, MIT Press, 1998. [9] Rappaport, T.S., Wireless Communications: Principles and Practice, Prentice-Hall Pub., Englewood Cliffs, NJ, 1996. [10] 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 aI., MIT Press, 1997. pp. 974-980.
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Learning to Parse Images Geoffrey E. Hinton and Zoubin Ghahramani Gatsby Computational Neuroscience Unit University College London London, United Kingdom WC1N 3AR {hinton,zoubin}@gatsby.ucl.ac.uk Vee Whye Tah Department of Computer Science University of Toronto Toronto, Ontario, Canada M5S 3G4 ywteh@cs.utoronto.ca Abstract We describe a class of probabilistic models that we call credibility networks. Using parse trees as internal representations of images, credibility networks are able to perform segmentation and recognition simultaneously, removing the need for ad hoc segmentation heuristics. Promising results in the problem of segmenting handwritten digits were obtained. 1 Introd uction The task of recognition has been the main focus of attention of statistical pattern recognition for the past 40 years. The paradigm problem is to classify an object from a vector of features extracted from the image. With the advent of backpropagation [1], the choice of features and the choice of weights to put on these features became part of a single, overall optimization and impressive performance was obtained for restricted but important tasks such as handwritten character identification [2]. A significant weakness of many current recognition systems is their reliance on a separate preprocessing stage that segments one object out of a scene and approximately normalizes it. Systems in which segmentation precedes recognition suffer from the fact that the segmenter does not know the shape of the object it is segmenting so it cannot use shape information to help it. Also, by segmenting an image, we remove the object to be recognized from the context in which it arises. Although this helps in removing the clutter present in the rest of the image, it might also reduce the ability to recognize an object correctly because the context in which an object arises gives a great deal of information about the nature of the object. Finally, each object can be described in terms of its parts, which can also be viewed as objects in their own right. This raises the question of how fine-grained the segmentations should be. In the words of David Marr: "Is a nose an object? Is a head one? ... What about a man on a horseback?" [3]. 464 G. E. Hinton, Z. Ghahramani and Y. W. Teh The successes of structural linguistics inspired an alternative approach to pattern recognition in which the paradigm problem was to parse an image using a hierarchical grammar of scenes and objects. Within linguistics, the structural approach was seen as an advance over earlier statistical approaches and for many years linguists eschewed probabilities, even though it had been known since the 1970's that a version of the EM algorithm could be used to fit stochastic context free grammars. Structural pattern recognition inherited the linguists aversion to probabilities and as a result it never worked very well for real data. With the advent of graphical models it has become clear that structure and probabilities can coexist. Moreover, the "explaining away" phenomenon that is central to inference in directed acyclic graphical models is exactly what is needed for performing inferences about possible segmentations of an image. In this paper we describe an image interpretation system which combines segmentation and recognition into the same inference process. The central idea is the use of parse trees of images. Graphical models called credibility networks which describe the joint distribution over the latent variables and over the possible parse trees are used. In section 2 we describe some current statistical models of image interpretation. In section 3 we develop credibility networks and in section 4 we derive useful learning and inference rules for binary credibility networks. In section 5 we demonstrate that binary credibility networks are useful in solving the problem of classifying and segmenting binary handwritten digits. Finally in section 6 we end with a discussion and directions for future research. 2 Related work Neal [4] introduced generative models composed of multiple layers of stochastic logistic units connected in a directed acyclic graph. In general, as each unit has multiple parents, it is intractable to compute the posterior distribution over hidden variables when certain variables are observed. However, Neal showed that Gibbs sampling can be used effectively for inference [4]. Efficient methods of approximating the posterior distribution were introduced later [5, 6, 7] and these approaches were shown to yield good density models for binary images of handwritten digits [8]. The problem with these models which make them inappropriate for modeling images is that they fail to respect the 'single-parent' constraint: in the correct interpretation of an image of opaque objects each object-part belongs to at most one object - images need parse trees, not parse DAGs. Multiscale models [9] are interesting generative models for images that use a fixed tree structure. Nodes high up in the tree control large blocks of the image while bottom level leaves correspond to individual pixels. Because a tree structure is used, it is easy to compute the exact posterior distribution over the latent (non-terminal) nodes given an image. As a result, the approach has worked much better than Markov random fields which generally involve an intractable partition function. A disadvantage is that there are serious block boundary artifacts, though overlapping trees can be used to smooth the transition from one block to another [10]. A more serious disadvantage is that the tree cannot possibly correspond to a parse tree because it is the same for every image. Zemel, Mozer and Hinton [11] proposed a neural network model in which the activities of neurons are used to represent the instantiation parameters of objects or their parts, Le. the viewpoint-dependent coordinate transformation between an object's intrinsic coordinate system and the image coordinate system. The weights on connections are then used to represent the viewpoint-invariant relationship between the instantiation parameters of a whole, rigid object and the instantiation parameLearning to Parse Images 465 ters of its parts. This model captures viewpoint invariance nicely and corresponds to the way viewpoint effects are handled in computer graphics, but there was no good inference procedure for hierarchical models and no systematic way of sharing modules that recognize parts of objects among multiple competing object models. Simard et al [12] noted that small changes in object instantiation parameters result in approximately linear changes in (real-valued) pixel intensities. These can be captured successfully by linear models. To model larger changes, many locally linear models can be pieced together. Hinton, Dayan and Revow [13] proposed a mixture of factor analyzers for this. Tipping and Bishop have recently shown how to make this approach much more computationally efficient [14]. To make the approach really efficient, however, it is necessary to have multiple levels of factor analyzers and to allow an analyzer at one level to be shared by several competing analyzers at the next level up. Deciding which subset of the analyzers at one level should be controlled by one analyzer at the level above is equivalent to image segmentation or the construction of part of a parse tree and the literature on linear models contains no proposals on how to achieve this. 3 A new approach to image interpretation We developed a class of graphical models called credibility networks in which the possible interpretations of an image are parse trees, with nodes representing objectparts and containing latent variables. Given a DAG, the possible parse trees of an image are constrained to be individual or collections of trees where each unit satisfies the single-parent constraint, with the leaves being the pixels of an image. Credibility networks describe a joint distribution over the latent variables and possible tree structures. The EM algorithm [15] can be used to fit credibility networks to data. Let i E I be a node in the graph. There are three random variables associated with i. The first is a multinomial variate ,xi = {.xij hEpa(i) which describes the parent of i from among the potential parents pa(i) : ,x . . _ {I if parent of i is j, 1J 0 if parent of i is not j. (1) The second is a binary variate Si which determines whether the object i l is present (Si = 1) or not (Si = 0). The third is the latent variables Xi that describe the pose and deformation of the object. Let A = {,xi : i E I}, S = {Si : i E I} and X = {Xi: i E I}. Each connection j -+ i has three parameters also. The first, Cij is an unnormalized prior probability that j is i's parent given that object j is present. The actual prior probability is C" S ' 7rij = 1J J (2) 2:kEPa(i) CikSk We assume there is always a unit 1 E pa(i) such that SI = 1. This acts as a default parent when no other potential parent is present and makes sure the denominator in (2) is never O. The second parameter, Pij, is the conditional probability that object i is present given that j is i's parent (,xij = 1). The third parameter tij characterizes the distribution of Xi given ,xij = 1 and Xj' Let 0 = {Cij,Pij, tij : i E I,j E pa(i)}. Using Bayes' rule the joint distribution over A, S and X given 0 is peA, S, XIO) = peA, SIO)p(XIA, S, 0). Note that A and S together define a parse tree for the image. Given the parse tree the distribution over latent variables p(XIA, S, 0) can be ITechnically this should be the object represented by node i. 466 G. E. Hinton, Z. Ghahramani and Y. W. Teh efficiently inferred from the image. The actual form of p(XIA, S, 0) is unimportant. The joint distribution over A and S is P(A,SIO) = II II (1I'ijP:j(1- Pij)I-8i )Aij (3) iEi jEpa(i) 4 Binary credibility networks The simulation results in section 5 are based on a simplified version of credibility networks in which the latent variables X are ignored. Notice that we can sum out A from the joint distribution (3), so that P(SIO) = II L 1I'ijP:j (1 - Pij)I-8i (4) iEi jEpa(i) Using Bayes' rule and dividing (3) by (4), we have P(AIS, 0) = II II ( CijSjP:j (1: Pij)I-8i 1-8') Aij (5) iEi jEpa(i) I:kEPa(i) cik s kPik(l - Pik) • Let rij = CijP:j (1 - Pij )1-8 .. We can view rij as the unnormalized posterior probability that j is i's parent given that object j is present. The actual posterior is the fraction in (5) : (6) Given some observations 0 c S, let 11 = S \ 0 be the hidden variables. We approximate the posterior distribution for 11 using a factored distribution Q(ll) = II O':i (1 - O'i)1-8i (7) iEi The variational free energy, F(Q,O) = EQ[-log P(SIO) + log Q(S)] is F(Q,O) = L(EQ[IOg L cijsj-Iog L CijSjP:j(1-Pij )1-8i]) + iEi jEpa(i) jEpa(i) L (O'i log O'i + (1 - O'i) log (1 - O'i)) (8) iEi The negative of the free energy -F is a lower bound on the log likelihood of generating the observations O. The variational EM algorithm improves this bound by iteratively improving -F with respect to Q (E-step) and to 0 (M-step). Let ch(i) be the possible children of i. The inference rules can be derived from (8) : EQ [log L CijSjPij -log L cijsj(l - Pij)] jEpa(i) jEpa(i) + L EQ [log L rljSj -log L CljSj J:::: lEch( i) jEpa(l) jEpa(I)' O'i = sigmoid (9) Let D be the training set and Q d be the mean field approximation to the posterior distribution over 11 given the training data (observation) dE D. Then the learning Learning to Parse Images 467 Figure 1: Sample images from the test set. The classes of the two digits in each image in a row are given to the left. rules are (10) (11) For an efficient implementation of credibility networks using mean field approximations, we still need to evaluate terms of the form E[logx] and ~[l/x] where x is a weighted sum of binary random variates. In our implementation we used the simplest approximations: E[logx] ~ 10gE[x] and E[l/x] ~ l/E[x]. Although biased the implementation works well enough in general. 5 Segmenting handwritten digits Hinton and Revow [16] used a mixture of factor analyzers model to segment and estimate the pose of digit strings. When the digits do not overlap, the model was able to identify the digits present and segment the image easily. The hard cases are those in which two or more digits overlap significantly. To assess the ability of credibility networks at segmenting handwritten digits, we used superpositions of digits at exactly the same location. This problem is much harder than segmenting digit strings in which digits partially overlap. The data used is a set of 4400 images of single digits from the classes 2, 3, 4 and 5 derived from the CEDAR CDROM 1 database [17]. Each image has size 16x16. The size of the credibility network is 256-64-4. The 64 middle layer units are meant to encode low level features, while each of the 4 top level units are meant to encode a digit class. We used 700 images of single digits from each class · to train the network. So it was not trained to segment images. During training we clamped at 1 the activation of the top layer unit corresponding to the class of the digit in the current image while fixing the rest at O. After training, the network was first tested on the 1600 images of single digits not in the training set. The predicted class of each image was taken to be the 468 G. E. Hinton, Z. Ghahramani and Y. W. Teh Figure 2: Segmentations of pairs of digits. (To make comparisons easier we show the overlapping image in both columns of a)-I).) class corresponding to the top layer unit with the highest activation. The error rate was 5.5%. We then showed the network 120 images of two overlapping digits from distinct classes. There were 20 images per combination of two classes. Some examples are given in Figure 1. The predicted classes of the two digits are chosen to be the corresponding classes of the 2 top layer units with the highest activations. A human subject (namely the third author) was tested on the same test set. The network achieved an error rate of 21.7% while the author erred on 19.2% of the images. We can in fact produce a segmentation of each image into an image for each class present. Recall that given the values of S the posterior probability of unit j being pixel i's parent is Wij. Then the posterior probability of pixel i belonging to digit class k is L:j EQ[WijWjk). This gives a simple way to segment the image. Figure 2 shows a number of segmentations. Note that for each pixel, the sum of the probabilities of the pixel belonging to each digit class is 1. To make the picture clearer, a white pixel means a probability of :::; .1 of belonging to a class, while black means ~ .6 probability, and the intensity of a gray pixel describes the size of the probability if it is between .1 and .6. Figures 2a) to 2f) shows successful segmentations, while Figure 2g) to 21) shows unsuccessful segmentations. 6 Discussion Using parse trees as the internal representations of images, credibility networks avoid the usual problems associated with a bottom-up approach to image interpretation. Segmentation can be carried out in a statistically sound manner, removing the need for hand crafted ad hoc segmentation heuristics. The granularity problem for segmentation is also resolved since credibility networks use parse trees as internal representations of images. The parse trees describe the segmentations of the image at every level of granularity, from individual pixels to the whole image. We plan to develop and implement credibility networks in which each latent variable Xi is a multivariate Gaussian, so that a node can represent the position, orientation and scale of a 2 or 3D object, and the conditional probability models on the links can represent the relationship between a moderately deformable object and its parts. Learning to Parse Images 469 Acknowledgments We thank Chris Williams, Stuart Russell and Phil Dawid for helpful discussions and NSERC and ITRC for funding. References [1] D. E. Rumelhart, G. E. Hinton, and R. J. Williams. Learning internal representations by error propagation. In D. E. Rumelhart, J. L. McClelland, and the PDP Research Group, editors, Parallel Distributed Processing: Explorations in The Microstructure of Cognition. Volume 1 : Foundations. The MIT Press, 1986. [2] Y. Le Cun, B. Boser, J. S. Denker, S. SoH a, R. E. Howard, and L. D. Jackel. Back-propagation applied to handwritten zip code recognition. Neural Computation, 1(4):541-551, 1989. [3] D. Marr. Vision: A Computational Investigation into the Human Representation and Processing of Visual Information. W. H. Freeman and company, San Francisco, 1980. [4] R. M. Neal. Connectionist learning of belief networks. Artificial Intelligence, 56:71113, 1992. [5] P. Dayan, G. E. Hinton, R. M. Neal, and R. S. Zemel. Helmholtz machines. Neural Computation, 7:1022-1037, 1995. [6] G. E. Hinton, P. Dayan, B. J. Frey, and R. M. Neal. The wake-sleep algorithm for self-organizing neural networks. Science, 268:1158-1161, 1995. [7] L. K. Saul and M. I. Jordan. Attractor dynamics in feedforward neural networks. Submitted for publication. [8] B. J. Frey, G. E. Hinton, and P. Dayan. Does the wake-sleep algorithm produce good density estimators? In D. Touretzky, M. Mozer, and M. Hasselmo, editors, Advances in Neural Information Processing Systems, volume 8. The MIT Press, 1995. [9] M. R. Luettgen and A. S. Willsky. Likelihood calculation for a class of multiscale stochastic models, with application to texture discrimination. IEEE Transactions on Image Processing, 4(2):194-207, 1995. [10] W. W. Irving, P. W. Fieguth, and A. S. Willsky. An overlapping tree approach to multiscale stochastic modeling and estimation. IEEE Transactions on Image Processing, 1995. [11] R. S. Zemel, M. C. Mozer, and G. E. Hinton. TRAFFIC: Recognizing objects using hierarchical reference frame transformations. In Advances in Neural Information Processing Systems, volume 2. Morgan Kaufmann Publishers, San Mateo CA, 1990. [12] P. Simard, Y. Le Cun, and J. Denker. Efficient pattern recognition using a new transformation distance. In S. Hanson, J. Cowan, and L. Giles, editors, Advances in Neural Information Processing Systems, volume 5. Morgan Kaufmann Publishers, San Mateo CA, 1992. [13] G. E. Hinton, P. Dayan, and M. Revow. Modeling the manifolds of images of handwritten digits. IEEE Transactions on Neural Networks, 8:65-74, 1997. [14] M. E. Tipping and C. M. Bishop. Mixtures of probabilistic principal component analysis. Technical Report NCRG/97/003, Aston University, Department of Computer Science and Applied Mathematics, 1997. [15] A.P. Dempster, N.M. Laird, and D.B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society B, 39:1-38, 1977. [16] G. E. Hinton and M. Revow. Using mixtures of factor analyzers for segmentation and pose estimation, 1997. [17] J. J. Hull. A database for handwritten text recognition research. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(5):550-554, 1994.
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Robust Recognition of Noisy and Superimposed Patterns via Selective Attention Soo-Young Lee Brain Science Research Center Korea Advanced Institute of Science & Technology Yusong-gu, Taejon 305-701 Korea sylee@ee.kaist.ac.kr Abstract Michael C. Mozer Department of Computer Science University of Colorado at Boulder Boulder, CO 80309 USA mozer@cs.colorado.edu In many classification tasks, recognition accuracy is low because input patterns are corrupted by noise or are spatially or temporally overlapping. We propose an approach to overcoming these limitations based on a model of human selective attention. The model, an early selection filter guided by top-down attentional control, entertains each candidate output class in sequence and adjusts attentional gain coefficients in order to produce a strong response for that class. The chosen class is then the one that obtains the strongest response with the least modulation of attention. We present simulation results on classification of corrupted and superimposed handwritten digit patterns, showing a significant improvement in recognition rates. The algorithm has also been applied in the domain of speech recognition, with comparable results. 1 Introduction In many classification tasks, recognition accuracy is low because input patterns are corrupted by noise or are spatially or temporally overlapping. Approaches have been proposed to make classifiers more robust to such perturbations, e.g., by requiring classifiers to have low input-to-output mapping sensitivity [1]. We propose an approach that is based on human selective attention. People use selective attention to focus on critical features of a stimulus and to suppress irrelevant features. It seems natural to incorporate a selective-attention mechanism into pattern recognition systems for noisy real world applications. Psychologists have for many years studied the mechanisms of selective attention (e.g., [2]-[4]). However, controversy still exists among competing theories, and only a few models are sufficiently well defmed to apply to engineering pattern recognition problems. Fukushima [5] has incorporated selective attention and attention-switching algorithms into his Neocognitron model, and has demonstrated good recognition performance on superimposed digits. However, the Neocognitron model has many unknown parameters which must be determined heuristically, and its performance is sensitive to the parameter values. Also, its computational requirements are prohibitively expensive for many realtime applications. Rao [6] has also recently introduced a selective attention model based 32 s. -Y. Lee and M C. Mozer on Kalman filters and demonstrated classifications of superimposed patterns. However, his model is based on linear systems, and a nonlinear extension is not straightforward. There being no definitive approach to incorporating selective attention into pattern recognition, we propose a novel approach and show it can improve recognition accuracy. 2 Psychological Views of Selective Attention The modern study of selective attention began with Broadbent [7]. Broadbent presented two auditory channels to subjects, one to each ear, and asked subjects to shadow one channel. He observed that although subjects could not recall most of what took place in the unshadowed channel, they could often recall the last few seconds of input on that channel. Therefore, he suggested that the brain briefly stores incoming stimuli but the stimulus information fades and is neither admitted to the conscious mind nor is encoded in a way that would permit later recollection, unless attention is directed toward it. This view is known as an early filtering or early selection model. Treisman [8] proposed a modification to this view in which the filter merely attenuates the input rather than absolutely preventing further analysis. Although late-selection and hybrid views of attention have been proposed, it is clear that early selection plays a significant role in human information processing [3]. The question about where attention acts in the stream of processing is independent of another important issue: what factors drive attention to select one ear or one location instead of another. Attention may be directed based on low-level stimulus features, such as the amplitude of a sound or the color of a visual stimulus. This type of attentional control is often called bottom up. Attention may also be directed based on expectations and object knowledge, e.g., to a location where critical task-relevant information is expected. This type of attentional control is often called top down. 3 A Multilayer Perceptron Architecture for Selective Attention We borrow the notion of an early selection filter with top-down control and integrate it into a multilayer perceptron (MLP) classifier, as depicted in Figure 1. The dotted box is a standard MLP classifier, and an attention layer with one-to-one connectivity is added in front of the input layer. Although we have depicted an MLP with a single hidden layer, our approach is applicable to general MLP architectures. The kth element of the input vector, denoted xk, is gated to the kth input of the MLP by an attention gain or filtering coefficient ak. Previously, the first author has shown a benefit of treating the ak's like ordinary adaptive parameters during training [9]-[12]. In the present work, we fix the attention gains at 1 during training, causing the architecture to behave as an ordinary MLP. However, we allow the gains to be adjusted during classification of test patterns. Our basic conjecture is that recognition accuracy may be improved if attention can suppress noise along irrelevant dimensions and enhance a weak signal along relevant dimensions. "Relevant" and "irrelevant" are determined by topdown control of attention. Essentially, we use knowledge in the trained MLP to determine which input dimensions are critical for classifying a test pattern. To be concrete, consider an MLP trained to classify handwritten digits. When a test pattern is presented, we can adjust the attentional gains via gradient descent so as to make the input as good an example of the class "0" as possible. We do this for each of the different output classes, "0" through "9", and choose the class for which the strongest response is obtained with the smallest Robust Pattern Recognition via Selective Attention 33 attentional modulation (the exact quantitative rule is presented below). The conjecture is that if the net can achieve a strong response for a class by making a small attentional modulation, that class is more likely to be correct than whichever class would have been selected without applying selective attention . .......................................................................................................................................................... _ ................ , : ! ~ ~ 1 Xl~----~~~~~--~~~~~~~~~~~~~yl 1 ~ ~ ~ l ~--~-+~~~~~~~~~~~~~~ i ~ Y3 ! ~------~~>E~~~~~~~~~~~~~~~~~. i • • x a : ! ~ ; 'l: w(l) h vJ2) /M 1 i ................................................ __ ........................................................................................................... _ ... _ ... ..! Figure 1: MLP architecture for selective attention The process of adjusting the attentional gains to achieve a strong response from a particular class-<:all it the attention class-proceeds as follows. First, a target output vector tS = [1'1 1'2· ··t'M f is defmed. For bipolar binary output representations, t,S = 1 is for the attention class and -1 for the others. Second, the attention gain ak's are set to I. Third, the attention gain ak's are adapted to minimize error ES == ~ L (t,S - y,)2 with the given , input x = [XI X2··· XN f and pre-trained and frozen synaptic weights W. The update rule is based on a gradient-descent algorithm with error back-propagation. At the (n+ J)'th iterative epoch, the attention gain a k is updated as (Ia) (lb) where E denotes the attention output error, ojl) thej'th attribute of the back-propagated error at the first hidden-layer, and WJ~) the synaptic weight between the input xk and the j'th neuron at the first hidden layer. Finally, " is a step size. The attention gains are thresholded to lie in [0, 1]. The application of selective attention to a test example is summarized as follows: Step 1: Apply a test input pattern to the trained MLP and compute output values. Step 2: For each of the classes with top m activation values, (I) Initialize all attention gain ak' s to 1 and set the target vector tS. (2) Apply the test pattern and attention gains to network and compute output. (3) Apply the selective attention algorithm in Eqs.( 1) to adapt the attention gains. (4) Repeat steps (2) and (3) until the attention process converges. (5) Compute an attention measure M on the asymptotic network state. 34 s.-Y. Lee and M. C. Mozer Step 3: Select the class with a minimum attention measure M as the recognized class. The attention measure is defined as (2a) D j = :f(Xk -xk )2/2N 2 2 = :fxk(l-ak) /2N (2b) Eo = 4[/i - y;(i)]2/2M , (2c) where D/ is the square of Euclidean distance between two input patterns before and after the application of selective attention and Eo is the output error after the application of selective attention. Here, D/ and Eo are normalized with the number of input pixels and number of output classes, respectively. The superscript s for attention classes is omitted for simplicity. To make the measure M a dimensionless quantity, one may normalize the D/ and Eo with the input energy (~kX~ ) and the training output error, respectively. However, it does not affect the selection process in Step 3. One can think of the attended input i as the minimal deformation of the test input needed to trigger the attended class, and therefore the Euclidean distance between x and i is a good measure for the classification confidence. In fact, D/ is basically the same quantity minimized by Rao [6]. However, the MLP classifier in our model is capable of nonlinear mapping between the input and output patterns. A nearest-neighbor classifier, with the training data as examples, could also be used to find the minimum-distance class. Our model with the MLP classifier computes a similar function without the large memory and computational requirements. The proposed selective attention algorithm was tested on recognition of noisy numeral patterns. The numeral database consists of samples of the handwritten digits (0 through 9) collected from 48 people, for a total of 480 samples. Each digit is encoded as a 16x16 binary pixel array. Roughly 16% of the pixels are black and coded as 1; white pixels are coded as O. Four experiments were conducted with different training sets of 280 training patterns each. A one hidden-layer MLP was trained by back propagation. The numbers of input, hidden, and output neurons were 256, 30, and to, respectively. Three noisy test patterns were generated from each training pattern by randomly flipping each pixel value with a probability Pt, and the 840 test patterns were presented to the network for classification. In Figure 2, the false recognition rate is plotted as a function of the number of candidates considered for the attentional manipulation, m. (Note that the run time of the algorithm is proportional to m, but that increasing m does not imply a more lax classification criterion, or additional external knowledge playing into the classification.) Results are shown for three different pixel inversion probabilities, Pt =0.05, 0.1, and 0.15. Considering the average 16% of black pixels in the data, the noisy input patterns with Pt= 0.15 correspond to a SNR of approximately 0 dB. For each condition in the figure, the false recognition rates for the four different training sets are marked with an '0', and the means are connected by the solid curve. Robust Pattern Recognition via Selective Attention 35 A standard MLP classifier corresponds to m = I (i.e., only the most active output of the MLP is considered as a candidate response). The false recognition rate is clearly lower when the attentional manipulation is used to select a response from the MLP (m > I). It appears that performance does not improve further by considering more than the top three candidates. 4 Attention Switching for Superimposed Patterns Suppose that we superimpose the binary input patterns for two different handwritten digits using the logical OR operator (the pixels corresponding to the black ink have logical value 1). Can we use attention to recognize the two patterns in sequence? This is an extreme case of a situation that is common in visual pattern recognition-where two patterns are spatially overlapping. We explore the following algorithm. First, one pattern is recognized with the selective attention process used in Section 3. Second, attention is switched from the recognized pattern to the remaining pixels in the image. Switching is accomplished by removing attention from the pixels of the recognized pattern: the attentional gain of an input is clamped to 0 following switching if and only if its value after the first-stage selective attention process was 1 (Le., that input was attended during the recognition of the first pattern); all other gains are set to 1. Third, the recognition process with selective attention is performed again to recognize the second pattern. The proposed selective attention and attention switching algorithm was tested for recognition of 2 superimposed numeral data. Again, four experiments were conducted with I/) Q) a; 2 a: 1.5 c:: o -..::; ·c Ol o o Q) a: 3l 0.5 ~ o l\ r---.. 2 3 4 5 Number of Candidates (a) Pr=O.05, 20r-~----~----~----~----~-. I/) Q) Cii a: 15 c:: ,g ·c g> 10 o Q) a: 3l 5 ~ OL-~----~----~----~----~~ 2 3 4 5 Number of Candidates (b) Pr=O.IO, 40r-~----~----~----~----~-. I/) Q) a; a: 30 c:: o :t: c:: Ol 20 8 Q) a: 3l 10 ~ OL-~----~----~----~----~~ 2 3 4 5 Number of Candidates (c) Pr=O.15, Figure 2: False recognition rates for noisy patterns as a function of the number of top candidates. Each binary pixel of training patterns is randomly inverted with a probability Pr. 36 s.-Y. Lee and M C. Mozer different training sets. For each experiment, 40 patterns were selected from 280 training patterns, and 720 test patterns were generated by superimposing pairs of patterns from different output classes. The test patterns were still binary. ~~ • I .-. •• ... ~ ~ ~ ~ [kJ ~ [9] [9] Et][3] ~[2J0[J ~~g]m Figure 3: Examples of Selective Attention and Attention Switching Figure 3 shows six examples of the selective attention and attention switching algorithm in action, each consisting of four panels in a horizontal sequence. The six examples were fonned by superimposing instances of the following digit pairs: (6,3), (9,0), (6,4), (9,3), (2,6), and (5,2). The fIrst panel for each example shows the superimposed pattern. The second panel shows the attended input i for the fIrst round classifIcation; because this input has continuous values, we have thresholded the values at 0.5 to facilitate viewing in the fIgure. The third panel shows the masking pattern for attention switching, generated by thresholding the input pattern at 1.0. The fourth panel sho~s the residual input pattern for the second round classifIcation. The attended input x has analog values, but thresholded by 0.5 to be shown in the second rectangles. Figure 3 shows that attention switching is done effectively, and the remaining input patterns to the second classifIer are quite visible. We compared perfonnance for three different methods. First, we simply selected the two MLP outputs with highest activity; this method utilizes neither selective attention. Second, we perfonned attention switching but did not apply selective attention (i.e., m=I). Third, we perfonned both attention switching and selective attention (with m=3). Table I summarizes the recognition rates for the fITst and the second patterns read out of the MLP for the three methods. As hypothesized, attention switching increases the recognition rate for the second pattern, and selective attention increases the recognition rate for both the fITSt and the second pattern. Table I: Recognition Rates (%) of Two Superimposed Numeral Patterns No selective attention or switching Switching only Switching & selective attention First Pattern 91. 3 91. 3 95.9 Second Pattern 62.7 75.4 77.4 Robust Pattern Recognition via Selective Attention 37 5 Conclusion In this paper, we demonstrated a selective-attention algorithm for noisy and superimposed patterns that obtains improved recognition rates. We also proposed a simple attention switching algorithm that utilizes the selective-attention framework to further improve performance on superimposed patterns. The algorithms are simple and easily implemented in feedforward MLPs. Although our experiments are preliminary, they suggest that attention-based algorithms will be useful for extracting and recognizing multiple patterns in a complex background. We have conducted further simulation studies supporting this conjecture in the domain of speech recognition, which we will integrate into this presentation if it is accepted at NIPS. Acknowledgements S.Y. Lee acknowledges supports from the Korean Ministry of Science and Technology. We thank Dr. Y. Le Cun for providing the handwritten digit database. References [1] Jeong D.G., and Lee, S.Y. (1996). Merging backpropagation and Hebbian learning rules for robust classification, Neural Networks, 9:1213-1222. [2] Cowan, N. (1997). Attention and Memory: An Integrated Framework, Oxford Univ. Press. [3] Pashler, H.E. (1998). The Psychology of Attention, MIT Press. [4] Parasuraman, R. (ed.) (1998). The Attentive Brain, MIT Press. [5] Fukushima, K. (1987). Neural network model for selective attention in visual pattern recognition and associative recall, Applied Optics, 26:4985-4992. [6] Rao, R.P.N. (1998). Correlates of attention in a model of dynamic visual recognition. In Neural Information Processing Systems 10, MIT Press. [7] Broadbent, D.E. (1958). Perception and Communication. Pergamon Press. [8] Treisman, A. (1960). Contextual cues in selective listening, Quarterly Journal of Experimental Psychology, 12:242-248. [9] Lee, H.J., Lee, S.Y. Lee, Shin, S.Y., and Koh, B.Y. (1991). TAG: A neural network model for large-scale optical implementation, Neural Computation, 3:135-143. [1O]Lee, S.Y., Jang, J.S., Shin, S.Y., & Shim, C.S. (1988). Optical Implementation of Associative Memory with Controlled Bit Significance, Applied Optics, 27:19211923. [11 ] Kruschke, J.K. (1992). ALCOVE: An Examplar-Based Connectionist Model of Category Learning, Psychological Review, 99:22-44. [12]Lee, S.Y., Kim, D.S., Abn, K.H., Jeong, J.H., Kim, H., Park, S.Y., Kim, L.Y., Lee, J.S., & Lee, H.Y. (1997). Voice Command II: a DSP implementation of robust speech recognition in real-world noisy environments, International Conference on Neural Information Processing, pp. 1051-1054, Dunedin, New Zealand.
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Bayesian Network Induction via Local Neighborhoods Dimitris Margaritis Department of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 D.Margaritis@cs.cmu.edu Sebastian Thrun Department of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 S. Thrun@cs.cmu.edu Abstract In recent years, Bayesian networks have become highly successful tool for diagnosis, analysis, and decision making in real-world domains. We present an efficient algorithm for learning Bayes networks from data. Our approach constructs Bayesian networks by first identifying each node's Markov blankets, then connecting nodes in a maximally consistent way. In contrast to the majority of work, which typically uses hill-climbing approaches that may produce dense and causally incorrect nets, our approach yields much more compact causal networks by heeding independencies in the data. Compact causal networks facilitate fast inference and are also easier to understand. We prove that under mild assumptions, our approach requires time polynomial in the size of the data and the number of nodes. A randomized variant, also presented here, yields comparable results at much higher speeds. 1 Introduction A great number of scientific fields today benefit from being able to automatically estimate the probability of certain quantities of interest that may be difficult or expensive to observe directly. For example, a doctor may be interested in estimating the probability of heart disease from indications of high blood pressure and other directly measurable quantities. A computer vision system may benefit from a probability distribution of buildings based on indicators of horizontal and vertical straight lines. Probability densities proliferate the sciences today and advances in its estimation are likely to have a wide impact on many different fields. Bayesian networks are a succinct and efficient way to represent a joint probability distribution among a set of variables. As such, they have been applied to fields such as those mentioned [Herskovits90][Agosta88]. Besides their ability for density estimation, their semantics lend them to what is sometimes loosely referred to as causal discovery, namely directional relationships among quantities involved. It has been widely accepted that the most parsimonious representation for a Bayesian net is one that closely represents the causal independence relationships that may exist. For these reasons, there has been great interest in automatically inducing the structure of Bayesian nets automatically from data, preferably also preserving the independence relationships in the process. Two research approaches have emerged. The first employs independence properties of the underlying network that produced the data in order to discover parts of its structure. This approach is mainly exemplified by the SGS and PC algorithms in [Spirtes93], as well 506 D. Margaritis and S. Thrun Figure 1: On the left, an example of a Markov blanket of variable X is shown. The members of the blanket are shown shaded. On the right, an example reconstruction of a 5 x 5 rectangular net of branching factor 3 by the algorithm presented in this paper using 20000 samples. Indicated by dotted lines are 3 directionality errors. as for restricted classes such as trees [Chow68] and poly trees [Rebane87]. The second approach is concerned more with data prediction, disregarding independencies in the data. It is typically identified with a greedy hill-climbing or best-first beam search in the space of legal structures, employing as a scoring function a form of data likelihood, sometimes penalized for network complexity. The result is a local maximum score network structure for representing the data, and is one of the more popular techniques used today. This paper presents an approach that belongs in the first category. It addresses the two main shortcomings of the prior work which, we believe, are preventing its use from becoming more widespread. These two disadvantages are: exponential execution times, and proneness to errors in dependence tests used. The former problem is addressed in this paper in two ways. One is by identifying the local neighborhood of each variable in the Bayesian net as a preprocessing step, in order to facilitate the recovery of the local structure around each variable in polynomial time under the assumption of bounded neighborhood size. The second, randomized version goes one step further, employing a user-specified number of randomized tests (constant or logarithmic) in order to ascertain the same result with high probability. The second disadvantage of this research approach, namely proneness to errors, is also addressed by the randomized version, by using multiple data sets (if available) and Bayesian accumulation of evidence. 2 The Grow-Shrink Markov Blanket Algorithm The concept of the Markov blanket of a variable or a set of variables is central to this paper. The concept itself is not new. For example, see [PearI88]. It is surprising, however, how little attention it has attracted for all its being a fundamental property of a Bayesian net. What is new in this paper is the introduction of the explicit use of this idea to effectively limit unnecessary computation, as well as a simple algorithm to compute it. The definition of a Markov blanket is as follows: denoting V as the set of variables and X HS Y as the conditional dependence of X and Y given the set S, the Markov blanket BL(X) ~ V of X E V is any set of variables such that for any Y E V - BL(X) - {X}, X ft-BL(x) Y. In other words, BL(X) completely shields variable X from any other variable in V . The notion of a minimal Markov blanket, called a Markov boundary, is also introduced in [PearI88] and its uniqueness shown under certain conditions. The Markov boundary is not unique in certain pathological situations, such as the equality of two variables. In our following discussion we will assume that the conditions necessary for its existence and uniqueness are satisfied and we will identify the Markov blanket with the Markov boundary, using the notation B (X) for the blanket of variable X from now on. It is also illuminating to mention that, in the Bayesian net framework, the Markov blanket of a node X is easily identifiable from the graph: it consists of all parents, children and parents of children of X. An example Markov blanket is shown in Fig. 1. Note that any of these nodes, say Y, is dependent with X given B (X) - {Y}. Bayesian Network Induction via Local Neighborhoods 507 1. S t- 0. 2. While:3 Y E V - {X} such that Y HS X, do S t- S U {Y}. [Growing phase] 3. While:3 YES such that Y ft-S-{Y} X, do S t- S - {Y}. [Shrinking phase] 4. B(X) t- S. Figure 2: The basic Markov blanket algorithm. The algorithm for the recovery of the Markov blanket of X is shown in Fig. 2. The idea behind step 2 is simple: as long as the Markov blanket property of X is violated (ie. there exists a variable in V that is dependent on X), we add it to the current set S until there are no more such variables. In this process however, there may be some variables that were added to S that were really outside the blanket. Such variables would have been rendered independent from X at a later point when "intervening" nodes of the underlying Bayesian net were added to S. This observation necessitates step 3, which identifies and removes those variables. The algorithm is efficient, requiring only O( n) conditional tests, making its running time O(n IDI), where n = IVI and D is the set of examples. For a detailed derivation of this bound as well as a formal proof of correctness, see [Margaritis99]. In practice one may try to minimize the number of tests in step 3 by heuristically ordering the variables in the loop of step 2, for example by ascending mutual information or probability of dependence between X and Y (as computed using the X2 test, see section 5). 3 Grow-Shrink (GS) Algorithm for Bayesian Net Induction The recovery of the local structure around each node is greatly facilitated by the knowledge of the nodes' Markov blankets. What would normally be a daunting task of employing dependence tests conditioned on an exponential number of subsets of large sets of variables-even though most of their members may be irrelevant-can now be focused on the Markov blankets of the nodes involved, making structure discovery much faster and more reliable. We present below the plain version of the GS algorithm that utilizes blanket information for inducing the structure of a Bayesian net. At a later point of this paper, we will present a robust, randomized version that has the potential of being faster and more reliable, as well as being able to operate in an "anytime" manner. In the following N (X) represents the direct neighbors of X. [ Compute Markov Blankets ] For all X E V, compute the Markov blanket B (X) . [ Compute Graph Structure] For all X E V and Y E B(X), determine Y to be a direct neighbor of X if X and Y are dependent given S for all S ~ T, where T is the smaller of B (X) - {Y} and B(Y) - {X}. [Orient Edges] For all X E V and YEN (X), orient Y -+ X if there exists a variable Z E N (X) - N (Y) - {Y} such that Y and Z are dependent given S U {X} for all S ~ U, where U is the smaller of B (Y) - {Z} and B (Z) - {Y}. [ Remove Cycles] Do the following while there exist cycles in the graph: 1. Compute the set of edges C = {X -+ Y such that X -+ Y is part of a cycle}. 2. Remove the edge in C that is part of the greatest number of cycles, and put it in R. 508 D. Margaritis and S. Thrun [ Reverse Edges] Insert each edge from R in the graph, reversed. [ Propagate Directions] For all X E V and Y E N(X) such that neither Y ~ X nor X ~ Y, execute the following rule until it no longer applies: If there exists a directed path from X to Y, orient X ~ Y . In the algorithm description above, step 2 determines which of the members of the blanket of each node are actually direct neighbors (parents and children). Assuming, without loss of generality, that B (X) - {Y} is the smaller set, if any of the tests are successful in separating (making independent) X from Y, the algorithm determines that there is no direct connection between them. That would happen when the conditioning set S includes all parents of X and no common children of X and Y. It is interesting to note that the motivation behind selecting the smaller set to condition on stems not only from computational efficiency but from reliability as well: a conditioning set S causes the data set to be split into 21S1 partitions; smaller conditioning sets cause the data set to be split into larger partitions and make dependence tests more reliable. Step 3 exploits the fact that two variables that have a common descendant become dependent when conditioning on a set that includes any such descendant. Since the direct neighbors of X and Y are known from step 2, we can determine whether a direct neighbor Y is a parent of X if there exists another node Z (which, coincidentally, is also a parent) such that any attempt to separate Y and Z by conditioning on a subset of the blanket of Y that includes X, fails (assuming that B(Y) is smaller than B(Z)). If the directionality is indeed Y ~ X ~ Z, there should be no such subset since, by conditioning on X, a permanent dependency path between Y and Z is created. This would not be the case if Y were a child of X. It is straightforward to show that the algorithm requires 0 (n2 + nb22b) conditional independence tests, where b = maxx(IB(X)I). Under the assumption that b is bounded by a constant, this algorithm is O( n 2) in the number of conditional independence tests. It is worthwhile to note that the time to compute a conditional independence test by a pass over the data set Dis O( n IDt) and not O(2IVI). An analysis and a formal proof of correctness of the algorithm is presented in [Margaritis99]. Discussion The main advantage of the algorithm comes through the use of Markov blankets to restrict the size of the conditioning sets. The Markov blankets may be usually wrong in the side of including too many nodes because they are represented by a disjunction of tests for all values of the conditioning set, on the same data. This emphasizes the importance of the "direct neighbors" step which removes nodes that were incorrectly added during the Markov blanket computation step by admitting variables whose dependence was shown high confidence in a large number of different tests. It is also possible that an edge direction is wrongly determined during step 3 due to nonrepresentative or noisy data. This may lead to directed cycles in the resulting graph. It is therefore necessary to remove those cycles by identifying the minimum set of edges than need to be reversed for all cycles to disappear. This problem is closely related [Margaritis99] to the Minimum Feedback Arc Set problem, which is concerned with identifying a minimum set of edges that need to be removed from a graph that possibly contains directed cycles, in order for all such cycles to disappear. Unfortunately, this problem is NP-complete in its generality [Junger85]. We introduce here a reasonable heuristic for its solution that is based on the number of cycles that an edge that is part of a cycle is involved in. Not all edge directions can be determined during the last two steps. For example, nodes with a single parent or multi-parent nodes (called colliders) whose parents are directly connected do not apply to step 3, and steps 4 and 5 are only concerned with already directed edges. Step 6 attempts to ameliorate that, through orienting edges in a way that does not introduce Bayesian Network Induction via Local Neighborhoods 509 a cycle, if the reverse direction necessarily does. It is not obvious that, for example, if the direction X -t Y produces a cycle in an otherwise acyclic graph, the opposite direction Y -t X will not also. However, this is the case. For the proof of this, see [Margaritis99]. The algorithm is similar to the SGS algorithm presented in [Spirtes93], but differs in a number of ways. Its main difference lies in the use of Markov blankets to dramatically improve performance (in many cases where the bounded blanket size assumptions hold). Its structure is similar to SGS, and the stability (frequently referred to as robustness in the following discussion) arguments presented in [Spirtes93] apply. Increased reliability stems from the use of smaller conditioning sets, leading to greater number of examples per test. The PC algorithm, also in [Spirtes93], differs from the GS algorithm in that it involves linear probing for a separator set, which makes it unnecessarily inefficient. 4 Randomized Version of the GS Algorithm The GS algorithm, as presented above, is appropriate for situations where the maximum Markov blanket of each of a set of variables is small. While it is reasonable to assume that in many real-life problems where high-level variables are involved this may be the case, other problems such as Bayesian image retrieval in computer vision, may employ finer representations. In these cases the variables used may depend in a direct manner on many others. For example, we may choose to use variables to characterize local texture in different parts of an image. If the resolution of the mapping from textures to variables is increasingly fine, direct dependencies among those variables may be plentiful and therefore the maximum Markov blanket size may be significant. Another problem that has plagued independence-test based algorithms for Bayesian net structure induction in general is that their decisions are based on a single or a few tests ("hard" decisions), making them prone to errors due to noise in the data. This also applies to the the GS algorithm. It would therefore be advantageous to employ multiple tests before deciding on a direct neighbor or the direction of an edge. The randomized version of the GS algorithm addresses these two problems. Both of them are tackled through randomized testing and Bayesian evidence accumulation. The problem of exponential running times in the maximum blanket size of steps 2 and 3 of the plain algorithm is overcome by replacing them by a series of tests, whose number may be specified by the user, with the members of the conditioning set chosen randomly from the smallest blanket of the two variables. Each such test provides evidence for or against the direct connection between the two variables, appropriately weighted by the probability that circumstances causing that event occur or not, and due to the fact that connectedness is the conjunction of more elementary events. This version of the algorithm is not shown here in detail due to space restrictions. Its operation follows closely the one of the plain GS version. The main difference lies in the usage of Bayesian updating of the posterior probability of a direct link (or a dependence through a collider) between a pair of variables X and Y using conditional dependence tests that take into account independent evidence. The posterior probability Pi of a link between X and Y after executing i dependence tests dj, j = 1, .. . , i is Pi-ldi Pi= ------------~---------Pi-ldi + (1 - Pi-d(G + 1 - dd where G == G(X, Y) = 1 - (4)ITI is a factor that takes values in the interval [0,1) and can be interpreted as the "(un)importance" of the truth of each test di , while T is the smaller of B(X) - {Y} and B(Y) - {X}. We can use this accumulated evidence to guide our decisions to the hypothesis that we feel most confident about. Besides being able to do that in a timely manner due to the user-specified number of tests, we also note how this approach also addresses the robustness problem mentioned above through the use of mUltiple weighted tests, and leaving for the end the "hard" decisions that involve a threshold (ie. comparing the posterior probability with a threshold, which in our case is ~) . 510 D. Margaritis and S. Thrun Kl-divergance verSUS number of samples 0.00015 r--~--C...-~--~;;::Pla-:-in-::G::::SCN:-_ -_--, Randomized GS8N .... - .. Hill-Clil"l"tling. score' data likelihood .. • ,. Hill-Glirnblng, soore: BIC .Q--00001 5e-05 4000 BOOO 12000 16000 20000 Nurrber of sarrples Edge errors versus number of sarrples 1~r---_--_-__ - __ -~ PlainGSBN Randomized GSBN .... ~ ... Direction errors versus number of sarrplss l00r-----------::p~lai~nG~Sr-BN~-~ Randomized GSBN .... ~ ... 125 I 100 .s ~ 75 GO ! SO w 25 ~.~ Hill-Clirrtling, score data likelihood . , .. Hill-Climbing, soore: BIC B Hill-Clirrbi~h~~i~~~~h~ .~_ 1? 75 ' .................................................... III- ....... .......... . ~ S ~ 50 i i5 25 o OL---~--~--~---~-~ o 4000 8000 12000 18000 20000 0 4000 8000 12000 18000 20000 Nurrber of sarrples Number of 5a!1l)les Figure 3: Results for a 5 x 5 rectangular net with branching factor 2 (in both directions, blanket size 8) as a function of the number of samples. On the top, KL-divergence is depicted for the plain GS, randomized GS, and hill-climbing algorithms. On the bottom, the percentage of edge and direction errors are shown. Note that certain edge error rates for the hill-climbing algorithm exceed 100%. 5 Results Throughout the algorithms presented in this paper we employ standard chi-square (X 2) conditional dependence tests (as is done also in [Spirtes93]) in order to compare the histograms P(X) and P(X I Y). The X2 test gives us a probability of the error of assuming that the two variables are dependent when in fact they are not (type II error of a dependence test), from which we can easily derive the probability that X and Y are dependent. There is an implicit confidence threshold T involved in each dependence test, indicating how certain we wish to be about the correctness of the test without unduly rejecting dependent pairs, something that is always possible in reality due to the presence of noise. In all experiments we used T = 0.95, which corresponds to a 95% confidence test. We test the effectiveness of the algorithms through the following procedure: we generate a random rectangular net of specified dimensions and up/down branching factor. A number of examples are drawn from that net using logic sampling and they are used as input to the algorithm under test. The resulting nets can be compared with the original ones along dimensions of KL-divergence and difference in edges and edge directionality. The KLdivergence was estimated using a Monte Carlo procedure. An example reconstruction was shown in the beginning of the paper, Fig. 1. Fig. 3 shows how the KL-divergence between the original and the reconstructed net as well as edge omissions/false additions/reversals as a function of number of samples used. It demonstrates two facts. First, that typical KL-divergence for both GS and hill-climbing algorithms is low (with hill-climbing slightly lower), which shows good performance for applications where prediction is of prime concern. Second, the number of incorrect edges and the errors in the directionality of the edges present is much higher for the hill-climbing algorithm, making it unsuitable for accurate Bayesian net reconstruction. Fig. 4 shows the effects of increasing the Markov blanket through an increasing branching factor. As expected, we see a dramatic (exponential) increase in execution time of the plain Bayesian Network Induction via Local Neighborhoods 511 Edge I Direction Errors versus Branching Factor 100r-------~~~----~~~----, Edge errors, plain GSBN ~ 90 Edge errors, randomized GSBN ---~--22000 20000 Execution Time versus Branching Factor Plain GSBN -Randomized GSBN ----K---80 Direction errors, plain GSBN Direction errors, randomized GSBN &--70 18000 16000 60 50 40 ~ 14000 ~ 12000 ~ 10000 i= 8000 30 ___ __ ___ _ __ _ ... ______ ___ .... ___ __ _ 6000 2°L==~==~~·-- -=----=-----=----~----------- ----l _ _ " _______ .olII ___ ____ __ • __ _ 10 -.,,""'-.. -----...... .. _ .• 4000 2000 .----O~------~--------~--------~ O~------~--------~------~ 2 3 4 5 2 3 4 5 Branching Factor Branching Factor Figure 4: Results for a 5 x 5 rectangular net from which 10000 samples were generated and used for reconstruction, versus increasing branching factor. On the left, errors are slowly increasing as expected, but comparable for the plain and randomized versions of the GS algorithm. On the right, corresponding execution times are shown. GS algorithm, though only a mild increase of the randomized version. The latter uses 200 (constant) conditional tests per decision, and its execution time increase can be attributed to the (quadratic) increase in the number of decisions. Note that the error percentages between the plain and the randomized version remain relatively close. The number of direction errors for the GS algorithm actually decreases due to the larger number of parents for each node (more "V" structures), which allows a greater number of opportunities to recover the directionality of an edge (using an increased number of tests). 6 Discussion In this paper we presented an efficient algorithm for computing the Markov blanket of a node and then used it in the two versions of the GS algorithm (plain and randomized) by exploiting the properties of the Markov blanket to facilitate fast reconstruction of the local neighborhood around each node, under assumptions of bounded neighborhood size. We also presented a randomized variant that has the advantages of faster execution speeds and added reconstruction robustness due to multiple tests and Bayesian accumulation of evidence. Simulation results demonstrate the reconstruction accuracy advantages of the algorithms presented here over hill-climbing methods. Additional results also show that the randomized version has a dramatical execution speed benefit over the plain one in cases where the assumption of bounded neighborhood does not hold, without significantly affecting the reconstruction error rate. References [Chow68] [Herskovits90] [Spirtes93] [PearI88] [Rebane87] [Verma90] [Agosta88] [Cheng97] [Margaritis99] [Jtinger85] C.K. Chow and C.N. Liu. Approximating discrete probability distributions with dependence trees. IEEE Transactions on Information Theory, 14, 1968. E.H. Herskovits and G.F. Cooper. Kutat6: An entropy-driven system for construction of probabilistic expert systems from databases. VAI-90. P. Spirtes, C. Glymour, and R. Scheines. Causation, Prediction, and Search, Springer, 1993. 1. Pearl. Probabilistic Reasoning in Intelligent Systems, Morgan Kaufmann, 1988. G. Rebane and J. Pearl. The recovery of causal poly-trees from statistical data. VAI-87. T.S. Verma, and J. Pearl. Equivalence and Synthesis of Causal Models. VAI-90. J.M. Agosta. The structure of Bayes networks for visual recognition. VAI-88. 1. Cheng, D.A. Bell, W. Liu, An algorithm for Bayesian network construction from data. AI and Statistics, 1997. D. Marg aritis , S. Thrun, Bayesian Network Induction via Local Neighborhoods. TR CMV-CS-99-134, forthcoming. M. Junger, Polyhedral combinatorics and the acyclic subdigraph problem, Heldermann, 1985.
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Spiking Boltzmann Machines Geoffrey E. Hinton Gatsby Computational Neuroscience Unit University College London London WCIN 3AR, UK hinton@gatsby. ucl. ac. uk Abstract Andrew D. Brown Department of Computer Science University of Toronto Toronto, Canada andy@cs.utoronto.ca We first show how to represent sharp posterior probability distributions using real valued coefficients on broadly-tuned basis functions. Then we show how the precise times of spikes can be used to convey the real-valued coefficients on the basis functions quickly and accurately. Finally we describe a simple simulation in which spiking neurons learn to model an image sequence by fitting a dynamic generative model. 1 Population codes and energy landscapes A perceived object is represented in the brain by the activities of many neurons, but there is no general consensus on how the activities of individual neurons combine to represent the multiple properties of an object. We start by focussing on the case of a single object that has multiple instantiation parameters such as position, velocity, size and orientation. We assume that each neuron has an ideal stimulus in the space of instantiation parameters and that its activation rate or probability of activation falls off monotonically in all directions as the actual stimulus departs from this ideal. The semantic problem is to define exactly what instantiation parameters are being represented when the activities of many such neurons are specified. Hinton, Rumelhart and McClelland (1986) consider binary neurons with receptive fields that are convex in instantiation space. They assume that when an object is present it activates all of the neurons in whose receptive fields its instantiation parameters lie. Consequently, if it is known that only one object is present, the parameter values of the object must lie within the feasible region formed by the intersection of the receptive fields of the active neurons. This will be called a conjunctive distributed representation. Assuming that each receptive field occupies only a small fraction of the whole space, an interesting property of this type of "coarse coding" is that the bigger the receptive fields, the more accurate the representation. However, large receptive fields lead to a loss of resolution when several objects are present simultaneously. When the sensory input is noisy, it is impossible to infer the exact parameters of objects so it makes sense for a perceptual system to represent the probability distribution across parameters rather than just a single best estimate or a feasible region. The full probability distribution is essential for correctly combining inforSpiking Boltzmann Machines E(x) P(X) 123 Figure 1: a) Energy landscape over a onedimensional space. Each neuron adds a dimple (dotted line) to the energy landscape (solid line). b) The corresponding probability density. Where dimples overlap the corresponding probability density becomes sharper. Since the dimples decay to zero, the location of a sharp probability peak is not affected by distant dimples and multimodal distributions can be represented. mation from different times or different Sources. One obvious way to represent this distribution (Anderson and van Essen, 1994) is to allow each neuron to represent a fairly compact probability distribution over the space of instantiation parameters and to treat the activity levels of neurons as (unnormalized) mixing proportions. The semantics of this disjunctive distributed representation is precise, but the percepts it allows are not because it is impossible to represent distributions that are sharper than the individual receptive fields and, in high-dimensional spaces, the individual fields must be broad in order to cover the space. Disjunctive representations are used in Kohonen's self-organizing map which is why it is restricted to very low dimensional latent spaces. The disjunctive model can be viewed as an attempt to approximate arbitrary smooth probability distributions by adding together probability distributions contributed by each active neuron. Coarse coding suggests a multiplicative approach in which the addition is done in the domain of energies (negative log probabilities). Each active neuron contributes an energy landscape over the whole space of instantiation parameters. The activity level of the neuron multiplies its energy landscape and the landscapes for all neurons in the population are added (Figure 1). If, for example, each neuron has a full covariance Gaussian tuning function, its energy landscape is a parabolic bowl whose curvature matrix is the inverse of the covariance matrix. The activity level of the neuron scales the inverse covariance matrix. If there are k instantiation parameters then only k + k(k + 1)/2 real numbers are required to span the space of means and inverse covariance matrices. So the real-valued activities of O(k2) neurons are sufficient to represent arbitrary full covariance Gaussian distributions over the space of instantiation parameters. Treating neural activities as multiplicative coefficients on additive contributions to energy landscapes has a number of advantages. Unlike disjunctive codes, vague distributions are represented by low activities so significant biochemical energy is only required when distributions are quite sharp. A central operation in Bayesian inference is to combine a prior term with a likelihood term or to combine two conditionally independent likelihood terms. This is trivially achieved by adding two energy landscapesl . lWe thank Zoubin Ghahramani for pointing out that another important operation, convolving a probability distribution with Gaussian noise, is a difficult non-linear operation on the energy landscape. 124 G. E. Hinton and A. D. Brown 2 Representing the coefficients on the basis functions To perform perception at video rates, the probability distributions over instantiation parameters need to be represented at about 30 frames per second. This seems difficult using relatively slow spiking neurons because it requires the real-valued multiplicative coefficients on the basis functions to be communicated accurately and quickly using all-or-none spikes. The trick is to realise that when a spike arrives at another neuron it produces a postsynaptic potential that is a smooth function of time. So from the perspective of the postsynaptic neuron, the spike has been convolved with a smooth temporal function. By adding a number of these smooth functions together, with appropriate temporal offsets, it is possible to represent any smoothly varying sequence of coefficient values on a basis function, and this makes it possible to represent the temporal evolution of probability distributions as shown in Figure 2. The ability to vary the location of a spike in the single dimension of time thus allows real-valued control of the representation of probability distributions over multiple spatial dimensions. a) .~ > " ~ 'iii OOiS Q) II: Encoded Value b) neuron 2 time I , , Time Figure 2: a)Two spiking neurons centered at 0 and 1 can represent the time-varying mean and standard deviation on a single spatial dimension. The spikes are first convolved with a temporal kernel and the resulting activity values are treated as exponents on Gaussian distributions centered at 0 and 1. The ratio of the activity values determines the mean and the sum of the activity values determines the inverse variance. b) The same method can be used for two (or more) spatial dimensions. Time flows from top to bottom. Each spike makes a contribution to the energy landscape that resembles an hourglass (thin lines). The waist of the hourglass corresponds to the time at which the spike has its strongest effect on some post-synaptic population. By moving the hourglasses in time, it is possible to get whatever temporal cross-sections are desired (thick lines) provided the temporal sampling rate is comparable to the time course of the effect of a spike. Our proposed use of spike timing to convey real values quickly and accurately does not require precise coincidence detection, sub-threshold oscillations, modifiable time delays, or any of the other paraphernalia that has been invoked to explain how the brain could make effective use of the single, real-valued degree of freedom in the timing of a spike (Hopfield, 1995). The coding scheme we have proposed would be far more convincing if we could show how it was learned and could demonstrate that it was effective in a simulation. There are two ways to design a learning algorithm for such spiking neurons. We could work in the relatively low-dimensional space of the instantiation parameters and design the learning to produce the right representations and interactions between representations in this space. Or we could treat this space as an implicit emergent property of the network and design the learning algorithm to optimize Spiking Boltzmann Machines 125 some objective function in the much higher-dimensional space of neural activities in the hope that this will create representations that can be understood using the implicit space of instantiation parameters. We chose the latter approach. 3 A learning algorithm for restricted Boltzmann machines Hinton (1999) describes a learning algorithm for probabilistic generative models that are composed of a number of experts. Each expert specifies a probability distribution over the visible variables and the experts are combined by multiplying these distributions together and renormalizing. (1) where d is a data vector in a discrete space, Om is all the parameters of individual model m, Pm(d\Om) is the probability of d under model m, and i is an index over all possible vectors in the data space. The coding scheme we have described is just a product of experts in which each spike is an expert. We first summarize the Product of Experts learning rule for a restricted Boltzmann machine (RBM) which consists of a layer of stochastic binary visible units connected to a layer of stochastic binary hidden units with no intralayer connections. We then extend RBM's to deal with temporal data. In an RBM, each hidden unit is an expert. When it is off it specifies a uniform distribution over the states of the visible units. When it is on, its weight to each visible unit specifies the log odds that the visible unit is on. Multiplying together the distributions specified by different hidden units is achieved by adding the log odds. Inference in an RBM is much easier than in a causal belief net because there is no explaining away. The hidden states, S j, are conditionally independent given the visible states, Si, and the distribution of Sj is given by the standard logistic function a: p(Sj = 1) = a(L:i WijSi). Conversely, the hidden states of an RBM are marginally dependent so it is easy for an RBM to learn population codes in which units may be highly correlated. It is hard to do this in causal belief nets with one hidden layer because the generative model of a causal belief net assumes marginal independence. An RBM can be trained by following the gradient of the log likelihood of the data: (2) where < SiSj >0 is the expected value of SiSj when data is clamped on the visible units and the hidden states are sampled from their conditional distribution given the data, and < SiSj >00 is the expected value of SiSj after prolonged Gibbs sampling that alternates between sampling from the conditional distribution of the hidden states given the visible states and vice versa. This learning rule not work well because the sampling noise in the estimate of < SiSj >00 swamps the gradient. It is far more effective to maximize the difference between the log likelihood of the data and the log likelihood of the one-step reconstructions of the data that are produced by first picking binary hidden states from their conditional distribution given the data and then picking binary visible states from their conditional distribution given the hidden states. The gradient of the log 126 G. E. Hinton and A. D. Brown likelihood of the one-step reconstructions is complicated because changing a weight changes the probability distribution of the reconstructions: + (3) where Ql is the distribution of the one-step reconstructions of the training data and Qoo is the equilibrium distribution (i.e. the stationary distribution of prolonged Gibbs sampling). Fortunately, the cumbersome third term is sufficiently small that ignoring it does not prevent the vector of weight changes from having a positive cosine with the true gradient of the difference of the log likelhoods so the following very simple learning rule works much better than Eq. 2. (4) 4 Restricted Boltzmann machines through time Using a restricted Boltzmann machine we can represent time by spatializing it, i.e. taking each visible unit, i, and hidden unit, j, and replicating them through time with the constraint that the weight WijT between replica t of i and replica t + T of j does not depend on t. To implement the desired temporal smoothing, we also force the weights to be a smooth function of T that has the shape of the temporal kernel, shown in Figure 3. The only remaining degree of freedom in the weights between replicas of i and replicas of j is the scale of the temporal kernel and it is this scale that is learned. The replicas of the visible and hidden units still form a bipartite graph and the probability distribution over the hidden replicas can be inferred exactly without considering data that lies further into the future than the width of the temporal kernel. One problem with the restricted Boltzmann machine when we spatialize time is that hidden units at one time step have no memory of their states at previous time steps; they only see the data. If we were to add undirected connections between hidden units at different time steps, then the architecture would return to a fully connected Boltzmann machine in which the hidden units are no longer conditionally independent given the data. A useful trick borrowed from Elman nets is to allow the hidden units to see their previous states, but to treat these observations like data that cannot be modified by future hidden states. Thus, the hidden states may still be inferred independently without resorting to Gibbs sampling. The connections between hidden layer weights also follow the time course of the temporal kernel. These connections act as a predictive prior over the hidden units. It is important to note that these forward connections are not required for the network to model a sequence, but only for the purposes of extrapolating into the future. Figure 3: The form of the temporal kernel. Spiking Boltzmann Machines Now the probability that Sj(t) = 1 given the states of the visible units is, P(Sj(t) = 1) = u (~W,jh,(t) + ~ W,;h,(t)) . 127 where hi(t) is the convolution of the history of visible unit i with the temporal kernel, 00 T=O and hk(t), the convolution of the hidden unit history, is computed similarly. 2 Learning the weights follows immediately from this formula for doing inference. In the positive phase the visible units are clamped at each time step and the posterior of the hidden units conditioned on the data is computed (we assume zero boundary conditions for time before t = 0). Then in the negative phase we sample from the posterior of the hidden units, and compute the distribution over the visible units at each time step given these hidden unit states. In each phase the correlations between the hidden and visible units are computed and the learning rule is, 00 00 AWij = L L r(7) ((Sj(t)Si(t - 7))0 - (Sj(t)Si(t - 7))1) . t=O T=O 5 Results We trained this network on a sequence of 8x8 synthetic images of a Gaussian blob moving in a circular path. In the following diagrams we display the time sequence of images as a matrix. Each row of the matrix represents a single image with its pixels stretched out into a vector in scanline order, and each column is the time course of a single pixel. The intensity f the pixel is represented by the area of the white patch. We used 20 hidden units. Figure 5a shows a segment (200 time steps) of the time series which was used in training. In this sequence the period of the blob is 80 time steps. Figure 5b shows how the trained model reconstructs the data after we sample from the hidden layer units. Once we have trained the model it is possible to do forecasting by clamping visible layer units for a segment of a sequence and then doing iterative Gibbs sampling to generate future points in the sequence. Figure 5c shows that given 50 time steps from the series, the model can predict reasonably far into the future, before the pattern dies out. One problem with these simulations is that we are treating the real valued intensities in the images as probabilities. While this works for the blob images, where the values can be viewed as the probabilities of pixels in a binary image being on, this is not true for more natural images. 6 Discussion In our initial simulations we used a causal sigmoid belief network (SBN) rather than a restricted Boltzmann machine. Inference in an SBN is much more difficult than in an RBM. It requires Gibbs sampling or severe approximations, and even if a temporal kernel is used to ensure that a replica of a hidden unit at one time 2Computing the conditional probability distribution over the visible units given the hidden states is done in a similar fashion, with the caveat that the weights in each direction must be symmetric. Thus, the convolution is done using the reverse kernel. 128 G. E. Hinton and A. D. Brown a) b) c) Figure 4: a) The original data, b) reconstruction of the data, and c) prediction of the data given 50 time steps of the sequence. The black line indicates where the prediction begins. has no connections to replicas of visible units at very different times, the posterior distribution of the hidden units still depends on data far in the future. The Gibbs sampling made our SBN simulations very slow and the sampling noise made the learning far less effective than in the RBM. Although the RBM simulations seem closer to biological plausibility, they too suffer from a major problem. To apply the learning procedure it is necessary to reconstruct the data from the hidden states and we do not know how to do this without interfering with the incoming datastream. In our simulations we simply ignored this problem by allowing a visible unit to have both an observed value and a reconstructed value at the same time. Acknowledgements We thank Zoubin Ghahramani, Peter Dayan, Rich Zemel, Terry Sejnowski and Radford Neal for helpful discussions. This research was funded by grants from the Gatsby Foundation and NSERC. References Anderson, C.H. & van Essen, D.C (1994). Neurobiological computational systems. In J.M Zureda, R.J. Marks, & C.J. Robinson (Eds.), Computational Intelligence Imitating Life 213-222. New York: IEEE Press. Hinton, G. E. (1999) Products of Experts. ICANN 99: Ninth international conference on Artificial Neural Networks, Edinburgh, 1-6. Hinton, G. E., McClelland, J. L., & Rumelhart, D. E. (1986) Distributed representations. In Rumelhart, D. E. and McClelland, J. L., editors, Parallel Distributed Processing: Explorations in the Microstructure of Cognition. Volume 1: Foundations, MIT Press, Cambridge, MA. Hopfield, J. (1995). Pattern recognition computation using action potential timing for stimulus representation. Nature, 376, 33-36.
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Actor-Critic Algorithms Vijay R. Konda John N. Tsitsiklis Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA, 02139. konda@mit.edu, jnt@mit.edu Abstract We propose and analyze a class of actor-critic algorithms for simulation-based optimization of a Markov decision process over a parameterized family of randomized stationary policies. These are two-time-scale algorithms in which the critic uses TD learning with a linear approximation architecture and the actor is updated in an approximate gradient direction based on information provided by the critic. We show that the features for the critic should span a subspace prescribed by the choice of parameterization of the actor. We conclude by discussing convergence properties and some open problems. 1 Introduction The vast majority of Reinforcement Learning (RL) [9J and Neuro-Dynamic Programming (NDP) [lJ methods fall into one of the following two categories: (a) Actor-only methods work with a parameterized family of policies. The gradient of the performance, with respect to the actor parameters, is directly estimated by simulation, and the parameters are updated in a direction of improvement [4, 5, 8, 13J. A possible drawback of such methods is that the gradient estimators may have a large variance. Furthermore, as the policy changes, a new gradient is estimated independently of past estimates. Hence, there is no "learning," in the sense of accumulation and consolidation of older information. (b) Critic-only methods rely exclusively on value function approximation and aim at learning an approximate solution to the Bellman equation, which will then hopefully prescribe a near-optimal policy. Such methods are indirect in the sense that they do not try to optimize directly over a policy space. A method of this type may succeed in constructing a "good" approximation of the value function, yet lack reliable guarantees in terms of near-optimality of the resulting policy. Actor-critic methods aim at combining the strong points of actor-only and criticonly methods. The critic uses an approximation architecture and simulation to learn a value function, which is then used to update the actor's policy parameters Actor-Critic Algorithms 1009 in a direction of performance improvement. Such methods, as long as they are gradient-based, may have desirable convergence properties, in contrast to criticonly methods for which convergence is guaranteed in very limited settings. They hold the promise of delivering faster convergence (due to variance reduction), when compared to actor-only methods. On the other hand, theoretical understanding of actor-critic methods has been limited to the case of lookup table representations of policies [6]. In this paper, we propose some actor-critic algorithms and provide an overview of a convergence proof. The algorithms are based on an important observation. Since the number of parameters that the actor has to update is relatively small (compared to the number of states), the critic need not attempt to compute or approximate the exact value function, which is a high-dimensional object. In fact, we show that the critic should ideally compute a certain "projection" of the value function onto a low-dimensional subspace spanned by a set of "basis functions," that are completely determined by the parameterization of the actor. Finally, as the analysis in [11] suggests for TD algorithms, our algorithms can be extended to the case of arbitrary state and action spaces as long as certain ergodicity assumptions are satisfied. We close this section by noting that ideas similar to ours have been presented in the simultaneous and independent work of Sutton et al. [10]. 2 Markov decision processes and parameterized family of RSP's Consider a Markov decision process with finite state space S, and finite action space A. Let 9 : S x A -t ffi. be a given cost function. A randomized stationary policy (RSP) is a mapping I-" that assigns to each state x a probability distribution over the action space A. We consider a set of randomized stationary policies JPl = {1-"9; e E ffi.n}, parameterized in terms of a vector e. For each pair (x, u) E S x A, 1-"9 (x, u) denotes the probability of taking action u when the state x is encountered, under the policy corresponding to e. Let PXy(u) denote the probability that the next state is y, given that the current state is x and the current action is u. Note that under any RSP, the sequence of states {Xn} and of state-action pairs {Xn' Un} of the Markov decision process form Markov chains with state spaces Sand S x A, respectively. We make the following assumptions about the family of policies JPl. (AI) For all xES and u E A the map e t-t 1-"9(X, u) is twice differentiable with bounded first, second derivatives. Furthermore, there exists a ffi.n_ valued function 'l/J9(X, u) such that \l1-"9(X, u) = 1-"9 (x, U)'l/J9(X, u) where the mapping e t-t 'l/J9(X, u) is bounded and has first bounded derivatives for any fixed x and u. (A2) For each e E ffi.n, the Markov chains {Xn} and {Xn, Un} are irreducible and aperiodic, with stationary probabilities 7r9(X) and 'T}9(X, u) = 7r9 (x) 1-"9 (x, u), respectively, under the RSP 1-"9. In reference to Assumption (AI), note that whenever 1-"9 (x, u) is nonzero we have \l1-"9(X, u) 'l/J9 (x, u) = ( ) = \lIn 1-"9 (x, u). 1-"9 x,u Consider the average cost function>. : ffi.n t-t ffi., given by >.(e) = L g(x, U)'T}9(X, u) . xES,uEA 1010 V. R. Konda and J. N. Tsitsiklis We are interested in minimizing >'(19) over all 19. For each 19 E Rn , let Ve : S t--7 R be the "differential" cost function, defined as solution of Poisson equation: >'(19) + Ve(x) = L I-'e(x,u) [g(X,U) + LPxY(U)Ve(Y)]. uEA Y Intuitively, Ve(x) can be viewed as the "disadvantage" of state x: it is the expected excess cost - on top of the average cost - incurred if we start at state x. It plays a role similar to that played by the more familiar value function that arises in total or discounted cost Markov decision problems. Finally, for every 19 E Rn, we define the q-function qe : S x A -+ R, by qe(x, u) = g(x, u) - >'(19) + LPxy(u)Ve(y). Y We recall the following result, as stated in [8]. (Different versions of this result have been established in [3, 4, 5].) Theorem 1. 8 '"' . 819. >'(19) = L..J 1]e(x, u)qe(x, u)1/;o(x, u) 1. X,U (1) where 1/;b (x, u) stands for the i th component of 1/;e . In [8], the quantity qe(x,u) in the above formula is interpreted as the expected excess cost incurred over a certain renewal period of the Markov chain {Xn, Un}, under the RSP I-'e, and is then estimated by means of simulation, leading to actoronly algorithms. Here, we provide an alternative interpretation of the formula in Theorem 1, as an inner product, and thus derive a different set of algorithms, which readily generalize to the case of an infinite space as well. For any 19 E Rn, we define the inner product (', .) e of two real valued functions q1 , q2 on S x A, viewed as vectors in RlsiIAI, by (q1, q2)e = L 1]e(x, U)q1 (x, U)q2(X, u). x,u With this notation we can rewrite the formula (1) as 8 . 819i >'(19) = (qe,1/;o)e, i = 1, ... ,n. Let 11 ·lle denote the norm induced by this inner product on RlsiIAI. For each 19 E Rn let we denote the span of the vectors {1/;b; 1 ::; i ::; n} in RISIIAI. (This is same as the set of all functions f on S x A of the form f(x ,u) = 2::7=1 ai1/;~(x , U), for some scalars a1,· . . ,an,) Note that although the gradient of >. depends on the q-function, which is a vector in a possibly very high dimensional space RlsiIAI, the dependence is only through its inner products with vectors in we. Thus, instead of "learning" the function qe, it would suffice to learn the projection of qe on the subspace We. Indeed, let rIe : RlsllAI t--7 We be the projection operator defined by rIeq = arg !llin Ilq - qlle. qEwe Since (qe ,1/;e)e = (rIeqe, 1/;e)e, it is enough to compute the projection of qe onto we. (2) Actor-Critic Algorithms 1011 3 Actor-critic algorithms We view actor critic-algorithms as stochastic gradient algorithms on the parameter space of the actor. When the actor parameter vector is 0, the job of the critic is to compute an approximation of the projection IIeqe of qe onto 'lie. The actor uses this approximation to update its policy in an approximate gradient direction. The analysis in [11, 12] shows that this is precisely what TD algorithms try to do, i.e., to compute the projection of an exact value function onto a subspace spanned by feature vectors. This allows us to implement the critic by using a TD algorithm. (Note, however, that other types of critics are possible, e.g., based on batch solution of least squares problems, as long as they aim at computing the same projection.) We note some minor differences with the common usage of TD. In our context, we need the projection of q-functions, rather than value functions. But this is easily achieved by replacing the Markov chain {xt} in [11, 12] by the Markov chain {Xn, Un}. A further difference is that [11, 12] assume that the control policy and the feature vectors are fixed. In our algorithms, the control policy as well as the features need to change as the actor updates its parameters. As shown in [6, 2], this need not pose any problems, as long as the actor parameters are updated on a slower time scale. We are now ready to describe two actor-critic algorithms, which differ only as far as the critic updates are concerned. In both variants, the critic is a TD algorithm with a linearly parameterized approximation architecture for the q-function, of the form m Q~(x, u) = I: r j 4>~(x, u), j=l where r = (rl, ... , rm) E ]Rm denotes the parameter vector of the critic. The features 4>~, j = 1, ... ,m, used by the critic are dependent on the actor parameter vector 0 and are chosen such that their span in ]RlsIIAI, denoted by <Pe, contains 'lI e. Note that the formula (2) still holds if IIe is redefined as projection onto <Pe as long as <Pe contains 'lie. The most straightforward choice would be to let m = n and 4>~ = 't/J~ for each i. Nevertheless, we allow the possibility that m > nand <Pe properly contains 'lie, so that the critic uses more features than that are actually necessary. This added flexibility may turn out to be useful in a number of ways: 1. It is possible for certain values of 0, the features 't/Je are either close to zero or are almost linearly dependent. For these values of 0, the operator IIe becomes ill-conditioned and the algorithms can become unstable. This might be avoided by using richer set of features 't/J~. 2. For the second algorithm that we propose (TD(a) a < 1) critic can only compute approximate - rather than exact - projection. The use of additional features can result in a reduction of the approximation error. Along with the parameter vector r, the critic stores some auxiliary parameters: these are a (scalar) estimate A, of the average cost, and an m-vector z which represents Sutton's eligibility trace [1, 9]. The actor and critic updates take place in the course of a simulation of a single sample path of the controlled Markov chain. Let rk, Zk, Ak be the parameters of the critic, and let Ok be the parameter vectpr of the actor, at time k. Let (Xk, Uk) be the state-action pair at that time. Let Xk+l be the new state, obtained after action Uk is applied. A new action Uk+l is generated according to the RSP corresponding to the actor parameter vector Ok. The critic carries out an update similar to the average cost temporal-difference method of [12]: Ak+l = Ak + 'Ydg(Xk, Uk) - Ak), 1012 V. R. Kanda and J. N. Tsitsiklis rk+l = rk + 1'k(9(Xk' Uk) - Ak + Q~~(Xk+l' Uk+l) - Q~~(Xk,Uk))Zk. (Here, 1'k is a positive stepsize parameter.) The two variants of the critic use different ways of updating Zk: TD(J) Critic: Let x* be a state in S. Zk+l Zk + ¢>91c (Xk+l' Uk+l) , if Xk+l ::/= x*, ¢>9,. (Xk+l, Uk+d, otherwise. TD(a) Critic, 0 ~ a < 1: Actor: Finally, the actor updates its parameter vector by letting (Jk+l = (Jk rhf(rk)Q~~ (Xk+l' Uk+l)1/!9,. (Xk+l' Uk+l). Here, 13k is a positive stepsize and r(rk) > 0 is a normalization factor satisfying: (A3) f(·) is Lipschitz continuous. (A4) There exists C > 0 such that C r(r) ~ 1 + Ilrll' The above presented algorithms are only two out of many variations. For instance, one could also consider "episodic" problems in which one starts from a given initial state and runs the process until a random termination time (at which time the process is reinitialized at x*), with the objective of minimizing the expected cost until termination. In this setting, the average cost estimate Ak is unnecessary and is removed from the critic update formula. If the critic parameter rk were to be reinitialized each time that x* is entered, one would obtain a method closely related to Williams' REINFORCE algorithm [13]. Such a method does not involve any value function learning, because the observations during one episode do not affect the critic parameter r during another episode. In contrast, in our approach, the observations from all past episodes affect current critic parameter r, and in this sense critic is "learning". This can be advantageous because, as long as (J is slowly changing, the observations from recent episodes carry useful information on the q-function under the current policy. 4 Convergence of actor-critic algorithms Since our actor-critic algorithms are gradient-based, one cannot expect to prove convergence to a globally optimal policy (within the given class of RSP's). The best that one could hope for is the convergence of '\l A((J) to zero; in practical terms, this will usually translate to convergence to a local minimum of A((J). Actually, because the T D(a) critic will generally converge to an approximation of the desired projection of the value function, the corresponding convergence result is necessarily weaker, only guaranteeing that '\l A((h) becomes small (infinitely often). Let us now introduce some further assumptions. Actor-Critic Algorithms (A5) For each 0 E ~n, we define an m x m matrix G(O) by G(O) = L1Jo(x,u)¢o(x,u)¢O(x,U)T. x,u 1013 We assume that G(O) is uniformly positive definite, that is, there exists some fl > 0 such that for all r E ~m and 0 E ~n rTG(O)r ~ fdlrW· (A6) We assume that the stepsize sequences bk}, {th} are positive, nonincreasing, and satisfy 15k > 0, Vk, L 15k = 00, L 15k < 00, k k where 15k stands for either /h or 'Yk. We also assume that 13k --+ o. 'Yk Note that the last assumption requires that the actor parameters be updated at a time scale slower than that of critic. Theorem 2. In an actor-critic algorithm with a TD(l) critic, liminf IIV'A(Ok)11 = 0 w.p. 1. k Furthermore, if {Od is bounded w.p. 1 then lim IIV' A(Ok)11 = 0 k w.p. 1. Theorem 3. For every f > 0, there exists a: sufficiently close to 1, such that liminfk IIV'A(Ok)11 ::; f w.p. 1. Note that the theoretical guarantees appear to be stronger in the case of the TD(l) critic. However, we expect that TD(a:) will perform better in practice because of much smaller variance for the parameter rk. (Similar issues arise when considering actor-only algorithms. The experiments reported in [7] indicate that introducing a forgetting factor a: < 1 can result in much faster convergence, with very little loss of performance.) We now provide an overview of the proofs of these theorems. Since 13k/'Yk --+ 0, the size of the actor updates becomes negligible compared to the size of the critic updates. Therefore the actor looks stationary, as far as the critic is concerned. Thus, the analysis in [1] for the TD(l) critic and the analysis in [12] for the TD(a:) critic (with a: < 1) can be used, with appropriate modifications, to conclude that the critic's approximation of IIokqok will be "asymptotically correct". If r(O) denotes the value to which the critic converges when the actor parameters are fixed at 0, then the update for the actor can be rewritten as Ok+l = Ok 13kr(r(Ok))Q~(Ok) (Xk+l, Uk+l)'l/JOk (Xk+1 , Uk+d + 13kek, where ek is an error that becomes asymptotically negligible. At this point, standard proof techniques for stochastic approximation algorithms can be used to complete the proof. 5 Conclusions The key observation in this paper is that in actor-critic methods, the actor parameterization and the critic parameterization need not, and should not be chosen 1014 V. R. Konda and J. N. Tsitsiklis independently. Rather, an appropriate approximation architecture for the critic is directly prescribed by the parameterization used in actor. Capitalizing on the above observation, we have presented a class of actor-critic algorithms, aimed at combining the advantages of actor-only and critic-only methods. In contrast to existing actor-critic methods, our algorithms apply to high-dimensional problems (they do not rely on lookup table representations), and are mathematically sound in the sense that they possess certain convergence properties. Acknowledgments: This research was partially supported by the NSF under grant ECS-9873451, and by the AFOSR under grant F49620-99-1-0320. References [1] D. P. Bertsekas and J. N. Tsitsiklis. Neurodynamic Programming. Athena Scientific, Belmont, MA, 1996. [2] V. S. Borkar. Stochastic approximation with two time scales. Systems and Control Letters, 29:291-294, 1996. [3] X. R. Cao and H. F. Chen. Perturbation realization, potentials, and sensitivity analysis of Markov processes. IEEE Transactions on Automatic Control, 42:1382-1393,1997. [4] P. W. Glynn. Stochastic approximation for monte carlo optimization. In Proceedings of the 1986 Winter Simulation Conference, pages 285-289, 1986. [5] T. Jaakola, S. P. Singh, and M. 1. Jordan. Reinforcement learning algorithms for partially observable Markov decision problems. In Advances in Neural Information Processing Systems, volume 7, pages 345- 352, San Francisco, CA, 1995. Morgan Kaufman. [6] V. R. Konda and V. S. Borkar. Actor-critic like learning algorithms for Markov decision processes. SIAM Journal on Control and Optimization, 38(1) :94-123, 1999. [7] P. Marbach. Simulation based optimization of Markov reward processes. PhD thesis, Massachusetts Institute of Technology, 1998. [8] P. Marbach and J. N. Tsitsiklis. Simulation-based optimization of Markov reward processes. Submitted to IEEE Transactions on Automatic Control. [9] R. Sutton and A. Barto. Reinforcement Learning: An Introduction. MIT Press, Cambridge, MA, 1995. [10] R. S. Sutton, D. McAllester, S. Singh, and Y. Mansour. Policy gradient methods for reinforcement learning with function approximation. In this proceedings. [11] J. N. Tsitsiklis and B. Van Roy. An analysis of temporal-difference learning with function approximation. IEEE Transactions on Automatic Control, 42(5):674-690, 1997. [12] J. N. Tsitsiklis and B. Van Roy. Average cost temporal-difference learning. Automatica, 35(11):1799-1808, 1999. [13] R. Williams. Simple statistical gradient following algorithms for connectionist reinforcement learning. Machine Learning, 8:229-256, 1992.
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Correctness of belief propagation in Gaussian graphical models of arbitrary topology Yair Weiss Computer Science Division UC Berkeley, 485 Soda Hall Berkeley, CA 94720-1776 Phone: 510-642-5029 yweiss@cs.berkeley.edu William T. Freeman Mitsubishi Electric Research Lab 201 Broadway Cambridge, MA 02139 Phone: 617-621-7527 freeman @merl.com Abstract Local "belief propagation" rules of the sort proposed by Pearl [15] are guaranteed to converge to the correct posterior probabilities in singly connected graphical models. Recently, a number of researchers have empirically demonstrated good performance of "loopy belief propagation"using these same rules on graphs with loops. Perhaps the most dramatic instance is the near Shannon-limit performance of "Turbo codes", whose decoding algorithm is equivalent to loopy belief propagation. Except for the case of graphs with a single loop, there has been little theoretical understanding of the performance of loopy propagation. Here we analyze belief propagation in networks with arbitrary topologies when the nodes in the graph describe jointly Gaussian random variables. We give an analytical formula relating the true posterior probabilities with those calculated using loopy propagation. We give sufficient conditions for convergence and show that when belief propagation converges it gives the correct posterior means for all graph topologies, not just networks with a single loop. The related "max-product" belief propagation algorithm finds the maximum posterior probability estimate for singly connected networks. We show that, even for non-Gaussian probability distributions, the convergence points of the max-product algorithm in loopy networks are maxima over a particular large local neighborhood of the posterior probability. These results help clarify the empirical performance results and motivate using the powerful belief propagation algorithm in a broader class of networks. Problems involving probabilistic belief propagation arise in a wide variety of applications, including error correcting codes, speech recognition and medical diagnosis. If the graph is singly connected, there exist local message-passing schemes to calculate the posterior probability of an unobserved variable given the observed variables. Pearl [15] derived such a scheme for singly connected Bayesian networks and showed that this "belief propagation" algorithm is guaranteed to converge to the correct posterior probabilities (or "beliefs"). Several groups have recently reported excellent experimental results by running algorithms 674 Y. Weiss and W T. Freeman equivalent to Pearl's algorithm on networks with loops [8, 13, 6]. Perhaps the most dramatic instance of this performance is for "Turbo code" [2] error correcting codes. These codes have been described as "the most exciting and potentially important development in coding theory in many years" [12] and have recently been shown [10, 11] to utilize an algorithm equivalent to belief propagation in a network with loops. Progress in the analysis of loopy belief propagation has been made for the case of networks with a single loop [17, 18, 4, 1]. For these networks, it can be shown that (1) unless all the compatabilities are deterministic, loopy belief propagation will converge. (2) The difference between the loopy beliefs and the true beliefs is related to the convergence rate of the messages the faster the convergence the more exact the approximation and (3) If the hidden nodes are binary, then the loopy beliefs and the true beliefs are both maximized by the same assignments, although the confidence in that assignment is wrong for the loopy beliefs. In this paper we analyze belief propagation in graphs of arbitrary topology, for nodes describing jointly Gaussian random variables. We give an exact formula relating the correct marginal posterior probabilities with the ones calculated using loopy belief propagation. We show that if belief propagation converges, then it will give the correct posterior means for all graph topologies, not just networks with a single loop. We show that the covariance estimates will generally be incorrect but present a relationship between the error in the covariance estimates and the convergence speed. For Gaussian or non-Gaussian variables, we show that the "max-product" algorithm, which calculates the MAP estimate in singly connected networks, only converges to points that are maxima over a particular large neighborhood of the posterior probability of loopy networks. 1 Analysis To simplify the notation, we assume the graphical model has been preprocessed into an undirected graphical model with pairwise potentials. Any graphical model can be converted into this form, and running belief propagation on the pairwise graph is equivalent to running belief propagation on the original graph [18]. We assume each node X i has a local observation Yi . In each iteration of belief propagation, each node X i sends a message to each neighboring X j that is based on the messages it received from the other neighbors, its local observation Yl and the pairwise potentials Wij(Xi , Xj) and Wii(Xi, Yi) . We assume the message-passing occurs in parallel. The idea behind the analysis is to build an unwrapped tree. The unwrapped tree is the graphical model which belief propagation is solving exactly when one applies the belief propagation rules in a loopy network [9, 20, 18]. It is constructed by maintaining the same local neighborhood structure as the loopy network but nodes are replicated so there are no loops. The potentials and the observations are replicated from the loopy graph. Figure 1 (a) shows an unwrapped tree for the diamond shaped graph in (b). By construction, the belief at the root node X-I is identical to that at node Xl in the loopy graph after four iterations of belief propagation. Each node has a shaded observed node attached to it, omitted here for clarity. Because the original network represents jointly Gaussian variables, so will the unwrapped tree. Since it is a tree, belief propagation is guaranteed to give the correct answer for the unwrapped graph. We can thus use Gaussian marginalization formulae to calculate the true mean and variances in both the original and the unwrapped networks. In this way, we calculate the accuracy of belief propagation for Gaussian networks of arbitrary topology. We assume that the joint mean is zero (the means can be added-in later). The joint distriCorrectness of Belief Propagation 675 Figure 1: Left: A Markov network with mUltiple loops. Right: The unwrapped network corresponding to this structure. bution of z = ( : ) is given by P(z) = ae-!zTVz, where V = (~:: ~::) . It is straightforward to construct the inverse covariance matrix V of the joint Gaussian that describes a given Gaussian graphical model [3]. Writing out the exponent of the joint and completing the square shows that the mean I-' of x, given the observations y, is given by: (1) and the covariance matrix C~IY of x given y is: C~IY = V~-;l. We will denote by C~dY the ith row of C~IY so the marginal posterior variance of Xi given the data is (72 (i) = C~i Iy (i). We will use - for unwrapped quantities. We scan the tree in breadth first order and denote by x the vector of values in the hidden nodes of the tree when so scanned. Simlarly, we denote by y the observed nodes scanned in the same order and Vn , V~y the inverse covariance matrices. Since we are scanning in breadth first order the last nodes are the leaf nodes and we denote by L the number of leaf nodes. By the nature of unwrapping, tL(1) is the mean of the belief at node Xl after t iterations of belief propagation, where t is the number of unwrappings. Similarly 0-2 (1) = 6~1Iy(1) is the variance of the belief at node Xl after t iterations. Because the data is replicated we can write y = Oy where O(i, j) = 1 if Yi is a replica of Yj and 0 otherwise. Since the potentials W(Xi' Yi) are replicated, we can write V~yO = OV~y. Since the W (Xi, X j) are also replicated and all non-leaf Xi have the same connectivity as the corresponding Xi, we can write V~~O = OVzz + E where E is zero in all but the last L rows. When these relationships between the loopy and unwrapped inverse covariance matrices are substituted into the loopy and unwrapped versions of equation I, one obtains the following expression, true for any iteration [19]: (2) where e is a vector that is zero everywhere but the last L components (corresponding to the leaf nodes). Our choice of the node for the root of the tree is arbitrary, so this applies to all nodes of the loopy network. This formula relates, for any node of a network with loops, the means calculated at each iteration by belief propagation with the true posterior means. Similarly when the relationship between the loopy and unwrapped inverse covariance matrices is substituted into the loopy and unwrapped definitions of C~IY we can relate the 676 Y Weiss and W T Freeman 0.5 0.4 ~ 0.3 ~ .~ 0.2 n; ~ 0.1 8 "t:> § 0 -0.1 -0.2 0 20 40 60 80 100 node Figure 2: The conditional correlation between the root node and all other nodes in the unwrapped tree of Fig. 1 after eight iterations. Potentials were chosen randomly. Nodes are presented in breadth first order so the last elements are the correlations between the root node and the leaf nodes. We show that if this correlation goes to zero, belief propagation converges and the loopy means are exact. Symbols plotted with a star denote correlations with nodes that correspond to the node Xl in the loopy graph. The sum of these correlations gives the correct variance of node Xl while loopy propagation uses only the first correlation. marginalized covariances calculated by belief propagation to the true ones [19]: -2 2 a (1) = a (1) + CZllyel - Czt/ye2 (3) where el is a vector that is zero everywhere but the last L components while e2 is equal to 1 for all nodes in the unwrapped tree that are replicas of Xl except for Xl. All other components of e2 are zero, Figure 2 shows Cz1lY for the diamond network in Fig. 1. We generated random potential functions and observations and calculated the conditional correlations in the unwrapped tree. Note that the conditional correlation decreases with distance in the tree we are scanning in breadth first order so the last L components correspond to the leaf nodes. As the number of iterations of loopy propagation is increased the size of the unwrapped tree increases and the conditional correlation between the leaf nodes and the root node decreases. From equations 2-3 it is clear that if the conditional correlation between the leaf nodes and the root nodes are zero for all sufficiently large unwrappings then (1) belief propagation converges (2) the means are exact and (3) the variances may be incorrect. In practice the conditional correlations will not actually be equal to zero for any finite unwrapping. In [19] we give a more precise statement: if the conditional correlation of the root node and the leaf nodes decreases rapidly enough then (1) belief propagation converges (2) the means are exact and (3) the variances may be incorrect. We also show sufficient conditions on the potentials III (Xi, X j) for the correlation to decrease rapidly enough: the rate at which the correlation decreases is determined by the ratio of off-diagonal and diagonal components in the quadratic fonn defining the potentials [19]. How wrong will the variances be? The tenn CZllye2 in equation 3 is simply the sum of many components of Cz11y. Figure 2 shows these components. The correct variance is the sum of all the components witHe the belief propagation variance approximates this sum with the first (and dominant) tenn. Whenever there is a positive correlation between the root node and other replicas of Xl the loopy variance is strictly less than the true variance the loopy estimate is overconfident. Correctness of Belief Propagation (a) ~07 e iDO.6 " ., ;;;05 fr ~04 '" ~03 0.2 0.1 677 SOR 20 30 40 50 60 iterations (b) Figure 3: (a) 25 x 25 graphical model for simulation. The unobserved nodes (unfilled) were connected to their four nearest neighbors and to an observation node (filled). (b) The error of the estimates of loopy propagation and successive over-relaxation (SOR) as a function of iteration. Note that belief propagation converges much faster than SOR. Note that when the conditional correlation decreases rapidly to zero two things happen. First, the convergence is faster (because CZdyel approaches zero faster). Second, the approximation error of the variances is smaller (because CZ1 /y e2 is smaller). Thus we have shown, as in the single loop case, quick convergence is correlated with good approximation. 2 Simulations We ran belief propagation on the 25 x 25 2D grid of Fig. 3 a. The joint probability was: (4) where Wij = 0 if nodes Xi, Xj are not neighbors and 0.01 otherwise and Wii was randomly selected to be 0 or 1 for all i with probability of 1 set to 0.2. The observations Yi were chosen randomly. This problem corresponds to an approximation problem from sparse data where only 20% of the points are visible. We found the exact posterior by solving equation 1. We also ran belief propagation and found that when it converged, the calculated means were identical to the true means up to machine precision. Also, as predicted by the theory, the calculated variances were too small the belief propagation estimate was overconfident. In many applications, the solution of equation 1 by matrix inversion is intractable and iterative methods are used. Figure 3 compares the error in the means as a function of iterations for loopy propagation and successive-over-relaxation (SOR), considered one of the best relaxation methods [16]. Note that after essentially five iterations loopy propagation gives the right answer while SOR requires many more. As expected by the fast convergence, the approximation error in the variances was quite small. The median error was 0.018. For comparison the true variances ranged from 0.01 to 0.94 with a mean of 0.322. Also, the nodes for which the approximation error was worse were indeed the nodes that converged slower. 678 Y. Weiss and W T Freeman 3 Discussion Independently, two other groups have recently analyzed special cases of Gaussian graphical models. Frey [7] analyzed the graphical model corresponding to factor analysis and gave conditions for the existence of a stable fixed-point. Rusmevichientong and Van Roy [14] analyzed a graphical model with the topology of turbo decoding but a Gaussian joint density. For this specific graph they gave sufficient conditions for convergence and showed that the means are exact. Our main interest in the Gaussian case is to understand the performance of belief propagation in general networks with multiple loops. We are struck by the similarity of our results for Gaussians in arbitrary networks and the results for single loops of arbitrary distributions [18]. First, in single loop networks with binary nodes, loopy belief at a node and the true belief at a node are maximized by the same assignment while the confidence in that assignment is incorrect. In Gaussian networks with multiple loops, the mean at each node is correct but the confidence around that mean may be incorrect. Second, for both singleloop and Gaussian networks, fast belief propagation convergence correlates with accurate beliefs. Third, in both Gaussians and discrete valued single loop networks, the statistical dependence between root and leaf nodes governs the convergence rate and accuracy. The two models are quite different. Mean field approximations are exact for Gaussian MRFs while they work poorly in sparsely connected discrete networks with a single loop. The results for the Gaussian and single-loop cases lead us to believe that similar results may hold for a larger class of networks. Can our analysis be extended to non-Gaussian distributions? The basic idea applies to arbitrary graphs and arbitrary potentials: belief propagation is performing exact inference on a tree that has the same local neighbor structure as the loopy graph. However, the linear algebra that we used to calculate exact expressions for the error in belief propagation at any iteration holds only for Gaussian variables. We have used a similar approach to analyze the related "max-product" belief propagation algorithm on arbitrary graphs with arbitrary distributions [5] (both discrete and continuous valued nodes). We show that if the max-product algorithm converges, the max-product assignment has greater posterior probability then any assignment in a particular large region around that assignment. While this is a weaker condition than a global maximum, it is much stronger than a simple local maximum of the posterior probability. The sum-product and max-product belief propagation algorithms are fast and parallelizable. Due to the well known hardness of probabilistic inference in graphical models, belief propagation will obviously not work for arbitrary networks and distributions. Nevertheless, a growing body of empirical evidence shows its success in many networks with loops. Our results justify applying belief propagation in certain networks with mUltiple loops. This may enable fast, approximate probabilistic inference in a range of new applications. References [1] S.M. Aji, G.B. Hom, and R.J. McEliece. On the convergence of iterative decoding on graphs with a single cycle. In Proc. 1998 ISIT, 1998. [2] c. Berrou, A. Glavieux, and P. Thitimajshima. Near Shannon limit error-correcting coding and decoding: Turbo codes. In Proc. IEEE International Communications Conference '93, 1993. [3] R. Cowell. Advanced inference in Bayesian networks. In M.1. Jordan, editor, Learning in Graphical Models. MIT Press, 1998. [4] G.D. Forney, F.R. Kschischang, and B. Marcus. Iterative decoding of tail-biting trellisses. preprint presented at 1998 Information Theory Workshop in San Diego, 1998. Correctness of Belief Propagation 679 [5] W. T. Freeman and Y. Weiss. On the fixed points of the max-product algorithm. Technical Report 99-39, MERL, 201 Broadway, Cambridge, MA 02139, 1999. [6] W.T. Freeman and E.C. Pasztor. Learning to estimate scenes from images. In M.S. Kearns, S.A. SoUa, and D.A. Cohn, editors, Adv. Neural Information Processing Systems I I. MIT Press, 1999. [7] B.J. Frey. Turbo factor analysis. In Adv. Neural Information Processing Systems 12. 2000. to appear. [8) Brendan J. Frey. Bayesian Networksfor Pattern Classification, Data Compression and Channel Coding. MIT Press, 1998. [9) R.G. Gallager. Low Density Parity Check Codes. MIT Press, 1963. [10) F. R. Kschischang and B. J. Frey. Iterative decoding of compound codes by probability propagation in graphical models. IEEE Journal on Selected Areas in Communication , 16(2):219-230, 1998. [11] R.J. McEliece, D.J.C. MackKay, and J.F. Cheng. Turbo decoding as as an instance of Pearl's 'belief propagation' algorithm. IEEE Journal on Selected Areas in Communication, 16(2): 140152,1998. [12J R.J. McEliece, E. Rodemich, and J.F. Cheng. The Turbo decision algorithm. In Proc. 33rd Allerton Conference on Communications, Control and Computing, pages 366-379, Monticello, IL, 1995. [I3J K.P. Murphy, Y. Weiss, and M.1. Jordan. Loopy belief propagation for approximate inference: an empirical study. In Proceedings of Uncertainty in AI, 1999. [14] Rusmevichientong P. and Van Roy B. An analysis of Turbo decoding with Gaussian densities. In Adv. Neural Information Processing Systems I2 . 2000. to appear. [15) Judea Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, 1988. [16J Gilbert Strang. Introduction to Applied Mathel1Ultics. Wellesley-Cambridge, 1986. [I7J Y. Weiss. Belief propagation and revision in networks with loops. Technical Report 1616, MIT AI lab, 1997. [18J Y. Weiss. Correctness of local probability propagation in graphical models with loops. Neural Computation, to appear, 2000. [19] Y. Weiss and W. T. Freeman. Loopy propagation gives the correct posterior means for Gaussians. Technical Report UCB.CSD-99-1046, Berkeley Computer Science Dept., 1999. www.cs.berkeley.edu yweiss/. [20J N. Wiberg. Codes and decoding on general graphs. PhD thesis, Department of Electrical Engineering, U. Linkoping, Sweden, 1996.
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Training Data Selection for Optimal Generalization in Trigonometric Polynomial Networks Masashi Sugiyama*and Hidemitsu Ogawa Department of Computer Science, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro-ku, Tokyo, 152-8552, Japan. sugi@cs. titeck. ac.jp Abstract In this paper, we consider the problem of active learning in trigonometric polynomial networks and give a necessary and sufficient condition of sample points to provide the optimal generalization capability. By analyzing the condition from the functional analytic point of view, we clarify the mechanism of achieving the optimal generalization capability. We also show that a set of training examples satisfying the condition does not only provide the optimal generalization but also reduces the computational complexity and memory required for the calculation of learning results. Finally, examples of sample points satisfying the condition are given and computer simulations are performed to demonstrate the effectiveness of the proposed active learning method. 1 Introduction Supervised learning is obtaining an underlying rule from training examples, and can be formulated as a function approximation problem. If sample points are actively designed, then learning can be performed more efficiently. In this paper, we discuss the problem of designing sample points, referred to as active learning, for optimal generalization. Active learning is classified into two categories depending on the optimality. One is global optimal, where a set of all training examples is optimal (e.g. Fedorov [3]). The other is greedy optimal, where the next training example to sample is optimal in each step (e.g. MacKay [5], Cohn [2], Fukumizu [4], and Sugiyama and Ogawa [10]). In this paper, we focus on the global optimal case and give a new active learning method in trigonometric polynomial networks. The proposed method does not employ any approximations in its derivation, so that it provides exactly the optimal generalization capability. Moreover, the proposed method reduces the computational complexity and memory required for the calculation of learning results. Finally, the effectiveness of the proposed method is demonstrated through computer simulations. ''http://ogawa-www.cs.titech.ac·jprsugi. Training Data Selection for Optimal Generalization 625 2 Formulation of supervised learning In this section, the supervised learning problem is formulated from the functional analytic point of view (see Ogawa [7]). Then, our learning criterion and model are described. 2.1 Supervised learning as an inverse problem Let us consider the problem of obtaining the optimal approximation to a target function f(x) of L variables from a set of M training examples. The training examples are made up of sample points Xm E V, where V is a subset of the Ldimensional Euclidean space R L, and corresponding sample values Ym E C: {(xm, Ym) 1 Ym = f(xm) + nm}~=l' (1) where Ym is degraded by zero-mean additive noise nm. Let nand Y be Mdimensional vectors whose m-th elements are nm and Ym, respectively. Y is called a sample value vector. In this paper, the target function f(x) is assumed to belong to a reproducing kernel Hilbert space H (Aronszajn [1]). If H is unknown, then it can be estimated by model selection methods (e.g. Sugiyama and Ogawa [9]). Let K(·,·) be the reproducing kernel of H. If a function 'l/Jm(x) is defined as 'l/Jm (x) = K (x, xm), then the value of f at a sample point Xm is expressed as f(xm) = (I, 'l/Jm), where (-,.) stands for the inner product. For this reason, 'l/Jm is called a sampling function. Let A be an operator defined as M A= ~ (em0~, (2) m=l where em is the m-th vector of the so-called standard basis in C M and (. 0 7) stands for the Neumann-Schatten productl. A is called a sampling operator. Then, the relationship between f and Y can be expressed as Y = Af +n. Let us denote a mapping from Y to a learning result fo by X: fo = Xy, (3) (4) where X is called a learning operator. Then, the supervised learning problem is reformulated as an inverse problem of obtaining X providing the best approximation fa to f under a certain learning criterion. 2.2 Learning criterion and model As mentioned above, function approximation is performed on the basis of a learning criterion. Our purpose of learning is to minimize the generalization error of the learning result fa measured by Je = Enllfo - f11 2 , (5) where En denotes the ensemble average over noise. In this paper, we adopt projection learning as our learning criterion. Let A*, R(A*), and PR(AO) be the adjoint operator of A, the range of A*, and the orthogonal projection operator onto R(A*), respectively. Then, projection learning is defined as follows. IFor any fixed 9 in a Hilbert space HI and any fixed f in a Hilbert space H2, the Neumann-Schatten product (f ® g) is an operator from HI to H2 defined by using any hE HI as (f®g)h = (h,g)f· 626 M Sugiyama and H. Ogawa Definition 1 (Projection learning) (Ogawa !6j) An operator X is called the projection learning operator if X minimizes the functional J p [X] = En II X nll 2 under the constraint XA = Pn(A*). It is well-known that Eq.(5) can be decomposed into the bias and variance: JG = IIPn(A*)f - fl12 + En11Xn112. (6) Eq.(6) implies that the projection learning criterion reduces the bias to a certain level and minimizes the variance. Let us consider the following function space. Definition 2 (Trigonometric polynomial space) Let x = (e(I),e(2), .. ·,e(L))T. For 1 :S l :S L, let Nl be a positive integer and Vl = [-7r,7r]. Then, a function space H is called a trigonometric polynomial space of order (N1 , N 2, ... , N L) if H is spanned by (7) (8) The dimension J.l of a trigonometric polynomial space of order (N1 , N 2 , ... , N L) is J.l = nf=1 (2Nl + 1), and the reproducing kernel of this space is expressed as where K, (~{l) , ~('}') = { L K(x, x') = II Kl(e(l), e(l)I), l=1 if e(l) -=1= e(l)' , if e(l) = e(l)I. 3 Active learning in trigonometric polynomial space (9) (10) The problem of active learning is to find a set {Xm}~=1 of sample points providing the optimal generalization capability. In this section, we give the optimal solution to the active learning problem in the trigonometric polynomial space. Let At be the Moore-Penrose generalized inverse 2 of A. Then, the following proposition holds. Proposition 1 If the noise covariance matrix Q is given as Q = a 2 I with a 2 > 0, then the projection learning operator X is expressed as X = At. Note that the sampling operator A is uniquely determined by {Xm}~=1 (see Eq.(2)). From Eq.(6), the bias of a learning result fo becomes zero for all f in H if and only if N(A) = {O}, where NO stands for the null space of an operator. For this reason, 2 An operator X is called the Moore-Penrose generalized inverse of an operator A if X satisfies AXA = A, XAX = X, (AX)'" = AX, and (XA)" = XA . Training Data Selection for Optimal Generalization 627 H Figure 1: Mechanism of noise suppression by Theorem 1. If a set {xm}~= l of sample points satisfies A* A = MI, then XAf = f, IIXntll = JMlln111, and Xn2 = o. we consider the case where a set {Xm}~=l of sample points satisfies N(A) = {o}. In this case, Eq.(6) is reduced to (11) which is equivalent to the noise variance in H. Consequently, the problem of active learning becomes the problem of finding a set {Xm }~= 1 of sample points minimizing Eq.(l1) under the constraint N(A) = {a}. First, we derive a condition for optimal generalization in terms of the sampling operator A. Theorem 1 Assume that the noise covariance matrix Q is given as Q = (721 with (72 > o. Then, Je in Eq.{11) is minimized under the constraint N(A) = {O} if and only if A*A =MI, (12) where I denotes the identity operator on H. In this case, the minimum value of Je is (72J.L/M, where J.L is the dimension of H . Eq.(12) implies that {:;k1jJm}~=l forms a pseudo orthonormal basis (Ogawa [8]) in H, which is an extension of orthonormal bases. The following lemma gives interpretation of Theorem 1. Lemma 1 When a set {Xm}~=l of sample points satisfies Eq.(12}, it holds that XAf f for all f E H, (13) IIAfl1 rullfll for all f E H, (14) IIXul1 { *llull for u E 'R.(A), (15) o for u E'R.(A).l. Eqs.(14) and (15) imply that kA becomes an isometry and VMX becomes a partial isometry with the initial space 'R.(A) , respectively. Let us decompose the noise n as n = nl + n2, where nl E 'R.(A) and n2 E 'R.(A).l. Then, the sample value vector y is rewritten as y = Af + nl + n2. It follows from Eq.(13) that the signal component Af is transformed into the original function f by X. From Eq.(15), X suppresses the magnitude of noise nl in 'R.(A) by k and completely removes the 628 • • -71" C .----. . 211" M ( a) Theorem 2 71" M Sugiyama and H. Ogawa • • • -71" C (b) Theorem 3 Figure 2: Two examples of sample points such that Condition (12) holds (1-£ = 3 and M = 6). noise n2 in R(A).l. This analysis is summarized in Fig.1. Note that Theorem 1 and its interpretation are valid for all Hilbert spaces such that K(x, x) is a constant for any x. In Theorem 1, we have given a necessary and sufficient condition to minimize Ja in terms of the sampling operator A. Now we give two examples of sample points {Xm };;:[=l such that Condition (12) holds. From here on, we focus on the case when the dimension L of the input x is 1 for simplicity. However, the following results can be easily scaled to the case when L > 1. Theorem 2 Let M 2: 1-£, where 1-£ is the dimension of H. Let c be an arbitrary constant such that -71" < c :::; -71" + ~. If a set {Xm};;:[=l of sample points is determined as (16) then Eq.(12) holds. Theorem 3 Let M = kl-£ where k is a positive integer. Let c be an arbitrary constant such that -71" :::; c :::; -71" + ~. If a set {Xm};;:[=l of sample points is determined as then Eq. (12) holds. 271" Xm =c+-r, 1-£ where r = m - 1 (mod 1-£), (17) Theorem 2 means that M sample points are fixed to 271" 1M intervals in the domain [-71",71"] and sample values are gathered once at each point (see Fig.2 (a» . In contrast, Theorem 3 means that 1-£ sample points are fixed to 271"11-£ intervals in the domain and sample values are gathered k times at each point (see Fig.2 (b». Now, we discuss calculation methods ofthe projection learning result fo(x). Let hm be the m-th column vector of the M-dimensional matrix (AA*)t. Then, for general sample points, the projection learning result fo(x) can be calculated as M fo(x) = L (y, hm)1/Jm(x). (18) m=l When we use the optimal sample points satisfying Condition (12), the following theorems hold. Theorem 4 When Eq.(12) holds, the projection learning result fo(x) can be calculated as 1 M fo(x) = M LYm1/Jm(X). (19) m=l Training Data Selection for Optimal Generalization 629 Theorem 5 When sample points are determined following Theorem 3, the projection learning result fo (x) can be calculated as 1 J.I. fo(x) = - LYp'I/Jp(x), I-' p=l 1 k where YP = k LYp+J.I.(q-l). q=l (20) In Eq.(18), the coefficient of 'l/Jm(x) is obtained by the inner product (y, hm). In contrast, it is replaced with Ym/M in Eq.(19), which implies that the Moore-Penrose generalized inverse of AA* is not required for calculating fo(x). This property is quite useful when the number M of training examples is very large since the calculation of the Moore-Penrose generalized inverse of high dimensional matrices is sometimes unstable. In Eq.(20), the number of basis functions is reduced to I-' and the coefficient of 'l/Jp(x) is obtained by Yp/I-', where YP is the mean sample values at xp. For general sample points, the computational complexity and memory required for calculating fo(x) by Eq.(18) are both O(M2). In contrast, Theorem 4 states that if a set of sample points satisfies Eq.(12), then both the computational complexity and memory are reduced to O(M). Hence, Theorem 1 and Theorem 4 do not only provide the optimal generalization but also reduce the computational complexity and memory. Moreover, if we determine sample points following Theorem 3 and calculate the learning result fo(x) by Theorem 5, then the computational complexity and memory are reduced to 0 (1-'). This is extremely efficient since I-' does not depend on the number M of training examples. The above results are shown in Tab.1. 4 Simulations In this section, the effectiveness of the proposed active learning method is demonstrated through computer simulations. Let H be a trigonometric polynomial space of order 100, and the noise covariance matrix Q be Q = I . Let us consider the following three sampling schemes. (A) Optimal sampling: Training examples are gathered following Theorem 3. (B) Experimental design: Eq.(2) in Cohn [2] is adopted as the active learning criterion. The value of this criterion is evaluated by 30 reference points. The sampling location is determined by multi-point-search with 3 candidates. (C) Passive learning: Training examples are given unilaterally. Fig.3 shows the relation between the number of training examples and the generalization error. The horizontal and vertical axes display the number of training examples and the generalization error Je measured by Eq.(5), respectively. The solid line shows the sampling scheme (A). The dashed and dotted lines denote the averages of 10 trials of the sampling schemes (B) and (C), respectively. When the number of training examples is 201, the generalization error of the sampling scheme (A) is 1 while the generalization errors of the sampling schemes (B) and (C) are 3.18 x 104 and 8.75 x 104, respectively. This graph illustrates that the proposed sampling scheme gives much better generalization capability than the sampling schemes (B) and (C) especially when the number of training examples is not so large. 5 Conclusion We proposed a new active learning method in the trigonometric polynomial space. The proposed method provides exactly the optimal generalization capability and 630 M Sugiyama and H Ogawa Table 1: Computational complexity 1 Or----,---:---,.-~---r---____." and memory required for projection learning. Computational Calculation Complexity methods and Memory Eq.(18) O(M2) Theorem 4 O(M) Theorem S§ O(J.L) § M = kJ.L where J.L is the dimension of Hand k is a positive integer. ... 8 E (jj c:: ,g 6 til .~ Cii (jj c:: 4 CI) Ol CI) .£: I- 2 Optimal sampling Experimental design Passive learning I I I I , I I I I I " .' ..... ' . . , ' . ..... '.;. .... ........ "- ........... - .... OL-----~------~-----~-----~ 300 400 500 600 The number of training examples Figure 3: Relation between the number of training examples and the generalization error. at the same time, it reduces the computational complexity and memory required for the calculation of learning results. The mechanism of achieving the optimal generalization was clarified from the functional analytic point of view. References [1] N. Aronszajn. Theory of reproducing kernels. Transactions on American Mathematical Society, 68:337-404, 1950. [2] D. Cohn, Neural network exploration using optimal experiment design. In J. Cowan et al. (Eds.), Advances in Neural Information Processing Systems 6, pp. 679-686. Morgan-Kaufmann Publishers Inc., San Mateo, CA, 1994. [3J V. V. Fedorov. Theory of Optimal Experiments. Academic Press, New York, 1972. [4] K. Fukumizu. Active learning in multilayer perceptrons. In D. Touretzky et al. (Eds.), Advances in Neural Information Processing Systems 8, pp. 295-301. The MIT Press, Cambridge, 1996. [5] D. MacKay. Information-based objective functions for active data selection. Neural Computation, 4(4):590-604, 1992. [6] H. Ogawa, Projection filter regularization of ill-conditioned problem. In Proceedings of SPIE, 808, Inverse Problems in Optics, pp. 189-196, 1987. [7] H, Ogawa. Neural network learning, generalization and over-learning. In Proceedings of the ICIIPS'92, International Conference on Intelligent Information Processing fj System, vol. 2, pp. 1-6, Beijing, China, 1992. [8] H. Ogawa. Theory of pseudo biorthogonal bases and its application. In Research Institute for Mathematical Science, RIMS Kokyuroku, 1067, Reproducing Kernels and their Applications, pp. 24-38, 1998. [9] M. Sugiyama and H. Ogawa. Functional analytic approach to model selectionSubspace information criterion. In Proceedings of 1999 Workshop on InformationBased Induction Sciences (IBIS'99), pp. 93-98, Syuzenji, Shizuoka, Japan, 1999 (Its complete version is available at ftp://ftp.cs.titech.ac.jp/pub/TR/99/TR990009.ps.gz). [10] M. Sugiyama and H. Ogawa. Incremental active learning in consideration of bias, Technical Report of IEICE, NC99-56, pp. 15-22, 1999 (Its complete version is available at ftp://ftp.cs.titech.ac.jp/pub/TR/99/TR99-001O.ps.gz).
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Image Recognition in Context: Application to Microscopic Urinalysis XuboSong* Department of Electrical and Computer Engineering Oregon Graduate Institute of Science and Technology Beaverton, OR 97006 xubosong@ece.ogi.edu Joseph Sill Department of Computation and Neural Systems California Institute of Technology Pasadena, CA 91125 joe@busy.work.caltech.edu Yaser Abu-Mostafa Department of Electrical Engineering California Institute of Technology Pasadena, CA 91125 yase r@work.caltech.edu Harvey Kasdan International Remote Imaging Systems, Inc. Chatsworth, CA 91311 Abstract We propose a new and efficient technique for incorporating contextual information into object classification. Most of the current techniques face the problem of exponential computation cost. In this paper, we propose a new general framework that incorporates partial context at a linear cost. This technique is applied to microscopic urinalysis image recognition, resulting in a significant improvement of recognition rate over the context free approach. This gain would have been impossible using conventional context incorporation techniques. 1 BACKGROUND: RECOGNITION IN CONTEXT There are a number of pattern recognition problem domains where the classification of an object should be based on more than simply the appearance of the object itself. In remote sensing image classification, where each pixel is part of ground cover, a pixel is more likely to be a glacier if it is in a mountainous area, than if surrounded by pixels of residential areas. In text analysis, one can expect to find certain letters occurring regularly in particular arrangement with other letters(qu, ee,est, tion, etc.). The information conveyed by the accompanying entities is referred to as contextual information. Human experts apply contextual information in their decision making [2][ 6]. It makes sense to design techniques and algorithms to make computers aggregate and utilize a more complete set of information in their decision making the way human experts do. In pattern recognition systems, however, *Author for correspondence 964 X B. Song, J Sill, Y. Abu-Mostafa and H. Kasdan the primary (and often only) source of information used to identify an object is the set of measurements, or features, associated with the object itself. Augmenting this information by incorporating context into the classification process can yield significant benefits. Consider a set of N objects Ti , i = 1, ... N. With each object we associate a class label Ci that is a member of a label set n = {1, ... , D} . Each object Ti is characterized by a set of measurements Xi E R P, which we call a feature vector. Many techniques [1][2][4J[6} incorporate context by conditioning the posterior probability of objects' identities on the joint features of all accompanying objects. i.e .• P(Cl, C2,··· , cNlxl , . . . , XN). and then maximizing it with respectto Cl, C2, . .. , CN. It can be shown thatp(cl,c2, . . . ,cNlxl, . . . ,xN) ex p(cllxl) ... p(CNlxN) (~c i ""'(N\ given p 1 •.. p CN certain reasonable assumptions. Once the context-free posterior probabilities p( Ci IXi) are known. e.g. through the use of a standard machine learning model such as a neural network, computing P(Cl, ... ,CNlxl, . . . ,XN) for all possible Cl, ... ,CN would entail (2N + 1)DN multiplications. and finding the maximum has complexity of DN. which is intractable for large Nand D. [2J Another problem with this formulation is the estimation of the high dimensional joint distribution p( Cl, ... , CN), which is ill-posed and data hungry. One way of dealing with these problems is to limit context to local regions. With this approach, only the pixels in a close neighborhood. or letters immediately adjacent are considered [4][6][7J. Such techniques may be ignoring useful information, and will not apply to situations where context doesn't have such locality, as in the case of microscopic urinalysis image recognition. Another way is to simplify the problem using specific domain knowledge [1], but this is only possible in certain domains. These difficulties motivate the efficient incorporation of partial context as a general framework, formulated in section 2. In section 3, we discuss microscopic urinalysis image recognition. and address the importance of using context for this application. Also in section 3, techniques are proposed to identify relevant context. Empirical results are shown in section 4. followed by discussions in section 5. 2 FORMULATION FOR INCORPORATION OF PARTIAL CONTEXT To avoid the exponential computational cost of using the identities of all accompanying objects directly as context, we use "partial context". denoted by A. It is called "partial" because it is derived from the class labels. as opposed to consisting of an explicit labelling of all objects. The physical definition of A depends on the problem at hand. In our application. A represents the presence or absence of certain classes. Then the posterior probability of an object Ti having class label Ci conditioned on its feature vector and the relevant context A is p(XiICi, A)P(Ci; A) P(Xi ; A) We assume that the feature distribution of an object depends only on its own class. i.e., p(xilci, A) = p(xi lci) . This assumption is roughly true for most real world problems. Then. Image Recognition in Context: Application to Microscopic Urinalysis 965 ( .1 . A) - p(xilci)p(Ci; A) _ ( .1 .)p(ciIA ) p(A)p(Xi) pC~Xt, -pCtXt p(xijJ~IIA) p(Ci) P(Xi; A) ()( p(cilxi) () = p(cilxi)P(Ci, A) P Ci where p(Ci, A) = p~(~j~) is called the context ratio, through which context plays its role. The context-sensitive posterior probability p( Ci lXi, A) is obtained through the context-free posterior probability p(cilxi) modified by the context ratio P(Ci, A). The partial-context maximum likelihood decision rule chooses class label Ci for element i such that Ci = argmaxp(cilxi, A) (I) Cj A systematic approach to identify relevant context A is addressed in section 3.3. The partial-context approach treats each element in a set individually, but with additional information from the context-bearing factor A. Once p(cilxi) are known for all i = 1, ... , N, and the context A is obtained, to maximize p(cilxi, A) from D possible values that Ci can take on and for all i, the total number of multiplications is 2N, and the complexity for finding the maximum is N D. Both are linear in N. The density estimation part is also trivial since it is very easy to estimate p(cIA). 3 MICROSCOPIC URINALYSIS 3.1 INTRODUCTION Urine is one of the most complex body fluid specimens: it potentially contains about 60 meaningful types of elements. Microscopic urinalysis detects the presence of elements that often provide early diagnostic information concerning dysfunction, infection, or inflammation of the kidneys and urinary tract. Thus this non-invasive technique can be of great value in clinical case management. Traditional manual microscopic analysis relies on human operators who read the samples visually and identify them, and therefore is time-consuming, labor-intensive and difficult to standardize. Automated microscopy of all specimens is more practical than manual microscopy, because it eliminates variation among different technologists. This variation becomes more pronounced when the same technologist examines increasing numbers of specimens. Also, it is less labor-intensive and thus less costly than manual microscopy. It also provides more consistent and accurate results. An automated urinalysis system workstation (The Y ellowI RI ST M, International Remote Imaging Systems, Inc.) has been introduced in numerous clinical laboratories for automated microscopy. Urine samples are processed and examined at lOOx (low power field) and 400x magnifications (high power field) with bright-field illumination. The Y ellowI RI ST M automated system collects video images of formed analytes in a stream of un centrifuged urine passing an optical assembly. Each image has one analyte in it. These images are given to a computer algorithm for automatic identification of analytes. Context is rich in urinalysis and plays a crucial role in analyte classification. Some combinations of analytes are more likely than others. For instance, the presence of bacteria indicates the presence of white blood cells, since bacteria tend to cause infection and thus trigger the production of more white blood cells. If amorphous crystals show up, they tend to show up in bunches and in all sizes. Therefore, if there are amorphous crystallook-alikes in various sizes, it is quite possible that they are amorphous crystals. Squamous epithelial cells can appear both flat or rolled up. If squamous epithelial cells in one form are detected, 966 X B. Song, J Sill, Y. Abu-Mostafa and H. Kasdan Table I: Features extracted from urine anylates images reature number ( 2 4 9 10 II 12 13 14 IS 16 reature desc:ription tht: m~an or hluc distribution the mean of gn...-cn dislrihulmn 15th paccnlile of £ray level hislo£ram 85 th percenlile of gray level hislogmm lh~ standard devia.tion \11' gray level intensity energy of the (.aplacian lransl\)rmalion of grey level image then it is likely that there are squamous epithelial cells in the other form. Utilizing such context is crucial for classification accuracy. The classes we are looking at are bacteria, calcium oxalate crystals, red blood cells, white blood cells, budding yeast, amorphous crystals, uric acid crystals, and artifacts. The task of automated microscopic urinalysis is, given a urine specimen that consists of up to a few hundred images of analytes, to classify each analyte into one of these classes. The automated urinalysis system we developed consists of three steps: image processing and feature extraction, learning and pattern recognition, and context incorporation. Figure 1 shows some example analyte images. Table 1 gives a list of features extracted from analyte images. 1 3.2 CONTEXT-FREE CLASSIFICATION The features are fed into a nonlinear feed-forward neural network with 16 inputs, 15 hidden units with sigmoid transfer functions, and 8 sigmoid output units. A cross-entropy error function is used in order to give the output a probability interpretation. Denote the input feature vector as x, the network outputs a D dimensional vector (D = 8 in our case) p = {p(dlx)} , d = 1, ... , D, where p(dlx) is p{dlx) = Prob( an analyte belongs to class dl feature x) The decision made at this stage is d{x) = argmax p(dlx) d 3.3 IDENTIFICATION OF RELEVANT PARTIAL CONTEXT Not all classes are relevant in terms of carrying contextual information. We propose three criteria based on which we can systematicalIy investigate the relevance of the class presence. To use these criteria, we need to know the folIowing distributions: the class prior distribution p(c) for c = 1, ... ,D; the conditional class distribution p{cIAd) for c = 1, ... ,D 1 >'1 and >'2 are respectively the larger and the smaller eigenvalues of the second moment matrix of an image. Image Recognition in Context: Application to Microscopic Urinalysis 967 and d = 1, . .. ,D; and the class presence prior distribution p(Ad) for d = 1, . . . ,D. Ad is a binary random variable indicating the presence of class d. Ad = 1 if class d is present, and Ad = 0 otherwise. All these distributions can be easily estimated from the database. The first criterion is the correlation coefficient between the presence of any two classes; the second one is the classical mutual information I(e; Ad) between the presence of a class Ad and the class probability pee), where I(e; Ad) is defined as I(e; Ad) = H(e) H(eIAd) where H(e) = 2:~1 p(e = i)ln(p(e = i)) is the entropy of the class priors and H(eIAd) = P(Ad = I)H(eIAd = 1)+P(Ad = O)H(eIAd = 0) is the conditional entropy of e conditioned on Ad. The third criterion is what we call the expected relative entropy D(eIIAd) between the presence ofa class Ad and the labeling probability pee), which we define as D(eIIAd) = P(Ad = I)D(p(e)llp(eIAd = 1)) + P(Ad = O)D(p(e)llp(eIAd = 0)) where D(p(e)llp(eIAd 1)) 2:~lP(e = ilAd = l)ln(p(c;/l~t)=l)) and D(p(e)llp(eIAd = 0)) = 2:~1 p(e = ilAd = O)ln(p(C;/l~t)=O)) According to the first criterion, one type of analyte is considered relevant to another if the absolute value of their correlation coefficient is beyond a certain threshold. It shows that uric acid crystals, budding yeast and calcium oxalate crystals are not relevant to any other types even by a generous threshold of 0.10. Similarly, the bigger the mutual information between the presence of a class and the class distribution, the more relevant this class is. Ranking the analyte types in terms of I(e; Ad) in a descending manner gives rise to the following list: bacteria, amorphous crystals, red blood cells, white blood cells, uric acid crystals, budding yeast and calcium oxalate crystals. Once again, ranking the analyte types in terms of D(eIIAd) in a descending manner gives rise to the following list: bacteria, red blood cells, amorphous crystals, white blood cells, calcium oxalate crystals, budding yeast and uric acid crystals. All three criteria lead to similar conclusions regarding the relevance of class presence - bacteria, red blood cells, amorphous crystals, and white blood cells are relevant, while calcium oxalate crystals, budding yeast and uric acid crystals are not. (Baed on prior knowledge, we discard artifacts from the outset as an irrelevant class.) 3.4 ALGORITHM FOR INCORPORATING PARTIAL CONTEXT Once the M relevant classes are identified, the following algorithm is used to incorporate partial context. Step 0 Estimate the priors p(eIAd) and pee), for e E {I, 2, .. . , D} and d E {I, 2, ... , D}. Step 1 For a given Xi, compute p(edxi) for ei = 1,2, . .. , Dusing whichever base machine learning model is preferred ( in our case, a neural network). Step 2 Let the M relevant classes be R1 , ..• , RM. According to the no-context p( ei IXi) and certain criteria for detecting the presence or absence of all the relevant classes, get ARI ,· ·· ,ARM' Step 3 Letp(ei lXi, Ao) = p(eilxi), where Ao is the null element. Incorporate context from each relevant class sequentially, i.e., for m = 1 to M, iteratively compute p(eiIARTn)p(AR"J p(eilxi; Ao, .. . , ARm_I ' ARTn) = p(eilxi' Ao,.· . , ARTn_J pee) Step 4 Recompute ARI , . . . ,ARM based on the new class labellings. Return to step 3 and repeat until algorithm converges.2 2Hence, the algorithm has an E-M flavor, in that it goes back and forth between finding the most 968 X B. Song, J. Sill, Y. Abu-Mostafa and H Kasdan amorphous crystals artifacts calcium oxalate crystals hyaline casts Figure I: Example of some of the analyte images. Step 5 Label the objects according to the final context-contammg p(cilxi, ARI'·· · ' ARM)' i.e., Ci = argmaxp(ciIXi, AR1 , ... , ARM) for i = 1, ... , N. Ci This algorithm is invariant with respect to the ordering of the M relevant classes in (Ai, ... , AM). The proof is omitted here. 4 RESULTS The algorithm using partia.1 context was tested on a database of 83 urine specimens, containing a total of 20,276 analyte images. Four classes are considered relevant according to the criteria described in section 3.3: bacteria, red blood cells, white blood cells and amorphous crystals. We measure two types of error: analyte-by-analyte error, and specimen diagnostic error. The average analyte-by-analyte error is reduced from 44.48% before using context to 36.66% after, resulting a relative error reduction of 17.6% (Table 2). The diagnosis for a specimen is either normal or abnormal. Tables 3 and 4 compare the diagnostic performance with and without using context, and Table 5 lists the relative changes. We can see using context significantly increases correct diagnosis for both normal and abnormal specimens, and reduces both false positives and false negatives. without context with context average element-by-element error 44.48 % 36.66 % Table 2: Comparison of using and not using contextual information for analyte-by-analyte error. probable class labels given the context and determining the context given the class labels. Image Recognition in Context: Application to Microscopic Urinalysis 969 estimated normal estimated abnormal truly normal 40.96 % 7.23 % truly abnormal 19.28 % 32.53 % Table 3: Diagnostic confusion matrix not using context estimated normal estimated abnormal truly normal 42.17 % 6.02% truly abnormal 16.87 % 34.94 % Table 4: Diagnostic confusion matrix using context estimated normal estimated abnormal truly normal +2.95 % -16.73 % truly abnormal - 12.50 % +7.41 % Table 5: Relative accuracy improvement (diagonal elements) and error reduction (off diagonal elements) in the diagnostic confusion matrix by using context. 5 CONCLUSIONS We proposed a novel framework that can incorporate context in a simple and efficient manner, avoiding exponential computation and high dimensional density estimation. The application of the partial context technique to microscopic urinalysis image recognition demonstrated the efficacy of the algorithm. This algorithm is not domain dependent, thus can be readily generalized to other pattern recognition areas. ACKNOWLEDGEMENTS The authors would like to thank Alexander Nicholson, Malik Magdon-Ismail, Amir Atiya at the Caltech Learning Systems Group for helpful discussions. References [I) Song, X.B. & SilU. & Abu-Mostafa & Harvey Kasdan, (1997) "Incorporating Contextual Information in White Blood Cell Identification", In M. Jordan, MJ. Kearns and S.A. Solla (eds.), Advances in Neural Information Processing Systems 7,1997, pp. 950-956. Cambridge, MA: MIT Press. [2] Song, Xubo (1999) "Contextual Pattern Recognition with Application to Biomedical Image Identification", Ph.D. Thesis, California Institute of Science and Technology. [3) Boehringer-Mannheim-Corporation, Urinalysis Today, Boehringer-Mannheim-Corporation, 1991. [4] Kittler, J.."Relaxation labelling", Pattern Recognition Theory and Applications, 1987, pp. 99108., Pierre A. Devijver and Josef Kittler, Editors, Springer-Verlag. [5] Kittler, J. & Illingworth, J., "Relaxation Labelling Algorithms - A Review", Image and Vision Computing, 1985, vol. 3, pp. 206-216. [6] Toussaint, G., "The Use of Context in Pattern Recognition", Pattern Recognition, 1978, vol. 10, pp. 189-204. [7] Swain, P. & Vardeman, S. & Tilton, J., "Contextual Classification of Multispectral Image Data", Pattern Recognition, 1981, Vol. 13, No.6, pp. 429-441.
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