We propose and analyze an algorithm that approximates solutions to the problem of optimal stopping in a discounted irreducible aperiodic Markovchain. The scheme involves the use of linear combinations offixed basis functions to approximate a Q-function. The weights of the linear combination are incrementally updated through an iterative process similar to Q-Iearning, involving simulation ofthe underlying Markov chain. Due to space limitations, we only provide an overview of a proof of convergence (with probability 1)and bounds on the approximation error. This is the first theoretical result that establishes the soundness of a Q-Iearninglike algorithmwhen combined with arbitrary linear function approximators tosolve a sequential decision problem.

The algorithm weanalyze performs online updating of a parameter vector during a single endless trajectory of an aperiodic irreducible finite state Markov chain. Results include convergence (with probability 1), a characterization of the limit of convergence, and a bound on the resulting approximation error. In addition to establishing new and stronger results than those previously available, our analysis is based on a new line of reasoning that provides new intuition about the dynamics of temporal-difference learning. Furthermore, we discuss the implications of two counterexamples with regards to the Significance of online updating and linearly parameterized function approximators. 1 INTRODUCTION The problem of predicting the expected long-term future cost (or reward) of a stochastic dynamic system manifests itself in both time-series prediction and control. Anexample in time-series prediction is that of estimating the net present value of a corporation, as a discounted sum of its future cash flows, based on the current state of its operations. In control, the ability to predict long-term future cost as a function of state enables the ranking of alternative states in order to guide decision-making. Indeed, such predictions constitute the cost-to-go function that is central to dynamic programming and optimal control (Bertsekas, 1995). Temporal-difference learning, originally proposed by Sutton (1988), is a method for approximating long-term future cost as a function of current state.

A Bayesian formulation of the problem leads to a clear concept of a solution whose computation, however, appears to entail an examination of an intractably-large number of hyperstates. This paper hassuggested extending the Gittins index approach (which applies with great power and elegance to the special class of multi-armed bandit processes) to general adaptive MDP's. The hope has been that if certain salient features of the value of information could be captured, even approximately, then one could be led to a reasonable method for avoiding certain defects of certainty-equivalence approaches (problems with identifiability, "metastability"). Obviously, positive evidence, in the form of empirical results from simulation experiments, would lend support to these ideas-work along these lines is underway. Local bandit approximation is but one approximate computational approach for problems of optimal learning and dual control. Most prominent in the literature of control theory is the "wide-sense" approach of [Bar-Shalom & Tse, 1976], which utilizes localquadratic approximations about nominal state/control trajectories. For certain problems, this method has demonstrated superior performance compared to a certainty-equivalence approach, but it is computationally very intensive and unwieldy, particularly for problems with controller dimension greater than one. One could revert to the view of the bandit problem, or general adaptive MDP, as simply a very large MDP defined over hyperstates, and then consider a some- Local Bandit Approximationfor Optimal Learning Problems 1025 what direct approach in which one performs approximate dynamic programming with function approximation over this domain-details of function-approximation, feature-selection, and "training" all become important design issues.

A central theme of computational vision research has been the realization thatreliable estimation of local scene properties requires propagating measurements across the image. Many authors have therefore suggested solving vision problems using architectures of locally connected units updating their activity in parallel. Unfortunately, theconvergence of traditional relaxation methods on such architectures has proven to be excruciatingly slow and in general they do not guarantee that the stable point will be a global minimum. In this paper we show that an architecture in which Bayesian Beliefs aboutimage properties are propagated between neighboring units yields convergence times which are several orders of magnitude fasterthan traditional methods and avoids local minima. In particular our architecture is non-iterative in the sense of Marr [5]: at every time step, the local estimates at a given location are optimal giventhe information which has already been propagated to that location. We illustrate the algorithm's performance on real images and compare it to several existing methods. 1 Theory The essence of our approach is shown in figure 1. Figure 1a shows the prototypical ill-posed problem: interpolation of a function from sparse data.

Images are ambiguous at each of many levels of a contextual hierarchy. Nevertheless,the high-level interpretation of most scenes is unambiguous, as evidenced by the superior performance of humans. Thisobservation argues for global vision models, such as deformable templates.Unfortunately, such models are computationally intractable for unconstrained problems. We propose a compositional modelin which primitives are recursively composed, subject to syntactic restrictions, to form tree-structured objects and object groupings. Ambiguity is propagated up the hierarchy in the form of multiple interpretations, which are later resolved by a Bayesian, equivalently minimum-description-Iength, cost functional.

This paper presents a new approach to speech recognition with hybrid HMM/ANN technology. While the standard approach to hybrid HMMI ANN systems is based on the use of neural networks as posterior probability estimators, the new approach is based on the use of mutual information neural networks trained with a special learning algorithm in order to maximize the mutual information between the input classes of the network and its resulting sequence of firing output neurons during training. It is shown in this paper that such a neural network is an optimal neural vector quantizer for a discrete hidden Markov model system trained on Maximum Likelihood principles. One of the main advantages of this approach is the fact, that such neural networks can be easily combined with HMM's of any complexity with context-dependent capabilities. It is shown that the resulting hybrid system achieves very high recognition rates, which are now already on the same level as the best conventional HMM systems with continuous parameters, and the capabilities of the mutual information neural networks are not yet entirely exploited.

Humans use visual as well as auditory speech signals to recognize spoken words. A variety of systems have been investigated for performing thistask. The main purpose of this research was to systematically comparethe performance of a range of dynamic visual features on a speechreading task. We have found that normalization ofimages to eliminate variation due to translation, scale, and planar rotation yielded substantial improvements in generalization performanceregardless of the visual representation used. In addition, the dynamic information in the difference between successive framesyielded better performance than optical-flow based approaches, and compression by local low-pass filtering worked surprisingly betterthan global principal components analysis (PCA). These results are examined and possible explanations are explored.

We describe the implementation of a hidden Markov model state decoding system, a component for a wordspotting speech recognition system.The key specification for this state decoder design is microwatt power dissipation; this requirement led to a continuoustime, analogcircuit implementation. We characterize the operation of a 10-word (81 state) state decoder test chip.

This paper discusses a probabilistic model-based approach to clustering sequences,using hidden Markov models (HMMs) . The problem can be framed as a generalization of the standard mixture model approach to clustering in feature space. Two primary issues are addressed. First, a novel parameter initialization procedure is proposed, and second, the more difficult problem of determining the number of clusters K, from the data, is investigated. Experimental resultsindicate that the proposed techniques are useful for revealing hidden cluster structure in data sets of sequences.

By adapting a framework used for supervised learning, we construct iterative algorithms that maximize the likelihood of the observations while also attempting to stay "close" to the current estimated parameters. We use a bound on the relative entropy between the two HMMs as a distance measure betweenthem. The result is new iterative training algorithms which are similar to the EM (Baum-Welch) algorithm for training HMMs. The proposed algorithms are composed of a step similar to the expectation step of Baum-Welch and a new update of the parameters which replaces the maximization (re-estimation) step. The algorithm takes only negligibly moretime per iteration and an approximated version uses the same expectation step as Baum-Welch.