It is widely accepted that the use of more compact representations than lookup tables is crucial to scaling reinforcement learning (RL) algorithms to real-world problems. Unfortunately almost all of the theory of reinforcement learning assumes lookup table representations. Inthis paper we address the pressing issue of combining function approximation and RL, and present 1) a function approximator basedon a simple extension to state aggregation (a commonly used form of compact representation), namely soft state aggregation, 2) a theory of convergence for RL with arbitrary, but fixed, soft state aggregation, 3) a novel intuitive understanding of the effect of state aggregation on online RL, and 4) a new heuristic adaptive state aggregation algorithm that finds improved compact representations by exploiting the non-discrete nature of soft state aggregation. Preliminary empirical results are also presented.

One of the fundamental properties that both neural networks and the central nervous system share is the ability to learn and generalize from examples. While this property has been studied extensively in the neural network literature it has not been thoroughly explored in human perceptual and motor learning. We have chosen a coordinate transformation system-the visuomotor map which transforms visual coordinates into motor coordinates-to study the generalization effects of learning new input-output pairs. Using a paradigm of computer controlled altered visual feedback, we have studied the generalization of the visuomotor map subsequent to both local and context-dependent remappings. A local remapping of one or two input-output pairs induced a significant global, yet decaying, change in the visuomotor map, suggesting a representation for the map composed of units with large functional receptive fields. Our study of context-dependent remappings indicated that a single point in visual space can be mapped to two different finger locations depending on a context variable-the starting point of the movement. Furthermore, as the context is varied there is a gradual shift between the two remappings, consistent with two visuomotor modules being learned and gated smoothly with the context. 1 Introduction The human central nervous system (CNS) receives sensory inputs from a multitude of modalities, each tuned to extract different forms of information from the 1126 Zoubin Ghahramani, Daniel M. Wolpert, Michael 1. Jordan

Statistical models of discrete time series have a wide range of applications, most notably to problems in speech recognition (Juang & Rabiner, 1991) and molecular biology (Baldi, Chauvin, Hunkapiller, & McClure, 1992). A common problem in these fields is to find a probabilistic model, and a set of model parameters, that 436 Lawrence K. Saul, Michael I. Jordan

It is widely accepted that the use of more compact representations than lookup tables is crucial to scaling reinforcement learning (RL) algorithms to real-world problems. Unfortunately almost all of the theory of reinforcement learning assumes lookup table representations. In this paper we address the pressing issue of combining function approximation and RL, and present 1) a function approximator based on a simple extension to state aggregation (a commonly used form of compact representation), namely soft state aggregation, 2) a theory of convergence for RL with arbitrary, but fixed, soft state aggregation, 3) a novel intuitive understanding of the effect of state aggregation on online RL, and 4) a new heuristic adaptive state aggregation algorithm that finds improved compact representations by exploiting the non-discrete nature of soft state aggregation. Preliminary empirical results are also presented.

For many types of learners one can compute the statistically "optimal" way to select data. We review how these techniques have been used with feedforward neural networks [MacKay, 1992; Cohn, 1994]. We then show how the same principles may be used to select data for two alternative, statistically-based learning architectures: mixtures of Gaussians and locally weighted regression. While the techniques for neural networks are expensive and approximate, the techniques for mixtures of Gaussians and locally weighted regression are both efficient and accurate.

An impedence controlled manipulandum for human movement studies.

Increasing attention has been paid to reinforcement learning algorithms in recent years, partly due to successes in the theoretical analysis of their behavior in Markov environments. If the Markov assumption is removed, however, neither generally the algorithms nor the analyses continue to be usable. We propose and analyze a new learning algorithm to solve a certain class of non-Markov decision problems. Our algorithm applies to problems in which the environment is Markov, but the learner has restricted access to state information. The algorithm involves a Monte-Carlo policy evaluation combined with a policy improvement method that is similar to that of Markov decision problems and is guaranteed to converge to a local maximum. The algorithm operates in the space of stochastic policies, a space which can yield a policy that performs considerably better than any deterministic policy. Although the space of stochastic policies is continuous-even for a discrete action space-our algorithm is computationally tractable.

An impedence controlled manipulandum for human movement studies.

Increasing attention has recently been paid to algorithms based on dynamic programming (DP) due to the suitability of DP for learning problems involving control. In stochastic environments where the system being controlled is only incompletely known, however, a unifying theoretical account of these methods has been missing. In this paper we relate DPbased learning algorithms to the powerful techniques of stochastic approximation via a new convergence theorem, enabling us to establish a class of convergent algorithms to which both TD("\) and Q-Iearning belong. 1 INTRODUCTION Learning to predict the future and to find an optimal way of controlling it are the basic goals of learning systems that interact with their environment. A variety of algorithms are currently being studied for the purposes of prediction and control in incompletely specified, stochastic environments. Here we consider learning algorithms defined in Markov environments. There are actions or controls (u) available for the learner that affect both the state transition probabilities, and the probability distribution for the immediate, state dependent costs (Ci(u)) incurred by the learner.

Real-world learning tasks may involve high-dimensional data sets with arbitrary patterns of missing data. In this paper we present a framework based on maximum likelihood density estimation for learning from such data set.s. VVe use mixture models for the density estimatesand make two distinct appeals to the Expectation Maximization (EM) principle (Dempster et al., 1977) in deriving a learning algorithm-EM is used both for the estimation of mixture componentsand for coping wit.h missing dat.a. The resulting algorithm is applicable t.o a wide range of supervised as well as unsupervised learning problems.