Recently, however, there have been some successful applications of neural networks in a totally different context - that of sequential decision making under uncertainty (stochastic control). Stochastic control problems have been studied extensively in the operations research and control theory literature for a long time, using the methodology of dynamic programming [Bertsekas, 1995]. In dynamic programming, the most important object is the cost-to-go (or value) junction, which evaluates the expected future 1046 B.V. ROY, 1. N. TSITSIKLIS

Department of Brain and Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 Abstract We present a new algorithm for associative reinforcement learning. Thealgorithm is based upon the idea of matching a network's output probability with a probability distribution derived from the environment's reward signal. This Probability Matching algorithm is shown to perform faster and be less susceptible to local minima than previously existing algorithms. We use Probability Matching totrain mixture of experts networks, an architecture for which other reinforcement learning rules fail to converge reliably on even simple problems. This architecture is particularly well suited for our algorithm as it can compute arbitrarily complex functions yet calculation of the output probability is simple. 1 INTRODUCTION The problem of learning associative networks from scalar reinforcement signals is notoriously difficult.

The controllers usually have no or only very little prior knowledge of the environment. While only local communication between controllers is allowed, the controllers must cooperate among themselves to achieve the common, global objective. Finding the optimal routing policy in such a distributed manner is very difficult. Moreover, since the environment is non-stationary, the optimal policy varies with time as a result of changes in network traffic and topology.

In Tesauro's 1992 landmark work on TD--gammon, he showed that the temporal dif--

Elucidation of the relationship between TD learning and dynamic programming (DP) has provided good theoretical insights (Barto et al., 1995). However, conventional TD algorithms were based on discrete-time, discrete-state formulations. In applying these algorithms to control problems, time, space and action had to be appropriately discretized using a priori knowledge or by trial and error. Furthermore, when a TD algorithm is used for neurobiological modeling, discrete-time operation is often very unnatural. There have been several attempts to extend TD-like algorithms to continuous cases. Bradtke et al. (1994) showed convergence results for DPbased algorithms for a discrete-time, continuous-state linear system with a quadratic cost. Bradtke and Duff (1995) derived TD-like algorithms for continuous-time, discrete-state systems (semi-Markov decision problems). Baird (1993) proposed the "advantage updating" algorithm by modifying Q-Iearning so that it works with arbitrary small time steps.

On large problems, reinforcement learning systems must use parameterized functionapproximators such as neural networks in order to generalize between similar situations and actions. In these cases there are no strong theoretical results on the accuracy of convergence, and computational resultshave been mixed. In particular, Boyan and Moore reported at last year's meeting a series of negative results in attempting to apply dynamic programming together with function approximation to simple control problems with continuous state spaces. In this paper, we present positive results for all the control tasks they attempted, and for one that is significantly larger. The most important differences are that we used sparse-coarse-coded function approximators (CMACs) whereas they used mostly global function approximators, and that we learned online whereas they learned offline. Boyan and Moore and others have suggested that the problems they encountered could be solved by using actual outcomes ("rollouts"), as in classical Monte Carlo methods, and as in the TD().) algorithm when).

We describe the reinforcement learning problem, motivate algorithms whichseek an approximation to the Q function, and present new convergence results for two such algorithms. 1 INTRODUCTION AND BACKGROUND Imagine an agent acting in some environment. At time t, the environment is in some state Xt chosen from a finite set of states. The agent perceives Xt, and is allowed to choose an action at from some finite set of actions. Meanwhile, the agent experiences a real-valued cost Ct, chosen from a distribution which also depends only on Xt and at and which has finite mean and variance. Such an environment is called a Markov decision process, or MDP.

Reinforcement learning has become one of the most actively studied learning frameworks in the area of intelligent autonomous agents. This article describes the results of a three-day meeting of leading researchers in this area that was sponsored by the National Science Foundation. Because reinforcement learning is an interdisciplinary topic, the workshop brought together researchers from a variety of fields, including machine learning, neural networks, AI, robotics, and operations research. The goals of the meeting were to (1) understand limitations of current reinforcement-learning systems and define promising directions for further research; (2) clarify the relationships between reinforcement learning and existing work in engineering fields, such as operations research; and (3) identify potential industrial applications of reinforcement learning.

Reinforcement learning has become one of the most actively studied learning frameworks in the area of intelligent autonomous agents. This article describes the results of a three-day meeting of leading researchers in this area that was sponsored by the National Science Foundation. Because reinforcement learning is an interdisciplinary topic, the workshop brought together researchers from a variety of fields, including machine learning, neural networks, AI, robotics, and operations research. Thirty leading researchers from the United States, Canada, Europe, and Japan, representing from many different universities, government, and industrial research laboratories participated in the workshop. The goals of the meeting were to (1) understand limitations of current reinforcement-learning systems and define promising directions for further research; (2) clarify the relationships between reinforcement learning and existing work in engineering fields, such as operations research; and (3) identify potential industrial applications of reinforcement learning.

This paper surveys the field of reinforcement learning from a computer-science perspective. It is written to be accessible to researchers familiar with machine learning. Both the historical basis of the field and a broad selection of current work are summarized. Reinforcement learning is the problem faced by an agent that learns behavior through trial-and-error interactions with a dynamic environment. The work described here has a resemblance to work in psychology, but differs considerably in the details and in the use of the word ``reinforcement.'' The paper discusses central issues of reinforcement learning, including trading off exploration and exploitation, establishing the foundations of the field via Markov decision theory, learning from delayed reinforcement, constructing empirical models to accelerate learning, making use of generalization and hierarchy, and coping with hidden state. It concludes with a survey of some implemented systems and an assessment of the practical utility of current methods for reinforcement learning.