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 has suggested 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 local quadratic 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.

Probability models can be used to predict outcomes and compensate for missing data, but even a perfect model cannot be used to make decisions unless the utility of the outcomes, or preferences between them, are also provided. This arises in many real-world problems, such as medical diagnosis, where the cost of the test as well as the expected improvement in the outcome must be considered. Relatively little work has been done on learning the utilities of outcomes for optimal decision making. In this paper, we show how temporal-difference reinforcement learning (TO(A» can be used to determine decision theoretic utilities within the context of a mixture model and apply this new approach to a problem in medical diagnosis. TO(A) learning of utilities reduces the number of tests that have to be done to achieve the same level of performance compared with the probability model alone, which results in significant cost savings and increased efficiency.

Policy iteration is known to have rapid and robust convergence properties, and for Markov tasks with lookup-table state-space representations, it is guaranteed to convergence to the optimal policy. Online Policy Improvement using Monte-Carlo Search 1069 In typical uses of policy iteration, the policy improvement step is an extensive off-line procedure. For example, in dynamic programming, one performs a sweep through all states in the state space. Reinforcement learning provides another approach to policy improvement; recently, several authors have investigated using RL in conjunction with nonlinear function approximators to represent the value functions and/or policies (Tesauro, 1992; Crites and Barto, 1996; Zhang and Dietterich, 1996). These studies are based on following actual state-space trajectories rather than sweeps through the full state space, but are still too slow to compute improved policies in real time.

Although TD-Gammon is one of the major successes in machine learning, it has not led to similar impressive breakthroughs in temporal difference learning for other applications or even other games. We were able to replicate some of the success of TD-Gammon, developing a competitive evaluation function on a 4000 parameter feed-forward neural network, without using back-propagation, reinforcement or temporal difference learning methods. Instead we apply simple hill-climbing in a relative fitness environment. These results and further analysis suggest that the surprising success of Tesauro's program had more to do with the co-evolutionary structure of the learning task and the dynamics of the backgammon game itself. 1 INTRODUCTION It took great chutzpah for Gerald Tesauro to start wasting computer cycles on temporal difference learning in the game of Backgammon (Tesauro, 1992). After all, the dream of computers mastering a domain by self-play or "introspection" had been around since the early days of AI, forming part of Samuel's checker player (Samuel, 1959) and used in Donald Michie's MENACE tictac-toe learner (Michie, 1961).

Closed-loop control relies on sensory feedback that is usually assumed to be free. But if sensing incurs a cost, it may be costeffective to take sequences of actions in open-loop mode. We describe a reinforcement learning algorithm that learns to combine open-loop and closed-loop control when sensing incurs a cost. Although we assume reliable sensors, use of open-loop control means that actions must sometimes be taken when the current state of the controlled system is uncertain. This is a special case of the hidden-state problem in reinforcement learning, and to cope, our algorithm relies on short-term memory. The main result of the paper is a rule that significantly limits exploration of possible memory states by pruning memory states for which the estimated value of information is greater than its cost. We prove that this rule allows convergence to an optimal policy.

We have calculated analytical expressions for how the bias and variance of the estimators provided by various temporal difference value estimation algorithms change with offline updates over trials in absorbing Markov chains using lookup table representations. We illustrate classes of learning curve behavior in various chains, and show the manner in which TD is sensitive to the choice of its stepsize and eligibility trace parameters. 1 INTRODUCTION A reassuring theory of asymptotic convergence is available for many reinforcement learning (RL) algorithms. What is not available, however, is a theory that explains the finite-term learning curve behavior of RL algorithms, e.g., what are the different kinds of learning curves, what are their key determinants, and how do different problem parameters effect rate of convergence. Answering these questions is crucial not only for making useful comparisons between algorithms, but also for developing hybrid and new RL methods. In this paper we provide preliminary answers to some of the above questions for the case of absorbing Markov chains, where mean square error between the estimated and true predictions is used as the quantity of interest in learning curves.

We propose and analyze an algorithm that approximates solutions to the problem of optimal stopping in a discounted irreducible aperiodic Markov chain. The scheme involves the use of linear combinations of fixed 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 of the 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 algorithm when combined with arbitrary linear function approximators to solve a sequential decision problem.

A new reinforcement learning architecture for nonlinear control is proposed. A direct feedback controller, or the actor, is trained by a value-gradient based controller, or the tutor. This architecture enables both efficient use of the value function and simple computation for real-time implementation. Good performance was verified in multidimensional nonlinear control tasks using Gaussian softmax networks.

Department of Computer Science, Carnegie Mellon University 5000 Forbes Ave, Pittsburgh, PA 15213 Abstract Dynamic Programming, Q-Iearning and other discrete Markov Decision Process solvers can be -applied to continuous d-dimensional state-spaces by quantizing the state space into an array of boxes. This is often problematic above two dimensions: a coarse quantization can lead to poor policies, and fine quantization is too expensive. Possible solutions are variable-resolution discretization, or function approximation by neural nets. A third option, which has been little studied in the reinforcement learning literature, is interpolation on a coarse grid. In this paper we study interpolation techniques that can result in vast improvements in the online behavior of the resulting control systems: multilinear interpolation, and an interpolation algorithm based on an interesting regular triangulation of d-dimensional space.

Department of Computer Science, Carnegie Mellon University 5000 Forbes Ave, Pittsburgh, PA 15213 Abstract Dynamic Programming, Q-Iearning and other discrete Markov Decision Process solvers can be -applied to continuous d-dimensional state-spaces by quantizing the state space into an array of boxes. This is often problematic above two dimensions: a coarse quantization can lead to poor policies, and fine quantization is too expensive. Possible solutions are variable-resolution discretization, or function approximation by neural nets. A third option, which has been little studied in the reinforcement learning literature, is interpolation on a coarse grid. In this paper we study interpolation techniques that can result in vast improvements in the online behavior of the resulting control systems: multilinear interpolation, and an interpolation algorithm based on an interesting regular triangulation of d-dimensional space.