Recent research on reinforcement learning has focused on algorithms basedon the principles of Dynamic Programming (DP). One of the most promising areas of application for these algorithms isthe control of dynamical systems, and some impressive results have been achieved. However, there are significant gaps between practice and theory. In particular, there are no con vergence proofsfor problems with continuous state and action spaces, or for systems involving nonlinear function approximators (such as multilayer perceptrons). This paper presents research applying DPbased reinforcement learning theory to Linear Quadratic Regulation (LQR),an important class of control problems involving continuous state and action spaces and requiring a simple type of nonlinear function approximator. We describe an algorithm based on Q-Iearning that is proven to converge to the optimal controller for a large class of LQR problems. We also describe a slightly different algorithm that is only locally convergent to the optimal Q-function, demonstrating one of the possible pitfalls of using a nonlinear function approximator with DPbased learning.

We present a new algorithm, Prioritized Sweeping, for efficient prediction and control of stochastic Markov systems. Incremental learning methods such as Temporal Differencing and Q-Iearning have fast real time performance. Classicalmethods are slower, but more accurate, because they make full use of the observations. Prioritized Sweeping aims for the best of both worlds. It uses all previous experiences both to prioritize important dynamicprogramming sweeps and to guide the exploration of statespace.

Platt's resource-allocation network (RAN) (Platt, 1991a, 1991b) is modified for a reinforcement-learning paradigm and to "restart" existing hidden units rather than adding new units. After restarting, unitscontinue to learn via back-propagation. The resulting restart algorithm is tested in a Q-Iearning network that learns to solve an inverted pendulum problem. Solutions are found faster on average with the restart algorithm than without it.

We present a new algorithm,prioritized sweeping, for efficient prediction and control of stochastic Markov systems. Incremental learning methods such as temporal differencing and Q-learning have real-time performance. Classical methods are slower, but more accurate, because they make full use of the observations. It uses all previous experiences both to prioritize important dynamic programming sweeps and to guide the exploration of state-space. We compare prioritized sweeping with other reinforcement learning schemes for a number of different stochastic optimal control problems.

Reinforcement learning is an effective technique for learning action policies in discrete stochastic environments, but its efficiency can decay exponentially with the size of the state space. In many situations significant portions of a large state space may be irrelevant to a specific goal and can be aggregated into a few, relevant, states. The U Tree algorithm generates a tree based state discretization that efficiently finds the relevant state chunks of large propositional domains. In this paper, we extend the U Tree algorithm to challenging domains with a continuous state space for which there is no initial discretization.

A method for transforming performance evaluation signals distal both in space and time into proximal signals usable by supervised learning algorithms, presented in [Jordan & Jacobs 90], is examined. A simple observation concerning differentiation through models trained with redundant inputs (as one of their networks is) explains a weakness in the original architecture and suggests a modification: an internal world model that encodes action-space exploration and, crucially, cancels input redundancy to the forward model is added. Learning time on an example task, cartpole balancing, is thereby reduced about 50 to 100 times. 1 INTRODUCTION In many learning control problems, the evaluation used to modify (and thus improve) control may not be available in terms of the controller's output: instead, it may be in terms of a spatial transformation of the controller's output variables (in which case we shall term it as being "distal in space"), or it may be available only several time steps into the future (termed as being "distal in time"). For example, control of a robot arm may be exerted in terms of joint angles, while evaluation may be in terms of the endpoint cartesian coordinates; furthermore, we may only wish to evaluate the endpoint coordinates reached after a certain period of time: the co- ·Current address: Computation and Neural Systems Program, California Institute of Technology, Pasadena CA.

Following Simon's (1969) observation that complex behaviour may simply be the reflection of a complex environment a number of researchers (eg.

TO('\) is an algorithm for adjusting the weights in a connectionist network which 259 260 Tesauro has the following form:

I present a modular network architecture and a learning algorithm based on incremental dynamic programming that allows a single learning agent to learn to solve multiple Markovian decision tasks (MDTs) with significant transfer of learning across the tasks. I consider a class of MDTs, called composite tasks, formed by temporally concatenating a number of simpler, elemental MDTs. The architecture is trained on a set of composite and elemental MDTs. The temporal structure of a composite task is assumed to be unknown and the architecture learns to produce a temporal decomposition. It is shown that under certain conditions the solution of a composite MDT can be constructed by computationally inexpensive modifications of the solutions of its constituent elemental MDTs. 1 INTRODUCTION Most applications of domain independent learning algorithms have focussed on learning single tasks. Building more sophisticated learning agents that operate in complex environments will require handling multiple tasks/goals (Singh, 1992). Research effort on the scaling problem has concentrated on discovering faster learning algorithms, and while that will certainly help, techniques that allow transfer of learning across tasks will be indispensable for building autonomous learning agents that have to learn to solve multiple tasks. In this paper I consider a learning agent that interacts with an external, finite-state, discrete-time, stochastic dynamical environment and faces multiple sequences of Markovian decision tasks (MDTs).

A method for transforming performance evaluation signals distal both in space and time into proximal signals usable by supervised learning algorithms, presented in [Jordan & Jacobs 90], is examined. A simple observation concerning differentiation through models trained with redundant inputs (as one of their networks is) explains a weakness in the original architecture and suggests a modification: an internal world model that encodes action-space exploration and, crucially, cancels input redundancy to the forward model is added. Learning time on an example task, cartpole balancing, is thereby reduced about 50 to 100 times. 1 INTRODUCTION In many learning control problems, the evaluation used to modify (and thus improve) control may not be available in terms of the controller's output: instead, it may be in terms of a spatial transformation of the controller's output variables (in which case we shall term it as being "distal in space"), or it may be available only several time steps into the future (termed as being "distal in time"). For example, control of a robot arm may be exerted in terms of joint angles, while evaluation may be in terms of the endpoint cartesian coordinates; furthermore, we may only wish to evaluate the endpoint coordinates reached after a certain period of time: the co- ·Current address: Computation and Neural Systems Program, California Institute of Technology, Pasadena CA.