Reinforcement Learning in Continuous Time and Space

• Published in: Neural computation Vol. 12; no. 1; pp. 219 - 245
• Main Author: Doya, Kenji
• Format: Journal Article
• Language: English
• Published: One Rogers Street, Cambridge, MA 02142-1209, USA MIT Press 01.01.2000
• Subjects: Algorithms; Applied sciences; Artificial Intelligence; Computer science; control theory; systems; Connectionism. Neural networks; Exact sciences and technology; Global analysis, analysis on manifolds; Learning; Learning and adaptive systems; Mathematics; Models, Psychological; Models, Statistical; Reinforcement (Psychology); Reward; Sciences and techniques of general use; Time Factors; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Bellman equation; Hamilton Jacobi equation; Continuous time systems; Pendulum; Neural network; Dynamical system; Learning algorithm
• ISSN: 0899-7667; 1530-888X
• DOI: 10.1162/089976600300015961

This article presents a reinforcement learning framework for continuous-time dynamical systems without a priori discretization of time, state, and action. Basedonthe Hamilton-Jacobi-Bellman (HJB) equation for infinite-horizon, discounted reward problems, we derive algorithms for estimating value functions and improving policies with the use of function approximators. The process of value function estimation is formulated as the minimization of a continuous-time form of the temporal difference (TD) error. Update methods based on backward Euler approximation and exponential eligibility traces are derived, and their correspondences with the conventional residual gradient, TD (0), and TD ( ) algorithms are shown. For policy improvement, two methods—a continuous actor-critic method and a value-gradient-based greedy policy—are formulated. As a special case of the latter, a nonlinear feedback control law using the value gradient and the model of the input gain is derived. The advantage updating, a model-free algorithm derived previously, is also formulated in the HJB-based framework. The performance of the proposed algorithms is first tested in a nonlinear control task of swinging a pendulum up with limited torque. It is shown in the simulations that (1) the task is accomplished by the continuous actor-critic method in a number of trials several times fewer than by the conventional discrete actor-critic method; (2) among the continuous policy update methods, the value-gradient-based policy with a known or learned dynamic model performs several times better than the actor-critic method; and (3) a value function update using exponential eligibility traces is more efficient and stable than that based on Euler approximation. The algorithms are then tested in a higher-dimensional task: cart-pole swing-up. This task is accomplished in several hundred trials using the value-gradient-based policy with a learned dynamic model.