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Machine Learning, 8, 279-292 (1992)
© 1992 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
Technical Note
Q,-Learning
CHRISTOPHER J.C.H. WATKINS
25b Framfield Road, Highbury, London N5 IUU, England
PETER DAYAN
Centre for Cognitive Science, University of Edinburgh, 2 Buccleuch Place, Edinburgh EH8 9EH, Scotland
Abstract. Q-learning (Watkins, 1989) is a simple way for agents to learn how to act optimally in controlled Markovian
domains. It amounts to an incremental method for dynamic programming which imposes limited computational
demands. It works by successively improving its evaluations of the quality of particular actions at particular states.
This paper presents and proves in detail a convergence theorem for Q,-learning based on that outlined in Watkins
(1989). We show that Q-learning converges to the optimum action-values with probability 1 so long as all actions
are repeatedly sampled in all states and the action-values are represented discretely. We also sketch extensions
to the cases of non-discounted, but absorbing, Markov environments, and where many Q values can be changed
each iteration, rather than just one.
Keywords. Q-learning, reinforcement learning, temporal differences, asynchronous dynamic programming
1. Introduction
Q-learning (Watkins, 1989) is a form of model-free reinforcement learning. It can also be
viewed as a method of asynchronous dynamic programming (DP). It provides agents with
the capability of learning to act optimally in Markovian domains by experiencing the con-
sequences of actions, without requiring them to build maps of the domains.
Learning proceeds similarly to Suttons (1984; 1988) method of temporal differences
(TD): an agent tries an action at a particular state, and evaluates its consequences in ter
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