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.