We study in this paper a Markov Decision Problem (MDP) with continuous state space and discrete decision variables. We propose an extension of the Q-learning algorithm introduced to solve this problem by Watkins in 1989 for completely discrete MDPs. Our algorithm relies on stochastic approximation and functional estimation, and uses kernels to locally update the Q-functions. We give a convergence proof for this algorithm under usual assumptions. Finally, we illustrate our algorithm by solving the classical moutain car task with continuous state space.

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