Q-Learning with Continuous State Spaces and Finite Decision Set

This paper aims to present an original technique in order to compute the optimal policy of a Markov decision problem with continuous state space and discrete decision variables. We propose an extension of the Q-learning algorithm introduced in 1989 by Watkins for discrete Markov decision problems. Our algorithm relies on stochastic approximation and functional estimation, and uses kernels to locally update the Q-functions. We state under mild assumptions a converge theorem for this algorithm. Finally, we illustrate our algorithm by solving two classical problems: the mountain car task and the puddle world task