Q-Learning for Continuous State and Action MDPs under Average Cost Criteria

For infinite-horizon average-cost criterion problems, there exist relatively few rigorous approximation and reinforcement learning results. In this paper, for such problems, we present several approximation and reinforcement learning results for Markov Decision Processes with standard Borel spaces. Toward this end, (i) we first provide a discretization based approximation method for fully observed Markov Decision Processes (MDPs) with continuous spaces under average cost criteria, and we provide error bounds for the approximations when the dynamics are only weakly continuous (for asymptotic convergence of errors as the grid sizes vanish) or Wasserstein continuous (with a rate in approximation as the grid sizes vanish) under certain ergodicity assumptions. In particular, we relax the total variation condition given in prior work to weak continuity as well as Wasserstein continuity conditions. (ii) We provide synchronous and asynchronous Q-learning algorithms for continuous spaces via quantization (where the quantized state is taken to be the actual state in corresponding Q-learning algorithms presented in the paper), and establish their convergence; for the former we utilize a span semi-norm approach and for the latter we use a direct contraction approach. (iii) We finally show that the convergence is to the optimal Q values of the finite approximate models constructed via quantization, which implies near optimality of the arrived solution. Our Q-learning convergence results and their convergence to near optimality are new for continuous spaces, and the proof method is new even for finite spaces, to our knowledge.