Asymptotic Optimality of Finite Approximations to Markov Decision Processes with Borel Spaces

Calculating optimal policies is known to be computationally difficult for Markov decision processes (MDPs) with Borel state and action spaces. This paper studies finite-state approximations of discrete time Markov decision processes with Borel state and action spaces, for both discounted and average costs criteria. The stationary policies thus obtained are shown to approximate the optimal stationary policy with arbitrary precision under quite general conditions for discounted cost and more restrictive conditions for average cost. For compact-state MDPs, we obtain explicit rate of convergence bounds quantifying how the approximation improves as the size of the approximating finite state space increases. Using information theoretic arguments, the order optimality of the obtained convergence rates is established for a large class of problems. We also show that, as a pre-processing step the action space can also be finitely approximated with sufficiently large number points; thereby, well known algorithms, such as value or policy iteration, Q-learning, etc., can be used to calculate near optimal policies.