Near Optimality of Quantized Policies in Stochastic Control Under Weak Continuity Conditions

Abstract: This paper studies the approximation of optimal control policies by quantized (discretized) policies for a very general class of Markov decision processes (MDPs). The problem is motivated by applications in networked control systems, computational methods for MDPs, and learning algorithms for MDPs. We consider the finite-action approximation of stationary policies for a discrete-time Markov decision process with discounted and average costs under a weak continuity assumption on the transition probability, which is a significant relaxation of conditions required in earlier literature. The discretization is constructive, and quantized policies are shown to approximate optimal deterministic stationary policies with arbitrary precision. The results are applied to the fully observed reduction of a partially observed Markov decision process, where weak continuity is a much more reasonable assumption than more stringent conditions such as strong continuity or continuity in total variation.