Another Look at Partially Observed Optimal Stochastic Control: Existence, Ergodicity, and Approximations Without Belief-Reduction

Abstract

We present an alternative view for the study of optimal control of partially observed Markov Decision Processes (POMDPs). We first revisit the traditional (and by now standard) separated-design method of reducing the problem to fully observed MDPs (belief-MDPs), and present conditions for the existence of optimal policies. Then, rather than working with this standard method, we define a Markov chain taking values in an infinite dimensional product space with the history process serving as the controlled state process and a further refinement in which the control actions and the state process are causally conditionally independent given the measurement/information process. We provide new sufficient conditions for the existence of optimal control policies under the discounted cost and average cost infinite horizon criteria. In particular, while in the belief-MDP reduction of POMDPs, weak Feller condition requirement imposes total variation continuity on either the system kernel or the measurement kernel, with the approach of this paper only weak continuity of both the transition kernel and the measurement kernel is needed (and total variation continuity is not) together with regularity conditions related to filter stability. For the discounted cost setup, we establish near optimality of finite window policies via a direct argument involving near optimality of quantized approximations for MDPs under weak Feller continuity, where finite truncations of memory can be viewed as quantizations of infinite memory with a uniform diameter in each finite window restriction under the product metric. For the average cost setup, we provide new existence conditions and also a general approach on how to initialize the randomness which we show to establish convergence to optimal cost. In the control-free case, our analysis leads to new and weak conditions for the existence and uniqueness of invariant probability measures for nonlinear filter processes, where we show that unique ergodicity of the measurement process and a measurability condition related to filter stability leads to unique ergodicity.