Set-based value operators for non-stationary Markovian environments

This paper analyzes finite state Markov Decision Processes (MDPs) with uncertain parameters in compact sets and re-examines results from robust MDP via set-based fixed point theory. We generalize the Bellman and policy evaluation operators to operators that contract on the space of value functions and denote them as \emph{value operators}. We generalize these value operators to act on the space of value function sets and denote them as \emph{set-based value operators}. We prove that these set-based value operators are contractions in the space of compact value function sets. Leveraging insights from set theory, we generalize the rectangularity condition for the Bellman operator from classic robust MDP literature to a \emph{containment condition} for a generic value operator, which is weaker and can be applied to a larger set of parameter-uncertain MDPs and contractive operators in dynamic programming and reinforcement learning. We prove that both the rectangularity condition and the containment condition sufficiently ensure that the set-based value operator's fixed point set contains its own supremum and infimum elements. For convex and compact sets of uncertain MDP parameters, we show equivalence between the classic robust value function and the supremum of the fixed point set of the set-based Bellman operator. Under dynamically changing MDP parameters in compact sets, we prove a set convergence result for value iteration, which otherwise may not converge to a single value function.