Anytime Guarantees for Reachability in Uncountable Markov Decision Processes

TL;DR

This paper develops an anytime algorithm for approximating reachability probabilities in uncountable state and action space MDPs with theoretical guarantees, under minimal assumptions, enabling reliable analysis of complex systems.

Contribution

It introduces two algorithms providing converging bounds for reachability probabilities with minimal assumptions, advancing the reliability and generality of analysis in continuous-state MDPs.

Findings

Provides converging lower bounds under weak assumptions

Offers converging upper bounds with stronger assumptions

Enables iterative approximation with known accuracy improvements

Abstract

We consider the problem of approximating the reachability probabilities in Markov decision processes (MDP) with uncountable (continuous) state and action spaces. While there are algorithms that, for special classes of such MDP, provide a sequence of approximations converging to the true value in the limit, our aim is to obtain an algorithm with guarantees on the precision of the approximation. As this problem is undecidable in general, assumptions on the MDP are necessary. Our main contribution is to identify sufficient assumptions that are as weak as possible, thus approaching the "boundary" of which systems can be correctly and reliably analyzed. To this end, we also argue why each of our assumptions is necessary for algorithms based on processing finitely many observations. We present two solution variants. The first one provides converging lower bounds under weaker assumptions than typical ones from previous works concerned with guarantees. The second one then utilizes stronger assumptions to additionally provide converging upper bounds. Altogether, we obtain an anytime algorithm, i.e. yielding a sequence of approximants with known and iteratively improving precision, converging to the true value in the limit. Besides, due to the generality of our assumptions, our algorithms are very general templates, readily allowing for various heuristics from literature in contrast to, e.g., a specific discretization algorithm. Our theoretical contribution thus paves the way for future practical improvements without sacrificing correctness guarantees.