Optimal Cost Almost-sure Reachability in POMDPs

TL;DR

This paper investigates the problem of minimizing expected total cost in POMDPs with almost-sure reachability, establishing complexity bounds and providing approximation algorithms with practical stopping criteria.

Contribution

It introduces the first decidability results for approximating optimal costs in POMDPs with positive costs and develops algorithms with proven bounds and practical performance.

Findings

Optimal cost approximation is decidable for positive costs.

The bounds on the optimal cost are double exponential.

The proposed algorithms perform well in practical examples.

Abstract

We consider partially observable Markov decision processes (POMDPs) with a set of target states and every transition is associated with an integer cost. The optimization objective we study asks to minimize the expected total cost till the target set is reached, while ensuring that the target set is reached almost-surely (with probability 1). We show that for integer costs approximating the optimal cost is undecidable. For positive costs, our results are as follows: (i) we establish matching lower and upper bounds for the optimal cost and the bound is double exponential; (ii) we show that the problem of approximating the optimal cost is decidable and present approximation algorithms developing on the existing algorithms for POMDPs with finite-horizon objectives. While the worst-case running time of our algorithm is double exponential, we also present efficient stopping criteria for the algorithm and show experimentally that it performs well in many examples of interest.