Optimal Cost Almost-sure Reachability in POMDPs
This addresses a fundamental optimization challenge in partially observable systems for applications like robotics and planning, though it is incremental as it builds on existing finite-horizon POMDP algorithms.
The paper tackles the problem of minimizing expected total cost to reach a target set almost surely in POMDPs with integer costs, showing that approximating optimal cost is undecidable for integer costs but decidable for positive costs, with double-exponential bounds and algorithms that perform well experimentally.
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.