Efficient Approximation of Optimal Control for Markov Games

TL;DR

This paper introduces higher-order approximation techniques for time-bounded reachability in continuous-time Markov decision processes and games, significantly reducing computational complexity while maintaining accuracy.

Contribution

It develops a sequence of approximation methods with higher accuracy orders, enabling practical algorithms for CTMDPs and CTMGs with theoretical guarantees.

Findings

Achieves higher accuracy approximations (O(ε^3), O(ε^4), O(ε^5))

Reduces the number of intervals needed for approximation

Provides the first practical algorithms for CTMGs

Abstract

We study the time-bounded reachability problem for continuous-time Markov decision processes (CTMDPs) and games (CTMGs). Existing techniques for this problem use discretisation techniques to break time into discrete intervals, and optimal control is approximated for each interval separately. Current techniques provide an accuracy of O(\epsilon^2) on each interval, which leads to an infeasibly large number of intervals. We propose a sequence of approximations that achieve accuracies of O(\epsilon^3), O(\epsilon^4), and O(\epsilon^5), that allow us to drastically reduce the number of intervals that are considered. For CTMDPs, the performance of the resulting algorithms is comparable to the heuristic approach given by Buckholz and Schulz, while also being theoretically justified. All of our results generalise to CTMGs, where our results yield the first practically implementable algorithms for this problem. We also provide positional strategies for both players that achieve similar error bounds.