Robust Control of Markov Decision Processes with Uncertain Transition Matrices

TLDR

Optimal solutions to Markov decision problems are highly sensitive to transition probabilities, and estimation errors limit their real‑world applicability. We aim to formulate a robust control problem for finite‑state, finite‑action Markov decision processes with uncertain transition matrices described by possibly nonconvex sets. We solve this problem with a robust dynamic‑programming algorithm and extend the results to other uncertainty sets, including finite‑valued transition matrices. We prove perfect duality, enabling solution via robust dynamic programming, and show that likelihood‑ or entropy‑based uncertainty sets yield statistically accurate uncertainty with almost the same computational cost as classical recursion, while a practical path‑planning example demonstrates that a robust strategy markedly improves worst‑case expected travel time even with crude uncertainty estimates.

Abstract

Optimal solutions to Markov decision problems may be very sensitive with respect to the state transition probabilities. In many practical problems, the estimation of these probabilities is far from accurate. Hence, estimation errors are limiting factors in applying Markov decision processes to real-world problems. We consider a robust control problem for a finite-state, finite-action Markov decision process, where uncertainty on the transition matrices is described in terms of possibly nonconvex sets. We show that perfect duality holds for this problem, and that as a consequence, it can be solved with a variant of the classical dynamic programming algorithm, the “robust dynamic programming” algorithm. We show that a particular choice of the uncertainty sets, involving likelihood regions or entropy bounds, leads to both a statistically accurate representation of uncertainty, and a complexity of the robust recursion that is almost the same as that of the classical recursion. Hence, robustness can be added at practically no extra computing cost. We derive similar results for other uncertainty sets, including one with a finite number of possible values for the transition matrices. We describe in a practical path planning example the benefits of using a robust strategy instead of the classical optimal strategy; even if the uncertainty level is only crudely guessed, the robust strategy yields a much better worst-case expected travel time.