TY - GEN

T1 - Anytime Guarantees for Reachability in Uncountable Markov Decision Processes.

AU - Grover, Kush

AU - Kretínský, Jan

AU - Meggendorfer, Tobias

AU - Weininger, Maximilian

PY - 2022/9/6

Y1 - 2022/9/6

N2 - 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.

AB - 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.

KW - Markov decision process

KW - Uncountable system

KW - anytime guarantee

KW - discrete-time Markov control process

KW - probabilistic verification

U2 - 10.4230/LIPIcs.CONCUR.2022.11

DO - 10.4230/LIPIcs.CONCUR.2022.11

M3 - Conference contribution/Paper

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 11:1-11:20

BT - 33rd International Conference on Concurrency Theory, CONCUR 2022

A2 - Klin, Bartek

A2 - Lasota, Slawomir

A2 - Muscholl, Anca

PB - Schloss Dagstuhl - Leibniz-Zentrum für Informatik

ER -