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DOI: 10.4230/LIPIcs.CONCUR.2022.11
Anytime Guarantees for Reachability in Uncountable Markov Decision Processes

Authors: Kush Grover, Jan Křetínský, Tobias Meggendorfer, and Maximilian Weininger

Published in: LIPIcs, Volume 243, 33rd International Conference on Concurrency Theory (CONCUR 2022)

Abstract
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.
Cite as

Kush Grover, Jan Křetínský, Tobias Meggendorfer, and Maximilian Weininger. Anytime Guarantees for Reachability in Uncountable Markov Decision Processes. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{grover_et_al:LIPIcs.CONCUR.2022.11,
  author =	{Grover, Kush and K\v{r}et{\'\i}nsk\'{y}, Jan and Meggendorfer, Tobias and Weininger, Maximilian},
  title =	{{Anytime Guarantees for Reachability in Uncountable Markov Decision Processes}},
  booktitle =	{33rd International Conference on Concurrency Theory (CONCUR 2022)},
  pages =	{11:1--11:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-246-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{243},
  editor =	{Klin, Bartek and Lasota, S{\l}awomir and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2022.11},
  URN =		{urn:nbn:de:0030-drops-170743},
  doi =		{10.4230/LIPIcs.CONCUR.2022.11},
  annote =	{Keywords: Uncountable system, Markov decision process, discrete-time Markov control process, probabilistic verification, anytime guarantee}
}
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