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Anytime Guarantees for Reachability in Uncountable Markov Decision Processes

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

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Author Details

Kush Grover
  • Technische Universität München, Germany
Jan Křetínský
  • Technische Universität München, Germany
Tobias Meggendorfer
  • Institute of Science and Technology Austria, Wien, Austria
Maximilian Weininger
  • Technische Universität München, Germany

Funding

  • Grover, Kush: The author has been supported by the DFG research training group GRK 2428 ConVeY.
  • Weininger, Maximilian: The author has been partially supported by DFG projects 383882557 Statistical Unbounded Verification (SUV) and 427755713 Group-By Objectives in Probabilistic Verification (GOPro).

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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) https://doi.org/10.4230/LIPIcs.CONCUR.2022.11

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Markov processes
  • Mathematics of computing → Continuous mathematics
  • Computing methodologies → Continuous models
Keywords
  • Uncountable system
  • Markov decision process
  • discrete-time Markov control process
  • probabilistic verification
  • anytime guarantee

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