Ali Shafiee

Ali Asadi, Krishnendu Chatterjee, Alipasha Montaseri et al.

A basic model in sequential decision making is the Markov decision process (MDP), which is extended to Robust MDPs (RMDPs) by allowing uncertainty in transition probabilities and optimizing against the worst-case transition probabilities from the uncertainty sets. The class of $(s, a)$-rectangular RMDPs with $L_p$ uncertainty sets provides a flexible and expressive model for such problems. We study this class of RMDPs with a discounted-sum cost criterion and a constant discount factor. The existence of an efficient algorithm for this class is a fundamental theoretical question in optimization and sequential decision making. Previous results only establish a strongly polynomial-time algorithm for $L_\infty$ uncertainty sets. In this work, our main results are as follows: (a)~we show that for any compact uncertainty set, the policy iteration algorithm for RMDPs is strongly polynomial with oracle access to solutions of Robust Markov chains (RMCs); (b)~we present strongly polynomial-time bounds on the policy iteration algorithm for RMCs with $L_1$ and $L_\infty$ uncertainty sets; and (c)~we establish hardness results for RMCs with $L_p$ uncertainty sets for integer $p$ satisfying $1<p<\infty$. Finally, motivated by our theoretical bounds, we present experimental results showing how fast policy iteration converges for RMDPs with $L_1$ and $L_\infty$ uncertainty sets.

Léonard Brice, Thomas A. Henzinger, Alipasha Montaseri et al.

We study concurrent graph games where n players cooperate against an opponent to reach a set of target states. Unlike traditional settings, we study distributed randomisation: team players do not share a source of randomness, and their private random sources are hidden from the opponent and from each other. We show that memoryless strategies are sufficient for the threshold problem (deciding whether there is a strategy for the team that ensures winning with probability that exceeds a threshold), a result that not only places the problem in the Existential Theory of the Reals (\exists\mathbb{R}) but also enables the construction of value iteration algorithms. We additionally show that the threshold problem is NP-hard. For the almost-sure reachability problem, we prove NP-completeness. We introduce Individually Randomised Alternating-time Temporal Logic (IRATL). This logic extends the standard ATL framework to reason about probability thresholds, with semantics explicitly designed for coalitions that lack a shared source of randomness. On the practical side, we implement and evaluate a solver for the threshold and almost-sure problem based on the algorithms that we develop.

Ali Asadi, Krishnendu Chatterjee, Ehsan Kafshdar Goharshady et al.

Robust Markov Decision Processes (RMDPs) generalize classical MDPs that consider uncertainties in transition probabilities by defining a set of possible transition functions. An objective is a set of runs (or infinite trajectories) of the RMDP, and the value for an objective is the maximal probability that the agent can guarantee against the adversarial environment. We consider (a) reachability objectives, where given a target set of states, the goal is to eventually arrive at one of them; and (b) parity objectives, which are a canonical representation for $ω$-regular objectives. The qualitative analysis problem asks whether the objective can be ensured with probability 1. In this work, we study the qualitative problem for reachability and parity objectives on RMDPs without making any assumption over the structures of the RMDPs, e.g., unichain or aperiodic. Our contributions are twofold. We first present efficient algorithms with oracle access to uncertainty sets that solve qualitative problems of reachability and parity objectives. We then report experimental results demonstrating the effectiveness of our oracle-based approach on classical RMDP examples from the literature scaling up to thousands of states.