Best-of-Both-Worlds for Heavy-Tailed Markov Decision Processes

TL;DR

This paper introduces new algorithms for heavy-tailed Markov Decision Processes that adaptively perform well in both adversarial and stochastic environments, achieving optimal regret bounds.

Contribution

The paper proposes the HT-FTRL-OM and HT-FTRL-UOB algorithms that attain Best-of-Both-Worlds guarantees for heavy-tailed MDPs, including unknown transition settings.

Findings

Achieves $ ilde{O}(T^{1/eta})$ regret in adversarial regimes.

Achieves $O( ext{log} T)$ regret in stochastic regimes.

Develops novel estimators and analysis techniques for heavy-tailed, adversarial, and stochastic environments.

Abstract

We investigate episodic Markov Decision Processes with heavy-tailed losses (HTMDPs). Existing approaches for HTMDPs are conservative in stochastic environments and lack adaptivity in adversarial regimes. In this work, we propose algorithms HT-FTRL-OM and HT-FTRL-UOB for HTMDPs that achieve Best-of-Both-Worlds (BoBW) guarantees: instance-independent regret in adversarial environments and logarithmic instance-dependent regret in self-bounding (including the stochastic case) environments. For the known transition setting, HT-FTRL-OM applies the Follow-The-Regularized-Leader (FTRL) framework over occupancy measures with novel skipping loss estimators, achieving a $\widetilde{{O}}(T^{1/\alpha})$ regret bound in adversarial regimes and a ${O}(\log T)$ regret in stochastic regimes. Building upon this framework, we develop a novel algorithm HT-FTRL-UOB to tackle the more challenging unknown-transition setting. Under a mild truncative nonnegativity condition on the loss distributions, this algorithm employs a pessimistic skipping loss estimator and achieves a $\widetilde{{O}}(T^{1/\alpha} + \sqrt{T})$ regret in adversarial regimes and a ${O}(\log^2(T))$ regret in stochastic regimes. Our analysis overcomes key barriers through several technical insights, including a local control mechanism for heavy-tailed shifted losses, a new suboptimal-mass propagation principle, and a novel regret decomposition that isolates transition uncertainty from heavy-tailed estimation errors and skipping bias.