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

arXiv:2602.01295v3 Announce Type: replace 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.

Source: arXiv cs.LG | 2026-05-18