No-Regret Online Reinforcement Learning with Adversarial Losses and Transitions

TL;DR

This paper introduces algorithms for online reinforcement learning in adversarial environments with both losses and transitions, achieving regret bounds that adapt to the level of adversarialness and environment difficulty.

Contribution

The work develops the first algorithms capable of handling adversarial transitions and losses with regret bounds that smoothly depend on adversarialness, including a black-box reduction to remove prior knowledge requirements.

Findings

Achieves $ ilde{O}( oot{T} + C^{ ext{P}})$ regret with adversarial transitions.

Provides a black-box reduction removing the need to know $C^{ ext{P}}$ beforehand.

Adapts to easier environments, achieving improved regret bounds in stochastic-like settings.

Abstract

Existing online learning algorithms for adversarial Markov Decision Processes achieve ${O}(\sqrt{T})$ regret after $T$ rounds of interactions even if the loss functions are chosen arbitrarily by an adversary, with the caveat that the transition function has to be fixed. This is because it has been shown that adversarial transition functions make no-regret learning impossible. Despite such impossibility results, in this work, we develop algorithms that can handle both adversarial losses and adversarial transitions, with regret increasing smoothly in the degree of maliciousness of the adversary. More concretely, we first propose an algorithm that enjoys $\widetilde{{O}}(\sqrt{T} + C^{\textsf{P}})$ regret where $C^{\textsf{P}}$ measures how adversarial the transition functions are and can be at most ${O}(T)$. While this algorithm itself requires knowledge of $C^{\textsf{P}}$, we further develop a black-box reduction approach that removes this requirement. Moreover, we also show that further refinements of the algorithm not only maintains the same regret bound, but also simultaneously adapts to easier environments (where losses are generated in a certain stochastically constrained manner as in Jin et al. [2021]) and achieves $\widetilde{{O}}(U + \sqrt{UC^{\textsf{L}}} + C^{\textsf{P}})$ regret, where $U$ is some standard gap-dependent coefficient and $C^{\textsf{L}}$ is the amount of corruption on losses.