Online Learning in MDPs with Partially Adversarial Transitions and Losses

We study reinforcement learning in MDPs whose transition function is stochastic at most steps but may behave adversarially at a fixed subset of $Λ$ steps per episode. This model captures environments that are stable except at a few vulnerable points. We introduce \emph{conditioned occupancy measures}, which remain stable across episodes even with adversarial transitions, and use them to design two algorithms. The first handles arbitrary adversarial steps and achieves regret $\tilde{O}(H S^Λ\sqrt{K S A^{Λ+1}})$, where $K$ is the number of episodes, $S$ is the number of state, $A$ is the number of actions and $H$ is the episode's horizon. The second, assuming the adversarial steps are consecutive, improves the dependence on $S$ to $\tilde{O}(H\sqrt{K S^{3} A^{Λ+1}})$. We further give a $K^{2/3}$-regret reduction that removes the need to know which steps are the $Λ$ adversarial steps. We also characterize the regret of adversarial MDPs in the \emph{fully adversarial} setting ($Λ=H-1$) both for full-information and bandit feedback, and provide almost matching upper and lower bounds (slightly strengthen existing lower bounds, and clarify how different feedback structures affect the hardness of learning).