Beating Adversarial Low-Rank MDPs with Unknown Transition and Bandit Feedback

We consider regret minimization in low-rank MDPs with fixed transition and adversarial losses. Previous work has investigated this problem under either full-information loss feedback with unknown transitions (Zhao et al., 2024), or bandit loss feedback with known transitions (Foster et al., 2022). First, we improve the $poly(d, A, H)T^{5/6}$ regret bound of Zhao et al. (2024) to $poly(d, A, H)T^{2/3}$ for the full-information unknown transition setting, where $d$ is the rank of the transitions, $A$ is the number of actions, $H$ is the horizon length, and $T$ is the number of episodes. Next, we initiate the study on the setting with bandit loss feedback and unknown transitions. Assuming that the loss has a linear structure, we propose both model-based and model-free algorithms achieving $poly(d, A, H)T^{2/3}$ regret, though they are computationally inefficient. We also propose oracle-efficient model-free algorithms with $poly(d, A, H)T^{4/5}$ regret. We show that the linear structure is necessary for the bandit case--without structure on the reward function, the regret has to scale polynomially with the number of states. This is contrary to the full-information case (Zhao et al., 2024), where the regret can be independent of the number of states even for unstructured reward functions.