Learning Adversarial MDPs with Bandit Feedback and Unknown Transition

We consider the problem of learning in episodic finite-horizon Markov decision processes with unknown transition function, bandit feedback, and adversarial losses. We propose an efficient algorithm that achieves $\mathcal{\tilde{O}}(L|X|^2\sqrt{|A|T})$ regret with high probability, where $L$ is the horizon, $|X|$ is the number of states, $|A|$ is the number of actions, and $T$ is the number of episodes. To the best of our knowledge, our algorithm is the first one to ensure {$\mathcal{\tilde{O}}(\sqrt{T})$} regret in this challenging setting. Our key technical contribution is to introduce an optimistic loss estimator that is inversely weighted by an $\textit{upper occupancy bound}$.