Abstract: We consider Markov Decision Processes (MDPs) where the rewards are unknown and may change in an adversarial manner. We provide an algorithm that achieves a regret bound of $O( \sqrt{\tau (\ln|S|+\ln|A|)T}\ln(T))$, where $S$ is the state space, $A$ is the action space, $\tau$ is the mixing time of the MDP, and $T$ is the number of periods. The algorithm's computational complexity is polynomial in $|S|$ and $|A|$. We then consider a setting often encountered in practice, where the state space of the MDP is too large to allow for exact solutions. By approximating the state-action occupancy measures with a linear architecture of dimension $d\ll|S|$, we propose a modified algorithm with a computational complexity polynomial in $d$ and independent of $|S|$. We also prove a regret bound for this modified algorithm, which to the best of our knowledge, is the first $\tilde{O}(\sqrt{T})$ regret bound in the large-scale MDP setting with adversarially changing rewards.