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Learning Markov decision processes (MDPs) in the presence of the adversary isa challenging problem in reinforcement learning (RL). In this paper, we studyRL in episodic MDPs with adversarial reward and full information feedback,where the unknown transition probability function is a linear function of agiven feature mapping, and the reward function can change arbitrarily episodeby episode. We propose an optimistic policy optimization algorithm POWERS andshow that it can achieve $\tilde{O}(dH\sqrt{T})$ regret, where $H$ is thelength of the episode, $T$ is the number of interactions with the MDP, and $d$is the dimension of the feature mapping. Furthermore, we also prove a matchinglower bound of $\tilde{\Omega}(dH\sqrt{T})$ up to logarithmic factors. Our keytechnical contributions are two-fold: (1) a new value function estimator basedon importance weighting; and (2) a tighter confidence set for the transitionkernel. They together lead to the nearly minimax optimal regret.

Near-optimal Policy Optimization Algorithms for Learning Adversarial Linear Mixture MDPs