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

Learning Markov decision processes (MDPs) in the presence of the adversary is a challenging problem in reinforcement learning (RL). In this paper, we study RL in episodic MDPs with adversarial reward and full information feedback, where the unknown transition probability function is a linear function of a given feature mapping, and the reward function can change arbitrarily episode by episode. We propose an optimistic policy optimization algorithm POWERS and show that it can achieve (O) over tilde (dH root T) regret, where H is the length of the episode, T is the number of interaction with the MDP, and d is the dimension of the feature mapping. Furthermore, we also prove a matching lower bound of (Omega) over tilde (dH root T) up to logarithmic factors. Our key technical contributions are two-fold: (1) a new value function estimator based on importance weighting; and (2) a tighter confidence set for the transition kernel. They together lead to the nearly minimax optimal regret.