Simultaneously Learning Stochastic and Adversarial Markov Decision Process with Linear Function Approximation
Abstract: Reinforcement learning (RL) has been commonly used in practice. To deal with the numerous states and actions in real applications, the function approximation method has been widely employed to improve the learning efficiency, among which the linear function approximation has attracted great interest both theoretically and empirically. Previous works on the linear Markov Decision Process (MDP) mainly study two settings, the stochastic setting where the reward is generated in a stochastic way and the adversarial setting where the reward can be chosen arbitrarily by an adversary. All these works treat these two environments separately. However, the learning agents often have no idea of how rewards are generated and a wrong reward type can severely disrupt the performance of those specially designed algorithms. So a natural question is whether an algorithm can be derived that can efficiently learn in both environments but without knowing the reward type. In this paper, we first consider such best-of-both-worlds problem for linear MDP with the known transition. We propose an algorithm and prove it can simultaneously achieve $O(\text{poly} \log K)$ regret in the stochastic setting and $O(\sqrt{K})$ regret in the adversarial setting where $K$ is the horizon. To the best of our knowledge, it is the first such result for linear MDP.
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