Improved Bayesian Regret Bounds for Thompson Sampling in Reinforcement Learning

In this paper, we prove the first Bayesian regret bounds for Thompson Sampling in reinforcement learning in a multitude of settings. We simplify the learning problem using a discrete set of surrogate environments, and present a refined analysis of the information ratio using posterior consistency. This leads to an upper bound of order eO(H p dl1T) in the time inhomogeneous reinforcement learning problem where H is the episode length and dl1 is the Kolmogorov l1 dimension of the space of environments. We then find concrete bounds of dl1 in a variety of settings, such as tabular, linear and finite mixtures, and discuss how how our results are either the first of their kind or improve the state-of-the-art.