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In this paper, we prove the first Bayesian regret bounds for ThompsonSampling in reinforcement learning in a multitude of settings. We simplify thelearning problem using a discrete set of surrogate environments, and present arefined analysis of the information ratio using posterior consistency. Thisleads to an upper bound of order $\widetilde{O}(H\sqrt{d_{l_1}T})$ in the timeinhomogeneous reinforcement learning problem where $H$ is the episode lengthand $d_{l_1}$ is the Kolmogorov $l_1-$dimension of the space of environments.We then find concrete bounds of $d_{l_1}$ in a variety of settings, such astabular, linear and finite mixtures, and discuss how how our results are eitherthe first of their kind or improve the state-of-the-art.

Improved Bayesian Regret Bounds for Thompson Sampling in Reinforcement Learning