  This paper studies the Bayesian regret of a variant of the Thompson-Sampling
algorithm for bandit problems. It builds upon the information-theoretic
framework of [Russo and Van Roy, 2015] and, more specifically, on the
rate-distortion analysis from [Dong and Van Roy, 2020], where they proved a
bound with regret rate of $O(d\sqrt{T \log(T)})$ for the $d$-dimensional linear
bandit setting. We focus on bandit problems with a metric action space and,
using a chaining argument, we establish new bounds that depend on the metric
entropy of the action space for a variant of Thompson-Sampling.
  Under suitable continuity assumption of the rewards, our bound offers a tight
rate of $O(d\sqrt{T})$ for $d$-dimensional linear bandit problems.
