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.

翻译：本文研究赌博机问题中汤普森采样变体的贝叶斯遗憾。该工作基于[Russo and Van Roy, 2015]的信息论框架，并具体延伸了[Dong and Van Roy, 2020]的率失真分析，后者对于$d$维线性赌博机设定证明了$O(d\sqrt{T \log(T)})$的遗憾率。我们聚焦于具有度量动作空间的赌博机问题，通过运用链式论证，为汤普森采样的变体建立了依赖于动作空间度量熵的新界。在奖励函数满足适当连续性假设的条件下，我们的界为$d$维线性赌博机问题提供了$O(d\sqrt{T})$的紧致率。