Chained Information-Theoretic Bounds and Tight Regret Rate for Linear Bandit Problems

Amaury Gouverneur, Borja Rodríguez Gálvez, Tobias Oechtering, Mikael Skoglund

Abstract:

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 & Van Roy, 2015] and, more specifically, on the rate-distortion analysis from [Dong & 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 action space's metric entropy for a Thompson Sampling variant.
Under suitable continuity assumption of the rewards, our bound offers a tight rate of $O(d\sqrt{T})$ for $d$-dimensional linear bandit problems.

Format: Long format (up to 8 pages + refs, appendix)

Publication Status: No

Submission Number: 15