Prior-free and prior-dependent regret bounds for Thompson Sampling

TL;DR

This paper analyzes the regret bounds of Thompson Sampling in stochastic multi-armed bandits, establishing optimal bounds for prior-free and prior-dependent cases, and demonstrating its advantages with certain priors.

Contribution

It proves that Thompson Sampling achieves optimal prior-free regret bounds and shows how specific priors can lead to uniformly bounded regret over time.

Findings

Thompson Sampling attains a regret bound of 14√(nK), which is optimal.

Existence of a prior distribution where any algorithm's regret is at least (1/20)√(nK).

With certain priors, Thompson Sampling's regret is bounded uniformly over time.

Abstract

We consider the stochastic multi-armed bandit problem with a prior distribution on the reward distributions. We are interested in studying prior-free and prior-dependent regret bounds, very much in the same spirit as the usual distribution-free and distribution-dependent bounds for the non-Bayesian stochastic bandit. Building on the techniques of Audibert and Bubeck [2009] and Russo and Roy [2013] we first show that Thompson Sampling attains an optimal prior-free bound in the sense that for any prior distribution its Bayesian regret is bounded from above by $14 \sqrt{n K}$. This result is unimprovable in the sense that there exists a prior distribution such that any algorithm has a Bayesian regret bounded from below by $\frac{1}{20} \sqrt{n K}$. We also study the case of priors for the setting of Bubeck et al. [2013] (where the optimal mean is known as well as a lower bound on the smallest gap) and we show that in this case the regret of Thompson Sampling is in fact uniformly bounded over time, thus showing that Thompson Sampling can greatly take advantage of the nice properties of these priors.