Further Optimal Regret Bounds for Thompson Sampling

TL;DR

This paper presents new, tighter regret bounds for Thompson Sampling in multi-armed bandit problems, including the first near-optimal problem-independent bound and an improved problem-dependent bound, using a novel martingale analysis.

Contribution

It introduces a new regret analysis for Thompson Sampling that achieves both optimal problem-dependent and near-optimal problem-independent bounds, solving an open problem.

Findings

Proves the optimal problem-dependent regret bound of (1+ε)∑i (ln T)/Δi + O(N/ε²).

Establishes the first near-optimal problem-independent bound of O(√NT ln T).

Provides a simple, extendable martingale-based analysis technique.

Abstract

Thompson Sampling is one of the oldest heuristics for multi-armed bandit problems. It is a randomized algorithm based on Bayesian ideas, and has recently generated significant interest after several studies demonstrated it to have better empirical performance compared to the state of the art methods. In this paper, we provide a novel regret analysis for Thompson Sampling that simultaneously proves both the optimal problem-dependent bound of $(1+\epsilon)\sum_i \frac{\ln T}{\Delta_i}+O(\frac{N}{\epsilon^2})$ and the first near-optimal problem-independent bound of $O(\sqrt{NT\ln T})$ on the expected regret of this algorithm. Our near-optimal problem-independent bound solves a COLT 2012 open problem of Chapelle and Li. The optimal problem-dependent regret bound for this problem was first proven recently by Kaufmann et al. [ALT 2012]. Our novel martingale-based analysis techniques are conceptually simple, easily extend to distributions other than the Beta distribution, and also extend to the more general contextual bandits setting [Manuscript, Agrawal and Goyal, 2012].