Further Optimal Regret Bounds for Thompson Sampling

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].

💡 Research Summary

Thompson Sampling (TS) has long been celebrated as a simple, Bayesian‑inspired heuristic for the stochastic multi‑armed bandit problem, yet its theoretical guarantees have lagged behind its empirical success. This paper delivers a unified regret analysis that simultaneously establishes the optimal problem‑dependent bound and the first near‑optimal problem‑independent bound for TS, thereby resolving an open question posed at COLT 2012 by Chapelle and Li.

Problem setting and algorithm.
The authors consider the classic $N$‑armed stochastic bandit with Bernoulli rewards of unknown means $\mu_i\in