On Regret Bounds of Thompson Sampling for Bayesian Optimization

Researchers Shion Takeno and Shogo Iwazaki have published a significant theoretical paper titled 'On Regret Bounds of Thompson Sampling for Bayesian Optimization' on arXiv. The 42-page work provides a rigorous mathematical analysis of Gaussian Process Thompson Sampling (GP-TS), a popular algorithm for optimizing expensive black-box functions. The paper addresses key gaps in the theoretical understanding of GP-TS, which has seen widespread practical use in fields like hyperparameter tuning and materials science, but whose theoretical guarantees have lagged behind alternatives like Gaussian Process Upper Confidence Bound (GP-UCB).

The core contribution is establishing several new regret bounds. First, the authors prove a regret lower bound for GP-TS, showing it suffers from a polynomial dependence on 1/δ with probability δ. More importantly, they derive an upper bound for the second moment of cumulative regret, which directly leads to an improved regret upper bound. The paper also provides expected lenient regret upper bounds and an improved cumulative regret upper bound dependent on the time horizon T. Along the way, the researchers provide useful technical lemmas, including a relaxation of necessary conditions from recent analyses to obtain these improved bounds.

This work represents a substantial step forward in the theoretical machine learning community's understanding of Thompson Sampling for Bayesian optimization. By providing these formal guarantees, the paper helps bridge the gap between the algorithm's empirical success and its theoretical foundations. The results give practitioners more confidence in using GP-TS and provide researchers with new tools for analyzing similar algorithms.