Posterior Sampling-Based Bayesian Optimization with Tighter Bayesian Regret Bounds

TL;DR

This paper demonstrates that the PIMS acquisition function in Bayesian optimization achieves tighter Bayesian regret bounds, avoids hyperparameter tuning, and effectively mitigates practical issues associated with GP-UCB and Thompson sampling.

Contribution

It introduces PIMS as a new acquisition function that attains the tighter BCR bound and addresses practical challenges in Bayesian optimization.

Findings

PIMS achieves the tighter BCR bound.

PIMS avoids hyperparameter tuning.

PIMS mitigates practical issues of existing methods.

Abstract

Among various acquisition functions (AFs) in Bayesian optimization (BO), Gaussian process upper confidence bound (GP-UCB) and Thompson sampling (TS) are well-known options with established theoretical properties regarding Bayesian cumulative regret (BCR). Recently, it has been shown that a randomized variant of GP-UCB achieves a tighter BCR bound compared with GP-UCB, which we call the tighter BCR bound for brevity. Inspired by this study, this paper first shows that TS achieves the tighter BCR bound. On the other hand, GP-UCB and TS often practically suffer from manual hyperparameter tuning and over-exploration issues, respectively. Therefore, we analyze yet another AF called a probability of improvement from the maximum of a sample path (PIMS). We show that PIMS achieves the tighter BCR bound and avoids the hyperparameter tuning, unlike GP-UCB. Furthermore, we demonstrate a wide range of experiments, focusing on the effectiveness of PIMS that mitigates the practical issues of GP-UCB and TS.