Regret Bounds for Expected Improvement Algorithms in Gaussian Process Bandit Optimization

TL;DR

This paper analyzes the theoretical convergence of expected improvement algorithms in Gaussian process bandit optimization, proposing variants that achieve regret bounds and faster convergence without requiring prior knowledge of certain parameters.

Contribution

The authors introduce a variant of EI with a standard incumbent and prove its convergence with regret bounds, also proposing an improved version that converges faster without needing prior parameter knowledge.

Findings

Proposed EI variant converges with a regret bound of O(γ_T√T).

Improved GP-EI algorithm achieves faster convergence than previous methods.

Empirical results demonstrate the effectiveness of the proposed algorithms.

Abstract

The expected improvement (EI) algorithm is one of the most popular strategies for optimization under uncertainty due to its simplicity and efficiency. Despite its popularity, the theoretical aspects of this algorithm have not been properly analyzed. In particular, whether in the noisy setting, the EI strategy with a standard incumbent converges is still an open question of the Gaussian process bandit optimization problem. We aim to answer this question by proposing a variant of EI with a standard incumbent defined via the GP predictive mean. We prove that our algorithm converges, and achieves a cumulative regret bound of $\mathcal O(\gamma_T\sqrt{T})$, where $\gamma_T$ is the maximum information gain between $T$ observations and the Gaussian process model. Based on this variant of EI, we further propose an algorithm called Improved GP-EI that converges faster than previous counterparts. In particular, our proposed variants of EI do not require the knowledge of the RKHS norm and the noise's sub-Gaussianity parameter as in previous works. Empirical validation in our paper demonstrates the effectiveness of our algorithms compared to several baselines.