Optimal Order Simple Regret for Gaussian Process Bandits

TL;DR

This paper derives a tighter, order-optimal bound on simple regret for Gaussian Process bandit algorithms, advancing understanding of exploration efficiency in non-convex, expensive optimization tasks.

Contribution

It introduces a new analysis that tightens the bounds on simple regret, including novel confidence intervals for GP models in RKHS.

Findings

Achieves an $ ilde{O}(\sqrt{rac{\gamma_N}{N}})$ simple regret bound

Proves the bound is order optimal up to logarithmic factors

Develops novel confidence intervals for GP models in RKHS

Abstract

Consider the sequential optimization of a continuous, possibly non-convex, and expensive to evaluate objective function $f$. The problem can be cast as a Gaussian Process (GP) bandit where $f$ lives in a reproducing kernel Hilbert space (RKHS). The state of the art analysis of several learning algorithms shows a significant gap between the lower and upper bounds on the simple regret performance. When $N$ is the number of exploration trials and $\gamma_N$ is the maximal information gain, we prove an $\tilde{\mathcal{O}}(\sqrt{\gamma_N/N})$ bound on the simple regret performance of a pure exploration algorithm that is significantly tighter than the existing bounds. We show that this bound is order optimal up to logarithmic factors for the cases where a lower bound on regret is known. To establish these results, we prove novel and sharp confidence intervals for GP models applicable to RKHS elements which may be of broader interest.