A Domain-Shrinking based Bayesian Optimization Algorithm with Order-Optimal Regret Performance

TL;DR

This paper introduces a novel Gaussian process-based Bayesian optimization algorithm that employs domain shrinking through tree-based region pruning, achieving order-optimal regret and significantly reducing computational complexity.

Contribution

The paper presents the first GP-based Bayesian optimization algorithm with proven order-optimal regret guarantees, utilizing domain shrinking for improved efficiency.

Findings

Achieves order-optimal regret performance up to poly-logarithmic factors.

Reduces computational complexity by a factor of O(T^{2d-1}) compared to GP-UCB.

Effectively concentrates queries in high-performing regions through iterative domain refinement.

Abstract

We consider sequential optimization of an unknown function in a reproducing kernel Hilbert space. We propose a Gaussian process-based algorithm and establish its order-optimal regret performance (up to a poly-logarithmic factor). This is the first GP-based algorithm with an order-optimal regret guarantee. The proposed algorithm is rooted in the methodology of domain shrinking realized through a sequence of tree-based region pruning and refining to concentrate queries in increasingly smaller high-performing regions of the function domain. The search for high-performing regions is localized and guided by an iterative estimation of the optimal function value to ensure both learning efficiency and computational efficiency. Compared with the prevailing GP-UCB family of algorithms, the proposed algorithm reduces computational complexity by a factor of $O(T^{2d-1})$ (where $T$ is the time horizon and $d$ the dimension of the function domain).