Asymptotic Performance of Time-Varying Bayesian Optimization

TL;DR

This paper provides the first theoretical analysis of Time-Varying Bayesian Optimization, establishing bounds and conditions under which the algorithm's regret diminishes over time for various kernel functions.

Contribution

It offers the first comprehensive theoretical bounds and conditions for the asymptotic performance of TVBO algorithms across all major stationary kernels.

Findings

Upper and lower bounds for cumulative regret are derived.

Conditions for no-regret property in TVBO are identified.

Analysis covers all major classes of stationary kernels.

Abstract

Time-Varying Bayesian Optimization (TVBO) is the go-to framework for optimizing a time-varying black-box objective function that may be noisy and expensive to evaluate, but its excellent empirical performance remains to be understood theoretically. Is it possible for the instantaneous regret of a TVBO algorithm to vanish asymptotically, and if so, when? We answer this question of great importance by providing upper bounds and algorithm-independent lower bounds for the cumulative regret of TVBO algorithms. In doing so, we provide important insights about the TVBO framework and derive sufficient conditions for a TVBO algorithm to have the no-regret property. To the best of our knowledge, our analysis is the first to cover all major classes of stationary kernel functions used in practice.