Theoretical Analysis of Bayesian Optimisation with Unknown Gaussian Process Hyper-Parameters

TL;DR

This paper provides a theoretical analysis of Bayesian optimisation with Gaussian processes when hyper-parameters are unknown, deriving regret bounds and offering guidelines for hyper-parameter estimation to improve optimisation performance.

Contribution

It introduces the first regret bounds for Bayesian optimisation with unknown hyper-parameters and offers practical guidelines for hyper-parameter estimation methods.

Findings

Derived a cumulative regret bound for Bayesian optimisation with unknown hyper-parameters.

Guidelines for designing hyper-parameter estimation methods are provided.

Simulation demonstrates the effectiveness of following the proposed guidelines.

Abstract

Bayesian optimisation has gained great popularity as a tool for optimising the parameters of machine learning algorithms and models. Somewhat ironically, setting up the hyper-parameters of Bayesian optimisation methods is notoriously hard. While reasonable practical solutions have been advanced, they can often fail to find the best optima. Surprisingly, there is little theoretical analysis of this crucial problem in the literature. To address this, we derive a cumulative regret bound for Bayesian optimisation with Gaussian processes and unknown kernel hyper-parameters in the stochastic setting. The bound, which applies to the expected improvement acquisition function and sub-Gaussian observation noise, provides us with guidelines on how to design hyper-parameter estimation methods. A simple simulation demonstrates the importance of following these guidelines.