How Bayesian Should Bayesian Optimisation Be?

TL;DR

This paper investigates whether a fully Bayesian approach to hyperparameter uncertainty in Gaussian process-based Bayesian optimization improves performance, comparing approximate inference methods across benchmark problems.

Contribution

It introduces and evaluates a fully Bayesian treatment of GP hyperparameters in BO, demonstrating its advantages with Expected Improvement and ARD kernels.

Findings

FBBO with EI and ARD kernel performs best in noise-free settings.

FBBO causes over-exploration with UCB, but not with EI.

Less performance difference among BO configurations under increased noise.

Abstract

Bayesian optimisation (BO) uses probabilistic surrogate models - usually Gaussian processes (GPs) - for the optimisation of expensive black-box functions. At each BO iteration, the GP hyperparameters are fit to previously-evaluated data by maximising the marginal likelihood. However, this fails to account for uncertainty in the hyperparameters themselves, leading to overconfident model predictions. This uncertainty can be accounted for by taking the Bayesian approach of marginalising out the model hyperparameters. We investigate whether a fully-Bayesian treatment of the Gaussian process hyperparameters in BO (FBBO) leads to improved optimisation performance. Since an analytic approach is intractable, we compare FBBO using three approximate inference schemes to the maximum likelihood approach, using the Expected Improvement (EI) and Upper Confidence Bound (UCB) acquisition functions paired with ARD and isotropic Matern kernels, across 15 well-known benchmark problems for 4 observational noise settings. FBBO using EI with an ARD kernel leads to the best performance in the noise-free setting, with much less difference between combinations of BO components when the noise is increased. FBBO leads to over-exploration with UCB, but is not detrimental with EI. Therefore, we recommend that FBBO using EI with an ARD kernel as the default choice for BO.