How Bayesian should Bayesian optimisation be?

Abstract

Bayesian optimisation (BO) uses probabilistic surrogate modelsusually 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 Matérn 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. CCS CONCEPTS• Theory of computation → Gaussian processes; Mathematical optimization; • Mathematics of computing → Bayesian computation.

MBORE: Multi-objective Bayesian Optimisation by Density-Ratio Estimation

“…Bar heights correspond to the proportion of times each model and scalarisation combination was the best on each test problem. As can be seen from the figure, MBORE with XGB and our novel PHC scalariser (15) has the best overall performance for both benchmarks. Surprisingly, MBORE with the MLP classification method performs worse than the best performing method on all the 63 WFG test problems.…”

“…In this work we use a Matérn 5/2 kernel, as recommended for modelling realistic functions [69,73]. The kernel's hyperparameters 𝜽 are learnt via maximising the log marginal likelihood [15,58].…”

“…𝑑 ∈ {20, 50, 100}, version of the WFG benchmark is also evaluated to assess optimisation performance in more challenging scenarios. Experiments are repeated for the scalarisation methods discussed in Section 2.2.1: augmented Tchebycheff (AT) [44], hypervolume improvement (HYPI) [57], dominance ranking (DomRank) [57], and our novel scalariser, Pareto hypervolume contribution (PHC) (15).…”

“…[0, 0, 0] 5 [3,5,7,9,11] [0, 0, 0, 0, 0] 10 [3,5,7,9,11,13,15,17,19,21] [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]…”

Insights into machine learning models from chemical physics: an energy landscapes approach (EL for ML)

“…136 Additionally, slice sampling, 137 adaptive importance sampling, 138 and entropy-based methods 139 have been used within Bayesian optimisation, where there is interest in moving beyond single-point estimates to fully-Bayesian approaches. 140,141 The main drawback with these methods is the high computational cost, and ideally they should only be employed when it will provide a substantial advantage over single point estimates. However, this condition cannot be known a priori .…”