Deterministic Global Optimization of the Acquisition Function in Bayesian Optimization: To Do or Not To Do?
Anastasia Georgiou, Daniel Jungen, Luise Kaven, Verena Hunstig, Constantine Frangakis, Ioannis Kevrekidis, Alexander Mitsos
TL;DR
<3-5 sentence high-level summary>The paper evaluates how the choice of inner solver for acquisition-function optimization in Bayesian Optimization affects the outer optimization performance, comparing a deterministic global solver (MAiNGO) against an informed local solver (ILS) and an informed multi-start stochastic global solver (IMS). Using Müller-Brown as the baseline and several additional benchmarks, the study analyzes convergence probabilities to globally near-optimal solutions and the iteration counts to convergence under fixed acquisition-function settings, highlighting how dataset diversity and exploration–exploitation balance shape outcomes. Key findings show that MAiNGO can dramatically reduce iteration counts in exploitative settings but may underperform when initialization is poor or exploration is needed, whereas IMS/ILS provide robustness through stochasticity. The work offers practical recommendations and underscores the potential for hybrid strategies to combine the strengths of deterministic and stochastic solvers in BO, particularly as problem dimensionality grows.
Abstract
Bayesian Optimization (BO) with Gaussian Processes relies on optimizing an acquisition function to determine sampling. We investigate the advantages and disadvantages of using a deterministic global solver (MAiNGO) compared to conventional local and stochastic global solvers (L-BFGS-B and multi-start, respectively) for the optimization of the acquisition function. For CPU efficiency, we set a time limit for MAiNGO, taking the best point as optimal. We perform repeated numerical experiments, initially using the Muller-Brown potential as a benchmark function, utilizing the lower confidence bound acquisition function; we further validate our findings with three alternative benchmark functions. Statistical analysis reveals that when the acquisition function is more exploitative (as opposed to exploratory), BO with MAiNGO converges in fewer iterations than with the local solvers. However, when the dataset lacks diversity, or when the acquisition function is overly exploitative, BO with MAiNGO, compared to the local solvers, is more likely to converge to a local rather than a global ly near-optimal solution of the black-box function. L-BFGS-B and multi-start mitigate this risk in BO by introducing stochasticity in the selection of the next sampling point, which enhances the exploration of uncharted regions in the search space and reduces dependence on acquisition function hyperparameters. Ultimately, suboptimal optimization of poorly chosen acquisition functions may be preferable to their optimal solution. When the acquisition function is more exploratory, BO with MAiNGO, multi-start, and L-BFGS-B achieve comparable probabilities of convergence to a globally near-optimal solution (although BO with MAiNGO may require more iterations to converge under these conditions).