A Hierarchical Expected Improvement Method for Bayesian Optimization

Abstract

The Expected Improvement (EI) method, proposed by Jones, Schonlau, andWelch, is a widely used Bayesian optimization method, which makes use of a fitted Gaussian process model for efficient black-box optimization. However, one key drawback of EI is that it is overly greedy in exploiting the fitted Gaussian process model for optimization, which results in suboptimal solutions even with large sample sizes. To address this, we propose a new hierarchical EI (HEI) framework, which makes use of a hierarchical Gaussian process model. HEI preserves a closed-form acquisition function, and corrects the over-greediness of EI by encouraging exploration of the optimization space. We then introduce hyperparameter estimation methods which allow HEI to mimic a fully Bayesian optimization procedure, while avoiding expensive Markov-chain Monte Carlo sampling steps. We prove the global convergence of HEI over a broad function space, and establish near-minimax convergence rates under certain prior specifications. Numerical experiments show the improvement of HEI over existing Bayesian optimization methods, for synthetic functions and a semiconductor manufacturing optimization problem. Supplementary materials for this article are available online.

KEYWORDS:

• Black-box optimization
• Experimental design
• Gaussian process
• Global optimization
• Hierarchical modeling
• Uncertainty quantification

Supplementary Materials

The online supplementary materials provide proofs of technical results in the paper.

Disclosure Statement

The authors report there are no competing interests to declare.

Notes

1 In numerical experiments, we use an independent uniform prior ${\theta }_l{\sim }^{\text{iid}}U[0,100]$ for this MAP estimate.

2 Stab-EI-UK requires the next query point $x_{n+1}$ to satisfy $s_n(x_{n+1})\ge \gamma \Vert s_n(x){\Vert }_{\infty }$. In our implementation, we randomly sample ${10}^{d+2}$ points to find $\Vert s_n(x){\Vert }_{\infty }$ and set $\gamma =\min (0.1d,0.8)$.

Additional information

Funding

This research is supported by ARO W911NF-17-1-0007, NSF DMS-1914632, NSF CSSI Frameworks 2004571, and NSF DMS 2210729.