Exploiting gradients and Hessians in Bayesian optimization and Bayesian quadrature

TL;DR

This paper extends Bayesian optimization and quadrature methods to incorporate derivative information, demonstrating that using gradients and Hessians significantly improves efficiency and accuracy in function learning, optimization, and integration tasks.

Contribution

It introduces novel techniques for leveraging derivatives in Gaussian process models for Bayesian optimization and quadrature, addressing previous ill-conditioning issues and demonstrating substantial performance gains.

Findings

Derivative information accelerates hyperparameter convergence.

Gradient-enhanced methods outperform standard Gaussian processes.

Applications show improved optimization and integration results.

Abstract

An exciting branch of machine learning research focuses on methods for learning, optimizing, and integrating unknown functions that are difficult or costly to evaluate. A popular Bayesian approach to this problem uses a Gaussian process (GP) to construct a posterior distribution over the function of interest given a set of observed measurements, and selects new points to evaluate using the statistics of this posterior. Here we extend these methods to exploit derivative information from the unknown function. We describe methods for Bayesian optimization (BO) and Bayesian quadrature (BQ) in settings where first and second derivatives may be evaluated along with the function itself. We perform sampling-based inference in order to incorporate uncertainty over hyperparameters, and show that both hyperparameter and function uncertainty decrease much more rapidly when using derivative information. Moreover, we introduce techniques for overcoming ill-conditioning issues that have plagued earlier methods for gradient-enhanced Gaussian processes and kriging. We illustrate the efficacy of these methods using applications to real and simulated Bayesian optimization and quadrature problems, and show that exploting derivatives can provide substantial gains over standard methods.