Bayesian Optimization with Gradients

TL;DR

This paper introduces a new Bayesian optimization algorithm, dKG, that effectively utilizes derivative information to reduce the number of function evaluations needed for optimizing complex, expensive functions.

Contribution

The paper develops the derivative-enabled knowledge-gradient (dKG) algorithm, which exploits derivatives in Bayesian optimization, offering theoretical guarantees and improved performance over derivative-free methods.

Findings

dKG achieves state-of-the-art results on various benchmarks.

The method handles noisy and incomplete derivative data.

It reduces the number of function evaluations needed for optimization.

Abstract

Bayesian optimization has been successful at global optimization of expensive-to-evaluate multimodal objective functions. However, unlike most optimization methods, Bayesian optimization typically does not use derivative information. In this paper we show how Bayesian optimization can exploit derivative information to decrease the number of objective function evaluations required for good performance. In particular, we develop a novel Bayesian optimization algorithm, the derivative-enabled knowledge-gradient (dKG), for which we show one-step Bayes-optimality, asymptotic consistency, and greater one-step value of information than is possible in the derivative-free setting. Our procedure accommodates noisy and incomplete derivative information, comes in both sequential and batch forms, and can optionally reduce the computational cost of inference through automatically selected retention of a single directional derivative. We also compute the d-KG acquisition function and its gradient using a novel fast discretization-free technique. We show d-KG provides state-of-the-art performance compared to a wide range of optimization procedures with and without gradients, on benchmarks including logistic regression, deep learning, kernel learning, and k-nearest neighbors.