Accelerating Look-ahead in Bayesian Optimization: Multilevel Monte Carlo is All you Need

TL;DR

This paper introduces a multilevel Monte Carlo approach to significantly accelerate multi-step look-ahead Bayesian optimization by reducing computational complexity in nested expectation calculations, applicable across various dimensions.

Contribution

The paper demonstrates that MLMC can improve the efficiency of multi-step look-ahead Bayesian optimization, achieving optimal convergence rates without smoothness assumptions.

Findings

MLMC achieves canonical Monte Carlo convergence rates for nested expectations.

The approach reduces computational complexity in high-dimensional Bayesian optimization.

Numerical experiments confirm the efficiency gains on benchmark problems.

Abstract

We leverage multilevel Monte Carlo (MLMC) to improve the performance of multi-step look-ahead Bayesian optimization (BO) methods that involve nested expectations and maximizations. Often these expectations must be computed by Monte Carlo (MC). The complexity rate of naive MC degrades for nested operations, whereas MLMC is capable of achieving the canonical MC convergence rate for this type of problem, independently of dimension and without any smoothness assumptions. Our theoretical study focuses on the approximation improvements for twoand three-step look-ahead acquisition functions, but, as we discuss, the approach is generalizable in various ways, including beyond the context of BO. Our findings are verified numerically and the benefits of MLMC for BO are illustrated on several benchmark examples. Code is available at https://github.com/Shangda-Yang/MLMCBO .