  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 .
