Constrained Two-step Look-Ahead Bayesian Optimization
Authors: Yunxiang Zhang, Xiangyu Zhang, Peter Frazier

Recent advances in computationally efficient non-myopic Bayesian optimization offer improved query efficiency over traditional myopic methods like expected improvement, with only a modest increase in computational cost. These advances have been largely limited to unconstrained BO methods with only a few exceptions which require heavy computation. For instance, one existing multi-step lookahead constrained BO method (Lam & Willcox, 2017) relies on computationally expensive unreliable brute force derivative-free optimization of a Monte Carlo rollout acquisition function. Methods that use the reparameterization trick for more efficient derivative-based optimization of non-myopic acquisition functions in the unconstrained setting, like sample average approximation and infinitesimal perturbation analysis, do not extend: constraints introduce discontinuities in the sampled acquisition function surface. Moreover, we argue here that being non-myopic is even more important in constrained problems because fear of violating constraints pushes myopic methods away from sampling the boundary between feasible and infeasible regions, slowing the discovery of optimal solutions with tight constraints. In this paper, we propose a computationally efficient two-step lookahead constrained Bayesian optimization acquisition function (2-OPT-C) supporting both sequential and batch settings. To enable fast acquisition function optimization, we develop a novel likelihood ratio-based unbiased estimator of the gradient of the two-step optimal acquisition function that does not use the reparameterization trick. In numerical experiments, 2-OPT-C typically improves query efficiency by 2x or more over previous methods, and in some cases by 10x or more.