Bayesian optimization (BO) methods choose sample points by optimizing an acquisition function derived from a statistical model of the objective. These acquisition functions are chosen to balance sampling regions with predicted good objective values against exploring regions where the objective is uncertain. Standard acquisition functions are myopic, considering only the impact of the next sample, but non-myopic acquisition functions may be more effective. In principle, one could model the sampling by a Markov decision process, and optimally choose the next sample by maximizing an expected reward computed by dynamic programming; however, this is infeasibly expensive. More practical approaches, such as rollout, consider a parametric family of sampling policies. In this paper, we show how to efficiently estimate rollout acquisition functions and their gradients, enabling stochastic gradient-based optimization of sampling policies.

Bayesian optimization (BO) is a class of sample-efficient global optimization methods, where a probabilistic model conditioned on previous observations is used to determine future evaluations via the optimization of an acquisition function. Most acquisition functions are myopic, meaning that they only consider the impact of the next function evaluation. Non-myopic acquisition functions consider the impact of the next $h$ function evaluations and are typically computed through rollout, in which $h$ steps of BO are simulated. These rollout acquisition functions are defined as $h$-dimensional integrals, and are expensive to compute and optimize. We show that a combination of quasi-Monte Carlo, common random numbers, and control variates significantly reduce the computational burden of rollout. We then formulate a policy-search based approach that removes the need to optimize the rollout acquisition function. Finally, we discuss the qualitative behavior of rollout policies in the setting of multi-modal objectives and model error.