  We propose a novel Bayesian method to solve the maximization of a
time-dependent expensive-to-evaluate stochastic oracle. We are interested in
the decision that maximizes the oracle at a finite time horizon, given a
limited budget of noisy evaluations of the oracle that can be performed before
the horizon. Our recursive two-step lookahead acquisition function for Bayesian
optimization makes nonmyopic decisions at every stage by maximizing the
expected utility at the specified time horizon. Specifically, we propose a
generalized two-step lookahead framework with a customizable \emph{value}
function that allows users to define the utility. We illustrate how lookahead
versions of classic acquisition functions such as the expected improvement,
probability of improvement, and upper confidence bound can be obtained with
this framework. We demonstrate the utility of our proposed approach on several
carefully constructed synthetic cases and a real-world quantum optimal control
problem.
