  We propose a novel Bayesian method to solve the maximization of a
time-dependent expensive-to-evaluate oracle. We are interested in the decision
that maximizes the oracle at a finite time horizon, when relatively few noisy
evaluations can be performed before the horizon. Our recursive, two-step
lookahead expected payoff ($\texttt{r2LEY}$) acquisition function makes
nonmyopic decisions at every stage by maximizing the estimated expected value
of the oracle at the horizon. $\texttt{r2LEY}$ circumvents the evaluation of
the expensive multistep (more than two steps) lookahead acquisition function by
recursively optimizing a two-step lookahead acquisition function at every
stage; unbiased estimators of this latter function and its gradient are
utilized for efficient optimization. $\texttt{r2LEY}$ is shown to exhibit
natural exploration properties far from the time horizon, enabling accurate
emulation of the oracle, which is exploited in the final decision made at the
horizon. To demonstrate the utility of $\texttt{r2LEY}$, we compare it with
time-dependent extensions of popular myopic acquisition functions via both
synthetic and real-world datasets.
