  Bayesian optimization is a sample-efficient approach to global optimization
that relies on theoretically motivated value heuristics (acquisition functions)
to guide its search process. Fully maximizing acquisition functions produces
the Bayes' decision rule, but this ideal is difficult to achieve since these
functions are frequently non-trivial to optimize. This statement is especially
true when evaluating queries in parallel, where acquisition functions are
routinely non-convex, high-dimensional, and intractable. We first show that
acquisition functions estimated via Monte Carlo integration are consistently
amenable to gradient-based optimization. Subsequently, we identify a common
family of acquisition functions, including EI and UCB, whose properties not
only facilitate but justify use of greedy approaches for their maximization.
