  Optimizing expensive-to-evaluate black-box functions of discrete (and
potentially continuous) design parameters is a ubiquitous problem in scientific
and engineering applications. Bayesian optimization (BO) is a popular,
sample-efficient method that leverages a probabilistic surrogate model and an
acquisition function (AF) to select promising designs to evaluate. However,
maximizing the AF over mixed or high-cardinality discrete search spaces is
challenging standard gradient-based methods cannot be used directly or
evaluating the AF at every point in the search space would be computationally
prohibitive. To address this issue, we propose using probabilistic
reparameterization (PR). Instead of directly optimizing the AF over the search
space containing discrete parameters, we instead maximize the expectation of
the AF over a probability distribution defined by continuous parameters. We
prove that under suitable reparameterizations, the BO policy that maximizes the
probabilistic objective is the same as that which maximizes the AF, and
therefore, PR enjoys the same regret bounds as the original BO policy using the
underlying AF. Moreover, our approach provably converges to a stationary point
of the probabilistic objective under gradient ascent using scalable, unbiased
estimators of both the probabilistic objective and its gradient. Therefore, as
the number of starting points and gradient steps increase, our approach will
recover of a maximizer of the AF (an often-neglected requisite for commonly
used BO regret bounds). We validate our approach empirically and demonstrate
state-of-the-art optimization performance on a wide range of real-world
applications. PR is complementary to (and benefits) recent work and naturally
generalizes to settings with multiple objectives and black-box constraints.
