  Bayesian optimization (BO) is a popular method for efficiently inferring
optima of an expensive black-box function via a sequence of queries. Existing
information-theoretic BO procedures aim to make queries that most reduce the
uncertainty about optima, where the uncertainty is captured by Shannon entropy.
However, an optimal measure of uncertainty would, ideally, factor in how we
intend to use the inferred quantity in some downstream procedure. In this
paper, we instead consider a generalization of Shannon entropy from work in
statistical decision theory (DeGroot 1962, Rao 1984), which contains a broad
class of uncertainty measures parameterized by a problem-specific loss function
corresponding to a downstream task. We first show that special cases of this
entropy lead to popular acquisition functions used in BO procedures such as
knowledge gradient, expected improvement, and entropy search. We then show how
alternative choices for the loss yield a flexible family of acquisition
functions that can be customized for use in novel optimization settings.
Additionally, we develop gradient-based methods to efficiently optimize our
proposed family of acquisition functions, and demonstrate strong empirical
performance on a diverse set of sequential decision making tasks, including
variants of top-$k$ optimization, multi-level set estimation, and sequence
search.
