H-Entropy Search: Generalizing Bayesian Optimization with a Decision-theoretic Uncertainty Measure

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 the H-entropy, 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 the H-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 for a variety of specialized optimization tasks, including variants of top-k estimation, level set estimation, and multi-valued search. For special cases of the loss and design space, we develop gradient-based methods to efficiently optimize our proposed family of acquisition functions, and demonstrate that the resulting BO procedure shows strong empirical performance on a diverse set of optimization tasks.