  Bayesian optimization has been successful at global optimization of
expensive-to-evaluate multimodal objective functions. However, unlike most
optimization methods, Bayesian optimization typically does not use derivative
information. In this paper we show how Bayesian optimization can exploit
derivative information to decrease the number of objective function evaluations
required for good performance. In particular, we develop a novel Bayesian
optimization algorithm, the derivative-enabled knowledge-gradient (dKG), for
which we show one-step Bayes-optimality, asymptotic consistency, and greater
one-step value of information than is possible in the derivative-free setting.
Our procedure accommodates noisy and incomplete derivative information, comes
in both sequential and batch forms, and can optionally reduce the computational
cost of inference through automatically selected retention of a single
directional derivative. We also compute the d-KG acquisition function and its
gradient using a novel fast discretization-free technique. We show d-KG
provides state-of-the-art performance compared to a wide range of optimization
procedures with and without gradients, on benchmarks including logistic
regression, deep learning, kernel learning, and k-nearest neighbors.
