  An exciting branch of machine learning research focuses on methods for
learning, optimizing, and integrating unknown functions that are difficult or
costly to evaluate. A popular Bayesian approach to this problem uses a Gaussian
process (GP) to construct a posterior distribution over the function of
interest given a set of observed measurements, and selects new points to
evaluate using the statistics of this posterior. Here we extend these methods
to exploit derivative information from the unknown function. We describe
methods for Bayesian optimization (BO) and Bayesian quadrature (BQ) in settings
where first and second derivatives may be evaluated along with the function
itself. We perform sampling-based inference in order to incorporate uncertainty
over hyperparameters, and show that both hyperparameter and function
uncertainty decrease much more rapidly when using derivative information.
Moreover, we introduce techniques for overcoming ill-conditioning issues that
have plagued earlier methods for gradient-enhanced Gaussian processes and
kriging. We illustrate the efficacy of these methods using applications to real
and simulated Bayesian optimization and quadrature problems, and show that
exploting derivatives can provide substantial gains over standard methods.
