Exploiting gradients and Hessians in Bayesian optimization and Bayesian quadrature
This work addresses the challenge of efficiently learning and optimizing costly functions in machine learning, representing an incremental improvement by incorporating derivative information into existing Bayesian frameworks.
The paper tackles the problem of optimizing and integrating expensive-to-evaluate functions by extending Bayesian optimization and Bayesian quadrature methods to incorporate first and second derivative information, showing that this approach reduces hyperparameter and function uncertainty more rapidly and provides substantial gains over standard methods.
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.