Title: Discretization-free Knowledge Gradient Methods for Bayesian Optimization,
Abstract: This paper studies Bayesian ranking and selection (R&S) problems with
correlated prior beliefs and continuous domains, i.e. Bayesian optimization
(BO). Knowledge gradient methods [Frazier et al., 2008, 2009] have been widely
studied for discrete R&S problems, which sample the one-step Bayes-optimal
point. When used over continuous domains, previous work on the knowledge
gradient [Scott et al., 2011, Wu and Frazier, 2016, Wu et al., 2017] often rely
on a discretized finite approximation. However, the discretization introduces
error and scales poorly as the dimension of domain grows. In this paper, we
develop a fast discretization-free knowledge gradient method for Bayesian
optimization. Our method is not restricted to the fully sequential setting, but
useful in all settings where knowledge gradient can be used over continuous
domains. We show how our method can be generalized to handle (i) batch of
points suggestion (parallel knowledge gradient); (ii) the setting where
derivative information is available in the optimization process
(derivative-enabled knowledge gradient). In numerical experiments, we
demonstrate that the discretization-free knowledge gradient method finds global
optima significantly faster than previous Bayesian optimization algorithms on
both synthetic test functions and real-world applications, especially when
function evaluations are noisy; and derivative-enabled knowledge gradient can
further improve the performances, even outperforming the gradient-based
optimizer such as BFGS when derivative information is available.