  The Expected Improvement (EI) method, proposed by Jones et al. (1998), is a
widely-used Bayesian optimization method, which makes use of a fitted Gaussian
process model for efficient black-box optimization. However, one key drawback
of EI is that it is overly greedy in exploiting the fitted Gaussian process
model for optimization, which results in suboptimal solutions even with large
sample sizes. To address this, we propose a new hierarchical EI (HEI)
framework, which makes use of a hierarchical Gaussian process model. HEI
preserves a closed-form acquisition function, and corrects the over-greediness
of EI by encouraging exploration of the optimization space. We then introduce
hyperparameter estimation methods which allow HEI to mimic a fully Bayesian
optimization procedure, while avoiding expensive Markov-chain Monte Carlo
sampling steps. We prove the global convergence of HEI over a broad function
space, and establish near-minimax convergence rates under certain prior
specifications. Numerical experiments show the improvement of HEI over existing
Bayesian optimization methods, for synthetic functions and a semiconductor
manufacturing optimization problem.
