Bayesian Optimization for Expensive Smooth-Varying Functions

Bayesian optimization (BO) is a powerful class of data-driven techniques for the maximization of expensive-to-evaluate objective functions. These techniques construct a Gaussian process (GP) regression for representing the objective function according to the latest available function evaluations and sequentially select samples and evaluate the function by maximizing an acquisition function. The primary assumption in most BO policies is that the objective function has a uniform level of smoothness over the input space, modeled by a kernel function. However, the uniform smoothness assumption is likely to be violated in a wide range of practical problems, primary domains in which the objective function is evaluated differently at various regions of input space (e.g., through different experiments, software, or approximators). This article develops a BO framework capable of optimizing expensive smooth-varying functions. Unlike the existing techniques that rely on a single GP model, the proposed framework constructs a set of local and global GP models to represent the objective function. The predictive mean and variance at any given sample in the input space are computed according to the posterior probabilities of the local and global GP models. Local and global models are adaptively controlled through a single parameter, which can be optimized along with other GP models' parameters during the optimization process. Using the predicted local and global values, the expected improvement acquisition function is employed as one of the possible acquisition functions for the selection process. The performance of the proposed framework is assessed extensively through two optimization benchmark problems.