Distance-distributed design for Gaussian process surrogates

A common challenge in computer experiments and related fields is to efficiently explore the input space using a small number of samples, i.e., the experimental design problem. Much of the recent focus in the computer experiment literature, where modeling is often via Gaussian process (GP) surrogates, has been on space-filling designs, via maximin distance, Latin hypercube, etc. However, it is easy to demonstrate empirically that such designs disappoint when the model hyperparameterization is unknown, and must be estimated from data observed at the chosen design sites. This is true even when the performance metric is prediction-based, or when the target of interest is inherently or eventually sequential in nature, such as in blackbox (Bayesian) optimization. Here we expose such inefficiencies, showing that in many cases purely random design is superior to higher-powered alternatives. We then propose a family of new schemes by reverse engineering the qualities of the random designs which give the best estimates of GP lengthscales. Specifically, we study the distribution of pairwise distances between design elements, and develop a numerical scheme to optimize those distances for a given sample size and dimension. We illustrate how our distance-based designs, and their hybrids with more conventional space-filling schemes, outperform in both static (one-shot design) and sequential settings.