Fast and Provably Accurate Sequential Designs using Hilbert Space Gaussian Processes

TL;DR

This paper introduces a fast, accurate method for sequential design in Gaussian process modeling using a Hilbert space approximation to efficiently evaluate the IMSE acquisition function.

Contribution

It develops a novel Hilbert space Gaussian process approximation that enables closed-form evaluation of the IMSE acquisition function, improving computational efficiency.

Findings

Achieves lower prediction error in numerical experiments.

Reduces computation time compared to existing methods.

Provides sharp bounds on approximation and acquisition function errors.

Abstract

Gaussian processes are widely used for accurate emulation of unknown surfaces in sequential design of expensive simulation experiments. Integrated mean squared error (IMSE) is an effective acquisition function for sequential designs based on Gaussian processes. However, existing approaches struggle with its implementation because the required integrals often lack closed-form expressions for most kernel functions. We propose a novel and computationally efficient Hilbert space Gaussian process approximation for the IMSE acquisition function, where a truncated eigenbasis representation of the integral enables closed-form evaluation. We establish sharp global non-asymptotic bounds for both the approximation error of isotropic kernels and the resulting error in the acquisition function. In a series of numerical experiments with $\gamma$-stabilizing, the proposed method achieves substantially lower prediction error and reduced computation time compared to existing benchmarks. These results demonstrate that the proposed Hilbert space Gaussian process framework provides an accurate and computationally efficient approach for Gaussian process based sequential design.