Relaxed Gaussian process interpolation: a goal-oriented approach to Bayesian optimization

TL;DR

This paper introduces relaxed Gaussian process (reGP) interpolation, a goal-oriented Bayesian optimization method that relaxes interpolation constraints outside interest ranges, leading to improved predictive distributions and convergence guarantees.

Contribution

The paper proposes reGP, a novel relaxation of Gaussian process interpolation, enhancing Bayesian optimization by focusing on goal-oriented predictive accuracy and providing theoretical convergence guarantees.

Findings

reGP improves predictive distributions in ranges of interest

Using reGP enhances Bayesian optimization performance

Theoretical convergence of the optimization algorithm is established

Abstract

This work presents a new procedure for obtaining predictive distributions in the context of Gaussian process (GP) modeling, with a relaxation of the interpolation constraints outside ranges of interest: the mean of the predictive distributions no longer necessarily interpolates the observed values when they are outside ranges of interest, but are simply constrained to remain outside. This method called relaxed Gaussian process (reGP) interpolation provides better predictive distributions in ranges of interest, especially in cases where a stationarity assumption for the GP model is not appropriate. It can be viewed as a goal-oriented method and becomes particularly interesting in Bayesian optimization, for example, for the minimization of an objective function, where good predictive distributions for low function values are important. When the expected improvement criterion and reGP are used for sequentially choosing evaluation points, the convergence of the resulting optimization algorithm is theoretically guaranteed (provided that the function to be optimized lies in the reproducing kernel Hilbert space attached to the known covariance of the underlying Gaussian process). Experiments indicate that using reGP instead of stationary GP models in Bayesian optimization is beneficial.