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.

翻译：本文提出了一种在高斯过程建模框架下获取预测分布的新方法，该方法在关注区间外放松了插值约束：当观测值位于关注区间外时，预测分布的均值不再必然插值这些观测值，而仅被约束保持在区间之外。这种称为松弛高斯过程插值的改进方法能够在关注区间内提供更优的预测分布，尤其适用于高斯过程模型的平稳性假设不成立的情形。该方法可视为一种面向目标的方法，在贝叶斯优化中具有特殊价值——例如在目标函数最小化问题中，对低函数值区域的准确预测分布至关重要。当采用期望改进准则与松弛高斯过程插值来序贯选择评估点时，所得优化算法的收敛性在理论上得到保证（前提是待优化函数位于与已知协方差函数对应的再生核希尔伯特空间内）。实验表明，在贝叶斯优化中使用松弛高斯过程模型相较于传统平稳高斯过程模型具有显著优势。