Goal-Oriented Lower-Tail Calibration of Gaussian Processes for Bayesian Optimization

• URL: http://arxiv.org/abs/2605.20145v1
• Date: Tue, 19 May 2026 17:32:25 GMT
• Title: Goal-Oriented Lower-Tail Calibration of Gaussian Processes for Bayesian Optimization
• Abstract summary: This article studies goal-oriented calibration of GP predictive distributions below a low threshold $t$ in the noiseless setting.<n>A framework for predictive reliability below $t$ is introduced, based on two notions of spatial calibration.<n>Building on this framework, we propose tcGP, a post-hoc method that calibrates GP predictive distributions below$t$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Bayesian optimization (BO) selects evaluation points for expensive black-box objectives using Gaussian process (GP) predictive distributions. Kernel choice and hyperparameter selection can lead to miscalibrated predictive distributions and an inappropriate exploration-exploitation trade-off. For minimization, sampling criteria such as expected improvement (EI) depend on the predictive distribution below the current best value, so lower-tail miscalibration directly affects the sampling decision. This article studies goal-oriented calibration of GP predictive distributions below a low threshold $t$ in the noiseless setting, for standard GP models with hyperparameters selected by maximum likelihood. A framework for predictive reliability below $t$ is introduced, based on two notions of spatial calibration: occurrence calibration over the design space and thresholded $μ$-calibration on sublevel sets of the form $\{x\in\mathbb{X}, f(x)\le t\}$. Building on this framework, we propose tcGP, a post-hoc method that calibrates GP predictive distributions below~$t$, and we show that the resulting EI-based global optimization algorithm remains dense in the design space. Experiments on standard benchmarks show improved lower-tail calibration and BO performance relative to standard GP models and globally calibrated GP models.

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