Goal-Oriented Lower-Tail Calibration of Gaussian Processes for Bayesian Optimization
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914580654981120 |
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| author | Pion, Aurélien Vazquez, Emmanuel |
| author_facet | Pion, Aurélien Vazquez, Emmanuel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20145 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Goal-Oriented Lower-Tail Calibration of Gaussian Processes for Bayesian Optimization Pion, Aurélien Vazquez, Emmanuel Machine Learning Methodology 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. |
| title | Goal-Oriented Lower-Tail Calibration of Gaussian Processes for Bayesian Optimization |
| topic | Machine Learning Methodology |
| url | https://arxiv.org/abs/2605.20145 |