# Gaussian Process Uniform Error Bounds with Unknown Hyperparameters for Safety-Critical Applications

arXiv: https://arxiv.org/abs/2109.02606 · HTML: https://mlsanity.com/paper/2109.02606/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 50 references that are in the map.
- First posted on arXiv 6 Sep 2021.

## Builds on (most cited first, 14 of 14)

- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Accurate Uncertainties for Deep Learning Using Calibrated Regression](https://mlsanity.com/paper/1807.00263/) (2018)
- [Safe Model-based Reinforcement Learning with Stability Guarantees](https://mlsanity.com/paper/1705.08551/) (2017)
- [On Kernelized Multi-armed Bandits](https://mlsanity.com/paper/1704.00445/) (2017)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Gaussian Process Kernels for Pattern Discovery and Extrapolation](https://mlsanity.com/paper/1302.4245/) (2013)
- [Gaussian Processes for Data-Efficient Learning in Robotics and Control](https://mlsanity.com/paper/1502.02860/) (2015)
- [Cautious Model Predictive Control using Gaussian Process Regression](https://mlsanity.com/paper/1705.10702/) (2017)
- [Adaptive and Safe Bayesian Optimization in High Dimensions via One-Dimensional Subspaces](https://mlsanity.com/paper/1902.03229/) (2019)
- [Uniform Error Bounds for Gaussian Process Regression with Application to Safe Control](https://mlsanity.com/paper/1906.01376/) (2019)
- [No-Regret Bayesian Optimization with Unknown Hyperparameters](https://mlsanity.com/paper/1901.03357/) (2019)
- [Practical and Rigorous Uncertainty Bounds for Gaussian Process Regression](https://mlsanity.com/paper/2105.02796/) (2021)
- [Deterministic error bounds for kernel-based learning techniques under bounded noise](https://mlsanity.com/paper/2008.04005/) (2020)
- [Mean Square Prediction Error of Misspecified Gaussian Process Models](https://mlsanity.com/paper/1811.06642/) (2018)

## Built on by (most cited first, 17 of 17)

- [Improved Regret Bounds for Gaussian Process Upper Confidence Bound in Bayesian Optimization](https://mlsanity.com/paper/2506.01393/) (2025)
- [Guaranteed Coverage Prediction Intervals with Gaussian Process Regression](https://mlsanity.com/paper/2310.15641/) (2023)
- [Can Learning Deteriorate Control? Analyzing Computational Delays in Gaussian Process-Based Event-Triggered Online Learning](https://mlsanity.com/paper/2305.08169/) (2023)
- [Safe and Stable Closed-Loop Learning for Neural-Network-Supported Model Predictive Control](https://mlsanity.com/paper/2409.10171/) (2024)
- [Bayesian Optimisation with Unknown Hyperparameters: Regret Bounds Logarithmically Closer to Optimal](https://mlsanity.com/paper/2410.10384/) (2024)
- [Sharp Calibrated Gaussian Processes](https://mlsanity.com/paper/2302.11961/) (2023)
- [Towards Safe Multi-Task Bayesian Optimization](https://mlsanity.com/paper/2312.07281/) (2023)
- [Multi-fidelity Gaussian process surrogate modeling for regression problems in physics](https://mlsanity.com/paper/2404.11965/) (2024)
- [Structure-Preserving Learning Using Gaussian Processes and Variational Integrators](https://mlsanity.com/paper/2112.05451/) (2021)
- [A Hierarchical Surrogate Model for Efficient Multi-Task Parameter Learning in Closed-Loop Control](https://mlsanity.com/paper/2508.12738/) (2025)
- [Personalized Rehabilitation Robotics based on Online Learning Control](https://mlsanity.com/paper/2110.00481/) (2021)
- [An Analysis of Safety Guarantees in Multi-Task Bayesian Optimization](https://mlsanity.com/paper/2503.08555/) (2025)
- [Error Bounds for Kernel-Based Linear System Identification with Unknown Hyperparameters](https://mlsanity.com/paper/2303.09842/) (2023)
- [Episodic Gaussian Process-Based Learning Control with Vanishing Tracking Errors](https://mlsanity.com/paper/2307.04415/) (2023)
- [Adaptive Individual Uncertainty under Out-Of-Distribution Shift with Expert-Routed Conformal Prediction](https://mlsanity.com/paper/2510.15233/) (2025)
- [Safe Bayesian Optimization for Uncertain Correlation Matrices in Linear Models of Co-Regionalization](https://mlsanity.com/paper/2605.13302/) (2026)
- [A Quadratic Order Reduction -- Gaussian Process Ordinary Differential Equation framework for the inference of Large Continuous Dynamical Systems](https://mlsanity.com/paper/2606.13063/) (2026)

## Abstract (arXiv)

Gaussian processes have become a promising tool for various safety-critical settings, since the posterior variance can be used to directly estimate the model error and quantify risk. However, state-of-the-art techniques for safety-critical settings hinge on the assumption that the kernel hyperparameters are known, which does not apply in general. To mitigate this, we introduce robust Gaussian process uniform error bounds in settings with unknown hyperparameters. Our approach computes a confidence region in the space of hyperparameters, which enables us to obtain a probabilistic upper bound for the model error of a Gaussian process with arbitrary hyperparameters. We do not require to know any bounds for the hyperparameters a priori, which is an assumption commonly found in related work. Instead, we are able to derive bounds from data in an intuitive fashion. We additionally employ the proposed technique to derive performance guarantees for a class of learning-based control problems. Experiments show that the bound performs significantly better than vanilla and fully Bayesian Gaussian processes.