# Uniform Error Bounds for Gaussian Process Regression with Application to Safe Control

arXiv: https://arxiv.org/abs/1906.01376 · HTML: https://mlsanity.com/paper/1906.01376/

## Key facts

- Cited by 64 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jun 2019.

## Built on by (most cited first, 30 of 64)

- [Practical and Rigorous Uncertainty Bounds for Gaussian Process Regression](https://mlsanity.com/paper/2105.02796/) (2021)
- [Scaling ResNets in the Large-depth Regime](https://mlsanity.com/paper/2206.06929/) (2022)
- [Gaussian Process Uniform Error Bounds with Unknown Hyperparameters for Safety-Critical Applications](https://mlsanity.com/paper/2109.02606/) (2021)
- [On Safety in Safe Bayesian Optimization](https://mlsanity.com/paper/2403.12948/) (2024)
- [Robust Uncertainty Bounds in Reproducing Kernel Hilbert Spaces: A Convex Optimization Approach](https://mlsanity.com/paper/2104.09582/) (2021)
- [A New Representation of Successor Features for Transfer across Dissimilar Environments](https://mlsanity.com/paper/2107.08426/) (2021)
- [Safe and Efficient Reinforcement Learning Using Disturbance-Observer-Based Control Barrier Functions](https://mlsanity.com/paper/2211.17250/) (2022)
- [Stopping Bayesian Optimization with Probabilistic Regret Bounds](https://mlsanity.com/paper/2402.16811/) (2024)
- [Collaborative and Distributed Bayesian Optimization via Consensus: Showcasing the Power of Collaboration for Optimal Design](https://mlsanity.com/paper/2306.14348/) (2023)
- [Minimizing UCB: a Better Local Search Strategy in Local Bayesian Optimization](https://mlsanity.com/paper/2405.15285/) (2024)
- [Residual Gaussian Process: A Tractable Nonparametric Bayesian Emulator for Multi-fidelity Simulations](https://mlsanity.com/paper/2104.03743/) (2021)
- [Can Learning Deteriorate Control? Analyzing Computational Delays in Gaussian Process-Based Event-Triggered Online Learning](https://mlsanity.com/paper/2305.08169/) (2023)
- [Uniform Error and Posterior Variance Bounds for Gaussian Process Regression with Application to Safe Control](https://mlsanity.com/paper/2101.05328/) (2021)
- [Power of Quantum Generative Learning](https://mlsanity.com/paper/2205.04730/) (2022)
- [Error Bounds For Gaussian Process Regression Under Bounded Support Noise With Applications To Safety Certification](https://mlsanity.com/paper/2408.09033/) (2024)
- [Formal Verification of Unknown Dynamical Systems via Gaussian Process Regression](https://mlsanity.com/paper/2201.00655/) (2021)
- [Towards Safe Multi-Task Bayesian Optimization](https://mlsanity.com/paper/2312.07281/) (2023)
- [Collaborative and Federated Black-box Optimization: A Bayesian Optimization Perspective](https://mlsanity.com/paper/2411.07523/) (2024)
- [Finite-Sample-Based Reachability for Safe Control with Gaussian Process Dynamics](https://mlsanity.com/paper/2505.07594/) (2025)
- [Toward L_∞-recovery of Nonlinear Functions: A Polynomial Sample Complexity Bound for Gaussian Random Fields](https://mlsanity.com/paper/2305.00322/) (2023)
- [Promises of Deep Kernel Learning for Control Synthesis](https://mlsanity.com/paper/2309.06569/) (2023)
- [Cooperative Learning with Gaussian Processes for Euler-Lagrange Systems Tracking Control under Switching Topologies](https://mlsanity.com/paper/2402.03048/) (2024)
- [Towards safe Bayesian optimization with Wiener kernel regression](https://mlsanity.com/paper/2411.02253/) (2024)
- [On the convergence rate of noisy Bayesian Optimization with Expected Improvement](https://mlsanity.com/paper/2501.09262/) (2025)
- [The Hardness of Validating Observational Studies with Experimental Data](https://mlsanity.com/paper/2503.14795/) (2025)
- [Optimal kernel regression bounds under energy-bounded noise](https://mlsanity.com/paper/2505.22235/) (2025)
- [Streaming Generated Gaussian Process Experts for Online Learning and Control: Extended Version](https://mlsanity.com/paper/2508.03679/) (2025)
- [Structure-Preserving Learning Using Gaussian Processes and Variational Integrators](https://mlsanity.com/paper/2112.05451/) (2021)
- [Safe Machine-Learning-supported Model Predictive Force and Motion Control in Robotics](https://mlsanity.com/paper/2303.04569/) (2023)
- [Learning-based Control for PMSM Using Distributed Gaussian Processes with Optimal Aggregation Strategy](https://mlsanity.com/paper/2307.13945/) (2023)

## Abstract (arXiv)

Data-driven models are subject to model errors due to limited and noisy training data. Key to the application of such models in safety-critical domains is the quantification of their model error. Gaussian processes provide such a measure and uniform error bounds have been derived, which allow safe control based on these models. However, existing error bounds require restrictive assumptions. In this paper, we employ the Gaussian process distribution and continuity arguments to derive a novel uniform error bound under weaker assumptions. Furthermore, we demonstrate how this distribution can be used to derive probabilistic Lipschitz constants and analyze the asymptotic behavior of our bound. Finally, we derive safety conditions for the control of unknown dynamical systems based on Gaussian process models and evaluate them in simulations of a robotic manipulator.