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

arXiv: https://arxiv.org/abs/2101.05328 · HTML: https://mlsanity.com/paper/2101.05328/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 57 references that are in the map.
- First posted on arXiv 13 Jan 2021.

## Builds on (most cited first, 13 of 13)

- [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)
- [Bayesian Optimization with Safety Constraints: Safe and Automatic Parameter Tuning in Robotics](https://mlsanity.com/paper/1602.04450/) (2016)
- [Learning-based Model Predictive Control for Safe Exploration](https://mlsanity.com/paper/1803.08287/) (2018)
- [Safe Controller Optimization for Quadrotors with Gaussian Processes](https://mlsanity.com/paper/1509.01066/) (2015)
- [Uniform Error Bounds for Gaussian Process Regression with Application to Safe Control](https://mlsanity.com/paper/1906.01376/) (2019)
- [Safe Learning of Regions of Attraction for Uncertain, Nonlinear Systems with Gaussian Processes](https://mlsanity.com/paper/1603.04915/) (2016)
- [An Empirical Evaluation of Deep Learning on Highway Driving](https://mlsanity.com/paper/1504.01716/) (2015)
- [Feedback Linearization based on Gaussian Processes with event-triggered Online Learning](https://mlsanity.com/paper/1911.06565/) (2019)
- [Gaussian Process bandits with adaptive discretization](https://mlsanity.com/paper/1712.01447/) (2017)
- [Provably Robust Learning-Based Approach for High-Accuracy Tracking Control of Lagrangian Systems](https://mlsanity.com/paper/1804.01031/) (2018)
- [Gaussian Process based Passivation of a Class of Nonlinear Systems with Unknown Dynamics](https://mlsanity.com/paper/1811.06648/) (2018)

## Built on by (most cited first, 7 of 7)

- [Residual Gaussian Process: A Tractable Nonparametric Bayesian Emulator for Multi-fidelity Simulations](https://mlsanity.com/paper/2104.03743/) (2021)
- [Toward L_∞-recovery of Nonlinear Functions: A Polynomial Sample Complexity Bound for Gaussian Random Fields](https://mlsanity.com/paper/2305.00322/) (2023)
- [Efficiently Computable Safety Bounds for Gaussian Processes in Active Learning](https://mlsanity.com/paper/2402.18260/) (2024)
- [Stochastic Cell Transmission Models of Traffic Networks](https://mlsanity.com/paper/2304.11654/) (2023)
- [Modeling Epidemic Spread: A Gaussian Process Regression Approach](https://mlsanity.com/paper/2312.09384/) (2023)
- [Quantifying Local Model Validity using Active Learning](https://mlsanity.com/paper/2406.07474/) (2024)
- [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)

In application areas where data generation is expensive, Gaussian processes are a preferred supervised learning model due to their high data-efficiency. Particularly in model-based control, Gaussian processes allow the derivation of performance guarantees using probabilistic model error bounds. To make these approaches applicable in practice, two open challenges must be solved i) Existing error bounds rely on prior knowledge, which might not be available for many real-world tasks. (ii) The relationship between training data and the posterior variance, which mainly drives the error bound, is not well understood and prevents the asymptotic analysis. This article addresses these issues by presenting a novel uniform error bound using Lipschitz continuity and an analysis of the posterior variance function for a large class of kernels. Additionally, we show how these results can be used to guarantee safe control of an unknown dynamical system and provide numerical illustration examples.