# Error Bounds For Gaussian Process Regression Under Bounded Support Noise With Applications To Safety Certification

arXiv: https://arxiv.org/abs/2408.09033 · HTML: https://mlsanity.com/paper/2408.09033/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 18 references that are in the map.
- First posted on arXiv 16 Aug 2024.

## Builds on (most cited first, 9 of 9)

- [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)
- [The Promises and Pitfalls of Deep Kernel Learning](https://mlsanity.com/paper/2102.12108/) (2021)
- [Uniform Error Bounds for Gaussian Process Regression with Application to Safe Control](https://mlsanity.com/paper/1906.01376/) (2019)
- [Deterministic error bounds for kernel-based learning techniques under bounded noise](https://mlsanity.com/paper/2008.04005/) (2020)
- [Formal Synthesis of Stochastic Systems via Control Barrier Certificates](https://mlsanity.com/paper/1905.04585/) (2019)
- [A Barrier Function Approach to Finite-Time Stochastic System Verification and Control](https://mlsanity.com/paper/1909.05109/) (2019)
- [Safety Guarantees for Neural Network Dynamic Systems via Stochastic Barrier Functions](https://mlsanity.com/paper/2206.07811/) (2022)
- [Promises of Deep Kernel Learning for Control Synthesis](https://mlsanity.com/paper/2309.06569/) (2023)

## Built on by (most cited first, 6 of 6)

- [Optimal kernel regression bounds under energy-bounded noise](https://mlsanity.com/paper/2505.22235/) (2025)
- [Learning-Based Shielding for Safe Autonomy under Unknown Dynamics](https://mlsanity.com/paper/2410.07359/) (2024)
- [Discovery of Probabilistic Dirichlet-to-Neumann Maps on Graphs](https://mlsanity.com/paper/2506.02337/) (2025)
- [Verification of Unknown Dynamical Systems via Autoencoder Latent Space](https://mlsanity.com/paper/2512.13593/) (2025)
- [Optimal uncertainty bounds for multivariate kernel regression under bounded noise: A Gaussian process-based dual function](https://mlsanity.com/paper/2603.16481/) (2026)
- [On Uniform Error Bounds for Kernel Regression under Non-Gaussian Noise](https://mlsanity.com/paper/2605.09757/) (2026)

## Abstract (arXiv)

Gaussian Process Regression (GPR) is a powerful and elegant method for learning complex functions from noisy data with a wide range of applications, including in safety-critical domains. Such applications have two key features: (i) they require rigorous error quantification, and (ii) the noise is often bounded and non-Gaussian due to, e.g., physical constraints. While error bounds for applying GPR in the presence of non-Gaussian noise exist, they tend to be overly restrictive and conservative in practice. In this paper, we provide novel error bounds for GPR under bounded support noise. Specifically, by relying on concentration inequalities and assuming that the latent function has low complexity in the reproducing kernel Hilbert space (RKHS) corresponding to the GP kernel, we derive both probabilistic and deterministic bounds on the error of the GPR. We show that these errors are substantially tighter than existing state-of-the-art bounds and are particularly well-suited for GPR with neural network kernels, i.e., Deep Kernel Learning (DKL). Furthermore, motivated by applications in safety-critical domains, we illustrate how these bounds can be combined with stochastic barrier functions to successfully quantify the safety probability of an unknown dynamical system from finite data. We validate the efficacy of our approach through several benchmarks and comparisons against existing bounds. The results show that our bounds are consistently smaller, and that DKLs can produce error bounds tighter than sample noise, significantly improving the safety probability of control systems.