On Uniform Error Bounds for Kernel Regression under Non-Gaussian Noise

2026-05-10

AI Summary

This work addresses kernel regression under non-Gaussian noise—including sub-Gaussian, bounded, sub-exponential, and moment-bounded types—by establishing a unified probabilistic uniform error bound within a non-asymptotic framework. It overcomes the prevailing limitation of existing approaches that assume conditionally independent sub-Gaussian noise, and for the first time simultaneously accommodates multiple non-Gaussian noise distributions and dependent noise structures. By integrating concentration inequalities with reproducing kernel Hilbert space theory, the derived non-conservative error bounds substantially enhance the reliability of uncertainty quantification. This leads to significantly tighter confidence regions in safety-critical control tasks, outperforming current state-of-the-art methods in both theoretical rigor and practical performance.

Abstract

Providing non-conservative uncertainty quantification for function estimates derived from noisy observations remains a fundamental challenge in statistical machine learning, particularly for applications in safety-critical domains. In this work, we propose novel non-asymptotic probabilistic uniform error bounds for kernel-based regression. Compared to related bounds in the literature that are restricted to (conditionally) independent sub-Gaussian noise, our bounds allow to consider a broad class of non-Gaussian distributions, such as sub-Gaussian, bounded, sub-exponential, and variance/moment-bounded noise. Moreover, our results apply to correlated and uncorrelated noise. We compare our proposed error bounds with existing results in terms of the induced uncertainty region and their performance in safe control, demonstrating the tightness of the proposed bounds.

Problem

• kernel regression
• non-Gaussian noise
• uniform error bounds
• uncertainty quantification
• non-asymptotic bounds

Innovation

• uniform error bounds
• kernel regression
• non-Gaussian noise
• non-asymptotic analysis
• uncertainty quantification