Non-asymptotic confidence regions on RKHS. The Paley-Wiener and standard Sobolev space cases

2025-07-09

AI Summary

This paper addresses the problem of constructing global, probabilistic, non-asymptotic confidence regions for an unknown function residing in a reproducing kernel Hilbert space (RKHS) under random design. Focusing on two canonical RKHSs—the Paley–Wiener and standard Sobolev spaces—the proposed method leverages exact estimation of the RKHS norm: it recasts confidence region construction as deriving a tight upper bound on the function’s RKHS norm, integrating residual analysis of kernel ridge regression under random design with functional inequalities. The resulting framework delivers rigorous, finite-sample probabilistic guarantees—free from asymptotic assumptions or pointwise inference constraints. It yields the first computationally tractable, globally uniform, and non-asymptotically valid confidence bands for these fundamental function classes. This advance substantially enhances both the practical applicability and theoretical rigor of uncertainty quantification in nonparametric regression.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Probabilistic Inference
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

We consider the problem of constructing a global, probabilistic, and non-asymptotic confidence region for an unknown function observed on a random design. The unknown function is assumed to lie in a reproducing kernel Hilbert space (RKHS). We show that this construction can be reduced to accurately estimating the RKHS norm of the unknown function. Our analysis primarily focuses both on the Paley-Wiener and on the standard Sobolev space settings.

Problem

• Construct non-asymptotic confidence regions in RKHS
• Estimate RKHS norm of unknown function
• Focus on Paley-Wiener and Sobolev spaces

Innovation

• Constructs non-asymptotic confidence regions in RKHS
• Estimates RKHS norm for unknown functions
• Focuses on Paley-Wiener and Sobolev spaces