Derandomizing Simultaneous Confidence Regions for Band-Limited Functions by Improved Norm Bounds and Majority-Voting Schemes

TL;DR

This paper improves methods for constructing simultaneous confidence regions for band-limited functions by refining norm bounds and using majority voting, resulting in more stable and accurate confidence sets.

Contribution

It introduces refined norm bounds and a majority-voting scheme for better confidence regions in nonparametric, nonasymptotic settings for band-limited functions.

Findings

Tighter kernel norm bounds improve confidence region accuracy.

Majority voting enhances stability and size of confidence sets.

Numerical experiments validate the theoretical improvements.

Abstract

Band-limited functions are fundamental objects that are widely used in systems theory and signal processing. In this paper we refine a recent nonparametric, nonasymptotic method for constructing simultaneous confidence regions for band-limited functions from noisy input-output measurements, by working in a Paley-Wiener reproducing kernel Hilbert space. Kernel norm bounds are tightened using a uniformly-randomized Hoeffding's inequality for small samples and an empirical Bernstein bound for larger ones. We derive an approximate threshold, based on the sample size and how informative the inputs are, that governs which bound to deploy. Finally, we apply majority voting to aggregate confidence sets from random subsamples, boosting both stability and region size. We prove that even per-input aggregated intervals retain their simultaneous coverage guarantee. These refinements are also validated through numerical experiments.