Reliable sampling-based RKHS norm estimation via superconvergence

TL;DR

This paper introduces a new sampling-based method for accurately estimating the RKHS norm of functions, which is crucial for error bounds in kernel methods used in control and system identification.

Contribution

The authors develop a theoretically grounded RKHS norm estimation technique leveraging superconvergence, applicable to various function classes with minimal prior knowledge.

Findings

The method achieves reliable RKHS norm estimates in numerical experiments.

It enhances the safety guarantees of learning-based control algorithms.

The approach is broadly applicable across relevant function classes.

Abstract

Kernel methods are one of the cornerstones of learning-based control, modern system identification, surrogate modelling, and related fields. A key advantage of this class of learning and function approximation methods is the availability of quantitative error bounds, which in turn play a key role in guaranteeing the safety of learned controllers and related learning-based algorithms. However, these error bounds rely on a particular property of the target function -- its reproducing kernel Hilbert space (RKHS) norm -- which is usually impossible to obtain in practice. Motivated by this severe shortcoming, we present a novel sampling-based RKHS norm estimation approach with a solid theoretical foundation, leveraging very recent advances in the theory of superconvergence in kernel methods. Our method is applicable to a broad range of practically relevant function classes and requires only reasonable prior knowledge about the target function. Extensive numerical experiments demonstrate the efficacy and practical applicability of the proposed method. By providing a reliable RKHS norm estimation approach, we remove a major obstacle to the practical deployment of learning-based control algorithms.