A short note on extension theorems and their connection to universal consistency in machine learning

TL;DR

This paper explores the theoretical foundations of extension theorems and their relation to universal consistency in machine learning, particularly focusing on kernel methods and their empirical success with common kernels.

Contribution

It provides a theoretical explanation for why classical kernels like Gaussian RBF and Sobolev perform well in practice, linking extension theorems to universal consistency.

Findings

Classical kernels often yield universally consistent estimators.

Extension theorems help explain the empirical success of common kernels.

Theoretical insights connect kernel choice to consistency in machine learning.

Abstract

Statistical machine learning plays an important role in modern statistics and computer science. One main goal of statistical machine learning is to provide universally consistent algorithms, i.e., the estimator converges in probability or in some stronger sense to the Bayes risk or to the Bayes decision function. Kernel methods based on minimizing the regularized risk over a reproducing kernel Hilbert space (RKHS) belong to these statistical machine learning methods. It is in general unknown which kernel yields optimal results for a particular data set or for the unknown probability measure. Hence various kernel learning methods were proposed to choose the kernel and therefore also its RKHS in a data adaptive manner. Nevertheless, many practitioners often use the classical Gaussian RBF kernel or certain Sobolev kernels with good success. The goal of this short note is to offer one possible theoretical explanation for this empirical fact.