From Kernel Methods to Neural Networks: A Unifying Variational Formulation

TL;DR

This paper introduces a unifying variational framework that connects kernel methods and neural networks, providing a comprehensive understanding of their solutions and approximation capabilities across different regularization norms.

Contribution

It develops a general regularization functional that encompasses kernel methods and neural networks, characterizes solutions for various norms, and explains neural network features like bias and skip connections.

Findings

Solutions include radial basis functions for Hilbertian norms.

Total-variation norm leads to two-layer neural networks with specific activation functions.

Framework guarantees universal approximation for a wide class of shallow networks.

Abstract

The minimization of a data-fidelity term and an additive regularization functional gives rise to a powerful framework for supervised learning. In this paper, we present a unifying regularization functional that depends on an operator and on a generic Radon-domain norm. We establish the existence of a minimizer and give the parametric form of the solution(s) under very mild assumptions. When the norm is Hilbertian, the proposed formulation yields a solution that involves radial-basis functions and is compatible with the classical methods of machine learning. By contrast, for the total-variation norm, the solution takes the form of a two-layer neural network with an activation function that is determined by the regularization operator. In particular, we retrieve the popular ReLU networks by letting the operator be the Laplacian. We also characterize the solution for the intermediate regularization norms $\|\cdot\|=\|\cdot\|_{L_p}$ with $p\in(1,2]$. Our framework offers guarantees of universal approximation for a broad family of regularization operators or, equivalently, for a wide variety of shallow neural networks, including the cases (such as ReLU) where the activation function is increasing polynomially. It also explains the favorable role of bias and skip connections in neural architectures.