From Kernel Methods to Neural Networks: A Unifying Variational Formulation
This work provides a theoretical framework that bridges kernel methods and neural networks, offering insights into neural architectures like bias and skip connections, but it is incremental as it builds on existing regularization theory.
The paper presents a unifying variational formulation for supervised learning that yields solutions as radial-basis functions for Hilbertian norms and as two-layer neural networks for total-variation norms, retrieving ReLU networks with the Laplacian operator, and offers universal approximation guarantees for a broad family of regularization operators.
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.