Enhanced Feature Learning via Regularisation: Integrating Neural Networks and Kernel Methods

TL;DR

This paper introduces Brownian Kernel Neural Networks (BKerNN), a novel regularisation-based feature learning method that combines neural networks and kernel methods, demonstrating improved robustness and performance over traditional approaches.

Contribution

The paper presents BKerNN, a new gradient-based method that integrates kernel ridge regression with neural network training, offering theoretical convergence guarantees and practical advantages.

Findings

BKerNN outperforms kernel ridge regression in experiments.

BKerNN shows competitive results compared to ReLU neural networks.

Theoretical risk convergence rate of O( (d/n)^{1/2} ) or n^{-1/6}.

Abstract

We propose a new method for feature learning and function estimation in supervised learning via regularised empirical risk minimisation. Our approach considers functions as expectations of Sobolev functions over all possible one-dimensional projections of the data. This framework is similar to kernel ridge regression, where the kernel is $\mathbb{E}_w ( k^{(B)}(w^\top x,w^\top x^\prime))$, with $k^{(B)}(a,b) := \min(|a|, |b|)\mathds{1}_{ab>0}$ the Brownian kernel, and the distribution of the projections $w$ is learnt. This can also be viewed as an infinite-width one-hidden layer neural network, optimising the first layer's weights through gradient descent and explicitly adjusting the non-linearity and weights of the second layer. We introduce a gradient-based computational method for the estimator, called Brownian Kernel Neural Network (BKerNN), using particles to approximate the expectation, where the positive homogeneity of the Brownian kernel \red{leads to improved robustness to local minima}. Using Rademacher complexity, we show that BKerNN's expected risk converges to the minimal risk with explicit high-probability rates of $O( \min((d/n)^{1/2}, n^{-1/6}))$ (up to logarithmic factors). Numerical experiments confirm our optimisation intuitions, and BKerNN outperforms kernel ridge regression, and favourably compares to a one-hidden layer neural network with ReLU activations in various settings and real data sets.