Enhanced Feature Learning via Regularisation: Integrating Neural Networks and Kernel Methods
This work addresses the problem of improving feature learning for supervised learning practitioners, though it appears incremental as it builds on existing kernel and neural network frameworks.
The paper tackles the problem of feature learning and function estimation in supervised learning by proposing a new method called Brownian Kernel Neural Network (BKerNN), which integrates neural networks and kernel methods through regularised empirical risk minimisation. The result shows that BKerNN outperforms kernel ridge regression and compares favourably to one-hidden layer ReLU neural networks in various settings, with theoretical convergence rates of O(min((d/n)^{1/2}, n^{-1/6})) up to logarithmic factors.
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.