Papers › Enhanced Feature Learning via Regularisation: Integrating Neural Networks and Kernel Methods
Enhanced Feature Learning via Regularisation: Integrating Neural Networks and Kernel Methods
Bertille Follain, Francis Bach
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 𝔼_w ( k⁽ᴮ⁾(w^⊤ x,w^⊤ x^')), with k⁽ᴮ⁾(a,b) := min(|a|, |b|)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.
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