The interpolation phase transition in neural networks: Memorization and generalization under lazy training

Abstract

Modern neural networks are often operated in a strongly overparametrized regime: they comprise so many parameters that they can interpolate the training set, even if actual labels are replaced by purely random ones. Despite this, they achieve good prediction error on unseen data: interpolating the training set does not lead to a large generalization error. Further, overparametrization appears to be beneficial in that it simplifies the optimization landscape. Here, we study these phenomena in the context of two-layers neural networks in the neural tangent (NT) regime. We consider a simple data model, with isotropic covariates vectors in d dimensions, and N hidden neurons. We assume that both the sample size n and the dimension d are large, and they are polynomially related. Our first main result is a characterization of the eigenstructure of the empirical NT kernel in the overparametrized regime $\mathit{N}\mathit{d}\gg \mathit{n}$. This characterization implies as a corollary that the minimum eigenvalue of the empirical NT kernel is bounded away from zero as soon as $\mathit{N}\mathit{d}\gg \mathit{n}$ and, therefore, the network can exactly interpolate arbitrary labels in the same regime.

Our second main result is a characterization of the generalization error of NT ridge regression including, as a special case, min-${\ell }_2$ norm interpolation. We prove that, as soon as $\mathit{N}\mathit{d}\gg \mathit{n}$, the test error is well approximated by the one of kernel ridge regression with respect to the infinite-width kernel. The latter is in turn well approximated by the error of polynomial ridge regression, whereby the regularization parameter is increased by a “self-induced” term related to the high-degree components of the activation function. The polynomial degree depends on the sample size and the dimension (in particular on $log\mathit{n}/log\mathit{d}$).