  Interpolators -- estimators that achieve zero training error -- have
attracted growing attention in machine learning, mainly because state-of-the
art neural networks appear to be models of this type. In this paper, we study
minimum $\ell_2$ norm ("ridgeless") interpolation in high-dimensional least
squares regression. We consider two different models for the feature
distribution: a linear model, where the feature vectors $x_i \in {\mathbb R}^p$
are obtained by applying a linear transform to a vector of i.i.d. entries, $x_i
= \Sigma^{1/2} z_i$ (with $z_i \in {\mathbb R}^p$); and a nonlinear model,
where the feature vectors are obtained by passing the input through a random
one-layer neural network, $x_i = \varphi(W z_i)$ (with $z_i \in {\mathbb R}^d$,
$W \in {\mathbb R}^{p \times d}$ a matrix of i.i.d. entries, and $\varphi$ an
activation function acting componentwise on $W z_i$). We recover -- in a
precise quantitative way -- several phenomena that have been observed in
large-scale neural networks and kernel machines, including the "double descent"
behavior of the prediction risk, and the potential benefits of
overparametrization.
