  We prove a precise geometric description of all one layer ReLU networks
$z(x;\theta)$ with a single linear unit and input/output dimensions equal to
one that interpolate a given dataset $\mathcal D=\{(x_i,f(x_i))\}$ and, among
all such interpolants, minimize the $\ell_2$-norm of the neuron weights. Such
networks can intuitively be thought of as those that minimize the mean-squared
error over $\mathcal D$ plus an infinitesimal weight decay penalty. We
therefore refer to them as ridgeless ReLU interpolants. Our description proves
that, to extrapolate values $z(x;\theta)$ for inputs $x\in (x_i,x_{i+1})$ lying
between two consecutive datapoints, a ridgeless ReLU interpolant simply
compares the signs of the discrete estimates for the curvature of $f$ at $x_i$
and $x_{i+1}$ derived from the dataset $\mathcal D$. If the curvature estimates
at $x_i$ and $x_{i+1}$ have different signs, then $z(x;\theta)$ must be linear
on $(x_i,x_{i+1})$. If in contrast the curvature estimates at $x_i$ and
$x_{i+1}$ are both positive (resp. negative), then $z(x;\theta)$ is convex
(resp. concave) on $(x_i,x_{i+1})$. Our results show that ridgeless ReLU
interpolants achieve the best possible generalization for learning $1d$
Lipschitz functions, up to universal constants.
