  In this paper, we consider one dimensional (shallow) ReLU neural networks in
which weights are chosen randomly and only the terminal layer is trained.
First, we mathematically show that for such networks L2-regularized regression
corresponds in function space to regularizing the estimate's second derivative
for fairly general loss functionals. For least squares regression, we show that
the trained network converges to the smooth spline interpolation of the
training data as the number of hidden nodes tends to infinity. Moreover, we
derive a novel correspondence between the early stopped gradient descent
(without any explicit regularization of the weights) and the smoothing spline
regression.
