  We investigate gradient descent training of wide neural networks and the
corresponding implicit bias in function space. For univariate regression, we
show that the solution of training a width-$n$ shallow ReLU network is within
$n^{- 1/2}$ of the function which fits the training data and whose difference
from the initial function has the smallest 2-norm of the second derivative
weighted by a curvature penalty that depends on the probability distribution
that is used to initialize the network parameters. We compute the curvature
penalty function explicitly for various common initialization procedures. For
instance, asymmetric initialization with a uniform distribution yields a
constant curvature penalty, and thence the solution function is the natural
cubic spline interpolation of the training data. \hj{For stochastic gradient
descent we obtain the same implicit bias result.} We obtain a similar result
for different activation functions. For multivariate regression we show an
analogous result, whereby the second derivative is replaced by the Radon
transform of a fractional Laplacian. For initialization schemes that yield a
constant penalty function, the solutions are polyharmonic splines. Moreover, we
show that the training trajectories are captured by trajectories of smoothing
splines with decreasing regularization strength.
