  We present a theoretical and empirical study of the gradient dynamics of
overparameterized shallow ReLU networks with one-dimensional input, solving
least-squares interpolation. We show that the gradient dynamics of such
networks are determined by the gradient flow in a non-redundant
parameterization of the network function. We examine the principal qualitative
features of this gradient flow. In particular, we determine conditions for two
learning regimes:kernel and adaptive, which depend both on the relative
magnitude of initialization of weights in different layers and the asymptotic
behavior of initialization coefficients in the limit of large network widths.
We show that learning in the kernel regime yields smooth interpolants,
minimizing curvature, and reduces to cubic splines for uniform initializations.
Learning in the adaptive regime favors instead linear splines, where knots
cluster adaptively at the sample points.
