Gradient Dynamics of Shallow Univariate ReLU Networks
This provides theoretical insights into learning regimes for shallow neural networks, which is incremental for researchers in optimization and deep learning theory.
The paper tackles the gradient dynamics of overparameterized shallow ReLU networks with one-dimensional input in least-squares interpolation, showing that the dynamics lead to two learning regimes—kernel and adaptive—which produce smooth interpolants minimizing curvature (e.g., cubic splines) or linear splines with knots clustering at sample points, respectively.
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.