Gradient Dynamics of Shallow Univariate ReLU Networks

TL;DR

This paper analyzes the gradient dynamics of overparameterized shallow ReLU networks with one-dimensional input, revealing how different initialization regimes lead to distinct interpolation behaviors such as cubic and linear splines.

Contribution

It provides a theoretical and empirical framework for understanding how initialization influences the learning regimes and resulting interpolants in shallow ReLU networks.

Findings

Kernel regime yields smooth cubic spline interpolants.

Adaptive regime produces linear splines with knots at data points.

Initialization conditions determine the transition between regimes.

Abstract

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.