Convergence of gradient based training for linear Graph Neural Networks
Dhiraj Patel, Anton Savostianov, Michael T. Schaub
TL;DR
This paper studies the convergence of gradient-based training for linear Graph Neural Networks (GNNs) in a semi-supervised node regression setting. It analyzes both gradient flow and gradient descent dynamics, proving that under a specific initialization the mean-squared loss $\mathcal{L}$ converges to the global minimum at an exponential rate, with the rate depending on the smallest nonzero singular value $\sigma_{\text{small}}((XS^H)_{*\mathcal{I}})$ and the initial weight singular values. The authors also characterize when the global minimum minimizes the total energy of the weight matrices and provide explicit formulations for the energy-optimal solution. Numerical experiments on synthetic graph models and a real-world CDC climate dataset validate the theory and illustrate how the graph shift operator and initialization affect convergence.
Abstract
Graph Neural Networks (GNNs) are powerful tools for addressing learning problems on graph structures, with a wide range of applications in molecular biology and social networks. However, the theoretical foundations underlying their empirical performance are not well understood. In this article, we examine the convergence of gradient dynamics in the training of linear GNNs. Specifically, we prove that the gradient flow training of a linear GNN with mean squared loss converges to the global minimum at an exponential rate. The convergence rate depends explicitly on the initial weights and the graph shift operator, which we validate on synthetic datasets from well-known graph models and real-world datasets. Furthermore, we discuss the gradient flow that minimizes the total weights at the global minimum. In addition to the gradient flow, we study the convergence of linear GNNs under gradient descent training, an iterative scheme viewed as a discretization of gradient flow.