Convergence of gradient based training for linear Graph Neural Networks

TL;DR

This paper analyzes the convergence behavior of gradient-based training methods for linear Graph Neural Networks, establishing exponential convergence to the global minimum and validating findings on synthetic and real datasets.

Contribution

It provides the first rigorous proof of exponential convergence for gradient flow in linear GNNs and explores the effects of initial weights and graph operators.

Findings

Gradient flow converges exponentially to the global minimum.

Convergence rate depends on initial weights and graph shift operator.

Results validated on synthetic and real-world datasets.

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.