Convergence of gradient based training for linear Graph Neural Networks
This provides foundational theoretical insights for researchers and practitioners using GNNs in domains like molecular biology and social networks, though it is incremental as it focuses on linear models.
The paper tackles the lack of theoretical understanding of Graph Neural Networks (GNNs) by proving that gradient flow training for linear GNNs with mean squared loss converges to the global minimum at an exponential rate, validated on synthetic and real-world datasets.
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.