How Graph Neural Networks Learn: Lessons from Training Dynamics in Function Space

A long-standing goal in deep learning has been to characterize the learning behavior of black-box models in a more interpretable manner. For graph neural networks (GNNs), considerable advances have been made in formalizing what functions they can represent, however it remains less clear whether and how GNNs learn desired functions during the optimization process. To fill this critical gap, we study the learning dynamics of GNNs in function space via the analytic framework of overparameterization. In particular, we find that the seemingly complicated training process of GNNs can be re-cast into a more familiar label propagation framework, due to the graph inductive bias implicit in this process. From this vantage point, we provide explanations for why the learned GNN functions successfully generalize and for their pathological behavior on heterophilic graphs, which are consistent with observations. Practically, sparsifying and implementing the learning dynamics lead to a minimalist semi-supervised learning algorithm with the efficiency of classic algorithms and the effectiveness of modern GNNs.