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A long-standing goal in deep learning has been to characterize the learningbehavior of black-box models in a more interpretable manner. For graph neuralnetworks (GNNs), considerable advances have been made in formalizing whatfunctions they can represent, but whether GNNs will learn desired functionsduring the optimization process remains less clear. To fill this gap, we studytheir training dynamics in function space. In particular, we find that thegradient descent optimization of GNNs implicitly leverages the graph structureto update the learned function, as can be quantified by a phenomenon which wecall \emph{kernel-graph alignment}. We provide theoretical explanations for theemergence of this phenomenon in the overparameterized regime and empiricallyvalidate it on real-world GNNs. This finding offers new interpretable insightsinto when and why the learned GNN functions generalize, highlighting theirlimitations in heterophilic graphs. Practically, we propose a parameter-freealgorithm that directly uses a sparse matrix (i.e. graph adjacency) to updatethe learned function. We demonstrate that this embarrassingly simple approachcan be as effective as GNNs while being orders-of-magnitude faster.

How Graph Neural Networks Learn: Lessons from Training Dynamics