Rademacher Meets Colors: More Expressivity, but at What Cost ?
This provides a theoretical explanation for a key problem in GNN research, addressing why more expressive models often generalize poorly, which is incremental but foundational for understanding GNN limitations.
The paper tackles the trade-off between expressivity and generalization in graph neural networks (GNNs) by linking the number of equivalence classes from Weisfeiler-Leman colorings to Rademacher complexity, showing that greater expressivity leads to higher complexity and weaker generalization guarantees, with stability under perturbations in color counts.
The expressive power of graph neural networks (GNNs) is typically understood through their correspondence with graph isomorphism tests such as the Weisfeiler-Leman (WL) hierarchy. While more expressive GNNs can distinguish a richer set of graphs, they are also observed to suffer from higher generalization error. This work provides a theoretical explanation for this trade-off by linking expressivity and generalization through the lens of coloring algorithms. Specifically, we show that the number of equivalence classes induced by WL colorings directly bounds the GNNs Rademacher complexity -- a key data-dependent measure of generalization. Our analysis reveals that greater expressivity leads to higher complexity and thus weaker generalization guarantees. Furthermore, we prove that the Rademacher complexity is stable under perturbations in the color counts across different samples, ensuring robustness to sampling variability across datasets. Importantly, our framework is not restricted to message-passing GNNs or 1-WL, but extends to arbitrary GNN architectures and expressivity measures that partition graphs into equivalence classes. These results unify the study of expressivity and generalization in GNNs, providing a principled understanding of why increasing expressive power often comes at the cost of generalization.