Title: On the Sample Complexity of GNNs
Primary area: learning theory
Keywords: Graph Neural Networks (GNNs), sample complexity, lower bounds

Abstract: Graph Neural Networks (GNNs) have demonstrated strong empirical performance across domains, yet their fundamental statistical behavior remains poorly understood. This paper presents a theoretical characterization of the sample complexity of ReLU-based GNNs. We establish tight minimax lower bounds on the generalization error, showing that for arbitrary graphs, *without structural assumptions* (i.e., in the worst case over admissible graphs), it scales as $\sqrt{\frac{\log d}{n}}$ with sample size $n$ and input dimension $d$, matching the $1/\sqrt{n}$ behavior known for feed-forward neural networks. Under structural graph assumptions—specifically, strong homophily and bounded spectral expansion—we derive a sharper lower bound of $\frac{d}{\log n}$. Empirical results on standard datasets (Cora, Reddit, QM9, Facebook) using GCN, GAT, and GraphSAGE support these theoretical predictions. Our findings establish fundamental limits on GNN generalization and underscore the role of graph structure in determining sample efficiency.

TL;DR: We prove tight minimax lower bounds on the sample complexity of GNNs, showing their generalization depends critically on graph structure and is often as slow as 1/log n.