Graph Neural Networks (GNNs) extend convolutional neural networks to operate on graphs. Despite their impressive performances in various graph learning tasks, the theoretical understanding of their generalization capability is still lacking. Previous GNN generalization bounds ignore the underlying graph structures, often leading to bounds that increase with the number of nodes -- a behavior contrary to the one experienced in practice. In this paper, we take a manifold perspective to establish the statistical generalization theory of GNNs on graphs sampled from a manifold in the spectral domain. As demonstrated empirically, we prove that the generalization bounds of GNNs decrease linearly with the size of the graphs in the logarithmic scale, and increase linearly with the spectral continuity constants of the filter functions. Notably, our theory explains both node-level and graph-level tasks. Our result has two implications: i) guaranteeing the generalization of GNNs to unseen data over manifolds; ii) providing insights into the practical design of GNNs, i.e., restrictions on the discriminability of GNNs are necessary to obtain a better generalization performance. We demonstrate our generalization bounds of GNNs using synthetic and multiple real-world datasets.

翻译：图神经网络（GNNs）将卷积神经网络扩展至图结构数据上运行。尽管在各种图学习任务中表现出色，但其泛化能力的理论理解仍然不足。以往的GNN泛化界忽略了底层图结构，往往导致边界随节点数量增加而增大——这与实际观察到的现象相悖。本文从流形视角出发，在谱域中建立了对从流形采样的图上GNN的统计泛化理论。通过实证验证，我们证明了GNN的泛化界在对数尺度上随图规模线性减小，并随滤波器函数的谱连续性常数线性增大。值得注意的是，我们的理论同时解释了节点级和图级任务。我们的结果具有两方面意义：i) 保证了GNN在流形上对未见数据的泛化能力；ii) 为GNN的实际设计提供了见解，即限制GNN的判别能力对于获得更好的泛化性能是必要的。我们使用合成数据集和多个真实数据集验证了所提出的GNN泛化界。