While graph convolutional networks show great practical promises, the theoretical understanding of their generalization properties as a function of the number of samples is still in its infancy compared to the more broadly studied case of supervised fully connected neural networks. In this article, we predict the performances of a single-layer graph convolutional network (GCN) trained on data produced by attributed stochastic block models (SBMs) in the high-dimensional limit. Previously, only ridge regression on contextual-SBM (CSBM) has been considered in Shi et al. 2022; we generalize the analysis to arbitrary convex loss and regularization for the CSBM and add the analysis for another data model, the neural-prior SBM. We also study the high signal-to-noise ratio limit, detail the convergence rates of the GCN and show that, while consistent, it does not reach the Bayes-optimal rate for any of the considered cases.

翻译：尽管图卷积网络在实际应用中展现出巨大潜力，但与广泛研究的全监督全连接神经网络相比，关于其泛化性能随样本数量变化的理论理解仍处于初期阶段。本文预测了在高维极限下，基于属性随机块模型（SBM）生成数据训练的单层图卷积网络（GCN）的性能表现。此前，Shi等人（2022）仅考虑了上下文SBM（CSBM）上的岭回归；我们将分析推广至CSBM的任意凸损失函数与正则化项，并新增对另一数据模型——神经先验SBM——的分析。我们还研究了高信噪比极限，详细推导了GCN的收敛速率，并证明尽管该网络具有一致性，但在所有考虑的场景中均未达到贝叶斯最优速率。