Asymptotic generalization error of a single-layer graph convolutional network
This work provides incremental theoretical insights into GCN generalization for graph data, addressing a gap compared to fully connected networks.
The authors tackled the theoretical understanding of generalization error for single-layer graph convolutional networks (GCNs) on attributed stochastic block models, predicting performance in high-dimensional limits and showing that GCNs are consistent but do not achieve Bayes-optimal rates.
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.