Asymptotic generalization error of a single-layer graph convolutional network

O Duranthon, Lenka Zdeborova

Keywords: GCN, SBM, CSBM, generalization, high-dimensional limit, replica

TL;DR: We give the exact prediction of the performance of a simple GCN on a model of data and we derive its optimal parameters.

Abstract:

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 derive the optimal parameters of the GCN. 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.

Submission Type: Full paper proceedings track submission (max 9 main pages).

Submission Number: 127