Semi-Supervised Learning on Graphs using Graph Neural Networks

TL;DR

This paper provides a theoretical framework for understanding the success of graph neural networks in semi-supervised learning, analyzing their error bounds, and validating findings with experiments.

Contribution

It introduces a sharp risk bound for GNNs with linear convolutions and deep ReLU readouts, clarifying how performance depends on label availability and graph structure.

Findings

Risk bounds explicitly depend on labeled data fraction and graph dependence

Convergence rates recover classical nonparametric behavior under full supervision

Numerical experiments validate theoretical predictions

Abstract

Graph neural networks (GNNs) work remarkably well in semi-supervised node regression, yet a rigorous theory explaining when and why they succeed remains lacking. To address this gap, we study an aggregate-and-readout model that encompasses several common message passing architectures: node features are first propagated over the graph then mapped to responses via a nonlinear function. For least-squares estimation over GNNs with linear graph convolutions and a deep ReLU readout, we prove a sharp non-asymptotic risk bound that separates approximation, stochastic, and optimization errors. The bound makes explicit how performance scales with the fraction of labeled nodes and graph-induced dependence. Approximation guarantees are further derived for graph-smoothing followed by smooth nonlinear readouts, yielding convergence rates that recover classical nonparametric behavior under full supervision while characterizing performance when labels are scarce. Numerical experiments validate our theory, providing a systematic framework for understanding GNN performance and limitations.