Topology-Aware PAC-Bayesian Generalization Analysis for Graph Neural Networks

TL;DR

This paper introduces a topology-aware PAC-Bayesian framework for analyzing the generalization of graph neural networks, explicitly incorporating graph structures to derive tighter bounds.

Contribution

It extends existing PAC-Bayesian bounds to graph-structured models by integrating spatial and spectral graph properties, providing a unified analysis of GNN generalization.

Findings

Derived a family of graph-structure-aware generalization bounds.

Bounds are tighter than existing PAC-Bayesian bounds for GNNs.

Framework unifies spatial and spectral perspectives in GNN analysis.

Abstract

Graph neural networks have demonstrated excellent applicability to a wide range of domains, including social networks, biological systems, recommendation systems, and wireless communications. Yet a principled theoretical understanding of their generalization behavior remains limited, particularly for graph classification tasks where complex interactions between model parameters and graph structure play a crucial role. Among existing theoretical tools, PAC-Bayesian norm-based generalization bounds provide a flexible and data-dependent framework; however, current results for GNNs often restrict the exploitation of graph structures. In this work, we propose a topology-aware PAC-Bayesian norm-based generalization framework for graph convolutional networks (GCNs) that extends a previously developed framework to graph-structured models. Our approach reformulates the derivation of generalization bounds as a stochastic optimization problem and introduces sensitivity matrices that measure the response of classification outputs with respect to structured weight perturbations. By imposing different structures on sensitivity matrices from both spatial and spectral perspectives, we derive a family of generalization error bounds with graph structures explicitly embedded. Such bounds could recover existing results as special cases, while yielding bounds that are tighter than state-of-the-art PAC-Bayesian bounds for GNNs. Notably, the proposed framework explicitly integrates graph structural properties into the generalization analysis, enabling a unified inspection of GNN generalization behavior from both spatial aggregation and spectral filtering viewpoints.