Understanding the Effect of GCN Convolutions in Regression Tasks

• URL: http://arxiv.org/abs/2410.20068v1
• Date: Sat, 26 Oct 2024 04:19:52 GMT
• Title: Understanding the Effect of GCN Convolutions in Regression Tasks
• Authors: Juntong Chen, Johannes Schmidt-Hieber, Claire Donnat, Olga Klopp,
• Abstract summary: Graph Convolutional Networks (GCNs) have become a pivotal method in machine learning for modeling functions over graphs. This paper provides a formal analysis of the impact of convolution operators on regression tasks over homophilic networks.
• Score: 8.299692647308323
• License:
• Abstract: Graph Convolutional Networks (GCNs) have become a pivotal method in machine learning for modeling functions over graphs. Despite their widespread success across various applications, their statistical properties (e.g. consistency, convergence rates) remain ill-characterized. To begin addressing this knowledge gap, in this paper, we provide a formal analysis of the impact of convolution operators on regression tasks over homophilic networks. Focusing on estimators based solely on neighborhood aggregation, we examine how two common convolutions - the original GCN and GraphSage convolutions - affect the learning error as a function of the neighborhood topology and the number of convolutional layers. We explicitly characterize the bias-variance trade-off incurred by GCNs as a function of the neighborhood size and identify specific graph topologies where convolution operators are less effective. Our theoretical findings are corroborated by synthetic experiments, and provide a start to a deeper quantitative understanding of convolutional effects in GCNs for offering rigorous guidelines for practitioners.

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