Generalization of Geometric Graph Neural Networks With Lipschitz Loss Functions

2024-09-08

IEEE Transactions on Signal Processing

AI Summary

This work investigates the generalization capability of geometric graph neural networks (GNNs) trained on geometric graphs formed by random sampling of points from a $d$-dimensional manifold, focusing on how the generalization gap—i.e., the difference between empirical and statistical risks—scales with sample size $n$ and manifold dimension $d$. Method: We propose the first theoretical framework ensuring generalization using only a single large-scale geometric graph, circumventing conventional reliance on multiple or small graphs. Leveraging Lipschitz loss assumptions and manifold sampling theory, we establish a non-asymptotic convergence guarantee from GNNs to manifold neural networks (MNNs). Results: We rigorously prove that the generalization gap decays as $O(sqrt{d/n})$. Extensive experiments on real-world geometric datasets validate this rate and demonstrate robust transferability of models trained on large graphs to previously unseen geometric graphs sampled from the same underlying manifold.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

Abstract

In this paper, we study the generalization capabilities of geometric graph neural networks (GNNs). We consider GNNs over a geometric graph constructed from a finite set of randomly sampled points over an embedded manifold with topological information captured. We prove a generalization gap between the optimal empirical risk and the optimal statistical risk of this GNN, which decreases with the number of sampled points from the manifold and increases with the dimension of the underlying manifold. This generalization gap ensures that the GNN trained on a graph on a set of sampled points can be utilized to process other unseen graphs constructed from the same underlying manifold. The most important observation is that the generalization capability can be realized with one large graph instead of being limited to the size of the graph as in previous results. The generalization gap is derived based on the non-asymptotic convergence result of a GNN on the sampled graph to the underlying manifold neural networks (MNNs). We verify this theoretical result with experiments on multiple real-world datasets.

Problem

• Study generalization of geometric GNNs on manifold-based graphs
• Prove generalization gap depends on sample size and manifold dimension
• Show generalization works with one large graph, not limited size

Innovation

• Geometric GNNs with Lipschitz loss functions
• Generalization gap decreases with sample points
• One large graph enables generalization capability