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Computer Science > Machine Learning

arXiv:2410.07829 (cs)
[Submitted on 10 Oct 2024]

Title:A note on the VC dimension of 1-dimensional GNNs

Authors:Noah Daniëls, Floris Geerts
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Abstract:Graph Neural Networks (GNNs) have become an essential tool for analyzing graph-structured data, leveraging their ability to capture complex relational information. While the expressivity of GNNs, particularly their equivalence to the Weisfeiler-Leman (1-WL) isomorphism test, has been well-documented, understanding their generalization capabilities remains critical. This paper focuses on the generalization of GNNs by investigating their Vapnik-Chervonenkis (VC) dimension. We extend previous results to demonstrate that 1-dimensional GNNs with a single parameter have an infinite VC dimension for unbounded graphs. Furthermore, we show that this also holds for GNNs using analytic non-polynomial activation functions, including the 1-dimensional GNNs that were recently shown to be as expressive as the 1-WL test. These results suggest inherent limitations in the generalization ability of even the most simple GNNs, when viewed from the VC dimension perspective.
Comments: 10 pages
Subjects: Machine Learning (cs.LG)
Cite as: arXiv:2410.07829 [cs.LG]
  (or arXiv:2410.07829v1 [cs.LG] for this version)
  https://doi.org/10.48550/arXiv.2410.07829

Submission history

From: Noah Daniëls [view email]
[v1] Thu, 10 Oct 2024 11:33:15 UTC (13 KB)
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