A note on the VC dimension of 1-dimensional GNNs
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929535651414016 |
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| author | Daniëls, Noah Geerts, Floris |
| author_facet | Daniëls, Noah Geerts, Floris |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07829 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on the VC dimension of 1-dimensional GNNs Daniëls, Noah Geerts, Floris Machine Learning 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. |
| title | A note on the VC dimension of 1-dimensional GNNs |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2410.07829 |