The complexity dynamics of grokking
We demonstrate the existence of a complexity phase transition in neural networks by studying the grokking phenomenon, where networks suddenly transition from memorization to generalization long after overfitting their training data. To characterize this phase transition, we introduce a theoretical f...
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| Published in: | Physica. D Vol. 482; p. 134859 |
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| Format: | Journal Article |
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01.11.2025
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| ISSN: | 0167-2789 |
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| Abstract | We demonstrate the existence of a complexity phase transition in neural networks by studying the grokking phenomenon, where networks suddenly transition from memorization to generalization long after overfitting their training data. To characterize this phase transition, we introduce a theoretical framework for measuring complexity based on rate–distortion theory and Kolmogorov complexity, which can be understood as principled lossy compression for networks. We find that properly regularized networks exhibit a sharp phase transition: complexity rises during memorization, then falls as the network discovers a simpler underlying pattern that generalizes. In contrast, unregularized networks remain trapped in a high-complexity memorization phase. We establish an explicit connection between our complexity measure and generalization bounds, providing a theoretical foundation for the link between lossy compression and generalization. Our framework achieves compression ratios 30-40× better than naïve approaches, enabling precise tracking of complexity dynamics. Finally, we introduce a regularization method based on spectral entropy that encourages networks toward low-complexity representations by penalizing their intrinsic dimension. |
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| AbstractList | We demonstrate the existence of a complexity phase transition in neural networks by studying the grokking phenomenon, where networks suddenly transition from memorization to generalization long after overfitting their training data. To characterize this phase transition, we introduce a theoretical framework for measuring complexity based on rate–distortion theory and Kolmogorov complexity, which can be understood as principled lossy compression for networks. We find that properly regularized networks exhibit a sharp phase transition: complexity rises during memorization, then falls as the network discovers a simpler underlying pattern that generalizes. In contrast, unregularized networks remain trapped in a high-complexity memorization phase. We establish an explicit connection between our complexity measure and generalization bounds, providing a theoretical foundation for the link between lossy compression and generalization. Our framework achieves compression ratios 30-40× better than naïve approaches, enabling precise tracking of complexity dynamics. Finally, we introduce a regularization method based on spectral entropy that encourages networks toward low-complexity representations by penalizing their intrinsic dimension. |
| ArticleNumber | 134859 |
| Author | Posner, Ingmar Sapora, Silvia Foerster, Jakob Hawes, Nick DeMoss, Branton |
| Author_xml | – sequence: 1 givenname: Branton orcidid: 0000-0001-6828-6787 surname: DeMoss fullname: DeMoss, Branton email: bdemoss@robots.ox.ac.uk – sequence: 2 givenname: Silvia surname: Sapora fullname: Sapora, Silvia – sequence: 3 givenname: Jakob surname: Foerster fullname: Foerster, Jakob – sequence: 4 givenname: Nick surname: Hawes fullname: Hawes, Nick – sequence: 5 givenname: Ingmar surname: Posner fullname: Posner, Ingmar |
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| Cites_doi | 10.1103/PhysRevE.111.014118 10.1007/978-1-4757-3860-5 10.1017/S0960129511000521 10.1145/168304.168306 10.1016/0005-1098(78)90005-5 10.1002/j.1538-7305.1948.tb00917.x 10.1103/PhysRevResearch.2.033312 10.1109/TIT.2010.2048491 |
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| Keywords | Algorithmic complexity Grokking Phase transition |
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