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Inversion dynamics of class manifolds in deep learning reveals tradeoffs underlying generalization

Nature Machine Intelligence volume 6, pages 40–47 (2024)Cite this article

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Abstract

To achieve near-zero training error in a classification problem, the layers of a feed-forward network have to disentangle the manifolds of data points with different labels to facilitate the discrimination. However, excessive class separation can lead to overfitting because good generalization requires learning invariant features, which involve some level of entanglement. We report on numerical experiments showing how the optimization dynamics finds representations that balance these opposing tendencies with a non-monotonic trend. After a fast segregation phase, a slower rearrangement (conserved across datasets and architectures) increases the class entanglement. The training error at the inversion is stable under subsampling and across network initializations and optimizers, which characterizes it as a property solely of the data structure and (very weakly) of the architecture. The inversion is the manifestation of tradeoffs elicited by well-defined and maximally stable elements of the training set called ‘stragglers’, which are particularly influential for generalization.

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Fig. 1: Non-monotonic learning dynamics.
Fig. 2: Stragglers shape the dynamics and influence generalization.
Fig. 3: Stragglers across datasets and architectures.

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Data availability

The datasets analysed during the current study are available in public repositories; links are in the corresponding publications54,55,56,57.

Code availability

The code produced and used in the current study59 is available on GitHub under GNU General Public License v.3 (GPL-3.0) at https://github.com/marco-gherardi/stragglers.

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Acknowledgements

P.R. acknowledges funding from the Fellini programme under the H2020-MSCA-COFUND action, grant no. 754496, INFN (IT).

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Authors and Affiliations

  1. Università degli Studi di Milano, Milan, Italy

    Simone Ciceri, Lorenzo Cassani & Marco Gherardi

  2. Università degli Studi di Torino and INFN, Sezione di Torino, Turin, Italy

    Matteo Osella & Filippo Valle

  3. Istituto Nazionale di Fisica Nucleare — Sezione di Milano, Milan, Italy

    Pietro Rotondo & Marco Gherardi

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Contributions

S.C. and M.G. discovered the stragglers. M.G., M.O. and P.R. conceived and designed the experiments. L.C., S.C., M.G. and F.V. performed the experiments. All authors analysed the results and wrote the paper. M.G. and M.O. supervised the analysis. M.G. coordinated the project.

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Correspondence to Marco Gherardi.

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Ciceri, S., Cassani, L., Osella, M. et al. Inversion dynamics of class manifolds in deep learning reveals tradeoffs underlying generalization. Nat Mach Intell 6, 40–47 (2024). https://doi.org/10.1038/s42256-023-00772-9

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