Double descent

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An example of the double descent phenomenon in a two-layer neural network: as the ratio of parameters to data points increases, the test error first falls, then rises, then falls again. The vertical line marks the "interpolation threshold" boundary between the underparametrized region (more data points than parameters) and the overparameterized region (more parameters than data points). Double descent in a two-layer neural network (Figure 3a from Rocks et al. 2022).png
An example of the double descent phenomenon in a two-layer neural network: as the ratio of parameters to data points increases, the test error first falls, then rises, then falls again. The vertical line marks the "interpolation threshold" boundary between the underparametrized region (more data points than parameters) and the overparameterized region (more parameters than data points).

Double descent in statistics and machine learning is the phenomenon where a model with a small number of parameters and a model with an extremely large number of parameters both have a small training error, but a model whose number of parameters is about the same as the number of data points used to train the model will have a much greater test error than one with a much larger number of parameters. [2] This phenomenon has been considered surprising, as it contradicts assumptions about overfitting in classical machine learning. [3]

Contents

History

Early observations of what would later be called double descent in specific models date back to 1989. [4] [5]

The term "double descent" was coined by Belkin et. al. [6] in 2019, [3] when the phenomenon gained popularity as a broader concept exhibited by many models. [7] [8] The latter development was prompted by a perceived contradiction between the conventional wisdom that too many parameters in the model result in a significant overfitting error (an extrapolation of the bias–variance tradeoff), [9] and the empirical observations in the 2010s that some modern machine learning techniques tend to perform better with larger models. [6] [10]

Theoretical models

Double descent occurs in linear regression with isotropic Gaussian covariates and isotropic Gaussian noise. [11]

A model of double descent at the thermodynamic limit has been analyzed using the replica trick, and the result has been confirmed numerically. [12]

Empirical examples

The scaling behavior of double descent has been found to follow a broken neural scaling law [13] functional form.

See also

References

  1. ↑ Rocks, Jason W. (2022). "Memorizing without overfitting: Bias, variance, and interpolation in overparameterized models". Physical Review Research. 4 (1) 013201. arXiv: 2010.13933 . Bibcode:2022PhRvR...4a3201R. doi:10.1103/PhysRevResearch.4.013201. PMID   36713351.
  2. ↑ "Deep Double Descent". OpenAI. 2019-12-05. Retrieved 2022-08-12.
  3. 1 2 Schaeffer, Rylan; Khona, Mikail; Robertson, Zachary; Boopathy, Akhilan; Pistunova, Kateryna; Rocks, Jason W.; Fiete, Ila Rani; Koyejo, Oluwasanmi (2023-03-24). "Double Descent Demystified: Identifying, Interpreting & Ablating the Sources of a Deep Learning Puzzle". arXiv: 2303.14151v1 [cs.LG].
  4. ↑ Vallet, F.; Cailton, J.-G.; Refregier, Ph (June 1989). "Linear and Nonlinear Extension of the Pseudo-Inverse Solution for Learning Boolean Functions" . Europhysics Letters. 9 (4): 315. Bibcode:1989EL......9..315V. doi:10.1209/0295-5075/9/4/003. ISSN   0295-5075.
  5. ↑ Loog, Marco; Viering, Tom; Mey, Alexander; Krijthe, Jesse H.; Tax, David M. J. (2020-05-19). "A brief prehistory of double descent". Proceedings of the National Academy of Sciences. 117 (20): 10625–10626. arXiv: 2004.04328 . Bibcode:2020PNAS..11710625L. doi: 10.1073/pnas.2001875117 . ISSN   0027-8424. PMC   7245109 . PMID   32371495.
  6. 1 2 Belkin, Mikhail; Hsu, Daniel; Ma, Siyuan; Mandal, Soumik (2019-08-06). "Reconciling modern machine learning practice and the bias-variance trade-off". Proceedings of the National Academy of Sciences. 116 (32): 15849–15854. arXiv: 1812.11118 . doi: 10.1073/pnas.1903070116 . ISSN   0027-8424. PMC   6689936 . PMID   31341078.
  7. ↑ Spigler, Stefano; Geiger, Mario; d'Ascoli, Stéphane; Sagun, Levent; Biroli, Giulio; Wyart, Matthieu (2019-11-22). "A jamming transition from under- to over-parametrization affects loss landscape and generalization". Journal of Physics A: Mathematical and Theoretical. 52 (47): 474001. arXiv: 1810.09665 . doi:10.1088/1751-8121/ab4c8b. ISSN   1751-8113.
  8. ↑ Viering, Tom; Loog, Marco (2023-06-01). "The Shape of Learning Curves: A Review". IEEE Transactions on Pattern Analysis and Machine Intelligence. 45 (6): 7799–7819. arXiv: 2103.10948 . Bibcode:2023ITPAM..45.7799V. doi:10.1109/TPAMI.2022.3220744. ISSN   0162-8828. PMID   36350870.
  9. ↑ Geman, Stuart; Bienenstock, Élie; Doursat, René (1992). "Neural networks and the bias/variance dilemma" (PDF). Neural Computation. 4: 1–58. doi:10.1162/neco.1992.4.1.1. S2CID   14215320.
  10. ↑ Preetum Nakkiran; Gal Kaplun; Yamini Bansal; Tristan Yang; Boaz Barak; Ilya Sutskever (29 December 2021). "Deep double descent: where bigger models and more data hurt". Journal of Statistical Mechanics: Theory and Experiment . 2021 (12). IOP Publishing Ltd and SISSA Medialab srl: 124003. arXiv: 1912.02292 . Bibcode:2021JSMTE2021l4003N. doi:10.1088/1742-5468/ac3a74. S2CID   207808916.
  11. ↑ Nakkiran, Preetum (2019-12-16). "More Data Can Hurt for Linear Regression: Sample-wise Double Descent". arXiv: 1912.07242v1 [stat.ML].
  12. ↑ Advani, Madhu S.; Saxe, Andrew M.; Sompolinsky, Haim (2020-12-01). "High-dimensional dynamics of generalization error in neural networks". Neural Networks. 132: 428–446. doi: 10.1016/j.neunet.2020.08.022 . ISSN   0893-6080. PMC   7685244 . PMID   33022471.
  13. ↑ Caballero, Ethan; Gupta, Kshitij; Rish, Irina; Krueger, David (2022). "Broken Neural Scaling Laws". International Conference on Learning Representations (ICLR), 2023.

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