In this work we establish an algorithm and distribution independent nonasymptotic trade-off between the model size, excess test loss, and training loss of linear predictors. Specifically, we show that models that perform well on the test data (have low excess loss) are either “classical”—have training loss close to the noise level—or are “modern”—have a much larger number of parameters compared to the minimum needed to fit the training data exactly. We also provide a more precise asymptotic analysis when the limiting spectral distribution of the whitened features is Marchenko–Pastur. Remarkably, while the Marchenko–Pastur analysis is far more precise near the interpolation peak, where the number of parameters is just enough to fit the training data, it coincides exactly with the distribution independent bound as the level of overparameterization increases.


  1. statistical learning theory
  2. overfitting
  3. linear regression
  4. overparametrization

MSC codes

  1. 62J05
  2. 62F12

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The authors would like to thank Amirhesam Abedsoltan for finding an error in a previous version of the proof of Theorem 2.1. Correcting the proof led to an improved lower bound which is now tight. We also thank the anonymous reviewers for insightful comments. We are grateful for support from the National Science Foundation (NSF) and the Simons Foundation for the Collaboration on the Theoretical Foundations of Deep Learning.


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Information & Authors


Published In

cover image SIAM Journal on Mathematics of Data Science
SIAM Journal on Mathematics of Data Science
Pages: 977 - 1004
ISSN (online): 2577-0187


Submitted: 9 December 2022
Accepted: 31 July 2023
Published online: 9 November 2023


  1. statistical learning theory
  2. overfitting
  3. linear regression
  4. overparametrization

MSC codes

  1. 62J05
  2. 62F12



Nikhil Ghosh Contact the author
Statistics Department, University of California, Berkeley, CA 95730 USA.
Mikhail Belkin
Halıcıoğlu Data Science Institute, University of California, San Diego, CA 92093 USA.

Funding Information

National Science Foundation (NSF): DMS-2031883, 814639, 1745640, IIS-1815697, CCF-2112665
Funding: This work was funded by NSF awards DMS-2031883 and 814639, with additional support from NSF grants 1745640, IIS-1815697, and CCF-2112665.

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