Uniform Convergence of Interpolators: Gaussian Width, Norm Bounds, and Benign Overfitting

Abstract

We consider interpolation learning in high-dimensional linear regression with Gaussian data, and prove a generic uniform convergence guarantee on the generalization error of interpolators in an arbitrary hypothesis class in terms of the class's Gaussian width. Applying the generic bound to Euclidean norm balls recovers the consistency result of Bartlett et al. (2020) for minimum-norm interpolators, and confirms a prediction of Zhou et al. (2020) for near-minimal-norm interpolators in the special case of Gaussian data. We demonstrate the generality of the bound by applying it to the simplex, obtaining a novel consistency result for minimum l1-norm interpolators (basis pursuit). Our results show how norm-based generalization bounds can explain and be used to analyze benign overfitting, at least in some settings.

arXiv commentv2: Minor changes only. As published at NeurIPS 2021: https://proceedings.neurips.cc/paper/2021/hash/ac9815bef801f58de83804bce86984ad-Abstract.html

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Cited by in corpus (4)

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• Tight bounds for minimum l1-norm interpolation of noisy data
• Foolish Crowds Support Benign Overfitting
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