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

Frederic Koehler, Lijia Zhou, Danica J. Sutherland, Nathan Srebro

Keywords: interpolation learning, uniform convergence, double descent, statistical learning theory

TL;DR: Uniform convergence of interpolating predictors can explain consistency for high-dimensional linear regression.

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 $\ell_1$-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.

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