Tight bounds for minimum l1-norm interpolation of noisy data

Guillaume Wang; Konstantin Donhauser; Fanny Yang

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arXiv · 2111.05987

Tight bounds for minimum l1-norm interpolation of noisy data

Guillaume Wang·Konstantin Donhauser·Fanny Yang

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Abstract

We provide matching upper and lower bounds of order $\sigma^2/\log(d/n)$ for the prediction error of the minimum $\ell_1$-norm interpolator, a.k.a. basis pursuit. Our result is tight up to negligible terms when $d \gg n$, and is the first to imply asymptotic consistency of noisy minimum-norm interpolation for isotropic features and sparse ground truths. Our work complements the literature on "benign overfitting" for minimum $\ell_2$-norm interpolation, where asymptotic consistency can be achieved only when the features are effectively low-dimensional.

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BibTeX RIS

Guillaume Wang, Konstantin Donhauser, Fanny Yang. 2021-11-10. Tight bounds for minimum l1-norm interpolation of noisy data. https://arxiv.org/abs/2111.05987

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