Tight bounds for minimum l1-norm interpolation of noisy data

11/10/2021

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by   Guillaume Wang, et al.

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We provide matching upper and lower bounds of order σ^2/log(d/n) for the prediction error of the minimum ℓ_1-norm interpolator, a.k.a. basis pursuit. Our result is tight up to negligible terms when d ≫ 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 ℓ_2-norm interpolation, where asymptotic consistency can be achieved only when the features are effectively low-dimensional.