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An evolving line of machine learning works observe empirical evidence thatsuggests interpolating estimators -- the ones that achieve zero training error-- may not necessarily be harmful. This paper pursues theoretical understandingfor an important type of interpolators: the minimum $\ell_{1}$-norminterpolator, which is motivated by the observation that several learningalgorithms favor low $\ell_1$-norm solutions in the over-parameterized regime.Concretely, we consider the noisy sparse regression model under Gaussiandesign, focusing on linear sparsity and high-dimensional asymptotics (so thatboth the number of features and the sparsity level scale proportionally withthe sample size).  We observe, and provide rigorous theoretical justification for, a curiousmulti-descent phenomenon; that is, the generalization risk of the minimum$\ell_1$-norm interpolator undergoes multiple (and possibly more than two)phases of descent and ascent as one increases the model capacity. Thisphenomenon stems from the special structure of the minimum $\ell_1$-norminterpolator as well as the delicate interplay between the over-parameterizedratio and the sparsity, thus unveiling a fundamental distinction in geometryfrom the minimum $\ell_2$-norm interpolator. Our finding is built upon an exactcharacterization of the risk behavior, which is governed by a system of twonon-linear equations with two unknowns.

Minimum $\ell_{1}$-norm interpolators: Precise asymptotics and multiple descent