An evolving line of machine learning works observe empirical evidence that suggests interpolating estimators—
the ones that achieve zero training error— may not necessarily be harmful. In this talk, we pursue theoretical understanding for an important type of interpolators: the minimum L1-norm interpolator, which is motivated by the observation that several learning algorithms favor low L1-norm solutions in the over-parameterized regime. Concretely, we consider the noisy sparse regression model under Gaussian design, focusing on linear sparsity and high-dimensional asymptotics (so that both the number of features and the sparsity level scale proportionally with the sample size).
We observe, and provide rigorous theoretical justification for, a curious multi-descent phenomenon; that is, the generalization risk of the minimum L1-norm interpolator undergoes multiple (and possibly more than two) phases of descent and ascent as one increases the model capacity. This phenomenon stems from the special structure of the minimum L1-norm interpolator as well as the delicate interplay between the over-parameterized ratio and the sparsity, thus unveiling a fundamental distinction in geometry from the minimum L2-norm interpolator. Our finding is built upon an exact characterization of the risk behavior, which is governed by a system of two non-linear equations with two unknowns.