The Lasso error is bounded iff its active set size is bounded away from n in the proportional regime

TL;DR

This paper analyzes the Lasso estimator in high-dimensional linear models without assuming sparsity, establishing that its error is bounded iff the number of selected variables stays away from the sample size, and revisits phase transitions in sparsity patterns.

Contribution

It provides a new proof of the phase transition for Lasso risk boundedness without fixed-point equations, extending analysis to small tuning parameters.

Findings

Lasso risk is bounded iff active set size is bounded away from n

Dense Lasso estimates cannot have constant risk

Phase transition characterizes sparsity patterns leading to unbounded risk

Abstract

This note develops an analysis of the Lasso \( \hat b\) in linear models without any sparsity or L1 assumption on the true regression vector, in the proportional regime where dimension \( p \) and sample \( n \) are of the same order. Under Gaussian design and covariance matrix with spectrum bounded away from 0 and $+\infty$, it is shown that the L2 risk is stochastically bounded if and only if the number of selected variables is bounded away from \( n \), in the sense that $$ (1-\|\hat b\|_0/n)^{-1} = O_P(1) \Longleftrightarrow \|\hat b- b^*\|_2 = O_P(1) $$ as \( n,p\to+\infty \). The right-to-left implication rules out constant risk for dense Lasso estimates (estimates with close to $n$ active variables), which can be used to discard tuning parameters leading to dense estimates. We then bring back sparsity in the picture, and revisit the precise phase transition characterizing the sparsity patterns of the true regression vector leading to unbounded Lasso risk -- or by the above equivalence to dense Lasso estimates. This precise phase transition was established by \citet{miolane2018distribution,celentano2020lasso} using fixed-point equations in an equivalent sequence model. An alternative proof of this phase transition is provided here using simple arguments without relying on the fixed-point equations or the equivalent sequence model. A modification of the well-known Restricted Eigenvalue argument allows to extend the analysis to any small tuning parameter of constant order, leading to a bounded risk on one side of the phase transition. On the other side of the phase transition, it is established the Lasso risk can be unbounded for a given sign pattern as soon as Basis Pursuit fails to recover that sign pattern in noiseless problems.