Unveiling low-dimensional patterns induced by convex non-differentiable regularizers

TL;DR

This paper investigates the asymptotic behavior of low-dimensional patterns induced by non-differentiable regularizers like Lasso and Fused Lasso in linear regression, providing new theoretical insights and practical procedures for pattern recovery.

Contribution

It introduces a novel convergence analysis for regularizer-induced patterns using Hausdorff distance and proposes procedures for consistent pattern recovery regardless of irrepresentability conditions.

Findings

Pattern convergence is established via Hausdorff distance.

Exact probability of true pattern recovery is derived.

Fused Lasso cannot reliably recover its clustering pattern without modifications.

Abstract

Popular regularizers with non-differentiable penalties, such as Lasso, Elastic Net, Generalized Lasso, or SLOPE, reduce the dimension of the parameter space by inducing sparsity or clustering in the estimators' coordinates. In this paper, we focus on linear regression and explore the asymptotic distributions of the resulting low-dimensional patterns when the number of regressors $p$ is fixed, the number of observations $n$ goes to infinity, and the penalty function increases at the rate of $\sqrt{n}$. While the asymptotic distribution of the rescaled estimation error can be derived by relatively standard arguments, convergence of patterns requires a separate proof, which is yet missing from the literature, even for the simplest case of Lasso. To fill this gap, we use the Hausdorff distance as a suitable mode of convergence for subdifferentials, resulting in the desired pattern convergence. Furthermore, we derive the exact limiting probability of recovering the true model pattern. This probability goes to 1 if and only if the penalty scaling constant diverges to infinity and the regularizer-specific asymptotic irrepresentability condition is satisfied. We then propose simple two-step procedures that asymptotically recover the model patterns, irrespective of whether the irrepresentability condition holds or not. Interestingly, our theory shows that Fused Lasso cannot reliably recover its own clustering pattern, even for independent regressors. It also demonstrates how this problem can be resolved by "concavifying" the Fused Lasso penalty coefficients. Additionally, sampling from the asymptotic error distribution facilitates comparisons between different regularizers. We provide short simulation studies showcasing an illustrative comparison between the asymptotic properties of Lasso, Fused Lasso, and SLOPE.