High-dimensional nonconvex lasso-type $M$-estimators

TL;DR

This paper develops a theoretical framework for high-dimensional $ ext{L}_1$-penalized $M$-estimators with nonconvex risks, demonstrating their convergence rates and providing conditions for practical applications like robust regression and classification.

Contribution

It introduces a new theory for nonconvex high-dimensional $M$-estimators with $ ext{L}_1$ penalty, establishing convergence rates and practical conditions.

Findings

Estimators achieve convergence rate $s_0\sqrt{rac{ ext{log}(nd)}{n}}$.

Provided sufficient conditions for the main assumptions.

Applied theory to robust linear regression, classification, and nonlinear least squares.

Abstract

This paper proposes a theory for $\ell_1$-norm penalized high-dimensional $M$-estimators, with nonconvex risk and unrestricted domain. Under high-level conditions, the estimators are shown to attain the rate of convergence $s_0\sqrt{\log(nd)/n}$, where $s_0$ is the number of nonzero coefficients of the parameter of interest. Sufficient conditions for our main assumptions are then developed and finally used in several examples including robust linear regression, binary classification and nonlinear least squares.