Convergence and sparsity of Lasso and group Lasso in high-dimensional generalized linear models

In this short paper, we investigate Lasso regularized generalized linear models in the “small n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}, large p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}” setting. While similar problems have been well-studied with SCAD penalty, the study of Lasso penalty is mostly restricted to the least squares loss function. Here we show the convergence rate of the Lasso penalized estimator as well as the sparsity property under suitable assumptions. We also extend the results to group Lasso regularized models when the variables are naturally grouped.