An Iterative Algorithm for Fitting Nonconvex Penalized Generalized Linear Models with Grouped Predictors

TL;DR

This paper introduces an iterative algorithm for fitting nonconvex penalized generalized linear models with grouped predictors, addressing high-dimensional data challenges, collinearity, and non-Gaussian models, with theoretical guarantees and practical applications.

Contribution

It proposes a simple, convergent algorithm for nonconvex penalized GLMs with grouped predictors, including novel penalty designs and parameter tuning strategies.

Findings

Algorithm guarantees convergence and tight scaling.

Improved performance in spectrum estimation and gene selection.

Effective handling of collinearity and non-Gaussian data.

Abstract

High-dimensional data pose challenges in statistical learning and modeling. Sometimes the predictors can be naturally grouped where pursuing the between-group sparsity is desired. Collinearity may occur in real-world high-dimensional applications where the popular $l_1$ technique suffers from both selection inconsistency and prediction inaccuracy. Moreover, the problems of interest often go beyond Gaussian models. To meet these challenges, nonconvex penalized generalized linear models with grouped predictors are investigated and a simple-to-implement algorithm is proposed for computation. A rigorous theoretical result guarantees its convergence and provides tight preliminary scaling. This framework allows for grouped predictors and nonconvex penalties, including the discrete $l_0$ and the `$l_0+l_2$' type penalties. Penalty design and parameter tuning for nonconvex penalties are examined. Applications of super-resolution spectrum estimation in signal processing and cancer classification with joint gene selection in bioinformatics show the performance improvement by nonconvex penalized estimation.