题    目：Iterative Reweighted Framework Based Algorithms for Sparse Linear Regression with Generalized Elastic Net Penalty

主讲人：丁彦昀 讲师

单    位：深圳职业技术大学

时    间：2025年1月17日 15:00

地    点：郑州校区九章学堂南楼C座302

摘    要：The elastic net penalty is frequently employed in high-dimensional statistics for parameter regression and variable selection. It is particularly beneficial compared to lasso when the number of predictors greatly surpasses the number of observations. However, empirical evidence has shown that the $\ell_q$-norm penalty (where $0 < q < 1$) often provides better regression compared to the $\ell_1$-norm penalty, demonstrating enhanced robustness in various scenarios. In this paper, we explore a generalized elastic net penalized model that employs a $\ell_r$-norm (where $r \geq 1$) in loss function to accommodate various types of noise, and employs a $\ell_q$-norm (where $0 < q < 1$) to replace the $\ell_1$-norm in elastic net penalty. We theoretically prove that the local minimizer of the proposed model is a generalized first-order stationary point, and then derive the computable lower bounds for the nonzero entries of this generalized stationary point. For the implementation, we utilize an iterative reweighted framework that leverages the locally Lipschitz continuous $\epsilon$-approximation, and subsequently propose two optimization algorithms. The first algorithm employs an alternating direction method of multipliers (ADMM), while the second utilizes a proximal majorization-minimization method (PMM), where the subproblems are addressed using the semismooth Newton method (SNN). We also perform extensive numerical experiments with both simulated and real data, showing that both algorithms demonstrate superior performance. Notably, the PMM-SSN is efficient than ADMM, even though the latter provides a simpler implementation.

简    介：丁彦昀，深圳职业技术大学，讲师。针对机器学习中的图像处理和信号恢复等问题，利用共轭梯度法、交替方向乘子法、半光滑牛顿法等做出了一系列的研究成果。在《iScience》（Cell子刊）、《Optimization》、《Optimization Methods and Software》、《Journal of Mathematical Imaging and Vision》和《Journal of Nonlinear and Variational Analysis》等国内外期刊发表SCI论文多篇。