Alternating Direction Method of Multipliers for ℓ1-ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{1}\hbox {-}\ell _{2}$$\end{document}-Regularized Logistic Regression Model

Logistic regression has been proved as a promising method for machine learning, which focuses on the problem of classification. In this paper, we present an ℓ1-ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{1}\hbox {-}\ell _{2}$$\end{document}-regularized logistic regression model, where the ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{1}$$\end{document}-norm is responsible for yielding a sparse logistic regression classifier and the ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{2}$$\end{document}-norm for keeping better classification accuracy. To solve the ℓ1-ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{1}\hbox {-}\ell _{2}$$\end{document}-regularized logistic regression model, we develop an alternating direction method of multipliers with embedding limited-Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method. Furthermore, we implement our model for binary classification problems by using real data examples selected from the University of California, Irvine Machines Learning Repository (UCI Repository). We compare our numerical results with those obtained by the well-known LIBSVM and SVM-Light software. The numerical results show that our ℓ1-ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{1}\hbox {-}\ell _{2}$$\end{document}-regularized logistic regression model achieves better classification and less CPU Time.