MM for Penalized Estimation

Abstract

Penalized estimation can conduct variable selection and parameter estimation simultaneously. The general framework is to minimize a loss function subject to a penalty designed to generate sparse variable selection. The majorization-minimization (MM) algorithm is a computational scheme for stability and simplicity, and the MM algorithm has been widely applied in penalized estimation. Much of the previous work have focused on convex loss functions such as generalized linear models. When data are contaminated with outliers, robust loss functions can generate more reliable estimates. Recent literature has witnessed a growing impact of nonconvex loss-based methods, which can generate robust estimation for data contaminated with outliers. This article investigates MM algorithm for penalized estimation, provide innovative optimality conditions and establish convergence theory with both convex and nonconvex loss functions. With respect to applications, we focus on several nonconvex loss functions, which were formerly studied in machine learning for regression and classification problems. Performance of the proposed algorithms are evaluated on simulated and real data including healthcare costs and cancer clinical status. Efficient implementations of the algorithms are available in the R package mpath in CRAN.

References in corpus (18)

• Rejoinder to "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Least Angle Regression
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Nearly unbiased variable selection under minimax concave penalty
• Coordinate descent algorithms for nonconvex penalized regression, with applications to biological feature selection
• Piecewise linear regularized solution paths
• Variable selection using MM algorithms
• Sparse least trimmed squares regression for analyzing high-dimensional large data sets
• Boosting in the presence of outliers: adaptive classification with non-convex loss functions
• Robust Parametric Classification and Variable Selection by a Minimum Distance Criterion
• MM Algorithms for Minimizing Nonsmoothly Penalized Objective Functions

Cited by in corpus (1)

• Unified Robust Estimation