Strong Rules for Nonconvex Penalties and Their Implications for Efficient Algorithms in High-Dimensional Regression

• Activity Number: 508
• Type: Contributed
• Date/Time: Wednesday, August 6, 2014 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistical Computing
• Author(s): Sangin Lee*+ and Patrick Breheny
• Companies: University of Iowa and University of Iowa
• Keywords: Coordinate descent algorithms ; Local convexity ; Nonconvex penalties ; Dimension reduction ; High-dimensional data

Abstract

We consider approaches for improving the efficiency of algorithms for fitting nonconvex penalized regression models such as SCAD and MCP in high dimensions. In particular, we develop rules for discarding variables during cyclic coordinate descent. This dimension reduction leads to a substantial improvement in the speed of these algorithms for high-dimensional problems. The rules we propose here eliminate a substantial fraction of the variables from the coordinate descent algorithm. Violations are quite rare, especially in the locally convex region of the solution path, and furthermore, may be easily detected and corrected by checking the Karush-Kuhn-Tucker conditions. We extend these rules to generalized linear models, as well as to other nonconvex penalties such as the L2-stabilized Mnet penalty, group MCP, and group SCAD. We explore three variants of the coordinate decent algorithm that incorporate these rules and study the efficiency of these algorithms in fitting models to both simulated data and on real data from a genome-wide association study.