Coordinate descent algorithms for nonconvex penalized regression, with applications to biological feature selection

TL;DR

This paper develops and analyzes coordinate descent algorithms for nonconvex penalized regression models like SCAD and MCP, demonstrating their theoretical convergence, computational efficiency, and practical advantages over lasso, especially in biological feature selection.

Contribution

It introduces fast coordinate descent algorithms for nonconvex penalties, proves their convergence, and applies convexity diagnostics to improve model stability and interpretability.

Findings

Coordinate descent algorithms are faster than existing methods.

MCP outperforms SCAD and lasso in simulations and data examples.

Convexity diagnostics help identify stable regions in parameter space.

Abstract

A number of variable selection methods have been proposed involving nonconvex penalty functions. These methods, which include the smoothly clipped absolute deviation (SCAD) penalty and the minimax concave penalty (MCP), have been demonstrated to have attractive theoretical properties, but model fitting is not a straightforward task, and the resulting solutions may be unstable. Here, we demonstrate the potential of coordinate descent algorithms for fitting these models, establishing theoretical convergence properties and demonstrating that they are significantly faster than competing approaches. In addition, we demonstrate the utility of convexity diagnostics to determine regions of the parameter space in which the objective function is locally convex, even though the penalty is not. Our simulation study and data examples indicate that nonconvex penalties like MCP and SCAD are worthwhile alternatives to the lasso in many applications. In particular, our numerical results suggest that MCP is the preferred approach among the three methods.