A Convex-Nonconvex Strategy for Grouped Variable Selection

TL;DR

This paper introduces a novel convex-penalized method called group GMC for grouped variable selection in linear regression, which reduces bias and avoids local optima issues of existing methods, with proven theoretical properties and superior empirical performance.

Contribution

It proposes the group GMC estimator, a convex alternative to nonconvex methods, with an efficient algorithm, theoretical error bounds, and demonstrated superior performance.

Findings

Outperforms existing methods in simulations

Maintains convexity while reducing bias

Provides theoretical error bounds

Abstract

This paper deals with the grouped variable selection problem. A widely used strategy is to augment the negative log-likelihood function with a sparsity-promoting penalty. Existing methods include the group Lasso, group SCAD, and group MCP. The group Lasso solves a convex optimization problem but is plagued by underestimation bias. The group SCAD and group MCP avoid this estimation bias but require solving a nonconvex optimization problem that may be plagued by suboptimal local optima. In this work, we propose an alternative method based on the generalized minimax concave (GMC) penalty, which is a folded concave penalty that maintains the convexity of the objective function. We develop a new method for grouped variable selection in linear regression, the group GMC, that generalizes the strategy of the original GMC estimator. We present an efficient algorithm for computing the group GMC estimator and also prove properties of the solution path to guide its numerical computation and tuning parameter selection in practice. We establish error bounds for both the group GMC and original GMC estimators. A rich set of simulation studies and a real data application indicate that the proposed group GMC approach outperforms existing methods in several different aspects under a wide array of scenarios.