Nonconvex Sparse Logistic Regression with Weakly Convex Regularization

TL;DR

This paper introduces a nonconvex sparse logistic regression approach using weakly convex regularization, which better induces sparsity than traditional methods, with theoretical analysis and practical algorithms demonstrated on real and synthetic data.

Contribution

It proposes a novel weakly convex regularization framework for sparse logistic regression, including convergence analysis and an iterative algorithm for practical implementation.

Findings

Weakly convex regularization improves sparsity over $ ext{L}_1$ norm.

The proposed method converges under certain conditions.

Numerical experiments validate the effectiveness of the approach.

Abstract

In this work we propose to fit a sparse logistic regression model by a weakly convex regularized nonconvex optimization problem. The idea is based on the finding that a weakly convex function as an approximation of the $\ell_0$ pseudo norm is able to better induce sparsity than the commonly used $\ell_1$ norm. For a class of weakly convex sparsity inducing functions, we prove the nonconvexity of the corresponding sparse logistic regression problem, and study its local optimality conditions and the choice of the regularization parameter to exclude trivial solutions. Despite the nonconvexity, a method based on proximal gradient descent is used to solve the general weakly convex sparse logistic regression, and its convergence behavior is studied theoretically. Then the general framework is applied to a specific weakly convex function, and a necessary and sufficient local optimality condition is provided. The solution method is instantiated in this case as an iterative firm-shrinkage algorithm, and its effectiveness is demonstrated in numerical experiments by both randomly generated and real datasets.