  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.
