A Unifying Framework of High-Dimensional Sparse Estimation with Difference-of-Convex (DC) Regularizations

Under the linear regression framework, we study the variable selection problem when the underlying model is assumed to have a small number of nonzero coefficients. Nonconvex penalties in specific forms are well studied in the literature for sparse estimation. Recent work pointed out that nearly all existing nonconvex penalties can be represented as differenceof-convex (DC) functions, which are the difference of two convex functions, while itself may not be convex. There is a large existing literature on optimization problems when their objectives and/or constraints involve DC functions. Efficient numerical solutions have been proposed. Under the DC framework, directional-stationary (d-stationary) solutions are considered, and they are usually not unique. In this paper, we show that under some mild conditions, a certain subset of d-stationary solutions in an optimization problem (with a DC objective) has some ideal statistical properties: namely, asymptotic estimation consistency, asymptotic model selection consistency, asymptotic efficiency. Our assumptions are either weaker than or comparable with those conditions that have been adopted in other existing works. This work shows that DC is a nice framework to offer a unified approach to these existing works where nonconvex penalties are involved. Our work bridges the communities of optimization and statistics.