In many learning tasks with structural properties, structured sparse modeling usually leads to better interpretability and higher generalization performance. While great efforts have focused on the convex regularization, recent studies show that nonconvex regularizers can outperform their convex counterparts in many situations. However, the resulting nonconvex optimization problems are still challenging, especially for the structured sparsity-inducing regularizers. In this paper, we propose a splitting method for solving nonconvex structured sparsity optimization problems. The proposed method alternates between a gradient step and an easily solvable proximal step, and thus enjoys low per-iteration computational complexity. We prove that the whole sequence generated by the proposed method converges to a critical point with at least sublinear convergence rate, relying on the Kurdyka-Łojasiewicz inequality. Experiments on both simulated and real-world data sets demonstrate the efficiency and efficacy of the proposed method.

The minimax concave plus penalty (MCP) has been demonstrated to be effective in nonconvex penalization for feature selection. In this paper we propose a novel construction approach for MCP. In particular, we show that MCP can be derived from a concave conjugate of the Euclidean distance function. This construction approach in turn leads us to an augmented Lagrange multiplier method for solving the penalized regression problem with MCP. In our method each tuning parameter corresponds to a feature, and these tuning parameters can be automatically updated. We also develop a d.c. (difference of convex functions) programming approach for the penalized regression problem. We find that the augmented Lagrange multiplier method degenerates into the d.c. programming method under specific conditions. Experimental analysis is conducted on a set of simulated data. The result is encouraging.