Non-convex sparsity-inducing penalties have recently received considerable attentions in sparse learning. Recent theoretical investigations have demonstrated their superiority over the convex counterparts in several sparse learning settings. However, solving the non-convex optimization problems associated with non-convex penalties remains a big challenge. A commonly used approach is the Multi-Stage (MS) convex relaxation (or DC programming), which relaxes the original non-convex problem to a sequence of convex problems. This approach is usually not very practical for large-scale problems because its computational cost is a multiple of solving a single convex problem. In this paper, we propose a General Iterative Shrinkage and Thresholding (GIST) algorithm to solve the nonconvex optimization problem for a large class of non-convex penalties. The GIST algorithm iteratively solves a proximal operator problem, which in turn has a closed-form solution for many commonly used penalties. At each outer iteration of the algorithm, we use a line search initialized by the Barzilai-Borwein (BB) rule that allows finding an appropriate step size quickly. The paper also presents a detailed convergence analysis of the GIST algorithm. The efficiency of the proposed algorithm is demonstrated by extensive experiments on large-scale data sets.

本文提出了一种新的GIST算法来解决大类非凸惩罚的非凸优化问题，该算法通过迭代解决近端算子问题，并在每个外部迭代中使用Barzilai-Borwein（BB）法来查找适当的步长，同时给出了详细的收敛分析。

一种用于非凸正则化优化问题的普适性迭代收缩与阈值算法