基于Huber损失和组Lasso惩罚问题的加速临近梯度算法

Accelerated Proximal Gradient Algorithm Based on Huber Loss and Group Lasso Penalty Problem

摘要: 在高维线性回归模型中，组稀疏恢复的做法是在原有线性模型的基础上增加一个组Lasso惩罚项，以诱导尽可能少的非零组，将问题转换为凸优化模型进行求解。本文研究Huber损失和组Lasoo 组合问题，其中惩罚项包含了组稀疏惩罚，惩罚项的目的是用于保证组元素稀疏性结构。首先，由于惩罚项是一个凸但不光滑函数，为了刻画组Lasso模型的最优性条件，给出了其经典次微分。其次，利用Nesterov加速技术提出了加速临近梯度算法来求解我们的模型。最后证明了所提出算法的收敛性。

Abstract: In the high-dimensional linear regression model, the method of group sparse recovery is to add a group Lasso penalty term to the original linear model so as to induce as few non-zero groups as possible, and then to transform the problem into a convex optimization model. This paper studies the group Lasso problem based on Huber loss, where the penalty term includes group sparsity penalty, and the purpose of the penalty term is to ensure the group sparsity structure of group element. Firstly, since the penalty term is a convex but not smooth function, in order to characterize the optimality conditions of the group Lasso model, its classical subdifferential is given. Secondly, an accelerated proximity gradient algorithm was proposed using Nesterov acceleration technology to solve our model. Finally, the convergence of the proposed algorithm was demonstrated.