标题： 零范数正则化鲁棒损失最小化：近端MM方法与统计误差界 显示英文标题

标题： Group zero-norm regularized robust loss minimization: proximal MM method and statistical error bound

摘要： 本研究聚焦于求解带有组零范数正则化的鲁棒损失最小化问题。我们提出了一种邻近Majorization-Minimization（PMM）算法来处理一类等价的凸差（DC）替代优化问题。首先，我们介绍了PMM方法的核心原理和迭代框架。在潜在函数满足Kurdyka-{\L }ojasiewicz（KL）性质假设下，我们建立了算法的全局收敛性，并刻画了其局部（次）线性收敛速度。此外，对于满足限制特征值条件的设计矩阵的线性观测模型，我们推导出了PMM生成的迭代序列（包括其极限点）与真实解之间的统计估计误差界。这些界不仅严格量化了算法的近似精度，还扩展了参考文献[57]中关于元素稀疏复合优化的先前结果。为了高效实现PMM框架，我们开发了一种邻近对偶半光滑牛顿法来解决关键子问题。在合成数据和UCI基准上的大量数值实验表明，与邻近交替方向乘子法（pADMM）相比，我们的PMM方法具有优越的计算效率。 显示英文摘要

摘要： This study focuses on solving group zero-norm regularized robust loss minimization problems. We propose a proximal Majorization-Minimization (PMM) algorithm to address a class of equivalent Difference-of-Convex (DC) surrogate optimization problems. First, we present the core principles and iterative framework of the PMM method. Under the Kurdyka-{\L}ojasiewicz (KL) property assumption of the potential function, we establish the global convergence of the algorithm and characterize its local (sub)linear convergence rate. Furthermore, for linear observation models with design matrices satisfying restricted eigenvalue conditions, we derive statistical estimation error bounds between the PMM-generated iterates (including their limit points) and the ground truth solution. These bounds not only rigorously quantify the approximation accuracy of the algorithm but also extend previous results on element-wise sparse composite optimization from reference [57]. To efficiently implement the PMM framework, we develop a proximal dual semismooth Newton method for solving critical subproblems. Extensive numerical experiments on both synthetic data and the UCI benchmark demonstrate the superior computational efficiency of our PMM method compared to the proximal Alternating Direction Method of Multipliers (pADMM).