论文标题
坡道损失SVM的近端操作员和最佳条件
Proximal Operator and Optimality Conditions for Ramp Loss SVM
论文作者
论文摘要
由于坡道损失的界限，支持坡道损失（称为$ l_r $ -SVM）的支持向量机已引起广泛的关注。但是，相应的优化问题是非凸，给定的Karush-Kuhn-Tucker（KKT）条件只是必要的条件。为了丰富$ L_R $ -SVM的最佳理论并深入统计性质，我们首先介绍和分析近端操作员的坡道损失，然后建立更强的最佳条件：PSTATIONITY：PSTATIONARITY：事实证明，这是$ l_r $ -svm的本地最小情况的一阶必需和充分的条件。最后，我们根据p-stationary点的概念来定义$ l_r $ support vectors，并表明所有$ l_r $ support vectors都属于支持超平面，该平面具有与硬利润率SVM的功能相同的功能。
Support vector machines with ramp loss (dubbed as $L_r$-SVM) have attracted wide attention due to the boundedness of ramp loss. However, the corresponding optimization problem is non-convex and the given Karush-Kuhn-Tucker (KKT) conditions are only the necessary conditions. To enrich the optimality theory of $L_r$-SVM and go deep into its statistical nature, we first introduce and analyze the proximal operator for ramp loss, and then establish a stronger optimality conditions: P-stationarity, which is proved to be the first-order necessary and sufficient conditions for local minimizer of $L_r$-SVM. Finally, we define the $L_r$ support vectors based on the concept of P-stationary point, and show that all $L_r$ support vectors fall into the support hyperplanes, which possesses the same feature as the one of hard margin SVM.
