标题： 鲁棒支持向量机的破裂点 显示英文标题

标题： Breakdown Point of Robust Support Vector Machine

摘要： 支持向量机（SVM）是解决分类问题的最成功的学习方法之一。尽管它广受欢迎，但SVM有一个严重的缺点，即对训练样本中的异常值敏感。错分类的惩罚由一种称为铰链损失的凸损失定义，而凸损失的无界性导致了对异常值的敏感性。为了应对异常值，提出了SVM的鲁棒变体，例如鲁棒异常检测算法和具有有界损失（称为斜坡损失）的SVM。本文提出了一种SVM的鲁棒变体，并从破裂点的角度研究了其鲁棒性。破裂点是一种鲁棒性度量，表示估计分类器仍能提供关于未污染数据信息的最大污染量。本文的主要贡献在于精确评估了鲁棒SVM的破裂点。对于学习参数（如我们算法中的正则化参数），我们推导出一个简单的公式，以保证分类器的鲁棒性。当通过交叉验证使用网格搜索确定学习参数时，我们的公式可以减少候选搜索点的数量。数值实验验证了所提出方法的鲁棒性。我们表明，鲁棒SVM的统计特性可以通过破裂点的理论分析很好地解释。 显示英文摘要

摘要： The support vector machine (SVM) is one of the most successful learning methods for solving classification problems. Despite its popularity, SVM has a serious drawback, that is sensitivity to outliers in training samples. The penalty on misclassification is defined by a convex loss called the hinge loss, and the unboundedness of the convex loss causes the sensitivity to outliers. To deal with outliers, robust variants of SVM have been proposed, such as the robust outlier detection algorithm and an SVM with a bounded loss called the ramp loss. In this paper, we propose a robust variant of SVM and investigate its robustness in terms of the breakdown point. The breakdown point is a robustness measure that is the largest amount of contamination such that the estimated classifier still gives information about the non-contaminated data. The main contribution of this paper is to show an exact evaluation of the breakdown point for the robust SVM. For learning parameters such as the regularization parameter in our algorithm, we derive a simple formula that guarantees the robustness of the classifier. When the learning parameters are determined with a grid search using cross validation, our formula works to reduce the number of candidate search points. The robustness of the proposed method is confirmed in numerical experiments. We show that the statistical properties of the robust SVM are well explained by a theoretical analysis of the breakdown point.