Breakdown Point of Robust Support Vector Machine

Abstract

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.