Nonconvex variants of support vector machines (SVMs) have been developed for various purposes. For example, robust SVMs attain robustness to outliers by using a nonconvex loss function, while extended -SVM (E -SVM) extends the range of the hyperparameter by introducing a nonconvex constraint. Here, we consider an extended robust support vector machine (ER-SVM), a robust variant of E -SVM. ER-SVM combines two types of nonconvexity from robust SVMs and E -SVM. Because of the two nonconvexities, the existing algorithm we proposed needs to be divided into two parts depending on whether the hyperparameter value is in the extended range or not. The algorithm also heuristically solves the nonconvex problem in the extended range. In this letter, we propose a new, efficient algorithm for ER-SVM. The algorithm deals with two types of nonconvexity while never entailing more computations than either E -SVM or robust SVM, and it finds a critical point of ER-SVM. Furthermore, we show that ER-SVM includes the existing robust SVMs as special cases. Numerical experiments confirm the effectiveness of integrating the two nonconvexities.

Financial risk measures have been used recently in machine learning. For example, -support vector machine ( -SVM) minimizes the conditional value at risk (CVaR) of margin distribution. The measure is popular in finance because of the subadditivity property, but it is very sensitive to a few outliers in the tail of the distribution. We propose a new classification method, extended robust SVM (ER-SVM), which minimizes an intermediate risk measure between the CVaR and value at risk (VaR) by expecting that the resulting model becomes less sensitive than -SVM to outliers. We can regard ER-SVM as an extension of robust SVM, which uses a truncated hinge loss. Numerical experiments imply the ER-SVM’s possibility of achieving a better prediction performance with proper parameter setting.