Ramp Loss Linear Programming Support Vector Machine

Abstract

The ramp loss is a robust but non-convex loss for classification. Compared with other non-convex losses, a local minimum of the ramp loss can be effectively found. The effectiveness of local search comes from the piecewise linearity of the ramp loss. Motivated by the fact that the $\ell_1$-penalty is piecewise linear as well, the $\ell_1$-penalty is applied for the ramp loss, resulting in a ramp loss linear programming support vector machine (ramp- LPSVM). The proposed ramp-LPSVM is a piecewise linear minimization problem and the related optimization techniques are applicable. Moreover, the $\ell_1$-penalty can enhance the sparsity. In this paper, the corresponding misclassification error and convergence behavior are discussed. Generally, the ramp loss is a truncated hinge loss. Therefore ramp-LPSVM possesses some similar properties as hinge loss SVMs. A local minimization algorithm and a global search strategy are discussed. The good optimization capability of the proposed algorithms makes ramp-LPSVM perform well in numerical experiments: the result of ramp-LPSVM is more robust than that of hinge SVMs and is sparser than that of ramp-SVM, which consists of the $\|\cdot\|_{\mathcal{K}} $-penalty and the ramp loss.