A Novel Loss Function-based Support Vector Machine for Binary Classification

The previous support vector machine(SVM) including $0/1$ loss SVM, hinge loss SVM, ramp loss SVM, truncated pinball loss SVM, and others, overlooked the degree of penalty for the correctly classified samples within the margin. This oversight affects the generalization ability of the SVM classifier to some extent. To address this limitation, from the perspective of confidence margin, we propose a novel Slide loss function ($\ells$) to construct the support vector machine classifier($\ells$-SVM). By introducing the concept of proximal stationary point, and utilizing the property of Lipschitz continuity, we derive the first-order optimality conditions for $\ells$-SVM. Based on this, we define the $\ells$ support vectors and working set of $\ells$-SVM. To efficiently handle $\ells$-SVM, we devise a fast alternating direction method of multipliers with the working set ($\ell_s$-ADMM), and provide the convergence analysis. The numerical experiments on real world datasets confirm the robustness and effectiveness of the proposed method.