A Novel Loss Function-based Support Vector Machine for Binary
  Classification
Yan Li Liping Zhang
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 ($\ell_s$) to construct the support
vector machine classifier($\ell_s$-SVM). By introducing the concept of proximal
stationary point, and utilizing the property of Lipschitz continuity, we derive
the first-order optimality conditions for $\ell_s$-SVM. Based on this, we
define the $\ell_s$ support vectors and working set of $\ell_s$-SVM. To
efficiently handle $\ell_s$-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.