Generalization performance of ? ? -support vector classifier based on conditional value-at-risk minimization
We extend the conditional geometric score (CGS) classifier of Gotoh and Takeda for binary linear classification to a nonlinear one, which we call the ? ? -support vector classifier (SVC), and investigate the equivalence between the ? ? -SVC and the (extended) ? ? -SVC. The CGS classifier has recently been found to be equivalent to the extended ? ? -SVC of Perez-Cruz et al. and, especially in the convex case, equivalent to the ? ? -SVC of Schlkopf et al. The CGS problem is to minimize a risk measure known as the conditional value-at-risk (? ? -CVaR). In this paper, we discuss theoretical aspects, mainly generalization performance, of the ? ? -SVC. The formula of a generalization error bound includes the ? ? -CVaR or a related quantity. It implies that the minimum ? ? -CVaR leads to a small generalization error bound of the ? ? -SVC. The viewpoint of CVaR minimization is useful to ensure the validity of not only the ? ? -SVC but also the (extended) ? ? -SVC.
conditional value-at-risk support vector machines nonconvex quadratic programming extended ? ? -svm ? ? ? ? ?