An error bound for L₁-norm support vector machine coefficients in ultra-high dimension

Comparing with the standard L₂-norm support vector machine (SVM), the L₁-norm SVM enjoys the nice property of simultaneously preforming classification and feature selection. In this paper, we investigate the statistical performance of L₁-norm SVM in ultra-high dimension, where the number of features p grows at an exponential rate of the sample size n. Different from existing theory for SVM which has been mainly focused on the generalization error rates and empirical risk, we study the asymptotic behavior of the coefficients of L₁-norm SVM. Our analysis reveals that the estimated L₁-norm SVM coefficients achieve near oracle rate, that is, with high probability, the L₂ error bound of the estimated L₁-norm SVM coefficients is of order O_p(√q log p/n), where q is the number of features with nonzero coefficients. Furthermore, we show that if the L₁-norm SVM is used as an initial value for a recently proposed algorithm for solving non-convex penalized SVM (Zhang et al., 2016b), then in two iterative steps it is guaranteed to produce an estimator that possesses the oracle property in ultra-high dimension, which in particular implies that with probability approaching one the zero coefficients are estimated as exactly zero. Simulation studies demonstrate the fine performance of L₁-norm SVM as a sparse classifier and its effectiveness to be utilized to solve non-convex penalized SVM problems in high dimension.