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 L1norm 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 L1norm SVM coefficients is of order Op( √ 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.