High-Dimensional Penalized Bernstein Support Vector Machines

Abstract: The support vector machines (SVM) is a powerful classifier used for binary classification to improve the prediction accuracy. However, the non-differentiability of the SVM hinge loss function can lead to computational difficulties in high dimensional settings. To overcome this problem, we rely on Bernstein polynomial and propose a new smoothed version of the SVM hinge loss called the Bernstein support vector machine (BernSVM), which is suitable for the high dimension $p >> n$ regime. As the BernSVM objective loss function is of the class $C^2$, we propose two efficient algorithms for computing the solution of the penalized BernSVM. The first algorithm is based on coordinate descent with maximization-majorization (MM) principle and the second one is IRLS-type algorithm (iterative re-weighted least squares). Under standard assumptions, we derive a cone condition and a restricted strong convexity to establish an upper bound for the weighted Lasso BernSVM estimator. Using a local linear approximation, we extend the latter result to penalized BernSVM with non convex penalties SCAD and MCP. Our bound holds with high probability and achieves a rate of order $\sqrt{s\log(p)/n}$, where $s$ is the number of active features. Simulation studies are considered to illustrate the prediction accuracy of BernSVM to its competitors and also to compare the performance of the two algorithms in terms of computational timing and error estimation. The use of the proposed method is illustrated through analysis of three large-scale real data examples.