  The Support Vector Machine (SVM) has been used in a wide variety of
classification problems. The original SVM uses the hinge loss function, which
is non-differentiable and makes the problem difficult to solve in particular
for regularized SVMs, such as with $\ell_1$-regularization. This paper
considers the Huberized SVM (HSVM), which uses a differentiable approximation
of the hinge loss function. We first explore the use of the Proximal Gradient
(PG) method to solving binary-class HSVM (B-HSVM) and then generalize it to
multi-class HSVM (M-HSVM). Under strong convexity assumptions, we show that our
algorithm converges linearly. In addition, we give a finite convergence result
about the support of the solution, based on which we further accelerate the
algorithm by a two-stage method. We present extensive numerical experiments on
both synthetic and real datasets which demonstrate the superiority of our
methods over some state-of-the-art methods for both binary- and multi-class
SVMs.
