  The linear Support Vector Machine (SVM) is a classic classification technique
in machine learning. Motivated by applications in modern high dimensional
statistics, we consider penalized SVM problems involving the minimization of a
hinge-loss function with a convex sparsity-inducing regularizer such as: the
L1-norm on the coefficients, its grouped generalization and the sorted
L1-penalty (aka Slope). Each problem can be expressed as a Linear Program (LP)
and is computationally challenging when the number of features and/or samples
is large -- the current state of algorithms for these problems is rather
nascent when compared to the usual L2-regularized linear SVM. To this end, we
propose new computational algorithms for these LPs by bringing together
techniques from (a) classical column (and constraint) generation methods and
(b) first order methods for non-smooth convex optimization -- techniques that
are rarely used together for solving large scale LPs. These components have
their respective strengths; and while they are found to be useful as separate
entities, they have not been used together in the context of solving large
scale LPs such as the ones studied herein. Our approach complements the
strengths of (a) and (b) -- leading to a scheme that seems to significantly
outperform commercial solvers as well as specialized implementations for these
problems. We present numerical results on a series of real and synthetic
datasets demonstrating the surprising effectiveness of classic
column/constraint generation methods in the context of challenging LP-based
machine learning tasks.
