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The linear Support Vector Machine (SVM) is a classic classification techniquein machine learning. Motivated by applications in modern high dimensionalstatistics, we consider penalized SVM problems involving the minimization of ahinge-loss function with a convex sparsity-inducing regularizer such as: theL1-norm on the coefficients, its grouped generalization and the sortedL1-penalty (aka Slope). Each problem can be expressed as a Linear Program (LP)and is computationally challenging when the number of features and/or samplesis large -- the current state of algorithms for these problems is rathernascent when compared to the usual L2-regularized linear SVM. To this end, wepropose new computational algorithms for these LPs by bringing togethertechniques from (a) classical column (and constraint) generation methods and(b) first order methods for non-smooth convex optimization -- techniques thatare rarely used together for solving large scale LPs. These components havetheir respective strengths; and while they are found to be useful as separateentities, they have not been used together in the context of solving largescale LPs such as the ones studied herein. Our approach complements thestrengths of (a) and (b) -- leading to a scheme that seems to significantlyoutperform commercial solvers as well as specialized implementations for theseproblems. We present numerical results on a series of real and syntheticdatasets demonstrating the surprising effectiveness of classiccolumn/constraint generation methods in the context of challenging LP-basedmachine learning tasks.

Solving L1-regularized SVMs and related linear programs: Revisiting the effectiveness of Column and Constraint Generation