Semismooth Support Vector Machines

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Semismooth Support Vector Machines (2000)

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by Michael C. Ferris , Todd S. Munson

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@MISC{Ferris00semismoothsupport,
    author = {Michael C. Ferris and Todd S. Munson},
    title = {Semismooth Support Vector Machines},
    year = {2000}
}

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Abstract

The linear support vector machine can be posed as a quadratic program in a variety of ways. In this paper, we look at a formulation using the two-norm for the misclassification error that leads to a positive definite quadratic program with a single equality constraint when the Wolfe dual is taken. The quadratic term is a small rank update to a positive definite matrix. We reformulate the optimality conditions as a semismooth system of equations using the Fischer-Burmeister function and apply a damped Newton method to solve the resulting problem. The algorithm is shown to converge from any starting point with a Q-quadratic rate of convergence. At each iteration, we use the Sherman-Morrison-Woodbury update formula to solve a single linear system of equations. Significant computational savings are realized as the inactive variables are identified and exploited during the solution process. Results for a 60 million variable problem are presented, demonstrating the e#ectiveness of the propos...

Keyphrases

semismooth support vector machine linear support vector machine solution process significant computational saving misclassification error q-quadratic rate positive definite quadratic program single linear system fischer-burmeister function optimality condition quadratic term quadratic program damped newton method variable problem sherman-morrison-woodbury update formula wolfe dual inactive variable semismooth system small rank update positive definite matrix single equality constraint

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