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A Two-Level Decomposition Framework Exploiting First and Second Order Information for SVM Training Problems

Giulio Galvan, Matteo Lapucci, Chih-Jen Lin, Marco Sciandrone; 22(23):1−38, 2021.


In this work we present a novel way to solve the sub-problems that originate when using decomposition algorithms to train Support Vector Machines (SVMs). State-of-the-art Sequential Minimization Optimization (SMO) solvers reduce the original problem to a sequence of sub-problems of two variables for which the solution is analytical. Although considering more than two variables at a time usually results in a lower number of iterations needed to train an SVM model, solving the sub-problem becomes much harder and the overall computational gains are limited, if any. We propose to apply the two-variables decomposition method to solve the sub-problems themselves and experimentally show that it is a viable and efficient way to deal with sub-problems of up to 50 variables. As a second contribution we explore different ways to select the working set and its size, combining first-order and second-order working set selection rules together with a strategy for exploiting cached elements of the Hessian matrix. An extensive numerical comparison shows that the method performs considerably better than state-of-the-art software.

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