A Linearly Convergent Linear-Time First-Order Algorithm for Support Vector Classification with a Core Set Result

INFORMS Journal on Computing · 2011 · Volume 23 · Issue 3

Abstract

We present a simple first-order approximation algorithm for the support vector classification problem. Given a pair of linearly separable data sets and ϵ ∈ (0,1), the proposed algorithm computes a separating hyperplane whose margin is within a factor of (1−ϵ) of that of the maximum-margin separating hyperplane. We discuss how our algorithm can be extended to nonlinearly separable and inseparable data sets. The running time of our algorithm is linear in the number of data points and in 1/ϵ. In particular, the number of support vectors computed by the algorithm is bounded above by O(ζ/ϵ) for all sufficiently small ϵ > 0, where ζ is the square of the ratio of the distances between the farthest and closest pairs of points in the two data sets. Furthermore, we establish that our algorithm exhibits linear convergence. Our computational experiments, presented in the online supplement, reveal that the proposed algorithm performs quite well on standard data sets in comparison with other first-order algorithms. We adopt the real number model of computation in our analysis.

Keywords

support vector machines, support vector classification, Frank–Wolfe algorithm, approximation algorithms, core sets, linear convergence

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