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Radius Margin Bounds for Support Vector Machines with the RBF Kernel

b88061{at}csie.ntu.edu.tw, Department of Computer Science and Information Engineering, National Taiwan University, Taipei 106, Taiwan

Wei-Chun Kao

b89106{at}csie.ntu.edu.tw, Department of Computer Science and Information Engineering, National Taiwan University, Taipei 106, Taiwan

Chia-Liang Sun

b88047{at}csie.ntu.edu.tw, Department of Computer Science and Information Engineering, National Taiwan University, Taipei 106, Taiwan

Li-Lun Wang

b7506054{at}csie.ntu.edu.tw, Department of Computer Science and Information Engineering, National Taiwan University, Taipei 106, Taiwan

Chih-Jen Lin

lincj{at}ntu.edu.tw, Department of Computer Science and Information Engineering, National Taiwan University, Taipei 106, Taiwan

An important approach for efficient support vector machine (SVM) model selection is to use differentiable bounds of the leave-one-out (loo) error. Past efforts focused on finding tight bounds of loo (e.g., radius margin bounds, span bounds). However, their practical viability is still not very satisfactory. Duan, Keerthi, and Poo (2003) showed that radius margin bound gives good prediction for L2-SVM, one of the cases we look at. In this letter, through analyses about why this bound performs well for L2-SVM, we show that finding a bound whose minima are in a region with small loo values may be more important than its tightness. Based on this principle, we propose modified radius margin bounds for L1-SVM (the other case) where the original bound is applicable only to the hard-margin case. Our modification for L1-SVM achieves comparable performance to L2-SVM. To study whether L1- or L2-SVM should be used, we analyze other properties, such as their differentiability, number of support vectors, and number of free support vectors. In this aspect, L1-SVM possesses the advantage of having fewer support vectors. Their implementations are also different, so we discuss related issues in detail.

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