Abstract
Comparing with the standard L2-norm support vector machine (SVM), the L1-norm SVM enjoys the nice property of simultaneously preforming classification and feature selection. In this paper, we investigate the statistical performance of L1-norm SVM in ultra-high dimension, where the number of features p grows at an exponential rate of the sample size n. Different from existing theory for SVM which has been mainly focused on the generalization error rates and empirical risk, we study the asymptotic behavior of the coefficients of L1-norm SVM. Our analysis reveals that the estimated L1-norm SVM coefficients achieve near oracle rate, that is, with high probability, the L2 error bound of the estimated L1-norm SVM coefficients is of order Op(√q log p/n), where q is the number of features with nonzero coefficients. Furthermore, we show that if the L1-norm SVM is used as an initial value for a recently proposed algorithm for solving non-convex penalized SVM (Zhang et al., 2016b), then in two iterative steps it is guaranteed to produce an estimator that possesses the oracle property in ultra-high dimension, which in particular implies that with probability approaching one the zero coefficients are estimated as exactly zero. Simulation studies demonstrate the fine performance of L1-norm SVM as a sparse classifier and its effectiveness to be utilized to solve non-convex penalized SVM problems in high dimension.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1-26 |
| Number of pages | 26 |
| Journal | Journal of Machine Learning Research |
| Volume | 17 |
| State | Published - Dec 1 2016 |
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Keywords
- Error bound
- Feature selection
- L-norm SVM
- Non-convex penalty
- Oracle property
- Support vector machine
- Ulta-high dimension
Cite this
An error bound for L1-norm support vector machine coefficients in ultra-high dimension. / Peng, Bo; Wang, Lan; Wu, Yichao.
In: Journal of Machine Learning Research, Vol. 17, 01.12.2016, p. 1-26.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - An error bound for L1-norm support vector machine coefficients in ultra-high dimension
AU - Peng, Bo
AU - Wang, Lan
AU - Wu, Yichao
PY - 2016/12/1
Y1 - 2016/12/1
N2 - Comparing with the standard L2-norm support vector machine (SVM), the L1-norm SVM enjoys the nice property of simultaneously preforming classification and feature selection. In this paper, we investigate the statistical performance of L1-norm SVM in ultra-high dimension, where the number of features p grows at an exponential rate of the sample size n. Different from existing theory for SVM which has been mainly focused on the generalization error rates and empirical risk, we study the asymptotic behavior of the coefficients of L1-norm SVM. Our analysis reveals that the estimated L1-norm SVM coefficients achieve near oracle rate, that is, with high probability, the L2 error bound of the estimated L1-norm SVM coefficients is of order Op(√q log p/n), where q is the number of features with nonzero coefficients. Furthermore, we show that if the L1-norm SVM is used as an initial value for a recently proposed algorithm for solving non-convex penalized SVM (Zhang et al., 2016b), then in two iterative steps it is guaranteed to produce an estimator that possesses the oracle property in ultra-high dimension, which in particular implies that with probability approaching one the zero coefficients are estimated as exactly zero. Simulation studies demonstrate the fine performance of L1-norm SVM as a sparse classifier and its effectiveness to be utilized to solve non-convex penalized SVM problems in high dimension.
AB - Comparing with the standard L2-norm support vector machine (SVM), the L1-norm SVM enjoys the nice property of simultaneously preforming classification and feature selection. In this paper, we investigate the statistical performance of L1-norm SVM in ultra-high dimension, where the number of features p grows at an exponential rate of the sample size n. Different from existing theory for SVM which has been mainly focused on the generalization error rates and empirical risk, we study the asymptotic behavior of the coefficients of L1-norm SVM. Our analysis reveals that the estimated L1-norm SVM coefficients achieve near oracle rate, that is, with high probability, the L2 error bound of the estimated L1-norm SVM coefficients is of order Op(√q log p/n), where q is the number of features with nonzero coefficients. Furthermore, we show that if the L1-norm SVM is used as an initial value for a recently proposed algorithm for solving non-convex penalized SVM (Zhang et al., 2016b), then in two iterative steps it is guaranteed to produce an estimator that possesses the oracle property in ultra-high dimension, which in particular implies that with probability approaching one the zero coefficients are estimated as exactly zero. Simulation studies demonstrate the fine performance of L1-norm SVM as a sparse classifier and its effectiveness to be utilized to solve non-convex penalized SVM problems in high dimension.
KW - Error bound
KW - Feature selection
KW - L-norm SVM
KW - Non-convex penalty
KW - Oracle property
KW - Support vector machine
KW - Ulta-high dimension
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M3 - Article
AN - SCOPUS:85011665519
VL - 17
SP - 1
EP - 26
JO - Journal of Machine Learning Research
JF - Journal of Machine Learning Research
SN - 1532-4435
ER -