### Abstract

Comparing with the standard L_{2}-norm support vector machine (SVM), the L_{1}-norm SVM enjoys the nice property of simultaneously preforming classification and feature selection. In this paper, we investigate the statistical performance of L_{1}-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 L_{1}-norm SVM. Our analysis reveals that the estimated L_{1}-norm SVM coefficients achieve near oracle rate, that is, with high probability, the L_{2} error bound of the estimated L_{1}-norm SVM coefficients is of order O_{p}(√q log p/n), where q is the number of features with nonzero coefficients. Furthermore, we show that if the L_{1}-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 L_{1}-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

_{1}-norm support vector machine coefficients in ultra-high dimension.

*Journal of Machine Learning Research*,

*17*, 1-26.

**An error bound for L _{1}-norm support vector machine coefficients in ultra-high dimension.** / Peng, Bo; Wang, Lan; Wu, Yichao.

Research output: Contribution to journal › Article

_{1}-norm support vector machine coefficients in ultra-high dimension',

*Journal of Machine Learning Research*, vol. 17, pp. 1-26.

_{1}-norm support vector machine coefficients in ultra-high dimension. Journal of Machine Learning Research. 2016 Dec 1;17:1-26.

}

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

UR - http://www.scopus.com/inward/record.url?scp=85011665519&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85011665519&partnerID=8YFLogxK

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 -