## Bernoulli

• Bernoulli
• Volume 17, Number 4 (2011), 1368-1385.

### Support vector machines with a reject option

#### Abstract

This paper studies $ℓ_1$ regularization with high-dimensional features for support vector machines with a built-in reject option (meaning that the decision of classifying an observation can be withheld at a cost lower than that of misclassification). The procedure can be conveniently implemented as a linear program and computed using standard software. We prove that the minimizer of the penalized population risk favors sparse solutions and show that the behavior of the empirical risk minimizer mimics that of the population risk minimizer. We also introduce a notion of classification complexity and prove that our minimizers adapt to the unknown complexity. Using a novel oracle inequality for the excess risk, we identify situations where fast rates of convergence occur.

#### Article information

Source
Bernoulli, Volume 17, Number 4 (2011), 1368-1385.

Dates
First available in Project Euclid: 4 November 2011

https://projecteuclid.org/euclid.bj/1320417508

Digital Object Identifier
doi:10.3150/10-BEJ320

Mathematical Reviews number (MathSciNet)
MR2854776

Zentralblatt MATH identifier
1243.68256

#### Citation

Wegkamp, Marten; Yuan, Ming. Support vector machines with a reject option. Bernoulli 17 (2011), no. 4, 1368--1385. doi:10.3150/10-BEJ320. https://projecteuclid.org/euclid.bj/1320417508

#### References

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