## Abstract

We consider the problem of learning a non-negative linear classifier with a ℓ_{1}-norm of at most k, and a fixed threshold, under the hinge-loss. This problem generalizes the problem of learning a k-monotone disjunction. We prove that we can learn efficiently in this setting, at a rate which is linear in both k and the size of the threshold, and that this is the best possible rate. We provide an efficient online learning algorithm that achieves the optimal rate, and show that in the batch case, empirical risk minimization achieves this rate as well. The rates we show are tighter than the uniform convergence rate, which grows with k^{2}.

Original language | American English |
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Pages (from-to) | 1275-1304 |

Number of pages | 30 |

Journal | Journal of Machine Learning Research |

Volume | 16 |

State | Published - Jul 2015 |

### Bibliographical note

Funding Information:Tong Zhang is supported by the following grants: NSF IIS1407939, NSF IIS1250985, and NIH R01AI116744.

Publisher Copyright:

© 2015 Sivan Sabato, Shai Shalev-Shwartz, Nathan Srebro, Daniel Hsu, and Tong Zhang.

## Keywords

- Empirical risk minimization
- Linear classifiers
- Monotone disjunctions
- Online learning
- Uniform convergence