### Abstract

We consider the problem of computing the Euclidean projection of a vector of length n onto a closed convex set including the ℓ_{1} and the specialized polyhedra employed in (Shalev-Shwartz & Singer, 2006). These problems have played building block roles in solving several ℓ_{1}-norm based sparse learning problems. Existing methods have a worst-case time complexity of O(n log n). In this paper, we propose to cast both Euclidean projections as root finding problems associated with specific auxiliary functions, which can be solved in linear time via bisection. We further make use of the special structure of the auxiliary functions, and propose an improved bisection algorithm. Empirical studies demonstrate that the proposed algorithms are much more efficient than the competing ones for computing the projections.

Original language | English (US) |
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Title of host publication | Proceedings of the 26th International Conference On Machine Learning, ICML 2009 |

Pages | 657-664 |

Number of pages | 8 |

State | Published - Dec 9 2009 |

Event | 26th International Conference On Machine Learning, ICML 2009 - Montreal, QC, Canada Duration: Jun 14 2009 → Jun 18 2009 |

### Publication series

Name | Proceedings of the 26th International Conference On Machine Learning, ICML 2009 |
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### Other

Other | 26th International Conference On Machine Learning, ICML 2009 |
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Country | Canada |

City | Montreal, QC |

Period | 6/14/09 → 6/18/09 |

### ASJC Scopus subject areas

- Artificial Intelligence
- Computer Networks and Communications
- Software

### Cite this

*Proceedings of the 26th International Conference On Machine Learning, ICML 2009*(pp. 657-664). (Proceedings of the 26th International Conference On Machine Learning, ICML 2009).