### Abstract

We consider the minimization of a smooth loss function regularized by the trace norm of the matrix variable. Such formulation finds applications in many machine learning tasks including multi-task learning, matrix classification, and matrix completion. The standard semidefinite programming formulation for this problem is computationally expensive. In addition, due to the non-smooth nature of the trace norm, the optimal first-order black-box method for solving such class of problems converges as O(1/√k), where k is the iteration counter. In this paper, we exploit the special structure of the trace norm, based on which we propose an extended gradient algorithm that converges as O(1/k). We further propose an accelerated gradient algorithm, which achieves the optimal convergence rate of O(1/k^{2}) for smooth problems. Experiments on multi-task learning problems demonstrate the efficiency of the proposed algorithms.

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

Pages | 457-464 |

Number of pages | 8 |

Publication status | Published - 2009 |

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

### 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 |

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### 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. 457-464)