### 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 |

State | 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 |

### Fingerprint

### 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)

**An accelerated gradient method for trace norm minimization.** / Ji, Shuiwang; Ye, Jieping.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 26th International Conference On Machine Learning, ICML 2009.*pp. 457-464, 26th International Conference On Machine Learning, ICML 2009, Montreal, QC, Canada, 6/14/09.

}

TY - GEN

T1 - An accelerated gradient method for trace norm minimization

AU - Ji, Shuiwang

AU - Ye, Jieping

PY - 2009

Y1 - 2009

N2 - 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/k2) for smooth problems. Experiments on multi-task learning problems demonstrate the efficiency of the proposed algorithms.

AB - 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/k2) for smooth problems. Experiments on multi-task learning problems demonstrate the efficiency of the proposed algorithms.

UR - http://www.scopus.com/inward/record.url?scp=71149103464&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=71149103464&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9781605585161

SP - 457

EP - 464

BT - Proceedings of the 26th International Conference On Machine Learning, ICML 2009

ER -