### Abstract

Several key problems in machine learning, such as feature selection and active learning, can be formulated as submodular set function maximization. We present herein a novel algorithm for maximizing a submodular set function under a cardinality constraint - the algorithm is based on a cutting-plane method and is implemented as an iterative small-scale binary-integer linear programming procedure. It is well known that this problem is NP-hard, and the approximation factor achieved by the greedy algorithm is the theoretical limit for polynomial time. As for (non-polynomial time) exact algorithms that perform reasonably in practice, there has been very little in the literature although the problem is quite important for many applications. Our algorithm is guaranteed to find the exact solution finitely many iterations, and it converges fast in practice due to the efficiency of the cutting-plane mechanism. Moreover, we also provide a method that produces successively decreasing upper-bounds of the optimal solution, while our algorithm provides successively increasing lower-bounds. Thus, the accuracy of the current solution can be estimated at any point, and the algorithm can be stopped early once a desired degree of tolerance is met. We evaluate our algorithm on sensor placement and feature selection applications showing good performance.

Original language | English |
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Title of host publication | Advances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference |

Pages | 916-924 |

Number of pages | 9 |

Publication status | Published - Dec 1 2009 |

Externally published | Yes |

Event | 23rd Annual Conference on Neural Information Processing Systems, NIPS 2009 - Vancouver, BC, Canada Duration: Dec 7 2009 → Dec 10 2009 |

### Publication series

Name | Advances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference |
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### Conference

Conference | 23rd Annual Conference on Neural Information Processing Systems, NIPS 2009 |
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Country | Canada |

City | Vancouver, BC |

Period | 12/7/09 → 12/10/09 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Information Systems

### Cite this

*Advances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference*(pp. 916-924). (Advances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference).

**Submodularity cuts and applications.** / Kawahara, Yoshinobu; Nagano, Kiyohito; Tsuda, Koji; Bilmes, Jeff A.

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

*Advances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference.*Advances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference, pp. 916-924, 23rd Annual Conference on Neural Information Processing Systems, NIPS 2009, Vancouver, BC, Canada, 12/7/09.

}

TY - GEN

T1 - Submodularity cuts and applications

AU - Kawahara, Yoshinobu

AU - Nagano, Kiyohito

AU - Tsuda, Koji

AU - Bilmes, Jeff A.

PY - 2009/12/1

Y1 - 2009/12/1

N2 - Several key problems in machine learning, such as feature selection and active learning, can be formulated as submodular set function maximization. We present herein a novel algorithm for maximizing a submodular set function under a cardinality constraint - the algorithm is based on a cutting-plane method and is implemented as an iterative small-scale binary-integer linear programming procedure. It is well known that this problem is NP-hard, and the approximation factor achieved by the greedy algorithm is the theoretical limit for polynomial time. As for (non-polynomial time) exact algorithms that perform reasonably in practice, there has been very little in the literature although the problem is quite important for many applications. Our algorithm is guaranteed to find the exact solution finitely many iterations, and it converges fast in practice due to the efficiency of the cutting-plane mechanism. Moreover, we also provide a method that produces successively decreasing upper-bounds of the optimal solution, while our algorithm provides successively increasing lower-bounds. Thus, the accuracy of the current solution can be estimated at any point, and the algorithm can be stopped early once a desired degree of tolerance is met. We evaluate our algorithm on sensor placement and feature selection applications showing good performance.

AB - Several key problems in machine learning, such as feature selection and active learning, can be formulated as submodular set function maximization. We present herein a novel algorithm for maximizing a submodular set function under a cardinality constraint - the algorithm is based on a cutting-plane method and is implemented as an iterative small-scale binary-integer linear programming procedure. It is well known that this problem is NP-hard, and the approximation factor achieved by the greedy algorithm is the theoretical limit for polynomial time. As for (non-polynomial time) exact algorithms that perform reasonably in practice, there has been very little in the literature although the problem is quite important for many applications. Our algorithm is guaranteed to find the exact solution finitely many iterations, and it converges fast in practice due to the efficiency of the cutting-plane mechanism. Moreover, we also provide a method that produces successively decreasing upper-bounds of the optimal solution, while our algorithm provides successively increasing lower-bounds. Thus, the accuracy of the current solution can be estimated at any point, and the algorithm can be stopped early once a desired degree of tolerance is met. We evaluate our algorithm on sensor placement and feature selection applications showing good performance.

UR - http://www.scopus.com/inward/record.url?scp=84858740625&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84858740625&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9781615679119

T3 - Advances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference

SP - 916

EP - 924

BT - Advances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference

ER -