### Abstract

A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for these local methods to find the global minimum is the proliferation of local minima with much higher error than the global minimum. Here we argue, based on results from statistical physics, random matrix theory, neural network theory, and empirical evidence, that a deeper and more profound difficulty originates from the proliferation of saddle points, not local minima, especially in high dimensional problems of practical interest. Such saddle points are surrounded by high error plateaus that can dramatically slow down learning, and give the illusory impression of the existence of a local minimum. Motivated by these arguments, we propose a new approach to second-order optimization, the saddle-free Newton method, that can rapidly escape high dimensional saddle points, unlike gradient descent and quasi-Newton methods. Weapply this algorithm to deep or recurrent neural network training, and provide numerical evidence for its superior optimization performance.

Original language | English (US) |
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Title of host publication | Advances in Neural Information Processing Systems |

Publisher | Neural information processing systems foundation |

Pages | 2933-2941 |

Number of pages | 9 |

Volume | 4 |

Edition | January |

State | Published - 2014 |

Event | 28th Annual Conference on Neural Information Processing Systems 2014, NIPS 2014 - Montreal, Canada Duration: Dec 8 2014 → Dec 13 2014 |

### Other

Other | 28th Annual Conference on Neural Information Processing Systems 2014, NIPS 2014 |
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Country | Canada |

City | Montreal |

Period | 12/8/14 → 12/13/14 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Networks and Communications
- Information Systems
- Signal Processing

### Cite this

*Advances in Neural Information Processing Systems*(January ed., Vol. 4, pp. 2933-2941). Neural information processing systems foundation.

**Identifying and attacking the saddle point problem in high-dimensional non-convex optimization.** / Dauphin, Yann N.; Pascanu, Razvan; Gulcehre, Caglar; Cho, Kyunghyun; Ganguli, Surya; Bengio, Yoshua.

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

*Advances in Neural Information Processing Systems.*January edn, vol. 4, Neural information processing systems foundation, pp. 2933-2941, 28th Annual Conference on Neural Information Processing Systems 2014, NIPS 2014, Montreal, Canada, 12/8/14.

}

TY - GEN

T1 - Identifying and attacking the saddle point problem in high-dimensional non-convex optimization

AU - Dauphin, Yann N.

AU - Pascanu, Razvan

AU - Gulcehre, Caglar

AU - Cho, Kyunghyun

AU - Ganguli, Surya

AU - Bengio, Yoshua

PY - 2014

Y1 - 2014

N2 - A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for these local methods to find the global minimum is the proliferation of local minima with much higher error than the global minimum. Here we argue, based on results from statistical physics, random matrix theory, neural network theory, and empirical evidence, that a deeper and more profound difficulty originates from the proliferation of saddle points, not local minima, especially in high dimensional problems of practical interest. Such saddle points are surrounded by high error plateaus that can dramatically slow down learning, and give the illusory impression of the existence of a local minimum. Motivated by these arguments, we propose a new approach to second-order optimization, the saddle-free Newton method, that can rapidly escape high dimensional saddle points, unlike gradient descent and quasi-Newton methods. Weapply this algorithm to deep or recurrent neural network training, and provide numerical evidence for its superior optimization performance.

AB - A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for these local methods to find the global minimum is the proliferation of local minima with much higher error than the global minimum. Here we argue, based on results from statistical physics, random matrix theory, neural network theory, and empirical evidence, that a deeper and more profound difficulty originates from the proliferation of saddle points, not local minima, especially in high dimensional problems of practical interest. Such saddle points are surrounded by high error plateaus that can dramatically slow down learning, and give the illusory impression of the existence of a local minimum. Motivated by these arguments, we propose a new approach to second-order optimization, the saddle-free Newton method, that can rapidly escape high dimensional saddle points, unlike gradient descent and quasi-Newton methods. Weapply this algorithm to deep or recurrent neural network training, and provide numerical evidence for its superior optimization performance.

UR - http://www.scopus.com/inward/record.url?scp=84928534967&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84928534967&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84928534967

VL - 4

SP - 2933

EP - 2941

BT - Advances in Neural Information Processing Systems

PB - Neural information processing systems foundation

ER -