### Abstract

We study the connection between the highly non-convex loss function of a simple model of the fully-connected feed-forward neural network and the Hamiltonian of the spherical spin-glass model under the assumptions of: i) variable independence, ii) redundancy in network parametrization, and iii) uniformity. These assumptions enable us to explain the complexity of the fully decoupled neural network through the prism of the results from random matrix theory. We show that for large-size decoupled networks the lowest critical values of the random loss function form a layered structure and they are located in a well-defined band lower-bounded by the global minimum. The number of local minima outside that band diminishes exponentially with the size of the network. We empirically verify that the mathematical model exhibits similar behavior as the computer simulations, despite the presence of high dependencies in real networks. We conjecture that both simulated annealing and SGD converge to the band of low critical points, and that all critical points found there are local minima of high quality measured by the test error. This emphasizes a major difference between large-and small-size networks where for the latter poor quality local minima have nonzero probability of being recovered. Finally, we prove that recovering the global minimum becomes harder as the network size increases and that it is in practice irrelevant as global minimum often leads to overfitting.

Original language | English (US) |
---|---|

Pages (from-to) | 192-204 |

Number of pages | 13 |

Journal | Journal of Machine Learning Research |

Volume | 38 |

State | Published - 2015 |

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### ASJC Scopus subject areas

- Artificial Intelligence
- Software
- Control and Systems Engineering
- Statistics and Probability

### Cite this

*Journal of Machine Learning Research*,

*38*, 192-204.

**The loss surfaces of multilayer networks.** / Choromanska, Anna; Henaff, Mikael; Mathieu, Michael; Ben Arous, Gerard; LeCun, Yann.

Research output: Contribution to journal › Article

*Journal of Machine Learning Research*, vol. 38, pp. 192-204.

}

TY - JOUR

T1 - The loss surfaces of multilayer networks

AU - Choromanska, Anna

AU - Henaff, Mikael

AU - Mathieu, Michael

AU - Ben Arous, Gerard

AU - LeCun, Yann

PY - 2015

Y1 - 2015

N2 - We study the connection between the highly non-convex loss function of a simple model of the fully-connected feed-forward neural network and the Hamiltonian of the spherical spin-glass model under the assumptions of: i) variable independence, ii) redundancy in network parametrization, and iii) uniformity. These assumptions enable us to explain the complexity of the fully decoupled neural network through the prism of the results from random matrix theory. We show that for large-size decoupled networks the lowest critical values of the random loss function form a layered structure and they are located in a well-defined band lower-bounded by the global minimum. The number of local minima outside that band diminishes exponentially with the size of the network. We empirically verify that the mathematical model exhibits similar behavior as the computer simulations, despite the presence of high dependencies in real networks. We conjecture that both simulated annealing and SGD converge to the band of low critical points, and that all critical points found there are local minima of high quality measured by the test error. This emphasizes a major difference between large-and small-size networks where for the latter poor quality local minima have nonzero probability of being recovered. Finally, we prove that recovering the global minimum becomes harder as the network size increases and that it is in practice irrelevant as global minimum often leads to overfitting.

AB - We study the connection between the highly non-convex loss function of a simple model of the fully-connected feed-forward neural network and the Hamiltonian of the spherical spin-glass model under the assumptions of: i) variable independence, ii) redundancy in network parametrization, and iii) uniformity. These assumptions enable us to explain the complexity of the fully decoupled neural network through the prism of the results from random matrix theory. We show that for large-size decoupled networks the lowest critical values of the random loss function form a layered structure and they are located in a well-defined band lower-bounded by the global minimum. The number of local minima outside that band diminishes exponentially with the size of the network. We empirically verify that the mathematical model exhibits similar behavior as the computer simulations, despite the presence of high dependencies in real networks. We conjecture that both simulated annealing and SGD converge to the band of low critical points, and that all critical points found there are local minima of high quality measured by the test error. This emphasizes a major difference between large-and small-size networks where for the latter poor quality local minima have nonzero probability of being recovered. Finally, we prove that recovering the global minimum becomes harder as the network size increases and that it is in practice irrelevant as global minimum often leads to overfitting.

UR - http://www.scopus.com/inward/record.url?scp=84954310140&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84954310140&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84954310140

VL - 38

SP - 192

EP - 204

JO - Journal of Machine Learning Research

JF - Journal of Machine Learning Research

SN - 1532-4435

ER -