### Abstract

We consider the problem of learning a one-hidden- layer neural network with non-overlapping con- volutional layer and ReLU activation, i.e., /(Z, w, a) = aJa(wTZj), in which both the convolutional weights w and the output weights a are parameters to be learned. When the labels are the outputs from a teacher network of the same architecture with fixed weights (w∗, a∗), we prove that with Gaussian input Z, there is a spurious local minimizer. Surprisingly, in the presence of the spurious local minimizer, gradient descent with weight normalization from randomly initialized weights can still be proven to recover the true parameters with constant probability, which can be boosted to probability 1 with multiple restarts. Wc also show that with constant probability, the same procedure could also converge to the spurious local minimum, showing that the local minimum plays a non-trivial role in the dynamics of gradient descent. Furthermore, a quantitative analysis shows that the gradient descent dynamics has two phases: it starts off slow, but converges much faster after several iterations.

Original language | English (US) |
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Title of host publication | 35th International Conference on Machine Learning, ICML 2018 |

Editors | Andreas Krause, Jennifer Dy |

Publisher | International Machine Learning Society (IMLS) |

Pages | 2142-2159 |

Number of pages | 18 |

ISBN (Electronic) | 9781510867963 |

State | Published - Jan 1 2018 |

Externally published | Yes |

Event | 35th International Conference on Machine Learning, ICML 2018 - Stockholm, Sweden Duration: Jul 10 2018 → Jul 15 2018 |

### Publication series

Name | 35th International Conference on Machine Learning, ICML 2018 |
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Volume | 3 |

### Other

Other | 35th International Conference on Machine Learning, ICML 2018 |
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Country | Sweden |

City | Stockholm |

Period | 7/10/18 → 7/15/18 |

### All Science Journal Classification (ASJC) codes

- Computational Theory and Mathematics
- Human-Computer Interaction
- Software

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## Cite this

*35th International Conference on Machine Learning, ICML 2018*(pp. 2142-2159). (35th International Conference on Machine Learning, ICML 2018; Vol. 3). International Machine Learning Society (IMLS).