### Abstract

Recent years have seen a flurry of activities in designing provably efficient nonconvex optimization procedures for solving statistical estimation problems. For various problems like phase retrieval or low-rank matrix completion, state-of-the-art non- convex procedures require proper regularization (e.g. trimming, regularized cost, projection) in order to guarantee fast convergence. When it comes to vanilla procedures such as gradient descent, however, prior theory either recommends highly conservative learning rates to avoid overshooting, or completely lacks performance guarantees. This paper uncovers a striking phenomenon in several nonconvex problems: even in the absence of explicit regularization, gradient descent follows a trajectory staying within a basin that enjoys nice geometry, consisting of points incoherent with the sampling mechanism. This "implicit regularization" feature allows gradient descent to proceed in a far more aggressive fashion without overshooting, which in turn results in substantial compu-tational savings. Focusing on two statistical estimation problems, i.e. solving random quadratic systems of equations and low-rank matrix completion, we establish that gradient descent achieves near-optimal statistical and computational guarantees without explicit regularization. As a byprod-uct, for noisy matrix completion, we demonstrate that gradient descent enables optimal control of both en try wise and spectral-norm errors.

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

Editors | Jennifer Dy, Andreas Krause |

Publisher | International Machine Learning Society (IMLS) |

Pages | 5264-5331 |

Number of pages | 68 |

ISBN (Electronic) | 9781510867963 |

State | Published - Jan 1 2018 |

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 | 8 |

### 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. 5264-5331). (35th International Conference on Machine Learning, ICML 2018; Vol. 8). International Machine Learning Society (IMLS).