## Abstract

Recent research shows that when Gradient Descent (GD) is applied to neural networks, the loss almost never decreases monotonically. Instead, the loss oscillates as gradient descent converges to its “Edge of Stability” (EoS). Here, we find a quantity that does decrease monotonically throughout GD training: the sharpness attained by the gradient flow solution (GFS)-the solution that would be obtained if, from now until convergence, we train with an infinitesimal step size. Theoretically, we analyze scalar neural networks with the squared loss, perhaps the simplest setting where the EoS phenomena still occur. In this model, we prove that the GFS sharpness decreases monotonically. Using this result, we characterize settings where GD provably converges to the EoS in scalar networks. Empirically, we show that GD monotonically decreases the GFS sharpness in a squared regression model as well as practical neural network architectures.

Original language | English |
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Pages (from-to) | 17684-17744 |

Number of pages | 61 |

Journal | Proceedings of Machine Learning Research |

Volume | 202 |

State | Published - 2023 |

Event | 40th International Conference on Machine Learning, ICML 2023 - Honolulu, United States Duration: 23 Jul 2023 → 29 Jul 2023 |

### Funding

Funders | Funder number |
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Blavatnik Family Foundation | |

European Research Executive Agency | |

European Commission | |

European Research Council | 101039436 |

Israel Science Foundation | 2486/21 |