### Abstract

We rigorously prove a central limit theorem for neural network models with a single hidden layer. The central limit theorem is proven in the asymptotic regime of simultaneously (A) large numbers of hidden units and (B) large numbers of stochastic gradient descent training iterations. Our result describes the neural network's fluctuations around its mean-field limit. The fluctuations have a Gaussian distribution and satisfy a stochastic partial differential equation. The proof relies upon weak convergence methods from stochastic analysis. In particular, we prove relative compactness for the sequence of processes and uniqueness of the limiting process in a suitable Sobolev space.

Original language | English (US) |
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Pages (from-to) | 1820-1852 |

Number of pages | 33 |

Journal | Stochastic Processes and their Applications |

Volume | 130 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2020 |

### ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics

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## Cite this

*Stochastic Processes and their Applications*,

*130*(3), 1820-1852. https://doi.org/10.1016/j.spa.2019.06.003