## Abstract

We analyze multilayer neural networks in the asymptotic regime of simultaneously (a) large network sizes and (b) large numbers of stochastic gradient descent training iterations. We rigorously establish the limiting behavior of the multilayer neural network output. The limit procedure is valid for any number of hidden layers, and it naturally also describes the limiting behavior of the training loss. The ideas that we explore are to (a) take the limits of each hidden layer sequentially and (b) characterize the evolution of parameters in terms of their initialization. The limit satisfies a system of deterministic integro-differential equations. The proof uses methods from weak convergence and stochastic analysis. We show that, under suitable assumptions on the activation functions and the behavior for large times, the limit neural network recovers a global minimum (with zero loss for the objective function).

Original language | English (US) |
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Pages (from-to) | 120-152 |

Number of pages | 33 |

Journal | Mathematics of Operations Research |

Volume | 47 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2022 |

Externally published | Yes |

## Keywords

- Deep learning
- Deep neural network
- Mean field limit
- Weak convergence

## ASJC Scopus subject areas

- Mathematics(all)
- Computer Science Applications
- Management Science and Operations Research