Mean Field Analysis of Deep Neural Networks

Justin Sirignano, Konstantinos Spiliopoulos

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)120-152
Number of pages33
JournalMathematics of Operations Research
Issue number1
StatePublished - Feb 2022


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

ASJC Scopus subject areas

  • General Mathematics
  • Computer Science Applications
  • Management Science and Operations Research


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