Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games I: The ergodic case

René Carmona, Mathieu Laurière

Research output: Contribution to journalArticlepeer-review

Abstract

We propose two algorithms for the solution of the optimal control of ergodic McKean-Vlasov dynamics. Both algorithms are based on approximations of the theoretical solutions by neural networks, the latter being characterized by their architecture and a set of parameters. This allows the use of modern machine learning tools, and efficient implementations of stochastic gradient descent. The first algorithm is based on the idiosyncrasies of the ergodic optimal control problem. We provide a mathematical proof of the convergence of the approximation scheme, and we analyze rigorously the approximation by controlling the different sources of error. The second method is an adaptation of the deep Galerkin method to the system of partial differential equations issued from the optimality condition. We demonstrate the efficiency of these algorithms on several numerical examples, some of them being chosen to show that our algorithms succeed where existing ones failed. We also argue that both methods can easily be applied to problems in dimensions larger than what can be found in the existing literature. Finally, we illustrate the fact that, although the first algorithm is specifically designed for mean field control problems, the second one is more general and can also be applied to the partial differential equation systems arising in the theory of mean field games.

Original languageEnglish (US)
Pages (from-to)1455-1485
Number of pages31
JournalSIAM Journal on Numerical Analysis
Volume59
Issue number3
DOIs
StatePublished - 2021

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Keywords

  • Ergodic mean field control
  • Ergodic mean field game
  • Machine learning
  • Numerical solution
  • Rate of convergence

Fingerprint

Dive into the research topics of 'Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games I: The ergodic case'. Together they form a unique fingerprint.

Cite this