Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games: I -- The Ergodic Case

We propose two algorithms for the solution of the optimal control of ergodic McKean-Vlasov dynamics. Both algorithms are based on the approximation 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 algorithm, 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.