Deep Learning for Population-Dependent Controls in Mean Field Control Problems

In this paper, we propose several approaches to learn optimal population-dependent controls, in order to solve mean field control problems (MFC). Such policies enable us to solve MFC problems with generic common noise. We analyze the convergence of the proposed approximation algorithms, particularly the N-particle approximation. The effectiveness of our algorithms is supported by three different experiments, including systemic risk, price impact and crowd motion. We first show that our algorithms converge to the correct solution in an explicitly solvable MFC problem. Then, we conclude by showing that population-dependent controls outperform state-dependent controls. Along the way, we show that specific neural network architectures can improve the learning further.