A Fast Proximal Gradient Method and Convergence Analysis for Dynamic Mean Field Planning

Abstract

In this paper, we propose an efficient and flexible algorithm to solve dynamic mean-field planning problems based on an accelerated proximal gradient method. Besides an easy-to-implement gradient descent step in this algorithm, a crucial projection step becomes solving an elliptic equation whose solution can be obtained by conventional methods efficiently. By induction on iterations used in the algorithm, we theoretically show that the proposed discrete solution converges to the underlying continuous solution as the grid size increases. Furthermore, we generalize our algorithm to mean-field game problems and accelerate it using multilevel and multigrid strategies. We conduct comprehensive numerical experiments to confirm the convergence analysis of the proposed algorithm, to show its efficiency and mass preservation property by comparing it with state-of-the-art methods, and to illustrates its flexibility for handling various mean-field variational problems.

arXiv comment38 pages, 9 figures

References in corpus (6)

• A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems
• Optimal Transport with Proximal Splitting
• Optimal-transport formulation of electronic density-functional theory
• Generalized Unnormalized Optimal Transport and its fast algorithms
• A mean-field game approach to price formation in electricity markets
• A Simple Multiscale Method for Mean Field Games