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

TL;DR

This paper introduces an efficient accelerated proximal gradient algorithm for dynamic mean-field planning, with proven convergence and extensions to mean-field games, demonstrating superior performance through numerical experiments.

Contribution

The paper presents a novel accelerated proximal gradient method with convergence guarantees for dynamic mean-field planning and its extension to mean-field games, enhancing computational efficiency.

Findings

Algorithm converges to continuous solution as grid size increases

Method outperforms state-of-the-art in efficiency and mass preservation

Flexible framework applicable to various mean-field variational problems

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.