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On the Generalized Conditional Gradient Method for Mean Field Games with Local Coupling Terms

Haruka Nakamura, Norikazu Saito

TL;DR

The paper addresses convergence of the generalized conditional gradient method for time-dependent second-order Mean Field Games with local coupling terms. It develops a refined analytical framework leveraging a Cole–Hopf transform to obtain uniform gradient bounds under a quadratic Hamiltonian, enabling explicit convergence rates in terms of the exploitability $\sigma_k$ and optimality gap $\varepsilon_k$ for both adaptive and predefined step-sizes. It proves existence and uniqueness of smooth MFG solutions in this local-coupling setting and provides a variational formulation when a potential structure is present, along with numerical experiments that validate the theoretical rates. The findings advance practical solvers for locally coupled MFGs and contribute rigorous regularity and convergence results that broaden the applicability of GCG to more realistic interaction models.

Abstract

We study the generalized conditional gradient (GCG) method for time-dependent second-order mean field games (MFG) with local coupling terms. While explicit convergence rates of the GCG method were previously established only for globally coupled interactions, the assumptions used there fail to cover typical local interactions such as congestion effects. To overcome this limitation, we introduce a refined analytical framework adapted to local couplings and derive explicit convergence estimates in terms of the exploitability and optimality gap. The key difficulty lies in establishing uniform bounds on the Hamilton--Jacobi--Bellman solutions; this is solved via the Cole--Hopf transformation under a standard quadratic Hamiltonian with a convection effect. We further provide numerical experiments demonstrating convergence behavior and confirming the theoretical rates. Additionally, the existence and uniqueness of smooth solutions to the MFG system with locally coupled interactions are established.

On the Generalized Conditional Gradient Method for Mean Field Games with Local Coupling Terms

TL;DR

The paper addresses convergence of the generalized conditional gradient method for time-dependent second-order Mean Field Games with local coupling terms. It develops a refined analytical framework leveraging a Cole–Hopf transform to obtain uniform gradient bounds under a quadratic Hamiltonian, enabling explicit convergence rates in terms of the exploitability and optimality gap for both adaptive and predefined step-sizes. It proves existence and uniqueness of smooth MFG solutions in this local-coupling setting and provides a variational formulation when a potential structure is present, along with numerical experiments that validate the theoretical rates. The findings advance practical solvers for locally coupled MFGs and contribute rigorous regularity and convergence results that broaden the applicability of GCG to more realistic interaction models.

Abstract

We study the generalized conditional gradient (GCG) method for time-dependent second-order mean field games (MFG) with local coupling terms. While explicit convergence rates of the GCG method were previously established only for globally coupled interactions, the assumptions used there fail to cover typical local interactions such as congestion effects. To overcome this limitation, we introduce a refined analytical framework adapted to local couplings and derive explicit convergence estimates in terms of the exploitability and optimality gap. The key difficulty lies in establishing uniform bounds on the Hamilton--Jacobi--Bellman solutions; this is solved via the Cole--Hopf transformation under a standard quadratic Hamiltonian with a convection effect. We further provide numerical experiments demonstrating convergence behavior and confirming the theoretical rates. Additionally, the existence and uniqueness of smooth solutions to the MFG system with locally coupled interactions are established.
Paper Structure (19 sections, 28 theorems, 139 equations, 11 figures, 1 table, 2 algorithms)

This paper contains 19 sections, 28 theorems, 139 equations, 11 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem settings
  3. Function Spaces
  4. Assumptions
  5. The well-posedness of MFG
  6. The well-posedness of the GCG method
  7. Main Results
  8. Variational formulations
  9. Step-size selection methods
  10. Explicit rates of convergence
  11. Proof of Theorem \ref{['LP23_prop23']}
  12. Proofs of Proposition \ref{['LP23_lem6']} and Theorem \ref{['LP23_thm7']}
  13. Proofs of Theorems \ref{['LP23_thm8-1']} and \ref{['LP23_thm8-2']}
  14. Proof of Theorem \ref{['thm_MFG_regularity']}
  15. Numerical experiments
  16. ...and 4 more sections

Key Result

Theorem 2.7

Suppose that (H), (f-B), (f-L), (f-M), and (TIV) are satisfied. Then, the MFG system MFG_eq admits a unique solution $(\overline{u},\overline{m})\in W^{1,2,q}(Q)\times W^{1,2,q}(Q)$. Moreover, $\overline{v}$ and $\overline{\gamma}$ defined as eq:vw are, respectively, functions of $\Theta$ and of $\G

Figures (11)

  • Figure 1: Density distribution $m$
  • Figure 2: Vector field of the control input at time $t = T$ (the scale represents the norm)
  • Figure 3: Adaptive stepsizes (left figure) and predefined stepsizes (right figure)
  • Figure 4: Step-sizes $\delta_k$
  • Figure 5: Density distribution (left figure) and control input (right figure)
  • ...and 6 more figures

Theorems & Definitions (65)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.7
  • Remark 2.8
  • Remark 2.9
  • Theorem 2.10
  • ...and 55 more