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

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.