On the convergence rates of generalized conditional gradient method for fully discretized Mean Field Games

We study convergence rates of the generalized conditional gradient (GCG) method applied to fully discretized Mean Field Games (MFG) systems. While explicit convergence rates of the GCG method have been established at the continuous PDE level, a rigorous analysis that simultaneously accounts for time-space discretization and iteration errors has been missing. In this work, we discretize the MFG system using finite difference method and analyze the resulting fully discrete GCG scheme. Under suitable structural assumptions on the Hamiltonian and coupling terms, we establish discrete maximum principles and derive explicit error estimates that quantify both discretization errors and iteration errors within a unified framework. Our estimates show how the convergence rates depend on the mesh sizes and the iteration number, and they reveal a non-uniform behavior with respect to the iteration. Moreover, we prove that higher convergence rates can be achieved under additional regularity assumptions on the solution. Numerical experiments are presented to illustrate the theoretical results and to confirm the predicted convergence behavior.

💡 Research Summary

This paper investigates the convergence behavior of the Generalized Conditional Gradient (GCG) algorithm when applied to fully discretized Mean Field Games (MFG) systems. While previous works have established explicit convergence rates for GCG at the continuous PDE level, no rigorous analysis has simultaneously accounted for both the time‑space discretization error and the iteration error inherent to the algorithm. The authors fill this gap by discretizing the MFG system with a finite‑difference scheme and deriving unified error estimates that capture the interplay between mesh sizes (Δt, Δx) and the iteration count k.

The model considered is a standard forward‑backward MFG on the torus Q = (0,T)×𝕋^d with Hamiltonian H(t,x,p)=½|p|²−h(t,x)·p, where h∈C^{1+α} is a given advection field. The coupling term is taken to be local, of the form f(t,x,m)=φ(m), with φ satisfying boundedness, Lipschitz continuity, monotonicity, and potential structure assumptions (denoted (f‑B), (f‑L), (f‑M), (f‑P)). Initial and terminal data are smooth (C^{2+α}). Under these structural hypotheses, the continuous MFG admits a classical solution with sufficient regularity.

The GCG method proceeds as follows: starting from an initial density m₀, at each iteration k the current density m_k defines γ_k = f(·,·,m_k). Solving the Hamilton‑Jacobi‑Bellman (HJB) equation with source γ_k yields u_k, which is then transformed via the Cole‑Hopf change of variables ϕ_k = exp(−u_k/2ν). The density is expressed as m_k = ϕ_k ψ_k, and ψ_k satisfies a linear parabolic equation coupled to ϕ_k. The new density m̃_k = ϕ_k ψ_k is combined with the previous iterate through a convex combination  m_{k+1} = (1−δ_k) m_k + δ_k m̃_k, where δ_k∈(0,1) is a step‑size. The authors focus on a predefined step‑size rule δ_k = k₂/(k₁+k₂) (with constants k₁≥k₂>0), which is computationally cheap and amenable to analysis.

For discretization, a uniform grid with time step Δt and spatial step Δx is introduced. Central differences approximate diffusion, while upwind or centered schemes handle the advection term h·∇. The resulting fully discrete scheme (the discrete GCG) preserves a discrete maximum principle: the numerical ϕ_k and ψ_k remain non‑negative and bounded uniformly in k. This principle is crucial for stability and for establishing error bounds.

The main theoretical contributions are:

1. Theorem 3.5 (Discrete Maximum Principle) – Guarantees non‑negativity and L^∞‑stability of the discrete iterates.
2. Theorem 3.6 (Unified Error Estimate) – Provides a bound of the form \