Convergence of the proximal fixed-point scheme for general mean-field games

Establish convergence to a mean-field Nash equilibrium for general (non-potential) mean-field games of the particle-based proximal fixed-point scheme that alternates, at each iteration, resampling trajectories using the current velocity field, performing a proximal descent update of particle trajectories to decrease the individual cost J(X; rho) with respect to the current population flow rho, and updating the velocity field via flow matching to the updated trajectories; specifically, show that the observed descent in the objective across iterations implies convergence to a fixed point.

Background

The paper introduces a particle-based proximal fixed-point algorithm for high-dimensional mean-field games. Each iteration consists of (i) resampling trajectories from the current velocity field, (ii) a proximal particle update that decreases the individual cost J(X; rho) under the current population flow, and (iii) a flow-matching regression step that fits a velocity field to the updated trajectories.

The authors prove descent properties for both the particle update and the flow-matching step and derive sublinear and linear convergence rates in the special case of optimal control (where couplings are independent of the population) under smoothness and strong convexity assumptions. However, for general mean-field games with couplings depending on the population distribution (non-potential settings), they do not establish that this descent leads to convergence to a fixed point, leaving the overall convergence analysis open.