Mean-Field Game Equilibria in Score-Based Diffusion Models: Convergence Rates, Nash Stability, and Adversarial Robustness

Keywords: Diffusion models, mean-field games, score-based generative models, Nash equilibrium, Hamilton-Jacobi-Bellman, adversarial robustness, Wasserstein distance, DDPM, DDIM

TL;DR: Training score-based diffusion models is formulated as a mean-field game; we prove equilibrium existence, $O(1/N)$ convergence, and adversarial robustness.

Abstract: We formulate the training of score-based diffusion models as a mean-field game (MFG) where an infinite population of denoising agents compete to minimize individual loss while collectively approximating the data distribution. This game-theoretic lens yields three main contributions. First, we prove existence and uniqueness of Nash equilibria for the MFG system under mild regularity conditions on the data distribution, establishing convergence at rate $O(1/N)$ in 2-Wasserstein distance as the number of discretization steps $N$ grows. Second, we derive a Hamilton-Jacobi-Bellman (HJB) equation governing the optimal score function and show that classical DDPM/DDIM training implicitly solves a forward-backward SDE system that characterizes the MFG equilibrium. Third, we prove that MFG equilibria are structurally stable against adversarial perturbations, providing the first game-theoretic robustness guarantee for diffusion models. We introduce MFG-Diffusion, a training algorithm that explicitly computes equilibrium strategies, demonstrating 8--12\% improvement in FID on CIFAR-10 and CelebA-HQ-256, with provable convergence guarantees. Our framework unifies score matching, denoising diffusion, and flow matching under a single game-theoretic umbrella and opens new avenues for defending against adversarial perturbations in diffusion-based systems.

Submission Number: 145