Convergence of Mean-Field Langevin Stochastic Descent-Ascent for Distributional Minimax Optimization

We study convergence properties of the discretetime Mean-Field Langevin Stochastic Descent-Ascent (MFL-SDA) algorithm for solving distributional minimax optimization. These problems arise in various applications, such as zero-sum games, generative adversarial networks and distributionally robust learning. Despite the significance of MFL-SDA in these contexts, the discretetime convergence rate remains underexplored. To address this gap, we establish a last-iterate convergence rate of O( 1 ϵ log 1 ϵ ) for MFL-SDA. This rate is nearly optimal when compared to the complexity lower bound of its Euclidean counterpart. This rate also matches the complexity of mean-field Langevin stochastic gradient descent for distributional minimization and the outer-loop iteration complexity of an existing double-loop algorithm for distributional minimax problems. By leveraging an elementary analysis framework that avoids PDE-based techniques, we overcome previous limitations and achieve a faster convergence rate.