Convergence of single-timescale mean-field Langevin descent-ascent for two-player zero-sum games

Let $T^d=(\mathbb{R}/\mathbb{Z})^d$ be the flat $d$-torus with Lebesgue measure, and let $\mathcal{P}(T^d)$ be the set of Borel probability measures on $T^d$. Fix $\beta>0$ and a smooth payoff $f\in C^\infty(T^d\times T^d)$. Define, for $(\mu,\nu)\in\mathcal{P}(T^d)\times\mathcal{P}(T^d)$,

where $H(\rho)=\int_{T^d} r\log r$ if $\rho$ has density $r$ w.r.t. Lebesgue measure and $H(\rho)=+\infty$ otherwise. In this setting $F_\beta$ has a unique saddle point $(\mu^\star,\nu^\star)$ (the entropy-regularized mixed Nash equilibrium).

Consider the single-timescale Wasserstein gradient descent-ascent (GDA) flow associated with $F_\beta$: $\mu_t$ follows the Wasserstein gradient flow that decreases $\mu\mapsto F_\beta(\mu,\nu_t)$ while $\nu_t$ follows the Wasserstein gradient flow that increases $\nu\mapsto F_\beta(\mu_t,\nu)$, using the same time parameter $t$. For instance, when $\mu_t=m_t\,dx$ and $\nu_t=n_t\,dy$ have smooth positive densities, writing $\Phi_{\nu}(x)=\int f(x,y)\,d\nu(y)$ and $\Psi_{\mu}(y)=\int f(x,y)\,d\mu(x)$, the formal PDE system is

with gradients and Laplacians on $T^d$.