A stochastic mirror descent algorithm for mean-field control problems

The paper builds a stochastic mirror descent method for continuous-time McKean–Vlasov control, where the dynamics depend on the law of the state, with measure-valued controls and a general Bregman regulariser. The dependence on the state law creates a coupled Wasserstein–Bregman stability problem and a McKean–Vlasov adjoint equation with no analogue in classical stochastic control.

The authors prove a first-order representation of the cost and its relative smoothness in two regimes, one covering linear-quadratic and more general non-quadratic models. A notion of relative semiconvexity along forward-backward trajectories gives energy dissipation, an \(O(1/n)\) rate at the convexity threshold and geometric convergence beyond it, without second-order flat derivatives.