A stochastic mirror descent algorithm for mean-field control problems
Abstract
We develop a stochastic mirror descent method for continuous-time McKean-Vlasov control with measure-valued controls and general Bregman regularisation. The dependence on the state law leads to a coupled Wasserstein-Bregman stability problem and to a McKean-Vlasov adjoint equation, which have no counterpart in classical stochastic control. We establish a first-order representation of the cost and prove its relative smoothness in two regimes: nonlinear dynamics with bounded cost derivatives, and affine state-law dynamics with linearly growing cost derivatives, the latter covering linear-quadratic (LQ) and more general nonquadratic models. Our regularity conditions in the control measure are formulated through directional finite increments and do not require second-order flat derivatives. We further introduce relative semiconvexity along admissible forward-backward trajectories, allowing compensation between the dynamics, running cost, and terminal cost beyond pointwise Hamiltonian convexity. This yields energy dissipation, an O(1/n) convergence rate at the convexity threshold, and geometric convergence when regularisation dominates relative nonconvexity. Explicit LQ criteria and numerical experiments illustrate the scope of the results.
Related
- Damped stochastic mirror descent for non-smooth ill-posed problems in Banach space
- Stochastic Mirror Descent under Dependent Observations
- Adaptive Schauder Stochastic Mirror Descent in Banach Spaces
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BibTeX
@misc{frikha2026stochastic,
title = {{A stochastic mirror descent algorithm for mean-field control problems}},
author = {Noufel Frikha and Aymen Haouas},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32303},
eprint = {2610.32303},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05834896v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A stochastic mirror descent algorithm for mean-field control problems
AU - Noufel Frikha
AU - Aymen Haouas
PY - 2026
DA - 2026/10/22
PB - HAL
DO - 10.48550/arxiv.2610.32303
UR - https://hal.science/hal-05834896v1
AB - We develop a stochastic mirror descent method for continuous-time McKean-Vlasov control with measure-valued controls and general Bregman regularisation. The dependence on the state law leads to a coupled Wasserstein-Bregman stability problem and to a McKean-Vlasov adjoint equation, which have no counterpart in classical stochastic control. We establish a first-order representation of the cost and prove its relative smoothness in two regimes: nonlinear dynamics with bounded cost derivatives, and affine state-law dynamics with linearly growing cost derivatives, the latter covering linear-quadratic (LQ) and more general nonquadratic models. Our regularity conditions in the control measure are formulated through directional finite increments and do not require second-order flat derivatives. We further introduce relative semiconvexity along admissible forward-backward trajectories, allowing compensation between the dynamics, running cost, and terminal cost beyond pointwise Hamiltonian convexity. This yields energy dissipation, an O(1/n) convergence rate at the convexity threshold, and geometric convergence when regularisation dominates relative nonconvexity. Explicit LQ criteria and numerical experiments illustrate the scope of the results.
ER -