A Sample-Wise Adjoint Regression Framework for Mean-Field Control with Connections to Adjoint Matching

Abstract

This work proposes a novel numerical approach for solving mean-field control (MFC) problems using an adjoint-based optimization framework motivated by the stochastic maximum principle (SMP). Rather than solving the adjoint processes $(Y_t,Z_t)$ explicitly, we construct sample-wise unbiased estimators of their discretized counterparts and use them to approximate the Hamiltonian control gradient through recursive regression. The control is then updated by a gradient-descent scheme. This approach differs from conventional deep-learning methods, which typically follow a ``discretize-then-optimize'' paradigm and directly optimize a globally parameterized control through the discretized objective. Numerical experiments demonstrate competitive, and in many cases improved performance compared with direct deep-learning approaches. The sample-wise and regression-based structure also supports scalable implementation in high-dimensional settings and makes the method attractive for generative modeling problems involving particle-based representations and distributional objectives. On the theoretical side, we establish a connection between a particular class of MFC problems and the \emph{adjoint matching} framework. Using the SMP under the mean field control setting, we further show that the self-consistent critical point of the resulting mean-field adjoint matching loss coincides with the optimal control.