Numerical Approximation for Path-Dependent McKean-Vlasov Control with Non-Asymptotic Error Estimates

Path-dependent McKean--Vlasov (MKV) control models large interacting populations with history-dependent dynamics and costs. This paper develops a unified approximation-and-learning framework for continuous time path-dependent MKV problem under open-loop controls. First, an Euler discretization scheme with piecewise-constant controls is shown to achieve a non-asymptotic error of $O(h^{1/4})$. Second, we establish a discrete dynamic programming principle and prove value equivalence between open-loop and history-dependent feedback controls, enabling optimization on a reduced filtration. Third, an interacting particle system is introduced to approximate the continuous-time value, yielding an overall error bound of $O(h^{1/4}) + O(M^{-\gamma})$ for $M$ particles and an explicitly given $\gamma > 0$. Finally, we propose a fully implementable neural-network policy-gradient method using pathwise features. Numerical experiments, including a path-dependent linear-quadratic benchmark, demonstrate the effectiveness of the algorithm.