Antithetic multilevel particle system sampling method for McKean-Vlasov SDEs

Let $\mu\in \mathcal{P}_2(\mathbb R^d)$, where $\mathcal{P}_2(\mathbb R^d)$ denotes the space of square integrable probability measures, and consider a Borel-measurable function $\Phi:\mathcal P_2(\mathbb R^d)\rightarrow \mathbb R$. IIn this paper we develop Antithetic Monte Carlo estimator (A-MLMC) for $\Phi(\mu)$, which achieves sharp error bound under mild regularity assumptions. The estimator takes as input the empirical laws $\mu^N = \frac1N \sum_{i=1}^{N}\delta_{X_i}$, where a) $(X_i)_{i=1}^N$ is a sequence of i.i.d samples from $\mu$ or b) $(X_i)_{i=1}^N$ is a system of interacting particles (diffusions) corresponding to a McKean-Vlasov stochastic differential equation (McKV-SDE). Each case requires a separate analysis. For a mean-field particle system, we also consider the empirical law induced by its Euler discretisation which gives a fully implementable algorithm. As by-products of our analysis, we establish a dimension-independent rate of uniform \textit{strong propagation of chaos}, as well as an $L^2$ estimate of the antithetic difference for i.i.d. random variables corresponding to general functionals defined on the space of probability measures.