Complexity of randomized algorithms for underdamped Langevin dynamics

We establish an information complexity lower bound of randomized algorithms for simulating underdamped Langevin dynamics. More specifically, we prove that the worst $L^2$ strong error is order $\Omega(\sqrt{d}\, N^{-3/2})$, solving a family $d$-dimensional dynamics, by any algorithm with only $N$ queries to $\nabla U$, driving Brownian motion and its weighted integration, respectively. The matches upper midpoint method recently proposed Shen Lee [NIPS 2019], in terms both parameters $d$.