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 of order Ω( √ d N -3/2 ), for solving a family of d-dimensional underdamped Langevin dynamics, by any randomized algorithm with only N queries to ∇U , the driving Brownian motion and its weighted integration, respectively.The lower bound we establish matches the upper bound for the randomized midpoint method recently proposed by Shen and Lee [NIPS 2019], in terms of both parameters N and d.