Convergence rate of randomized midpoint Langevin Monte Carlo

The randomized midpoint Langevin Monte Carlo (RLMC), introduced by Shen and Lee (2019), is a variant of classical Unadjusted Langevin Algorithm. It was shown in the literature that the RLMC is an efficient algorithm for approximating high-dimensional probability distribution $\pi$. In this paper, we establish the exponential ergodicity of RLMC with constant step-size. Moreover, we design a dereasing-step size RLMC and provide its convergence rate in terms of a functional class distance.