Convergence rate of randomized midpoint Langevin Monte Carlo

TL;DR

The paper analyzes randomized midpoint Langevin Monte Carlo (RLMC) for sampling from $\pi(dx)=Z^{-1}e^{-U(x)}dx$. It proves exponential ergodicity of RLMC with a constant step size $\eta\in(0,m/L^2]$, establishing convergence to an invariant measure $\pi_\eta$ in the weighted metric $d_{TV,V}$ under $V(x)=1+|x|^2$, by verifying irreducibility, strong Feller, and a Lyapunov drift. It then introduces a decreasing-step RLMC with steps $\gamma_n$ and proves convergence rates in the function-distance $d_{\mathcal{G}}$ to both the coupled diffusion and the target $\pi$, yielding an overall rate of $O(\gamma_n^{1/2})$ for approximating $\pi$. The results hinge on a detailed comparison between the continuous Langevin diffusion and the RLMC discretization via a random midpoint Euler–Maruyama scheme, including precise one-step error bounds and moment estimates. The theoretical findings provide practical guarantees on the accuracy of RLMC-based sampling with both fixed and diminishing step sizes, with implications for high-dimensional Bayesian computation and Monte Carlo methods.

Abstract

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 $π$. 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.

Theorems & Definitions (24)

• Remark 2.2
• Theorem 2.3
• Remark 2.4
• Theorem 2.6
• Definition 3.1: Accessible set, small set, irreducibility
• Definition 3.2: Hairer ha
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• proof : Proof of Theorem \ref{['ms']}