On the Ergodicity, Bias and Asymptotic Normality of Randomized Midpoint Sampling Method

TL;DR

This paper investigates the probabilistic properties of the randomized midpoint method for Langevin diffusions, revealing its bias, asymptotic normality, and implications for numerical integration accuracy.

Contribution

It provides the first detailed analysis of the stationary distribution bias and asymptotic normality of the randomized midpoint discretization for Langevin diffusions.

Findings

Stationary distribution is biased away from the target unless step-size tends to zero.

Asymptotic normality holds for numerical integration with the method.

The method offers specific advantages and disadvantages compared to other discretizations.

Abstract

The randomized midpoint method, proposed by [SL19], has emerged as an optimal discretization procedure for simulating the continuous time Langevin diffusions. Focusing on the case of strong-convex and smooth potentials, in this paper, we analyze several probabilistic properties of the randomized midpoint discretization method for both overdamped and underdamped Langevin diffusions. We first characterize the stationary distribution of the discrete chain obtained with constant step-size discretization and show that it is biased away from the target distribution. Notably, the step-size needs to go to zero to obtain asymptotic unbiasedness. Next, we establish the asymptotic normality for numerical integration using the randomized midpoint method and highlight the relative advantages and disadvantages over other discretizations. Our results collectively provide several insights into the behavior of the randomized midpoint discretization method, including obtaining confidence intervals for numerical integrations.