On the Ergodicity, Bias and Asymptotic Normality of Randomized Midpoint Sampling Method
This work provides theoretical insights into a discretization method for computational statistics, but it is incremental as it builds on prior analysis of the randomized midpoint method.
The paper analyzes the randomized midpoint method for simulating Langevin diffusions, showing that it produces a biased stationary distribution requiring step-size reduction for unbiasedness, and establishes asymptotic normality to enable confidence intervals for numerical integration.
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.