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.
翻译:由[ SL19] 提议的随机中点法已成为模拟Langevin连续扩散时间的最佳离散程序。我们在本文件中重点分析了随机中点分解法中多放和少放的Langevin扩散的几种概率性。我们首先将离散链的固定分布特征描述为不断的分解,并表明它与目标分布有偏差。值得注意的是,步数需要降到零,以获得无症状的不公正性。接下来,我们用随机中点法为数字集成确立了无症状的常态,并突出了相对于其他分解法的相对优缺点。我们的结果共同为随机中点分解法的行为提供了一些洞察力,包括为数字集成获得信任间隔。