论文标题
关于随机中点采样方法的终身性，偏见和渐近正态性
On the Ergodicity, Bias and Asymptotic Normality of Randomized Midpoint Sampling Method
论文作者
论文摘要
[SL19]提出的随机中点方法已成为模拟连续时间langevin扩散的最佳离散程序。在本文中，我们重点介绍了强率和平滑电势的情况，我们分析了过度阻尼和引导不足的Langevin扩散的随机中值离散方法的几种概率特性。我们首先表征了以恒定的踏板尺寸离散化获得的离散链的固定分布，并表明它偏离了目标分布。值得注意的是，阶梯大小需要达到零才能获得渐近的无偏见。接下来，我们使用随机中点方法建立了用于数值集成的渐近正态性，并突出了与其他离散化相比的相对优势和缺点。我们的结果共同提供了对随机中点离散方法行为的几种见解，包括获得数值集成的置信区间。
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.
