发表机构
College of Computing and Data Science Nanyang Technological University; Department of Statistics University of Warwick（南洋理工大学计算与数据科学学院; 华威大学统计学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明随机梯度Langevin动力学在Wasserstein距离下的渐近偏差界，强凸与Lipschitz条件下，四阶矩得$h$阶界，二阶矩仅得$h^{1/2}$阶，并指出二阶矩不足以保证均匀$h$阶界。
AI 中文摘要
我们证明了在二阶Wasserstein距离下随机梯度Langevin动力学的渐近偏差界。我们假设负对数密度是强凸的且具有Lipschitz梯度，并且随机梯度估计器是无偏的，其误差满足均方Lipschitz条件。在随机梯度误差的四阶矩假设下，界为$h$阶；在仅二阶矩假设下，界为$h^{1/2}$阶，其中$h$为步长。一个尖峰噪声示例表明，仅二阶矩假设不足以获得对固定方差噪声分布一致成立的$h$阶界。
英文摘要
We prove asymptotic bias bounds for stochastic gradient Langevin dynamics in Wasserstein distance of order two. We assume that the negative log-density is strongly convex with a Lipschitz gradient, and that the stochastic gradient estimator is unbiased with an error satisfying a mean-square Lipschitz condition. The bounds are of order $h$ under a fourth moment assumption on the stochastic gradient error and of order $h^{1/2}$ under only a second moment assumption, where $h$ is the stepsize. A spiked-noise example shows that a second moment assumption alone is insufficient for a bound of order $h$ that is uniform over noise distributions with a fixed variance.
Comments8 pages, 1 figure
