We study the Inexact Langevin Dynamics (ILD), Inexact Langevin Algorithm (ILA), and Score-based Generative Modeling (SGM) when utilizing estimated score functions for sampling. Our focus lies in establishing stable biased convergence guarantees in terms of the Kullback-Leibler (KL) divergence. To achieve these guarantees, we impose two key assumptions: 1) the target distribution satisfies the log-Sobolev inequality (LSI), and 2) the score estimator exhibits a bounded Moment Generating Function (MGF) error. Notably, the MGF error assumption we adopt is more lenient compared to the $L^\infty$ error assumption used in existing literature. However, it is stronger than the $L^2$ error assumption utilized in recent works, which often leads to unstable bounds. We explore the question of how to obtain a provably accurate score estimator that satisfies the MGF error assumption. Specifically, we demonstrate that a simple estimator based on kernel density estimation fulfills the MGF error assumption for sub-Gaussian target distribution, at the population level.

翻译：我们研究在利用估计分数函数进行采样时，非精确朗之万动力学（ILD）、非精确朗之万算法（ILA）以及基于分数的生成模型（SGM）。我们的重点在于建立关于库尔贝克-莱布勒（KL）散度的稳定有偏收敛保证。为实现这些保证，我们施加两个关键假设：1）目标分布满足对数-索博列夫不等式（LSI），2）分数估计器具有有界矩生成函数（MGF）误差。值得注意的是，我们采用的MGF误差假设相较于现有文献中使用的$L^\infty$误差假设更为宽松，但比近期工作中使用的$L^2$误差假设更强，后者常导致不稳定的界限。我们探讨如何获得满足MGF误差假设的可证明准确的分数估计器这一问题。具体而言，我们证明：在总体水平上，基于核密度估计的简单估计器对于次高斯目标分布满足MGF误差假设。