  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.
