Convergence of the Inexact Langevin Algorithm and Score-based Generative Models in KL Divergence
This work addresses the theoretical stability of sampling methods in machine learning, providing incremental improvements in error assumptions for practitioners in generative modeling.
The paper tackles the problem of establishing stable convergence guarantees for inexact Langevin algorithms and score-based generative models when using estimated score functions, by proving biased convergence in KL divergence under a log-Sobolev inequality and a bounded moment generating function error assumption, which is shown to be achievable with a kernel density estimator for sub-Gaussian targets.
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.