Convergence of the Inexact Langevin Algorithm and Score-based Generative Models in KL Divergence

• URL: http://arxiv.org/abs/2211.01512v2
• Date: Fri, 2 Jun 2023 14:57:31 GMT
• Title: Convergence of the Inexact Langevin Algorithm and Score-based Generative Models in KL Divergence
• Authors: Kaylee Yingxi Yang, Andre Wibisono
• Abstract summary: We study the Inexact Langevin Dynamics (ILD), Inexact Langevin Algorithm (ILA), and Score-based Generative Modeling (SGM) when utilizing estimated score functions for sampling.
• Score: 4.974890682815778
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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