Convergence of the Inexact Langevin Algorithm in KL Divergence with Application to Score-based Generative Models
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866908917932490752 |
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| author | Yang, Kaylee Yingxi Wibisono, Andre |
| author_facet | Yang, Kaylee Yingxi Wibisono, Andre |
| contents | Motivated by the increasingly popular Score-based Generative Modeling (SGM), we study the Inexact Langevin Dynamics (ILD) and Inexact Langevin Algorithm (ILA) where a score function estimate is used in place of the exact score. We establish {\em 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, and 2) the error of score estimator exhibits a sub-Gaussian tail, referred to as Moment Generating Function (MGF) error assumption. Under the stronger $L^\infty$ score error assumption, we obtain a stable convergence bound in Rényi divergence. We also generalize the proof technique to SGM, and derive a stable convergence bound in KL divergence. In addition, we explore the question of how to obtain a provably accurate score estimator. We demonstrate that a simple estimator based on kernel density estimation fulfills the MGF error assumption for sub-Gaussian target distributions, at the population level. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_01512 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Convergence of the Inexact Langevin Algorithm in KL Divergence with Application to Score-based Generative Models Yang, Kaylee Yingxi Wibisono, Andre Machine Learning Statistics Theory Motivated by the increasingly popular Score-based Generative Modeling (SGM), we study the Inexact Langevin Dynamics (ILD) and Inexact Langevin Algorithm (ILA) where a score function estimate is used in place of the exact score. We establish {\em 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, and 2) the error of score estimator exhibits a sub-Gaussian tail, referred to as Moment Generating Function (MGF) error assumption. Under the stronger $L^\infty$ score error assumption, we obtain a stable convergence bound in Rényi divergence. We also generalize the proof technique to SGM, and derive a stable convergence bound in KL divergence. In addition, we explore the question of how to obtain a provably accurate score estimator. We demonstrate that a simple estimator based on kernel density estimation fulfills the MGF error assumption for sub-Gaussian target distributions, at the population level. |
| title | Convergence of the Inexact Langevin Algorithm in KL Divergence with Application to Score-based Generative Models |
| topic | Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2211.01512 |