Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions

• URL: http://arxiv.org/abs/2211.01916v1
• Date: Thu, 3 Nov 2022 15:51:00 GMT
• Title: Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions
• Authors: Hongrui Chen, Holden Lee, Jianfeng Lu
• Abstract summary: We provide convergence guarantees with complexity for any data distribution with second-order moment. Our result does not rely on any log-concavity or functional inequality assumption. Our theoretical analysis provides comparison between different discrete approximations and may guide the choice of discretization points in practice.
• Score: 9.953088581242845
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we focus on the theoretical analysis of diffusion-based generative modeling. Under an $L^2$-accurate score estimator, we provide convergence guarantees with polynomial complexity for any data distribution with second-order moment, by either employing an early stopping technique or assuming smoothness condition on the score function of the data distribution. Our result does not rely on any log-concavity or functional inequality assumption and has a logarithmic dependence on the smoothness. In particular, we show that under only a finite second moment condition, approximating the following in KL divergence in $\epsilon$-accuracy can be done in $\tilde O\left(\frac{d^2 \log^2 (1/\delta)}{\epsilon^2}\right)$ steps: 1) the variance-$\delta$ Gaussian perturbation of any data distribution; 2) data distributions with $1/\delta$-smooth score functions. Our theoretical analysis also provides quantitative comparison between different discrete approximations and may guide the choice of discretization points in practice.

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