A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models

• URL: http://arxiv.org/abs/2408.02320v1
• Date: Mon, 5 Aug 2024 09:02:24 GMT
• Title: A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models
• Authors: Gen Li, Yuting Wei, Yuejie Chi, Yuxin Chen,
• Abstract summary: We develop non-asymptotic convergence theory for a diffusion-based sampler. We prove that $d/varepsilon$ are sufficient to approximate the target distribution to within $varepsilon$ total-variation distance. Our results also characterize how $ell$ score estimation errors affect the quality of the data generation processes.
• Score: 45.60426164657739
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Diffusion models, which convert noise into new data instances by learning to reverse a diffusion process, have become a cornerstone in contemporary generative modeling. In this work, we develop non-asymptotic convergence theory for a popular diffusion-based sampler (i.e., the probability flow ODE sampler) in discrete time, assuming access to $\ell_2$-accurate estimates of the (Stein) score functions. For distributions in $\mathbb{R}^d$, we prove that $d/\varepsilon$ iterations -- modulo some logarithmic and lower-order terms -- are sufficient to approximate the target distribution to within $\varepsilon$ total-variation distance. This is the first result establishing nearly linear dimension-dependency (in $d$) for the probability flow ODE sampler. Imposing only minimal assumptions on the target data distribution (e.g., no smoothness assumption is imposed), our results also characterize how $\ell_2$ score estimation errors affect the quality of the data generation processes. In contrast to prior works, our theory is developed based on an elementary yet versatile non-asymptotic approach without the need of resorting to SDE and ODE toolboxes.

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