We provide the first convergence guarantees for the Consistency Models (CMs), a newly emerging type of one-step generative models that can generate comparable samples to those generated by Diffusion Models. Our main result is that, under the basic assumptions on score-matching errors, consistency errors and smoothness of the data distribution, CMs can efficiently sample from any realistic data distribution in one step with small $W_2$ error. Our results (1) hold for $L^2$-accurate score and consistency assumption (rather than $L^\infty$-accurate); (2) do note require strong assumptions on the data distribution such as log-Sobelev inequality; (3) scale polynomially in all parameters; and (4) match the state-of-the-art convergence guarantee for score-based generative models (SGMs). We also provide the result that the Multistep Consistency Sampling procedure can further reduce the error comparing to one step sampling, which support the original statement of "Consistency Models, Yang Song 2023". Our result further imply a TV error guarantee when take some Langevin-based modifications to the output distributions.

翻译：本文首次提供了一致性模型（Consistency Models，CMs）的收敛性保证。一致性模型是一种新兴的单步生成模型，能够生成与扩散模型样本质量相当的样本。我们的主要结论是：在评分匹配误差、一致性误差以及数据分布平滑性的基本假设下，一致性模型能够以较小的$W_2$误差，在单步内高效地从任意实际数据分布中采样。我们的结果：（1）适用于$L^2$精度评分与一致性假设（而非$L^\infty$精度）；（2）无需对数据分布施加对数Sobolev不等式等强假设；（3）所有参数均呈多项式缩放；（4）与基于得分的生成模型（SGMs）的最新收敛性保证相匹配。我们还证明，与单步采样相比，多步一致性采样流程可进一步降低误差，这支持了"一致性模型，Yang Song 2023"的原始论述。进一步地，当对输出分布采取基于Langevin的修正时，我们的结果蕴含了TV误差保证。