Score-based diffusion models have achieved remarkable empirical performance in the field of machine learning and artificial intelligence for their ability to generate high-quality new data instances from complex distributions. Improving our understanding of diffusion models, including mainly convergence analysis for such models, has attracted a lot of interests. Despite a lot of theoretical attempts, there still exists significant gap between theory and practice. Towards to close this gap, we establish an iteration complexity at the order of $d^{1/3}\varepsilon^{-2/3}$, which is better than $d^{5/12}\varepsilon^{-1}$, the best known complexity achieved before our work. This convergence analysis is based on a randomized midpoint method, which is first proposed for log-concave sampling (Shen and Lee, 2019), and then extended to diffusion models by Gupta et al. (2024). Our theory accommodates $\varepsilon$-accurate score estimates, and does not require log-concavity on the target distribution. Moreover, the algorithm can also be parallelized to run in only $O(\log^2(d/\varepsilon))$ parallel rounds in a similar way to prior works.

本研究针对扩散模型在收敛性分析上的理论与实践之间的差距进行探讨，提出了一种新的迭代复杂度 $d^{1/3}\varepsilon^{-2/3}$，优于之前已知的最佳复杂度 $d^{5/12}\varepsilon^{-1}$。通过随机中点方法，这项研究为扩散模型提供了收敛性分析，并能在无对数凹性的限制下，实现高效的并行运算。

扩散概率模型的改进收敛速率