Diffusion models, which convert noise into new data instances by learning to reverse a Markov diffusion process, have become a cornerstone in contemporary generative modeling. While their practical power has now been widely recognized, the theoretical underpinnings remain far from mature. In this work, we develop a suite of non-asymptotic theory towards understanding the data generation process of diffusion models in discrete time, assuming access to $\ell_2$-accurate estimates of the (Stein) score functions. For a popular deterministic sampler (based on the probability flow ODE), we establish a convergence rate proportional to $1/T$ (with $T$ the total number of steps), improving upon past results; for another mainstream stochastic sampler (i.e., a type of the denoising diffusion probabilistic model), we derive a convergence rate proportional to $1/\sqrt{T}$, matching the state-of-the-art theory. Imposing only minimal assumptions on the target data distribution (e.g., no smoothness assumption is imposed), our results 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 resorting to toolboxes for SDEs and ODEs. Further, we design two accelerated variants, improving the convergence to $1/T^2$ for the ODE-based sampler and $1/T$ for the DDPM-type sampler, which might be of independent theoretical and empirical interest.

翻译：扩散模型通过学习逆转马尔可夫扩散过程将噪声转化为新数据实例，已成为当代生成建模的基石。尽管其实际能力已被广泛认可，但其理论基础仍远未成熟。本文在假设能获得(Stein)评分函数的ℓ₂精确估计的前提下，发展了关于离散时间扩散模型数据生成过程的非渐进理论。针对一种流行的确定性采样器(基于概率流ODE)，我们建立了与总步数T成反比(即1/T)的收敛速率，改进了以往的结果；针对另一种主流随机采样器(即去噪扩散概率模型的一种变体)，我们推导出与1/√T成正比的收敛速率，达到了当前最优理论水平。在仅对目标数据分布施加极简假设(例如不要求光滑性)的条件下，我们的研究刻画了ℓ₂评分估计误差如何影响数据生成质量。与先前工作不同，我们基于一种基础而通用的非渐进方法发展理论，无需借助SDE和ODE的数学工具。此外，我们设计了两种加速变体，将基于ODE的采样器收敛速率提升至1/T²，将DDPM型采样器提升至1/T，这可能在理论和实践上均具有独立价值。