The denoising diffusion model emerges recently as a powerful generative technique that converts noise into data. Theoretical convergence guarantee has been mainly studied for continuous-time diffusion models, and has been obtained for discrete-time diffusion models only for distributions with bounded support in the literature. In this paper, we establish the convergence guarantee for substantially larger classes of distributions under discrete-time diffusion models and further improve the convergence rate for distributions with bounded support. In particular, we first establish the convergence rates for both smooth and general (possibly non-smooth) distributions having finite second moment. We then specialize our results to a number of interesting classes of distributions with explicit parameter dependencies, including distributions with Lipschitz scores, Gaussian mixture distributions, and distributions with bounded support. We further propose a novel accelerated sampler and show that it improves the convergence rates of the corresponding regular sampler by orders of magnitude with respect to all system parameters. For distributions with bounded support, our result improves the dimensional dependence of the previous convergence rate by orders of magnitude. Our study features a novel analysis technique that constructs tilting factor representation of the convergence error and exploits Tweedie's formula for handling Taylor expansion power terms.

翻译：去噪扩散模型近年来作为一种将噪声转化为数据的强大生成技术而兴起。其理论收敛保证主要针对连续时间扩散模型进行了研究，而在文献中，仅针对具有有界支撑的分布获得了离散时间扩散模型的收敛保证。本文针对离散时间扩散模型下更大类别的分布建立了收敛保证，并进一步改进了有界支撑分布的收敛速率。具体而言，我们首先针对具有有限二阶矩的光滑分布和一般（可能非光滑）分布建立了收敛速率。随后，我们将结果推广到若干具有显式参数依赖性的有趣分布类别，包括具有Lipschitz分数的分布、高斯混合分布以及有界支撑分布。我们进一步提出了一种新型加速采样器，并证明其相对于所有系统参数而言，将相应常规采样器的收敛速率提升了多个数量级。对于有界支撑分布，我们的结果将先前收敛速率中的维度依赖性改进了多个数量级。我们的研究采用了一种新颖的分析技术，即构造收敛误差的倾斜因子表示，并利用Tweedie公式处理泰勒展开的高阶项。