Accelerated diffusion models hold the potential to significantly enhance the efficiency of standard diffusion processes. Theoretically, these models have been shown to achieve faster convergence rates than the standard $\mathcal O(1/\epsilon^2)$ rate of vanilla diffusion models, where $\epsilon$ denotes the target accuracy. However, current theoretical studies have established the acceleration advantage only for restrictive target distribution classes, such as those with smoothness conditions imposed along the entire sampling path or with bounded support. In this work, we significantly broaden the target distribution classes with a new accelerated stochastic DDPM sampler. In particular, we show that it achieves accelerated performance for three broad distribution classes not considered before. Our first class relies on the smoothness condition posed only to the target density $q_0$, which is far more relaxed than the existing smoothness conditions posed to all $q_t$ along the entire sampling path. Our second class requires only a finite second moment condition, allowing for a much wider class of target distributions than the existing finite-support condition. Our third class is Gaussian mixture, for which our result establishes the first acceleration guarantee. Moreover, among accelerated DDPM type samplers, our results specialized for bounded-support distributions show an improved dependency on the data dimension $d$. Our analysis introduces a novel technique for establishing performance guarantees via constructing a tilting factor representation of the convergence error and utilizing Tweedie's formula to handle Taylor expansion terms. This new analytical framework may be of independent interest.

翻译：加速扩散模型具有显著提升标准扩散过程效率的潜力。理论上，这些模型已被证明能够实现比标准扩散模型$\mathcal O(1/\epsilon^2)$更快的收敛速率，其中$\epsilon$表示目标精度。然而，当前的理论研究仅在受限的目标分布类别中确立了加速优势，例如在整个采样路径上施加平滑性条件或具有有界支撑的分布类别。在本工作中，我们通过一种新的加速随机DDPM采样器显著扩展了目标分布类别。具体而言，我们证明了该采样器在三个先前未被考虑的广泛分布类别中实现了加速性能。我们的第一类分布仅要求目标密度$q_0$满足平滑性条件，这比现有要求整个采样路径上所有$q_t$均满足平滑性的条件更为宽松。第二类分布仅需有限二阶矩条件，相比现有的有限支撑条件，允许更广泛的目标分布类别。第三类分布为高斯混合分布，我们的结果为此建立了首个加速保证。此外，在针对有界支撑分布的特化结果中，我们的分析显示出对数据维度$d$依赖关系的改进。我们的分析引入了一种新技术，通过构建收敛误差的倾斜因子表示并利用Tweedie公式处理泰勒展开项来建立性能保证。这一新的分析框架可能具有独立的研究价值。