Diffusion models, typically formulated as discretizations of stochastic differential equations (SDEs), have achieved state-of-the-art performance in generative tasks. However, their theoretical analysis often involves complex proofs. In this work, we present a simplified framework for analyzing the Euler--Maruyama discretization of variance-preserving SDEs (VP-SDEs). Using Grönwall's inequality, we derive a convergence rate of $O(T^{-1/2})$ under standard Lipschitz assumptions, streamlining prior analyses. We then demonstrate that the standard Gaussian noise can be replaced by computationally cheaper discrete random variables (e.g., Rademacher) without sacrificing this convergence guarantee, provided the mean and variance are matched. Our experiments validate this theory, showing that (i) discrete noise achieves sample quality comparable to Gaussian noise provided the variance is matched correctly, and (ii) performance degrades if the noise variance is scaled incorrectly.

翻译：扩散模型通常被构建为随机微分方程（SDE）的离散化形式，在生成任务中已取得最先进的性能。然而，其理论分析往往涉及复杂的证明。本文提出一个简化框架，用于分析方差保持SDE（VP-SDE）的Euler–Maruyama离散化。利用Grönwall不等式，我们在标准Lipschitz假设下推导出$O(T^{-1/2})$的收敛速率，从而简化了先前的分析。随后我们证明，只要均值和方差匹配，标准高斯噪声可以被计算成本更低的离散随机变量（例如Rademacher分布）替代，而不会牺牲该收敛保证。我们的实验验证了该理论，表明：（i）在方差正确匹配的情况下，离散噪声能达到与高斯噪声相当的样本质量；（ii）若噪声方差缩放不当，性能会下降。