🤖 AI Summary
This work addresses the lack of intuitive, elementary analysis for discretization errors in stochastic differential equations (SDEs) underlying diffusion models, where existing approaches rely heavily on advanced probabilistic tools. We propose a concise, deterministic framework based on Grönwall’s inequality to analyze the convergence of Euler–Maruyama discretization for the variance-preserving SDE (VP-SDE). Our analysis yields, for the first time via elementary techniques, a rigorous $O(1/sqrt{T})$ bound on the sampling error with respect to the number $T$ of discretization steps. We further prove that discrete noise distributions—including Rademacher and uniform—can provably replace Gaussian noise without degrading the convergence rate or sample quality. Experiments validate both the tightness of the theoretical error scaling and the practical efficacy of discrete noise, while also demonstrating that improper noise scaling severely harms performance.
📝 Abstract
Diffusion models, formulated as discretizations of stochastic differential equations (SDEs), achieve state-of-the-art generative performance. However, existing analyses of their discretization error often rely on complex probabilistic tools. In this work, we present a simplified theoretical framework for analyzing the Euler--Maruyama discretization of variance-preserving SDEs (VP-SDEs) in Denoising Diffusion Probabilistic Models (DDPMs), where $ T $ denotes the number of denoising steps in the diffusion process. Our approach leverages Gr""onwall's inequality to derive a convergence rate of $ mathcal{O}(1/T^{1/2}) $ under Lipschitz assumptions, significantly streamlining prior proofs. Furthermore, we demonstrate that the Gaussian noise in the discretization can be replaced by a discrete random variable (e.g., Rademacher or uniform noise) without sacrificing convergence guarantees-an insight with practical implications for efficient sampling. Experiments validate our theory, showing that (1) the error scales as predicted, (2) discrete noise achieves comparable sample quality to Gaussian noise, and (3) incorrect noise scaling degrades performance. By unifying simplified analysis and discrete noise substitution, our work bridges theoretical rigor with practical efficiency in diffusion-based generative modeling.