提出新方法分析扩散模型采样误差，给出更优的理论保证。
Wasserstein bounds for denoising diffusion probabilistic models via the Föllmer process
- 将扩散采样视为Föllmer过程的离散化，而非传统反向奥恩斯坦-乌伦贝克过程。
- 在多种方差调度下，得到维数与步数最优的Wasserstein误差上界。
- 适用于一般对数凹分布，即使缺乏运输不等式也能达到最优误差界。
本文研究了去噪扩散概率模型（DDPMs）在2-Wasserstein距离下的采样误差上界。贡献有三：(i) 在分数函数满足一般Lipschitz条件且方差调度覆盖余弦调度等广泛情形下，建立了在维度和步数上均最优的紧上界，复现并推广了文献中若干已知的紧界；(ii) 证明了此类Lipschitz条件蕴含目标分布的对数Sobolev不等式，从而导出二次运输成本不等式；在已有工作覆盖的设定下，结合近期关于几何型方差调度下KL散度的紧误差界，可推得近乎最优的Wasserstein界（仅差一个对数因子）；(iii) 对于一般的对数凹目标分布，即使目标分布不满足二次运输成本不等式，最优的Wasserstein误差界仍可实现。分析基于将DDPM采样器视为Föllmer过程的离散化，而非传统的反向奥恩斯坦-乌伦贝克过程。
原文摘要 · Abstract (English)
This paper studies sampling error bounds for denoising diffusion probabilistic models (DDPMs) in the 2-Wasserstein distance. Our contributions are threefold. (i) Under general Lipschitz-type conditions on the score function and for a broad class of variance schedules, including the cosine schedule, we establish sharp upper bounds that are optimal in both the dimension and the number of steps, and recover several sharp error bounds previously obtained in the literature. (ii) We prove that the same Lipschitz-type conditions, which encompass those commonly imposed on the (learned) score, imply a logarithmic Sobolev inequality and hence a quadratic transportation cost inequality for the DDPM. As a consequence, in settings covered by existing work, an optimal Wasserstein bound, up to a logarithmic factor, follows from the recently obtained sharp error bound in the Kullback-Leibler divergence under geometric-type variance schedules. (iii) We show that for general log-concave target distributions, the optimal Wasserstein error bound remains attainable even without a quadratic transportation cost inequality for the target. Our analysis is based on viewing the DDPM sampler as a discretization of the Föllmer process rather than the conventional reverse Ornstein-Uhlenbeck process.
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