Conditional diffusion models serve as the foundation of modern image synthesis and find extensive application in fields like computational biology and reinforcement learning. In these applications, conditional diffusion models incorporate various conditional information, such as prompt input, to guide the sample generation towards desired properties. Despite the empirical success, theory of conditional diffusion models is largely missing. This paper bridges this gap by presenting a sharp statistical theory of distribution estimation using conditional diffusion models. Our analysis yields a sample complexity bound that adapts to the smoothness of the data distribution and matches the minimax lower bound. The key to our theoretical development lies in an approximation result for the conditional score function, which relies on a novel diffused Taylor approximation technique. Moreover, we demonstrate the utility of our statistical theory in elucidating the performance of conditional diffusion models across diverse applications, including model-based transition kernel estimation in reinforcement learning, solving inverse problems, and reward conditioned sample generation.

本文通过提出一种锐利的统计理论，通过采用条件扩散模型进行分布估计，以实现对数据分布平滑性的自适应样本复杂度界限，并与极小max求解下界相匹配。通过对条件分数函数的近似结果揭示了理论的关键发展，该结果依赖于一种新型扩散泰勒逼近技术。此外，在求解逆问题和奖励条件样本生成等多种应用中，我们还展示了统计理论的实用性。

揭示具有无分类器指导的条件扩散模型：一个尖锐的统计理论