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.

翻译：条件扩散模型是现代图像合成的基础，并在计算生物学和强化学习等领域得到广泛应用。在这些应用中，条件扩散模型通过融入提示输入等各类条件信息，引导样本生成朝向期望的属性。尽管取得了经验上的成功，但条件扩散模型的理论基础仍大量缺失。本文通过提出一种基于条件扩散模型进行分布估计的精确统计理论来填补这一空白。我们的分析给出了一个样本复杂度界，该界自适应于数据分布的平滑度，并匹配极小化最优下界。理论发展的关键在于条件得分函数的逼近结果，该结果依赖于一种新颖的扩散泰勒逼近技术。此外，我们展示了该统计理论在阐明条件扩散模型于多种应用中的性能方面的实用性，这些应用包括强化学习中基于模型的转移核估计、求解逆问题以及奖励条件样本生成。