  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.
