Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theory

TL;DR

This paper develops a rigorous statistical theory for conditional diffusion models, providing bounds on sample complexity and insights into their performance across various applications like image synthesis and reinforcement learning.

Contribution

It introduces a sharp theoretical framework for conditional diffusion models, including a novel diffused Taylor approximation for the conditional score function.

Findings

Sample complexity bound matches minimax lower bound

The theory explains performance in inverse problems and reinforcement learning

Approximation techniques improve understanding of conditional score functions

Abstract

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.