Theory on Score-Mismatched Diffusion Models and Zero-Shot Conditional Samplers
Yuchen Liang, Peizhong Ju, Yingbin Liang, Ness Shroff
TL;DR
This paper develops a first set of non-asymptotic, dimension-aware KL guarantees for score-mismatched diffusion models and zero-shot conditional samplers, proving that asymptotic bias scales with the accumulated score mismatch. By introducing a tilting-factor analysis, it establishes explicit dependence on data dimension for targets with finite second (and sixth) moments, and provides convergence guarantees for zero-shot samplers applied to linear conditional models. A key contribution is the Bias-Optimal DDNM (BO-DDNM) sampler, which minimizes the asymptotic bias and improves convergence in Gaussian and Gaussian-mixture target settings, especially under measurement noise. The results are complemented by numerical studies, and the framework encompasses popular zero-shot samplers such as CCDF and DDNM, offering principled guidance for designing efficient, training-free conditional diffusion methods with theoretical guarantees.
Abstract
The denoising diffusion model has recently emerged as a powerful generative technique, capable of transforming noise into meaningful data. While theoretical convergence guarantees for diffusion models are well established when the target distribution aligns with the training distribution, practical scenarios often present mismatches. One common case is in the zero-shot conditional diffusion sampling, where the target conditional distribution is different from the (unconditional) training distribution. These score-mismatched diffusion models remain largely unexplored from a theoretical perspective. In this paper, we present the first performance guarantee with explicit dimensional dependencies for general score-mismatched diffusion samplers, focusing on target distributions with finite second moments. We show that score mismatches result in an asymptotic distributional bias between the target and sampling distributions, proportional to the accumulated mismatch between the target and training distributions. This result can be directly applied to zero-shot conditional samplers for any conditional model, irrespective of measurement noise. Interestingly, the derived convergence upper bound offers useful guidance for designing a novel bias-optimal zero-shot sampler in linear conditional models that minimizes the asymptotic bias. For such bias-optimal samplers, we further establish convergence guarantees with explicit dependencies on dimension and conditioning, applied to several interesting target distributions, including those with bounded support and Gaussian mixtures. Our findings are supported by numerical studies.