Tightening the Score Matching Gap for Diffusion Models

Diffusion models (DMs) are a state-of-the-art generative method to approximately sample from an unknown distribution. Their training and evaluation primarily rely on an Evidence Lower Bound (ELBO), which relates the Kullback-Leibler (KL) divergence of model samples to the score matching loss along the path, which serves as a tractable surrogate. The difference between sample quality and the score matching loss produced by this bound leads to the score matching gap, which is known to be tight in the worst-case but not descriptive of sample quality in general. In this work, we provide a theoretical analysis of this gap, developing tighter bounds for three metrics: KL divergence, reverse KL divergence, and Wasserstein distance, effectively exploiting the regularity of the class of score estimators. Our results suggest that the quality of the score approximation has more impact on closing the score matching gap for low noise scales. To obtain these bounds, our key technical insight is to exploit the contraction properties of the backward processes. In particular, we rely on entropy flows, logarithmic Sobolev inequalities and reflection couplings, rigorously linking the ergodicity of the Langevin diffusion to the score matching gap problem.

Diffusion models are a state-of-the-art and highly popular generative methodelling technique. They are trained using a technique called "score matching," but the link between training error and actual sample quality has been poorly understood and in the worst case, it can be very loose. We sharpened this link by exploiting smoothness in the score network showing that errors at low noise levels matter far more than at high ones, helping explain when diffusion models can be trusted and where training/validation effort should focus.