Provably Reliable Classifier Guidance via Cross-Entropy Control

TL;DR

This paper provides a theoretical analysis linking classifier training via cross-entropy loss to the effectiveness of guidance in diffusion models, offering bounds on guidance error and insights into classifier selection.

Contribution

It establishes the first quantitative connection between classifier training quality and guidance accuracy in diffusion models using cross-entropy control.

Findings

Controlling cross-entropy at each diffusion step bounds guidance error.

Classifiers with low conditional KL divergence induce low guidance vector error.

Provides theoretical guidelines for selecting classifiers for effective diffusion guidance.

Abstract

Classifier-guided diffusion models generate conditional samples by augmenting the reverse-time score with the gradient of the log-probability predicted by a probabilistic classifier. In practice, this classifier is usually obtained by minimizing an empirical loss function. While existing statistical theory guarantees good generalization performance when the sample size is sufficiently large, it remains unclear whether such training yields an effective guidance mechanism. We study this question in the context of cross-entropy loss, which is widely used for classifier training. Under mild smoothness assumptions on the classifier, we show that controlling the cross-entropy at each diffusion model step is sufficient to control the corresponding guidance error. In particular, probabilistic classifiers achieving conditional KL divergence $\varepsilon^2$ induce guidance vectors with mean squared error $\widetilde O(d \varepsilon )$, up to constant and logarithmic factors. Our result yields an upper bound on the sampling error of classifier-guided diffusion models and bears resemblance to a reverse log-Sobolev--type inequality. To the best of our knowledge, this is the first result that quantitatively links classifier training to guidance alignment in diffusion models, providing both a theoretical explanation for the empirical success of classifier guidance, and principled guidelines for selecting classifiers that induce effective guidance.