Provably Reliable Classifier Guidance through Cross-entropy Error Control

TL;DR

The work addresses when training a classifier with cross-entropy yields reliable classifier-guided diffusion. It establishes a sharp positive result: under OU diffusion smoothing and mild $C^2$ regularity, a classifier with conditional KL error $\varepsilon^2$ induces a small guidance error, with a bound that scales like $\tilde{O}(\frac{d\varepsilon}{\sigma_t^2})$ (up to factors involving $P_{data}(y)$); this leads to sampling guarantees with complexity $\tilde{O}\left(\frac{d\log^2(1/\delta)}{\varepsilon_{score}^2 + \varepsilon_{guide}^2}\right)$, matching the unconditional case. A complementary negative result shows that, without smoothness, small KL divergence does not ensure correct guidance, via explicit constructions where $D_{KL}=O(1/n)$ yet the guidance error remains bounded away from zero or even diverges. The experiments verify both the necessity of smoothness and the predicted dependence of guidance on cross-entropy error and noise level, offering principled guidelines for classifier design in diffusion-based conditional generation. Overall, the paper provides a rigorous link between classifier training and diffusion-guided sampling, enabling principled classifier selection and reliable conditional generation with provable guarantees.

Abstract

Classifier-guided diffusion models generate conditional samples by augmenting the reverse-time score with the gradient of a learned classifier, yet it remains unclear whether standard classifier training procedures yield effective diffusion guidance. We address this gap by showing that, under mild smoothness assumptions on the classifiers, controlling the cross-entropy error at each diffusion step also controls the error of the resulting guidance vectors: classifiers achieving conditional KL divergence $\varepsilon^2$ from the ground-truth conditional label probabilities induce guidance vectors with mean squared error $\widetilde{O}(d \varepsilon )$. Our result yields an upper bound on the sampling error under classifier guidance and bears resemblance to a reverse log-Sobolev-type inequality. Moreover, we show that the classifier smoothness assumption is essential, by constructing simple counterexamples demonstrating that, without it, control of the guidance vector can fail for almost all distributions. To our knowledge, our work establishes the first quantitative link between classifier training and guidance alignment, yielding both a theoretical foundation for classifier guidance and principled guidelines for classifier selection.

Figures (6)

• Figure 1: Theorem \ref{['thm:kl-not-score']} Numerical Validation (Controlling classification error does not guarantee effective guidance.)Top row (Regime 1, $\delta_n = 1/n$): KL divergence converges to zero (left), but guidance error remains constant at $\approx 0.39$ (right), demonstrating $\Theta(1)$ behavior with no convergence. Bottom row (Regime 2, $\delta_n = 1/\sqrt{n}$): KL divergence converges to zero (left), but guidance error diverges linearly (right), demonstrating $\Theta(n)$ growth.
• Figure 2: Validation of Theorem \ref{['theorem: good-classifier-preserve-guidance']} at $v=0.1$ ($\sigma_v^2=0.181$).
• Figure 3: Validation of Theorem \ref{['theorem: good-classifier-preserve-guidance']} at $v=0.01$ ($\sigma_v^2=0.020$).
• Figure 4: Validation of Theorem \ref{['theorem: good-classifier-preserve-guidance']} at $v=0.05$ ($\sigma_v^2=0.095$).
• Figure 5: Validation of Theorem \ref{['theorem: good-classifier-preserve-guidance']} at $v=0.3$ ($\sigma_v^2=0.451$).

Theorems & Definitions (36)

• Theorem 3.1
• Remark 3.1
• Theorem 3.2
• Remark 3.2
• Lemma 3.1
• Theorem 3.3
• Theorem 3.4
• Corollary 3.1
• Theorem 2.1: Small classification error does not imply effective guidance
• proof : Proof of \ref{['prop-appendix:kl-not-score']}