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Provable Efficiency of Guidance in Diffusion Models for General Data Distribution

Gen Li, Yuchen Jiao

TL;DR

The paper tackles the theoretical foundations of diffusion-model guidance under general data distributions, showing that nonzero guidance decreases the average inverse classifier probability, i.e., improves overall sample quality, even when uniform improvements for all samples do not hold. Central to the analysis is a continuous-time SDE perspective and a martingale property: the inverse conditional probability $p_{c|X_t}(c|X_t)^{-1}$ forms a martingale, and classifier-guided terms push the expectation lower via Ito calculus. The authors provide a rigorous main theorem, accompany it with numerical validations on Gaussian Mixture Models and ImageNet, and discuss discretization and score-estimation robustness. The results justify diffusion guidance in a general setting and point to future work on adaptive guidance to balance per-sample effects and further connect to IS-based quality metrics.

Abstract

Diffusion models have emerged as a powerful framework for generative modeling, with guidance techniques playing a crucial role in enhancing sample quality. Despite their empirical success, a comprehensive theoretical understanding of the guidance effect remains limited. Existing studies only focus on case studies, where the distribution conditioned on each class is either isotropic Gaussian or supported on a one-dimensional interval with some extra conditions. How to analyze the guidance effect beyond these case studies remains an open question. Towards closing this gap, we make an attempt to analyze diffusion guidance under general data distributions. Rather than demonstrating uniform sample quality improvement, which does not hold in some distributions, we prove that guidance can improve the whole sample quality, in the sense that the average reciprocal of the classifier probability decreases with the existence of guidance. This aligns with the motivation of introducing guidance.

Provable Efficiency of Guidance in Diffusion Models for General Data Distribution

TL;DR

The paper tackles the theoretical foundations of diffusion-model guidance under general data distributions, showing that nonzero guidance decreases the average inverse classifier probability, i.e., improves overall sample quality, even when uniform improvements for all samples do not hold. Central to the analysis is a continuous-time SDE perspective and a martingale property: the inverse conditional probability forms a martingale, and classifier-guided terms push the expectation lower via Ito calculus. The authors provide a rigorous main theorem, accompany it with numerical validations on Gaussian Mixture Models and ImageNet, and discuss discretization and score-estimation robustness. The results justify diffusion guidance in a general setting and point to future work on adaptive guidance to balance per-sample effects and further connect to IS-based quality metrics.

Abstract

Diffusion models have emerged as a powerful framework for generative modeling, with guidance techniques playing a crucial role in enhancing sample quality. Despite their empirical success, a comprehensive theoretical understanding of the guidance effect remains limited. Existing studies only focus on case studies, where the distribution conditioned on each class is either isotropic Gaussian or supported on a one-dimensional interval with some extra conditions. How to analyze the guidance effect beyond these case studies remains an open question. Towards closing this gap, we make an attempt to analyze diffusion guidance under general data distributions. Rather than demonstrating uniform sample quality improvement, which does not hold in some distributions, we prove that guidance can improve the whole sample quality, in the sense that the average reciprocal of the classifier probability decreases with the existence of guidance. This aligns with the motivation of introducing guidance.
Paper Structure (25 sections, 4 theorems, 69 equations, 2 figures, 1 table)

This paper contains 25 sections, 4 theorems, 69 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Our Contributions
  4. Comparison with prior works when restricted to specific distributions:
  5. Organization.
  6. Background
  7. Diffusion models
  8. Guidance
  9. Continuous time limit
  10. Main results
  11. Numerical validation
  12. Gaussian Mixture Model:
  13. ImageNet dataset:
  14. Analysis
  15. Proof of Lemma \ref{['lem:invariance']}
  16. ...and 10 more sections

Key Result

Lemma 1

It can be shown that for $0 \le \tau \le t \le 1-\delta$, and if $Y_{\delta} \sim p_{X_{1-\delta}\,|\, c}$, then

Figures (2)

  • Figure 1: Experimental results on GMM. left: Ratio of samples with improved classifier probabilities for different guidance scales $w$; right: Expectation of $-p_{c\,|\, X_0}(1\,|\, Y_1^w)^{-1}$ for varying $w$.
  • Figure 2: Experimental results on ImageNet dataset. left: Ratio of samples with improved classifier probabilities for different guidance scales $w$; right: Expectation of $-p_{c\,|\, X_0}(1\,|\, Y_1^w)^{-1}$ for varying $w$.

Theorems & Definitions (5)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • proof : Proof of Theorem \ref{['thm:main']}
  • Theorem 2