# What does guidance do? A fine-grained analysis in a simple setting

arXiv: https://arxiv.org/abs/2409.13074 · HTML: https://mlsanity.com/paper/2409.13074/

## Key facts

- Cited by 48 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 29 of its 35 references that are in the map.
- First posted on arXiv 19 Sep 2024.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 29 of 29)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Guiding a Diffusion Model with a Bad Version of Itself](https://mlsanity.com/paper/2406.02507/) (2024)
- [Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization](https://mlsanity.com/paper/2308.03686/) (2023)
- [Reduce, Reuse, Recycle: Compositional Generation with Energy-Based Diffusion Models and MCMC](https://mlsanity.com/paper/2302.11552/) (2023)
- [Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions](https://mlsanity.com/paper/2211.01916/) (2022)
- [Convergence of score-based generative modeling for general data distributions](https://mlsanity.com/paper/2209.12381/) (2022)
- [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023)
- [The probability flow ODE is provably fast](https://mlsanity.com/paper/2305.11798/) (2023)
- [Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models](https://mlsanity.com/paper/2306.09251/) (2023)
- [Classifier-Free Guidance is a Predictor-Corrector](https://mlsanity.com/paper/2408.09000/) (2024)
- [Accelerating Convergence of Score-Based Diffusion Models, Provably](https://mlsanity.com/paper/2403.03852/) (2024)
- [A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models](https://mlsanity.com/paper/2408.02320/) (2024)
- [Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theory](https://mlsanity.com/paper/2403.11968/) (2024)
- [Learning Mixtures of Gaussians Using the DDPM Objective](https://mlsanity.com/paper/2307.01178/) (2023)
- [Critical windows: non-asymptotic theory for feature emergence in diffusion models](https://mlsanity.com/paper/2403.01633/) (2024)
- [Theoretical Insights for Diffusion Guidance: A Case Study for Gaussian Mixture Models](https://mlsanity.com/paper/2403.01639/) (2024)
- [Learning general Gaussian mixtures with efficient score matching](https://mlsanity.com/paper/2404.18893/) (2024)
- [Learning Mixtures of Gaussians Using Diffusion Models](https://mlsanity.com/paper/2404.18869/) (2024)
- [Analysis of learning a flow-based generative model from limited sample complexity](https://mlsanity.com/paper/2310.03575/) (2023)
- [Diffusion Posterior Sampling is Computationally Intractable](https://mlsanity.com/paper/2402.12727/) (2024)

## Built on by (most cited first, 30 of 48)

- [Low-dimensional adaptation of diffusion models: Convergence in total variation](https://mlsanity.com/paper/2501.12982/) (2025)
- [Stochastic Runge-Kutta Methods: Provable Acceleration of Diffusion Models](https://mlsanity.com/paper/2410.04760/) (2024)
- [Debiasing Guidance for Discrete Diffusion with Sequential Monte Carlo](https://mlsanity.com/paper/2502.06079/) (2025)
- [pi-Flow: Policy-Based Few-Step Generation via Imitation Distillation](https://mlsanity.com/paper/2510.14974/) (2025)
- [Dimension-Free Convergence of Diffusion Models for Approximate Gaussian Mixtures](https://mlsanity.com/paper/2504.05300/) (2025)
- [DriftLite: Lightweight Drift Control for Inference-Time Scaling of Diffusion Models](https://mlsanity.com/paper/2509.21655/) (2025)
- [Rectified Diffusion Guidance for Conditional Generation](https://mlsanity.com/paper/2410.18737/) (2024)
- [How much is a noisy image worth? Data Scaling Laws for Ambient Diffusion](https://mlsanity.com/paper/2411.02780/) (2024)
- [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025)
- [A Closer Look at Model Collapse: From a Generalization-to-Memorization Perspective](https://mlsanity.com/paper/2509.16499/) (2025)
- [Feedback Guidance of Diffusion Models](https://mlsanity.com/paper/2506.06085/) (2025)
- [Stage-wise Dynamics of Classifier-Free Guidance in Diffusion Models](https://mlsanity.com/paper/2509.22007/) (2025)
- [Learn to Guide Your Diffusion Model](https://mlsanity.com/paper/2510.00815/) (2025)
- [Improving Classifier-Free Guidance in Masked Diffusion: Low-Dim Theoretical Insights with High-Dim Impact](https://mlsanity.com/paper/2507.08965/) (2025)
- [Adaptive Diffusion Guidance via Stochastic Optimal Control](https://mlsanity.com/paper/2505.19367/) (2025)
- [Composition and Alignment of Diffusion Models using Constrained Learning](https://mlsanity.com/paper/2508.19104/) (2025)
- [What Exactly Does Guidance Do in Masked Discrete Diffusion Models](https://mlsanity.com/paper/2506.10971/) (2025)
- [Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation](https://mlsanity.com/paper/2506.11039/) (2025)
- [REG: Rectified Gradient Guidance for Conditional Diffusion Models](https://mlsanity.com/paper/2501.18865/) (2025)
- [Provable Efficiency of Guidance in Diffusion Models for General Data Distribution](https://mlsanity.com/paper/2505.01382/) (2025)
- [Theory on Score-Mismatched Diffusion Models and Zero-Shot Conditional Samplers](https://mlsanity.com/paper/2410.13746/) (2024)
- [Towards a Golden Classifier-Free Guidance Path via Foresight Fixed Point Iterations](https://mlsanity.com/paper/2510.21512/) (2025)
- [Conditional Diffusion Models with Classifier-Free Gibbs-like Guidance](https://mlsanity.com/paper/2505.21101/) (2025)
- [AudioMoG: Guiding Audio Generation with Mixture-of-Guidance](https://mlsanity.com/paper/2509.23727/) (2025)
- [Steering diffusion models with quadratic rewards: a fine-grained analysis](https://mlsanity.com/paper/2602.16570/) (2026)
- [Emergence of Distortions in High-Dimensional Guided Diffusion Models](https://mlsanity.com/paper/2602.00716/) (2026)
- [Feature-guided score diffusion for sampling conditional densities](https://mlsanity.com/paper/2410.11646/) (2024)
- [SURGE: Approximation and Training Free Particle Filter for Diffusion Surrogate](https://mlsanity.com/paper/2605.18745/) (2026)
- [Are we really tilting? The mechanics of reward guidance in flow and diffusion models](https://mlsanity.com/paper/2606.02884/) (2026)
- [A solvable generative model with a linear, one-step denoiser](https://mlsanity.com/paper/2411.17807/) (2024)

## Abstract (arXiv)

The use of guidance in diffusion models was originally motivated by the premise that the guidance-modified score is that of the data distribution tilted by a conditional likelihood raised to some power. In this work we clarify this misconception by rigorously proving that guidance fails to sample from the intended tilted distribution. Our main result is to give a fine-grained characterization of the dynamics of guidance in two cases, (1) mixtures of compactly supported distributions and (2) mixtures of Gaussians, which reflect salient properties of guidance that manifest on real-world data. In both cases, we prove that as the guidance parameter increases, the guided model samples more heavily from the boundary of the support of the conditional distribution. We also prove that for any nonzero level of score estimation error, sufficiently large guidance will result in sampling away from the support, theoretically justifying the empirical finding that large guidance results in distorted generations. In addition to verifying these results empirically in synthetic settings, we also show how our theoretical insights can offer useful prescriptions for practical deployment.