# Analytic Distribution of Classifier-Free Guidance for Schedule Design

arXiv: https://arxiv.org/abs/2607.19725 · HTML: https://mlsanity.com/paper/2607.19725/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 30 of its 36 references that are in the map.
- First posted on arXiv 22 Jul 2026.

## Placed in

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

## Builds on (most cited first, 30 of 30)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (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)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DreamFusion: Text-to-3D using 2D Diffusion](https://mlsanity.com/paper/2209.14988/) (2022)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [eDiff-I: Text-to-Image Diffusion Models with an Ensemble of Expert Denoisers](https://mlsanity.com/paper/2211.01324/) (2022)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [AudioLDM: Text-to-Audio Generation with Latent Diffusion Models](https://mlsanity.com/paper/2301.12503/) (2023)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Guiding a Diffusion Model with a Bad Version of Itself](https://mlsanity.com/paper/2406.02507/) (2024)
- [Applying Guidance in a Limited Interval Improves Sample and Distribution Quality in Diffusion Models](https://mlsanity.com/paper/2404.07724/) (2024)
- [CFG++: Manifold-constrained Classifier Free Guidance for Diffusion Models](https://mlsanity.com/paper/2406.08070/) (2024)
- [Eliminating Oversaturation and Artifacts of High Guidance Scales in Diffusion Models](https://mlsanity.com/paper/2410.02416/) (2024)
- [Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts](https://mlsanity.com/paper/2503.02819/) (2025)
- [Classifier-Free Guidance is a Predictor-Corrector](https://mlsanity.com/paper/2408.09000/) (2024)
- [Classifier-free Guidance with Adaptive Scaling](https://mlsanity.com/paper/2502.10574/) (2025)
- [Stage-wise Dynamics of Classifier-Free Guidance in Diffusion Models](https://mlsanity.com/paper/2509.22007/) (2025)
- [C2FG: Control Classifier-Free Guidance via Score Discrepancy Analysis](https://mlsanity.com/paper/2603.08155/) (2026)

## Built on by (most cited first, 2 of 2)

- [Classifier-Free Guidance in Flow Matching: Non-Autonomous Potentials, Overshoot, and Posterior-Mean Control](https://mlsanity.com/paper/2609.24287/) (2026)
- [GuidanceNFT: NFT-Based Reward Fine-Tuning of Guided Diffusion Models](https://mlsanity.com/paper/2610.15856/) (2026)

## Abstract (arXiv)

Classifier-free guidance (CFG) is the default mechanism for conditional generation in diffusion models, but the distribution sampled by its deterministic guided dynamics is not captured by the usual product-distribution heuristic p0ω q01-ω. We analyze CFG through the probability flow ODE and derive exact analytic path-integral representations of the induced distributions for both constant and time-dependent guidance. The resulting formulas show that CFG modifies pt0 by an exponential path-integral correction, and that a time-dependent schedule enters this correction through the weight ω(t)-1. This characterization explains how score discrepancies accumulate along sampling trajectories and motivates Distribution-Guided CFG (DG-CFG), a schedule that balances timestep contributions while accounting for signal strength and low-noise score-error amplification. A toy model with analytic scores closely verifies the predicted distributions. Across Stable Diffusion~1.5, Stable Diffusion~2.1, and Stable Diffusion~XL, DG-CFG yields a stronger diversity--fidelity trade-off and robustly mitigates the saturation and quality degradation caused by strong constant or heuristic guidance. Complete NFE experiments on Stable Diffusion~1.5 and Stable Diffusion~2.1 confirm that these gains persist across sampling budgets, while fixed-quality experiments on both backbones show that DG-CFG reaches target metrics with fewer sampling steps.