# Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theory

arXiv: https://arxiv.org/abs/2403.11968 · HTML: https://mlsanity.com/paper/2403.11968/

## Key facts

- Cited by 63 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 48 of its 99 references that are in the map.
- First posted on arXiv 18 Mar 2024.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)
- Deep reinforcement learning methods / [RL with diffusion and flow models](https://mlsanity.com/deeprl/rl-with-diffusion-and-flow-models/)

## Builds on (most cited first, 30 of 48)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Human Motion Diffusion Model](https://mlsanity.com/paper/2209.14916/) (2022)
- [Planning with Diffusion for Flexible Behavior Synthesis](https://mlsanity.com/paper/2205.09991/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Plug and Play Language Models: A Simple Approach to Controlled Text Generation](https://mlsanity.com/paper/1912.02164/) (2019)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Is Conditional Generative Modeling all you need for Decision-Making?](https://mlsanity.com/paper/2211.15657/) (2022)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [IDQL: Implicit Q-Learning as an Actor-Critic Method with Diffusion Policies](https://mlsanity.com/paper/2304.10573/) (2023)
- [Imitating Human Behaviour with Diffusion Models](https://mlsanity.com/paper/2301.10677/) (2023)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Goal-Conditioned Imitation Learning using Score-based Diffusion Policies](https://mlsanity.com/paper/2304.02532/) (2023)
- [Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization](https://mlsanity.com/paper/2308.03686/) (2023)
- [Structure-based Drug Design with Equivariant Diffusion Models](https://mlsanity.com/paper/2210.13695/) (2022)
- [Diffusion-based Time Series Imputation and Forecasting with Structured State Space Models](https://mlsanity.com/paper/2208.09399/) (2022)
- [Solving Inverse Problems with Latent Diffusion Models via Hard Data Consistency](https://mlsanity.com/paper/2307.08123/) (2023)
- [Convergence of score-based generative modeling for general data distributions](https://mlsanity.com/paper/2209.12381/) (2022)
- [Convergence for score-based generative modeling with polynomial complexity](https://mlsanity.com/paper/2206.06227/) (2022)
- [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023)
- [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023)
- [Diff-TTS: A Denoising Diffusion Model for Text-to-Speech](https://mlsanity.com/paper/2104.01409/) (2021)
- [Protein Design with Guided Discrete Diffusion](https://mlsanity.com/paper/2305.20009/) (2023)
- [Symbolic Music Generation with Diffusion Models](https://mlsanity.com/paper/2103.16091/) (2021)
- [The probability flow ODE is provably fast](https://mlsanity.com/paper/2305.11798/) (2023)

## Built on by (most cited first, 30 of 63)

- [An Overview of Diffusion Models: Applications, Guided Generation, Statistical Rates and Optimization](https://mlsanity.com/paper/2404.07771/) (2024)
- [A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models](https://mlsanity.com/paper/2408.02320/) (2024)
- [Gradient Guidance for Diffusion Models: An Optimization Perspective](https://mlsanity.com/paper/2404.14743/) (2024)
- [What does guidance do? A fine-grained analysis in a simple setting](https://mlsanity.com/paper/2409.13074/) (2024)
- [Low-dimensional adaptation of diffusion models: Convergence in total variation](https://mlsanity.com/paper/2501.12982/) (2025)
- [On Statistical Rates of Conditional Diffusion Transformers: Approximation, Estimation and Minimax Optimality](https://mlsanity.com/paper/2411.17522/) (2024)
- [U-Nets as Belief Propagation: Efficient Classification, Denoising, and Diffusion in Generative Hierarchical Models](https://mlsanity.com/paper/2404.18444/) (2024)
- [A Theoretical Analysis of Discrete Flow Matching Generative Models](https://mlsanity.com/paper/2509.22623/) (2025)
- [Learn to Guide Your Diffusion Model](https://mlsanity.com/paper/2510.00815/) (2025)
- [Provable Separations between Memorization and Generalization in Diffusion Models](https://mlsanity.com/paper/2511.03202/) (2025)
- [Conditional Diffusion Models Based Conditional Independence Testing](https://mlsanity.com/paper/2412.11744/) (2024)
- [Towards a unified framework for guided diffusion models](https://mlsanity.com/paper/2512.04985/) (2025)
- [Enhancing Accuracy in Generative Models via Knowledge Transfer](https://mlsanity.com/paper/2405.16837/) (2024)
- [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)
- [On Flow Matching KL Divergence](https://mlsanity.com/paper/2511.05480/) (2025)
- [Diffusion Transformer Captures Spatial-Temporal Dependencies: A Theory for Gaussian Process Data](https://mlsanity.com/paper/2407.16134/) (2024)
- [Provable Sample-Efficient Transfer Learning Conditional Diffusion Models via Representation Learning](https://mlsanity.com/paper/2502.04491/) (2025)
- [Dynamic data generation and dynamic portfolio selection: an application of a score-based diffusion model](https://mlsanity.com/paper/2507.09916/) (2025)
- [Inference-Time Alignment for Diffusion Models via Variationally Stable Doob's Matching](https://mlsanity.com/paper/2601.06514/) (2026)
- [Diffusion on Graph: Augmentation of Graph Structure for Node Classification](https://mlsanity.com/paper/2503.12563/) (2025)
- [Emergence of Distortions in High-Dimensional Guided Diffusion Models](https://mlsanity.com/paper/2602.00716/) (2026)
- [Model Free Prediction with Uncertainty Assessment](https://mlsanity.com/paper/2405.12684/) (2024)
- [Differentially Private Kernel Density Estimation](https://mlsanity.com/paper/2409.01688/) (2024)
- [Can Diffusion Models Disentangle? A Theoretical Perspective](https://mlsanity.com/paper/2504.00220/) (2025)
- [Schr"odinger bridge for generative AI: Soft-constrained formulation and convergence analysis](https://mlsanity.com/paper/2510.11829/) (2025)
- [Automatic Scene Generation: State-of-the-Art Techniques, Models, Datasets, Challenges, and Future Prospects](https://mlsanity.com/paper/2410.01816/) (2024)
- [Diffusion Models with Double Guidance: Generate with aggregated datasets](https://mlsanity.com/paper/2505.13213/) (2025)
- [Improved Sample Complexity For Diffusion Model Training Without Empirical Risk Minimizer Access](https://mlsanity.com/paper/2505.18344/) (2025)
- [Autoregressive regularized score-based diffusion models for multi-scenarios fluid flow prediction](https://mlsanity.com/paper/2505.24145/) (2025)

## Abstract (arXiv)

Conditional diffusion models serve as the foundation of modern image synthesis and find extensive application in fields like computational biology and reinforcement learning. In these applications, conditional diffusion models incorporate various conditional information, such as prompt input, to guide the sample generation towards desired properties. Despite the empirical success, theory of conditional diffusion models is largely missing. This paper bridges this gap by presenting a sharp statistical theory of distribution estimation using conditional diffusion models. Our analysis yields a sample complexity bound that adapts to the smoothness of the data distribution and matches the minimax lower bound. The key to our theoretical development lies in an approximation result for the conditional score function, which relies on a novel diffused Taylor approximation technique. Moreover, we demonstrate the utility of our statistical theory in elucidating the performance of conditional diffusion models across diverse applications, including model-based transition kernel estimation in reinforcement learning, solving inverse problems, and reward conditioned sample generation.