# Gradient Guidance for Diffusion Models: An Optimization Perspective

arXiv: https://arxiv.org/abs/2404.14743 · HTML: https://mlsanity.com/paper/2404.14743/

## Key facts

- Cited by 61 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 59 of its 72 references that are in the map.
- First posted on arXiv 23 Apr 2024.

## Placed in

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

## Builds on (most cited first, 30 of 59)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [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)
- [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)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [RePaint: Inpainting using Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.09865/) (2022)
- [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)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [ImageReward: Learning and Evaluating Human Preferences for Text-to-Image Generation](https://mlsanity.com/paper/2304.05977/) (2023)
- [Training Diffusion Models with Reinforcement Learning](https://mlsanity.com/paper/2305.13301/) (2023)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Model](https://mlsanity.com/paper/2212.00490/) (2022)
- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Is Conditional Generative Modeling all you need for Decision-Making?](https://mlsanity.com/paper/2211.15657/) (2022)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [Sora: A Review on Background, Technology, Limitations, and Opportunities of Large Vision Models](https://mlsanity.com/paper/2402.17177/) (2024)
- [DPOK: Reinforcement Learning for Fine-tuning Text-to-Image Diffusion Models](https://mlsanity.com/paper/2305.16381/) (2023)
- [Directly Fine-Tuning Diffusion Models on Differentiable Rewards](https://mlsanity.com/paper/2309.17400/) (2023)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Universal Guidance for Diffusion Models](https://mlsanity.com/paper/2302.07121/) (2023)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)

## Built on by (most cited first, 30 of 61)

- [MMaDA: Multimodal Large Diffusion Language Models](https://mlsanity.com/paper/2505.15809/) (2025)
- [Derivative-Free Guidance in Continuous and Discrete Diffusion Models with Soft Value-Based Decoding](https://mlsanity.com/paper/2408.08252/) (2024)
- [Fine-Tuning Discrete Diffusion Models via Reward Optimization with Applications to DNA and Protein Design](https://mlsanity.com/paper/2410.13643/) (2024)
- [Rectified Diffusion: Straightness Is Not Your Need in Rectified Flow](https://mlsanity.com/paper/2410.07303/) (2024)
- [Scaling Image and Video Generation via Test-Time Evolutionary Search](https://mlsanity.com/paper/2505.17618/) (2025)
- [Dynamic Search for Inference-Time Alignment in Diffusion Models](https://mlsanity.com/paper/2503.02039/) (2025)
- [Efficient Diversity-Preserving Diffusion Alignment via Gradient-Informed GFlowNets](https://mlsanity.com/paper/2412.07775/) (2024)
- [Training-Free Guidance Beyond Differentiability: Scalable Path Steering with Tree Search in Diffusion and Flow Models](https://mlsanity.com/paper/2502.11420/) (2025)
- [DriftLite: Lightweight Drift Control for Inference-Time Scaling of Diffusion Models](https://mlsanity.com/paper/2509.21655/) (2025)
- [On Statistical Rates of Conditional Diffusion Transformers: Approximation, Estimation and Minimax Optimality](https://mlsanity.com/paper/2411.17522/) (2024)
- [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025)
- [Adding Conditional Control to Diffusion Models with Reinforcement Learning](https://mlsanity.com/paper/2406.12120/) (2024)
- [InstructG2I: Synthesizing Images from Multimodal Attributed Graphs](https://mlsanity.com/paper/2410.07157/) (2024)
- [CoDe: Blockwise Control for Denoising Diffusion Models](https://mlsanity.com/paper/2502.00968/) (2025)
- [LinEAS: End-to-end Learning of Activation Steering with a Distributional Loss](https://mlsanity.com/paper/2503.10679/) (2025)
- [Diffusion Models as Network Optimizers: Explorations and Analysis](https://mlsanity.com/paper/2411.00453/) (2024)
- [Source-Guided Flow Matching](https://mlsanity.com/paper/2508.14807/) (2025)
- [Physics-Informed Distillation of Diffusion Models for PDE-Constrained Generation](https://mlsanity.com/paper/2505.22391/) (2025)
- [HardFlow: Hard-Constrained Sampling for Flow-Matching Models via Trajectory Optimization](https://mlsanity.com/paper/2511.08425/) (2025)
- [LoRA-MGPO: Mitigating Double Descent in Low-Rank Adaptation via Momentum-Guided Perturbation Optimization](https://mlsanity.com/paper/2502.14538/) (2025)
- [Neuro-Symbolic Generative Diffusion Models for Physically Grounded, Robust, and Safe Generation](https://mlsanity.com/paper/2506.01121/) (2025)
- [Nonlinear Assimilation via Score-based Sequential Langevin Sampling](https://mlsanity.com/paper/2411.13443/) (2024)
- [Gradient-Free Classifier Guidance for Diffusion Model Sampling](https://mlsanity.com/paper/2411.15393/) (2024)
- [Projected Coupled Diffusion for Test-Time Constrained Joint Generation](https://mlsanity.com/paper/2508.10531/) (2025)
- [Refining Alignment Framework for Diffusion Models with Intermediate-Step Preference Ranking](https://mlsanity.com/paper/2502.01667/) (2025)
- [HOSIG: Full-Body Human-Object-Scene Interaction Generation with Hierarchical Scene Perception](https://mlsanity.com/paper/2506.01579/) (2025)
- [Finite Difference Flow Optimization for RL Post-Training of Text-to-Image Models](https://mlsanity.com/paper/2603.12893/) (2026)
- [Latent Diffusion Models for Controllable RNA Sequence Generation](https://mlsanity.com/paper/2409.09828/) (2024)
- [A First-order Generative Bilevel Optimization Framework for Diffusion Models](https://mlsanity.com/paper/2502.08808/) (2025)
- [Tree-Guided Diffusion Planner](https://mlsanity.com/paper/2508.21800/) (2025)

## Abstract (arXiv)

Diffusion models have demonstrated empirical successes in various applications and can be adapted to task-specific needs via guidance. This paper studies a form of gradient guidance for adapting a pre-trained diffusion model towards optimizing user-specified objectives. We establish a mathematical framework for guided diffusion to systematically study its optimization theory and algorithmic design. Our theoretical analysis spots a strong link between guided diffusion models and optimization: gradient-guided diffusion models are essentially sampling solutions to a regularized optimization problem, where the regularization is imposed by the pre-training data. As for guidance design, directly bringing in the gradient of an external objective function as guidance would jeopardize the structure in generated samples. We investigate a modified form of gradient guidance based on a forward prediction loss, which leverages the information in pre-trained score functions and provably preserves the latent structure. We further consider an iteratively fine-tuned version of gradient-guided diffusion where guidance and score network are both updated with newly generated samples. This process mimics a first-order optimization iteration in expectation, for which we proved O(1/K) convergence rate to the global optimum when the objective function is concave. Our code will be released at https://github.com/yukang123/GGDMOptim.git.