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arXiv · 2404.14743

Gradient Guidance for Diffusion Models: An Optimization Perspective

Yingqing Guo·Hui Yuan·Yukang Yang·Minshuo Chen·Mengdi Wang
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Abstract

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.

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Yingqing Guo, Hui Yuan, Yukang Yang, Minshuo Chen, Mengdi Wang. 2024-04-23. Gradient Guidance for Diffusion Models: An Optimization Perspective. https://arxiv.org/abs/2404.14743

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