Title: Gradient Guidance for Diffusion Models: An Optimization Perspective

Abstract: Diffusion models have demonstrated empirical successes in various applications and can be adapted to task-specific needs via guidance. This paper introduces a form of gradient guidance for adapting or fine-tuning diffusion models towards user-specified optimization objectives. We study the theoretic aspects of a guided score-based sampling process, linking the gradient-guided diffusion model to first-order optimization. We show that adding gradient guidance to the sampling process of a pre-trained diffusion model is essentially equivalent to solving a regularized optimization problem, where the regularization term acts as a prior determined by the pre-training data. Diffusion models are able to learn data's latent subspace, however, explicitly adding the gradient of an external objective function to the sample process would jeopardize the structure in generated samples. To remedy this issue, we consider a modified form of gradient guidance based on a forward prediction loss, which leverages the pre-trained score function to preserve the latent structure in generated samples. We further consider an iteratively fine-tuned version of gradient-guided diffusion where one can query gradients at newly generated data points and update the score network using new 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.