This paper introduces a form of gradient guidance for adapting or fine-tuning diffusion models to achieve user-specified optimization objectives. The authors study the theoretical aspects of a guided score-based sampling process, linking it to first-order optimization. They show that adding gradient guidance to the sampling process of a pre-trained diffusion model is equivalent to solving a regularized optimization problem, where the regularization term acts as a prior determined by the pre-training data. However, directly adding the gradient of an external objective function to the sample process can compromise the structure in generated samples. To address this, the authors propose 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. They also introduce an iteratively fine-tuned version of gradient-guided diffusion, where gradients at newly generated data points are used to update the score network, mimicking a first-order optimization iteration. The paper proves that this approach converges to the global optimum with a rate of $\mathcal{O}(1/K)$ when the objective function is concave. The findings suggest that this novel gradient guidance not only preserves the latent subspace structure of the data but also ensures fast convergence towards the optimal solution. Numerical experiments support these theoretical findings.This paper introduces a form of gradient guidance for adapting or fine-tuning diffusion models to achieve user-specified optimization objectives. The authors study the theoretical aspects of a guided score-based sampling process, linking it to first-order optimization. They show that adding gradient guidance to the sampling process of a pre-trained diffusion model is equivalent to solving a regularized optimization problem, where the regularization term acts as a prior determined by the pre-training data. However, directly adding the gradient of an external objective function to the sample process can compromise the structure in generated samples. To address this, the authors propose 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. They also introduce an iteratively fine-tuned version of gradient-guided diffusion, where gradients at newly generated data points are used to update the score network, mimicking a first-order optimization iteration. The paper proves that this approach converges to the global optimum with a rate of $\mathcal{O}(1/K)$ when the objective function is concave. The findings suggest that this novel gradient guidance not only preserves the latent subspace structure of the data but also ensures fast convergence towards the optimal solution. Numerical experiments support these theoretical findings.