Diffusion models have emerged as a powerful tool rivaling GANs in generating high-quality samples with improved fidelity, flexibility, and robustness. A key component of these models is to learn the score function through score matching. Despite empirical success on various tasks,  it remains unclear whether gradient-based algorithms can learn the score function with a provable accuracy. As a first step toward answering this question, this paper establishes a mathematical framework for analyzing score estimation using neural networks trained by gradient descent. Our analysis covers both the optimization and the generalization aspects of the learning procedure. In particular, we propose a parametric form to formulate the denoising score-matching problem as a regression with noisy labels. Compared to the standard supervised learning setup, the score-matching problem introduces distinct challenges, including unbounded inputs, vector-valued outputs, and an additional time variable, preventing existing techniques from being applied directly. In this paper, we show that with a properly designed neural network architecture, the score function can be accurately approximated by a reproducing kernel Hilbert space induced by neural tangent kernels. Furthermore, by applying an early-stopping rule for gradient descent and leveraging certain coupling arguments between neural network training and kernel regression, we establish  the first generalization error (sample complexity) bounds for learning the score function despite the presence of noise in the observations. Our analysis is grounded in a novel parametric form of the neural network and an innovative connection between score matching and regression analysis, facilitating the application of advanced statistical and optimization techniques.
Neural Network-Based Score Estimation in Diffusion Models: Optimization and Generalization
Yinbin Han Meisam Razaviyayn Renyuan Xu