Approximation and Generalization Abilities of Score-based Neural Network Generative Models for Sub-Gaussian Distributions

This paper studies the approximation and generalization abilities of score-based neural network generative models (SGMs) in estimating an unknown distribution $P_0$ from $n$ i.i.d. observations in $d$ dimensions. Assuming merely that $P_0$ is $\alpha$-sub-Gaussian, we prove that for any time step $t \in [t_0, n^{\mathcal{O}(1)}]$, where $t_0 > \mathcal{O}(\alpha^2n^{-2/d}\log n)$, there exists a deep ReLU neural network with width $\leq \mathcal{O}(n^{\frac{3}{d}}\log_2n)$ and depth $\leq \mathcal{O}(\log^2n)$ that can approximate the scores with $\tilde{\mathcal{O}}(n^{-1})$ mean square error and achieve a nearly optimal rate of $\tilde{\mathcal{O}}(n^{-1}t_0^{-d/2})$ for score estimation, as measured by the score matching loss. Our framework is universal and can be used to establish convergence rates for SGMs under milder assumptions than previous work. For example, assuming further that the target density function $p_0$ lies in Sobolev or Besov classes, with an appropriately early stopping strategy, we demonstrate that neural network-based SGMs can attain nearly minimax convergence rates up to logarithmic factors. Our analysis removes several crucial assumptions, such as Lipschitz continuity of the score function or a strictly positive lower bound on the target density.