We give an improved theoretical analysis of score-based generative modeling. Under a score estimate with small $L^2$ error (averaged across timesteps), we provide efficient convergence guarantees for any data distribution with second-order moment, by either employing early stopping or assuming smoothness condition on the score function of the data distribution. Our result does not rely on any log-concavity or functional inequality assumption and has a logarithmic dependence on the smoothness. In particular, we show that under only a finite second moment condition, approximating the following in reverse KL divergence in $\epsilon$-accuracy can be done in $\tilde O\left(\frac{d \log (1/\delta)}{\epsilon}\right)$ steps: 1) the variance-$\delta$ Gaussian perturbation of any data distribution; 2) data distributions with $1/\delta$-smooth score functions. Our analysis also provides a quantitative comparison between different discrete approximations and may guide the choice of discretization points in practice.

翻译：本文对基于得分的生成建模进行了改进的理论分析。在得分估计具有较小$L^2$误差（跨时间步平均）的条件下，我们为任何具有二阶矩的数据分布提供了高效的收敛保证，可通过采用早期停止或假设数据分布得分函数的光滑性条件实现。我们的结果不依赖任何对数凹性或函数不等式假设，且对光滑性具有对数依赖性。特别地，我们证明在仅有限二阶矩条件下，以下目标在逆向KL散度下达到$\epsilon$精度所需的步数为$\tilde O\left(\frac{d \log (1/\delta)}{\epsilon}\right)$：1）任何数据分布经过方差为$\delta$的高斯扰动；2）具有$1/\delta$光滑得分函数的数据分布。我们的分析还提供了不同离散近似之间的定量比较，并可能指导实践中离散点的选择。