We provide full theoretical guarantees for the convergence behaviour of diffusion-based generative models under the assumption of strongly log-concave data distributions while our approximating class of functions used for score estimation is made of Lipschitz continuous functions. We demonstrate via a motivating example, sampling from a Gaussian distribution with unknown mean, the powerfulness of our approach. In this case, explicit estimates are provided for the associated optimization problem, i.e. score approximation, while these are combined with the corresponding sampling estimates. As a result, we obtain the best known upper bound estimates in terms of key quantities of interest, such as the dimension and rates of convergence, for the Wasserstein-2 distance between the data distribution (Gaussian with unknown mean) and our sampling algorithm. Beyond the motivating example and in order to allow for the use of a diverse range of stochastic optimizers, we present our results using an $L^2$-accurate score estimation assumption, which crucially is formed under an expectation with respect to the stochastic optimizer and our novel auxiliary process that uses only known information. This approach yields the best known convergence rate for our sampling algorithm.

翻译：我们针对强对数凹数据分布下的扩散生成模型的收敛行为提供了完整的理论保证，同时用于分数估计的近似函数类别由利普希茨连续函数构成。通过一个从均值未知的高斯分布中取样的激励性示例，我们展示了方法的强大之处。在此情况下，我们为相关的优化问题（即分数近似）提供了明确估计，并将其与相应的取样估计结合。由此，我们获得了数据分布（均值未知的高斯分布）与取样算法之间的Wasserstein-2距离在关键感兴趣量（如维度和收敛速率）方面的已知最佳上界估计。在激励性示例之外，为允许使用多样化的随机优化器，我们基于$L^2$精确的分数估计假设呈现结果——该假设关键地是在随机优化器期望下形成的，并辅以仅使用已知信息的新型辅助过程。这一方法为我们的取样算法提供了已知的最佳收敛速率。