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：我们为扩散生成模型在强对数凹数据分布下的收敛行为提供了完整的理论保证，其中用于分数估计的函数近似类由利普希茨连续函数构成。通过一个激励性示例——从均值未知的高斯分布中采样，我们展示了该方法的高效性。在该情形下，我们给出了相关优化问题（即分数近似）的显式估计，并将其与相应的采样估计相结合。由此，我们获得了数据分布（均值未知的高斯分布）与采样算法之间Wasserstein-2距离的已知最优上界估计，其中涉及关键感兴趣的量化指标，如维度和收敛速率。超越该激励示例，为允许使用多样化的随机优化器，我们基于$L^2$精确分数估计假设给出了结果，该假设关键地建立在关于随机优化器及仅利用已知信息的新型辅助过程的期望之上。这一方法为我们所提出的采样算法带来了已知最优收敛速率。