We provide full theoretical guarantees for the convergence behaviour of diffusion-based generative models under the assumption of strongly logconcave 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：我们针对数据分布满足强对数凹假设下的扩散生成模型收敛行为提供了完整的理论保证，其中用于分数估计的逼近函数类由Lipschitz连续函数构成。通过一个从均值未知高斯分布采样的激励性示例，我们展示了方法的强大效能。在该案例中，我们为关联的优化问题（即分数逼近）提供了显式估计，并将其与相应的采样估计相结合。由此，我们获得了关于数据分布（均值未知的高斯分布）与采样算法之间Wasserstein-2距离的关键量化指标（如维度和收敛速度）的最佳已知上界估计。为超越此激励性示例并支持使用多种随机优化器，我们基于$L^2$精度分数估计假设呈现结果——该假设的关键在于利用随机优化器的期望信息以及我们仅依赖已知信息的新型辅助过程。该方法为采样算法带来了已知最优收敛速度。