Enhanced convergence guarantees of score-based generative models in W₂-distance beyond log-concavity

Although score-based generative models (SGMs) have achieved remarkable success in real-world sample generation tasks, it is still far from sufficient to understand them from a mathematical perspective. In this paper, we aim to provide enhanced convergence guarantees for two existing SGMs in W₂-distance beyond log-concavity. More precisely, we develop a novel framework of {{black}discretization} error analysis for Euler-Maruyama (EM) and Poisson midpoint (PM) time discretization schemes for SGMs. Under a non-log-concavity condition, we show that Õ(√d/ε) iterations suffice to approximate the target distributions in $ε$-accuracy for the classical EM-based SGM, considerably improving upon the existing iteration complexity Õ(d/ε²). Notably, the novel framework of error analysis also enables us to establish an Õ(√d/ε2/3) iteration complexity for the PM-based SGM in $ε$-accuracy, significantly outperforming current state-of-the-art Wasserstein convergence guarantees for SDE-based diffusion samplers.