Enhanced convergence guarantees of score-based generative models in $\mathcal{W}_2$-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 $\mathcal{W}_2$-distance beyond log-concavity. More precisely, we develop a novel framework of error analysis for Euler-Maruyama (EM) and Poisson midpoint (PM) time discretization schemes for SGMs. Under a non-log-concavity condition, we show that $\tilde{\mathcal{O}}(\sqrt{d}/\epsilon)$ iterations suffice to approximate the target distributions in $\epsilon$-accuracy for the classical EM-based SGM, considerably improving upon the existing iteration complexity $\tilde{\mathcal{O}}(d/\epsilon^2)$. Notably, the novel framework of error analysis enables us to establish an $\tilde{\mathcal{O}}(\sqrt{d}/\epsilon^{2/3})$ iteration complexity for the PM-based SGM in $\epsilon$-accuracy, significantly outperforming current state-of-the-art Wasserstein convergence guarantees for SDE-based diffusion samplers.