We study the asymptotic error of score-based diffusion model sampling in large-sample scenarios from a non-parametric statistics perspective. We show that a kernel-based score estimator achieves an optimal mean square error of $\widetilde{O}\left(n^{-1} t^{-\frac{d+2}{2}}(t^{\frac{d}{2}} \vee 1)\right)$ for the score function of $p_0*\mathcal{N}(0,t\boldsymbol{I}_d)$, where $n$ and $d$ represent the sample size and the dimension, $t$ is bounded above and below by polynomials of $n$, and $p_0$ is an arbitrary sub-Gaussian distribution. As a consequence, this yields an $\widetilde{O}\left(n^{-1/2} t^{-\frac{d}{4}}\right)$ upper bound for the total variation error of the distribution of the sample generated by the diffusion model under a mere sub-Gaussian assumption. If in addition, $p_0$ belongs to the nonparametric family of the $\beta$-Sobolev space with $\beta\le 2$, by adopting an early stopping strategy, we obtain that the diffusion model is nearly (up to log factors) minimax optimal. This removes the crucial lower bound assumption on $p_0$ in previous proofs of the minimax optimality of the diffusion model for nonparametric families.

翻译：我们从非参数统计的角度研究了大规模场景下基于分数的扩散模型采样的渐近误差。我们证明，对于 $p_0*\mathcal{N}(0,t\boldsymbol{I}_d)$ 的分数函数，基于核的分数估计器能达到 $\widetilde{O}\left(n^{-1} t^{-\frac{d+2}{2}}(t^{\frac{d}{2}} \vee 1)\right)$ 的最优均方误差，其中 $n$ 和 $d$ 分别代表样本量和维度，$t$ 的上下界由 $n$ 的多项式给出，且 $p_0$ 为任意亚高斯分布。由此，在仅满足亚高斯假设的条件下，我们得到扩散模型生成样本分布的总变分误差的上界为 $\widetilde{O}\left(n^{-1/2} t^{-\frac{d}{4}}\right)$。此外，若 $p_0$ 属于 $\beta\le 2$ 的 $\beta$-Sobolev 空间非参数族，通过采用早停策略，我们得出扩散模型几乎（至多相差对数因子）达到最小最大最优。这去除了以往非参数族扩散模型最小最大最优性证明中关于 $p_0$ 的关键下界假设。