We show that $L^2$-accurate score estimation, in the absence of strong assumptions on the data distribution, is computationally hard even when sample complexity is polynomial in the relevant problem parameters. Our reduction builds on the result of Chen et al. (ICLR 2023), who showed that the problem of generating samples from an unknown data distribution reduces to $L^2$-accurate score estimation. Our hard-to-estimate distributions are the "Gaussian pancakes" distributions, originally due to Diakonikolas et al. (FOCS 2017), which have been shown to be computationally indistinguishable from the standard Gaussian under widely believed hardness assumptions from lattice-based cryptography (Bruna et al., STOC 2021; Gupte et al., FOCS 2022).

翻译：我们证明，在对数据分布没有强假设的情况下，即使样本复杂度与相关问题的参数呈多项式关系，$L^2$精度下的分数估计在计算上仍然是困难的。我们的归约建立在Chen等人（ICLR 2023）的结果之上，他们表明从未知数据分布生成样本的问题可归约为$L^2$精度下的分数估计。我们的难估计分布是“高斯薄饼”分布，该分布最初由Diakonikolas等人（FOCS 2017）提出，并且已被证明在基于格密码学（Bruna等人，STOC 2021；Gupte等人，FOCS 2022）广泛认可的难度假设下，与标准高斯分布在计算上不可区分。