In this paper, we introduce the $\ell_p^p$-error metric (for $p \geq 2$) when answering linear queries under the constraint of differential privacy. We characterize such an error under $(\epsilon,\delta)$-differential privacy. Before this paper, tight characterization in the hardness of privately answering linear queries was known under $\ell_2^2$-error metric (Edmonds et al., STOC 2020) and $\ell_p^2$-error metric for unbiased mechanisms (Nikolov and Tang, ITCS 2024). As a direct consequence of our results, we give tight bounds on answering prefix sum and parity queries under differential privacy for all constant $p$ in terms of the $\ell_p^p$ error, generalizing the bounds in Henzinger et al. (SODA 2023) for $p=2$.

翻译：本文在差分隐私约束下回答线性查询时，引入了$\ell_p^p$误差度量（其中$p \geq 2$）。我们刻画了在$(\epsilon,\delta)$-差分隐私下此类误差的界。在本文之前，关于私有回答线性查询的困难度紧界刻画仅在$\ell_2^2$误差度量（Edmonds等人，STOC 2020）以及针对无偏机制的$\ell_p^2$误差度量（Nikolov与Tang，ITCS 2024）下已知。作为我们结果的直接推论，我们针对所有常数$p$，在$\ell_p^p$误差意义下，给出了差分隐私中回答前缀和查询与奇偶查询的紧界，从而推广了Henzinger等人（SODA 2023）在$p=2$时的结果。