Optimality of Matrix Mechanism on $\ell_p^p$-metric

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 in the natural add/remove model. Before this paper, tight characterization in the hardness of privately answering linear queries was known under $\ell_2^2$-error metric (Edmonds et al. 2020) and $\ell_p^2$-error metric for unbiased mechanisms in the substitution model (Nikolov et al. 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 Hhenzinger et al. for $p=2$.