We study the optimal design of additive mechanisms for vector-valued queries under $ε$-differential privacy (DP). Given only the sensitivity of a query and a norm-monotone cost function measuring utility loss, we ask which noise distribution minimizes expected cost among all additive $ε$-DP mechanisms. Using convex rearrangement theory, we show that this infinite-dimensional optimization problem admits a reduction to a one-dimensional compact and convex family of radially symmetric distributions whose extreme points are the staircase distributions. As a consequence, we prove that for any dimension, any norm, and any norm-monotone cost function, there exists an $ε$-DP staircase mechanism that is optimal among all additive mechanisms. This result resolves a conjecture of Geng, Kairouz, Oh, and Viswanath, and provides a geometric explanation for the emergence of staircase mechanisms as extremal solutions in differential privacy.

翻译：我们研究在$ε$-差分隐私（DP）约束下针对向量值查询的加性机制最优设计问题。在仅给定查询敏感度及衡量效用损失的范数单调代价函数条件下，我们探讨在所有满足$ε$-DP的加性机制中，何种噪声分布能使期望代价最小化。通过凸重排理论，我们证明该无限维优化问题可归约至一个一维紧凸的径向对称分布族，其极值点恰为阶梯分布。由此我们证明：对于任意维度、任意范数及任意范数单调代价函数，总存在一个$ε$-DP阶梯机制在所有加性机制中达到最优。该结果证实了Geng、Kairouz、Oh和Viswanath的猜想，并为阶梯机制作为差分隐私极值解的出现提供了几何解释。