We derive the optimal $\epsilon$-differentially private mechanism for single
real-valued query function under a very general utility-maximization (or
cost-minimization) framework. The class of noise probability distributions in
the optimal mechanism has {\em staircase-shaped} probability density functions
which are symmetric (around the origin), monotonically decreasing and
geometrically decaying. The staircase mechanism can be viewed as a {\em
geometric mixture of uniform probability distributions}, providing a simple
algorithmic description for the mechanism. Furthermore, the staircase mechanism
naturally generalizes to discrete query output settings as well as more
abstract settings. We explicitly derive the optimal noise probability
distributions with minimum expectation of noise amplitude and power. Comparing
the optimal performances with those of the Laplacian mechanism, we show that in
the high privacy regime ($\epsilon$ is small), Laplacian mechanism is
asymptotically optimal as $\epsilon \to 0$; in the low privacy regime
($\epsilon$ is large), the minimum expectation of noise amplitude and minimum
noise power are $\Theta(\Delta e^{-\frac{\epsilon}{2}})$ and $\Theta(\Delta^2
e^{-\frac{2\epsilon}{3}})$ as $\epsilon \to +\infty$, while the expectation of
noise amplitude and power using the Laplacian mechanism are
$\frac{\Delta}{\epsilon}$ and $\frac{2\Delta^2}{\epsilon^2}$, where $\Delta$ is
the sensitivity of the query function. We conclude that the gains are more
pronounced in the low privacy regime.

本文提出了单一实值查询函数的最优 ε- 差分隐私机制，其噪声概率分布类具有楼梯形状、对称性、单调递减和几何衰减，可以被视为均匀概率分布的几何混合，并将此机制自然地推广到了离散查询输出环境和更抽象的环境中。通过将最优性能与拉普拉斯机制的性能进行比较，我们得出结论，当 ε 趋近于 0 时，拉普拉斯机制是渐近最优的；而在 ε 趋近于 +∞时，最小噪声振幅和最小噪声功率分别为 Θ(Δe^-ε/2) 和 Θ(Δ^2 e^-2ε/3)，而拉普拉斯机制的噪声振幅和功率期望分别为 Δ/ε 和 2Δ^2/ε^2，在低隐私保护级别下获得更明显的收益。