We propose a unified optimization framework for designing continuous and discrete noise distributions that ensure differential privacy (DP) by minimizing R\'enyi DP, a variant of DP, under a cost constraint. R\'enyi DP has the advantage that by considering different values of the R\'enyi parameter $\alpha$, we can tailor our optimization for any number of compositions. To solve the optimization problem, we reduce it to a finite-dimensional convex formulation and perform preconditioned gradient descent. The resulting noise distributions are then compared to their Gaussian and Laplace counterparts. Numerical results demonstrate that our optimized distributions are consistently better, with significant improvements in $(\varepsilon, \delta)$-DP guarantees in the moderate composition regimes, compared to Gaussian and Laplace distributions with the same variance.

翻译：我们提出了一个统一的优化框架，用于设计连续和离散噪声分布，该框架通过在成本约束下最小化Rényi差分隐私（DP）——一种DP的变体——来确保差分隐私。Rényi DP的优势在于，通过考虑Rényi参数$\alpha$的不同取值，我们可以针对任意次数的组合优化进行定制。为了解决该优化问题，我们将其简化为有限维凸规划形式，并执行预处理梯度下降。随后，将所得噪声分布与其对应的高斯分布和拉普拉斯分布进行比较。数值结果表明，在相同方差条件下，与高斯分布和拉普拉斯分布相比，我们优化得到的分布在中等组合次数范围内，其$(\varepsilon, \delta)$-DP保证始终更优，且具有显著提升。