We consider the task of constructing confidence intervals with differential privacy. We propose two private variants of the non-parametric bootstrap, which privately compute the median of the results of multiple "little" bootstraps run on partitions of the data and give asymptotic bounds on the coverage error of the resulting confidence intervals. For a fixed differential privacy parameter $\epsilon$, our methods enjoy the same error rates as that of the non-private bootstrap to within logarithmic factors in the sample size $n$. We empirically validate the performance of our methods for mean estimation, median estimation, and logistic regression with both real and synthetic data. Our methods achieve similar coverage accuracy to existing methods (and non-private baselines) while providing notably shorter ($\gtrsim 10$ times) confidence intervals than previous approaches.

翻译：我们研究在差分隐私约束下构建置信区间的任务。我们提出了两种非参数自助法的隐私变体，这些方法通过计算数据分区上运行的多个"小型"自助法结果的中位数来隐私地估计参数，并给出了所得置信区间覆盖误差的渐近界。对于固定的差分隐私参数$\epsilon$，我们的方法在样本量$n$的对数因子范围内，实现了与非隐私自助法相同的误差率。我们通过真实数据和合成数据，在均值估计、中位数估计和逻辑回归任务中实证验证了方法的性能。我们的方法在达到与现有方法（及非隐私基线）相近覆盖精度的同时，能够提供显著更短（$\gtrsim 10$倍）的置信区间。