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：我们考虑在差分隐私下构建置信区间的任务。我们提出了非参数自助法的两种隐私变体，该方法在数据划分上运行多个“小”自助法结果的中间值，并给出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$\epsilon$，我们的方法享有与非私密自助法相同的误差率，仅在样本大小$n$上达到对数因子范围内的差异。我们通过真实数据和合成数据对均值估计、中位数估计和逻辑回归的经验性能进行了验证。我们的方法在实现与现有方法（以及非私密基线）相似的覆盖精度的同时，提供了比先前方法显著更短（约$\gtrsim 10$倍）的置信区间。