Based on binary inquiries, we developed an algorithm to estimate population quantiles under Local Differential Privacy (LDP). By self-normalizing, our algorithm provides asymptotically normal estimation with valid inference, resulting in tight confidence intervals without the need for nuisance parameters to be estimated. Our proposed method can be conducted fully online, leading to high computational efficiency and minimal storage requirements with $\mathcal{O}(1)$ space. We also proved an optimality result by an elegant application of one central limit theorem of Gaussian Differential Privacy (GDP) when targeting the frequently encountered median estimation problem. With mathematical proof and extensive numerical testing, we demonstrate the validity of our algorithm both theoretically and experimentally.

翻译：基于二进制查询，我们提出了一种在局部差分隐私（LDP）框架下估计总体分位数的算法。通过自标准化技术，该算法可生成渐近正态的估计结果并实现有效推断，从而在无需估计多余参数的情况下获得紧致置信区间。所提方法完全支持在线计算，具有极高的计算效率与最低的存储需求（仅需$\mathcal{O}(1)$空间复杂度）。针对常见的中位数估计问题，我们通过高斯差分隐私（GDP）中心极限定理的优雅应用，证明了该算法的最优性。通过数学证明与大规模数值实验，我们从理论和实验两个层面验证了算法的有效性。