Asymptotic Anytime-Valid Quantile Inference under Local Differential Privacy
Abstract
Sequential quantile inference is difficult under local differential privacy because every record is randomized before reaching the analyst and the limiting quantile variance depends on an unknown density. We develop an online procedure that combines randomized response with dynamically chained parallel stochastic gradient descent (P-SGD). The resulting Polyak--Ruppert estimator admits a strong Gaussian approximation. A cross-chain quadratic statistic, computed entirely from private iterates, consistently estimates the limiting variance without a separate online density estimator. These results yield asymptotic confidence sequences and, under polynomial chain growth, asymptotic time-uniform coverage. Arm-wise constructions support locally private quantile best-arm identification, time-uniform simple-regret bounds, and sequential A/B tests of quantile treatment effects. Simulations and salary-data analyses illustrate the finite-sample behavior and practical use of the proposed methods.
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BibTeX
@misc{cai2026asymptotic,
title = {{Asymptotic Anytime-Valid Quantile Inference under Local Differential Privacy}},
author = {Leheng Cai and Qirui Hu and Shuyuan Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21338},
eprint = {2609.21338},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21338},
}
RIS
TY - UNPB
TI - Asymptotic Anytime-Valid Quantile Inference under Local Differential Privacy
AU - Leheng Cai
AU - Qirui Hu
AU - Shuyuan Wu
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21338
UR - https://arxiv.org/abs/2609.21338
AB - Sequential quantile inference is difficult under local differential privacy because every record is randomized before reaching the analyst and the limiting quantile variance depends on an unknown density. We develop an online procedure that combines randomized response with dynamically chained parallel stochastic gradient descent (P-SGD). The resulting Polyak--Ruppert estimator admits a strong Gaussian approximation. A cross-chain quadratic statistic, computed entirely from private iterates, consistently estimates the limiting variance without a separate online density estimator. These results yield asymptotic confidence sequences and, under polynomial chain growth, asymptotic time-uniform coverage. Arm-wise constructions support locally private quantile best-arm identification, time-uniform simple-regret bounds, and sequential A/B tests of quantile treatment effects. Simulations and salary-data analyses illustrate the finite-sample behavior and practical use of the proposed methods.
ER -