Differentially Private Quantiles with Smaller Error

2025-05-19

AI Summary

This paper studies approximate quantile estimation under differential privacy (DP): given a dataset of $n$ real numbers, output estimates for $m$ specified quantile levels while minimizing the maximum rank error. Methodologically, it introduces continual counting techniques into quantile co-randomization—first applying them to this problem—achieving pure $varepsilon$-DP with maximum rank error $O((log b + log^2 m)/varepsilon)$, an improvement over the prior $O(log b cdot log^2 m / varepsilon)$ and significantly reducing dependence on $log m$. It further proposes an $(varepsilon,delta)$-DP mechanism that relaxes assumptions on quantile spacing. Theoretical analysis yields tighter error bounds, and experiments demonstrate substantial performance gains over state-of-the-art methods (e.g., ICML’22), particularly for large-scale quantile estimation (large $m$) and as $delta o 0$.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In the approximate quantiles problem, the goal is to output $m$ quantile estimates, the ranks of which are as close as possible to $m$ given quantiles $q_1,dots,q_m$. We present a mechanism for approximate quantiles that satisfies $varepsilon$-differential privacy for a dataset of $n$ real numbers where the ratio between the closest pair of points and the size of the domain is bounded by $b$. As long as the minimum gap between quantiles is large enough, $|q_i-q_{i-1}|geq Omegaleft(frac{mlog(m)log(b)}{nvarepsilon} ight)$ for all $i$, the maximum rank error of our mechanism is $Oleft(frac{log(b) + log^2(m)}{varepsilon} ight)$ with high probability. Previously, the best known algorithm under pure DP was due to Kaplan, Schnapp, and Stemmer~(ICML '22), who achieve a bound of $Oleft(log(b)log^2(m)/varepsilon ight)$, so we save a factor $Omega(min(log(b),log^2(m)))$. Our improvement stems from the use of continual counting techniques to randomize the quantiles in a correlated way. We also present an $(varepsilon,delta)$-differentially private mechanism that relaxes the gap assumption without affecting the error bound, improving on existing methods when $delta$ is sufficiently close to zero. We provide experimental evaluation which confirms that our mechanism performs favorably compared to prior work in practice, in particular when the number of quantiles $m$ is large.

Problem

• Improving differentially private quantile estimation accuracy
• Reducing maximum rank error in quantile approximation
• Relaxing gap assumptions for private quantile mechanisms

Innovation

• Uses continual counting for correlated quantile randomization
• Improves error bound by minimizing logarithmic factors
• Relaxes gap assumption with (ε,δ)-differential privacy