Title: Differentially Private Quantiles with Smaller Error
Venue: NeurIPS 2025
Accepted: yes
Keywords: differential privacy

quantile estimation
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 $0 \leq q_1 \leq\dots \leq q_m \leq 1$. 
We present a mechanism for approximate quantiles that satisfies $\varepsilon$-differential privacy for a dataset of $n$ real numbers where the ratio between the distance between the closest pair of points and the size of the domain is bounded by $\psi$.
As long as the minimum gap between quantiles is sufficiently large, $|q_i-q_{i-1}|\geq \Omega\left(\frac{m\log(m)\log(\psi)}{n\varepsilon}\right)$ for all $i$, the maximum rank error of our mechanism is $O\left(\frac{\log(\psi) + \log^2(m)}{\varepsilon}\right)$ with high probability.
Previously, the best known algorithm under pure DP was due to Kaplan, Schnapp, and Stemmer (ICML '22), who achieved a bound of $O\left(\frac{\log(\psi)\log^2(m) + \log^3(m)}{\varepsilon}\right)$. 
Our improvement stems from the use of continual counting techniques which allows the quantiles to be randomized in a correlated manner.
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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a novel mechanism for computing approximate quantiles under $\\epsilon$-differential privacy (DP) and $(\\epsilon, \\delta)$-DP, achieving improved error bounds compared to prior work, notably Kaplan et al. (ICML '22). The authors leverage continual counting techniques to reduce the maximum rank error, particularly when the number of quantiles $m$ is large. The paper also includes experimental validation, showing practical improvements.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- Improved Error Bounds: The proposed mechanism achieves a maximum rank error of $O\\left(\\frac{\\log(b) + \\log^2(m)}{\\epsilon}\\right)$ for $\\epsilon$-DP, improving over Kaplan et al.'s $O\\left(\\frac{\\log(b)\\log^2(m)}{\\epsilon}\\right)$ by a factor of $\\Omega\\left(\\min(\\log(b), \\log^2(m))\\right)$. This is a meaningful theoretical advancement, especially for large $m$.\n- Novel Approach: The use of continual counting to randomize quantiles in a correlated manner is innovative and distinguishes this work from prior divide-and-conquer strategies.\n- Experimental Validation: The experiments confirm the theoretical improvements, showing a factor of 2 improvement in accuracy over Kaplan et al. when estimating 200 quantiles with $\\epsilon=1$ and $\\delta=10^{-16}$ under substitute adjacency.\n- $(\\epsilon, \\delta)$-DP Mechanism: The relaxation of the gap assumption in the $(\\epsilon, \\delta)$-DP mechanism, with error independent of $\\delta$, is a notable contribution, particularly when $\\delta$ is small. I have questions on this part that are detailed below.\n\nWeaknesses:\nSee my questions below.\n\nQUESTIONS:\n1. Minimum Gap Requirement:\n   The minimum gap assumption, $\\left|q_i - q_{i-1}\\right| \\geq \\Omega\\left(\\frac{m \\log(m) \\log(b)}{n \\varepsilon}\\right)$, is a critical condition for the $\\varepsilon$-DP mechanism but differs in nature from Kaplan et al.'s requirement, which pertains to the spacing of data points. Here, the constraint is on the quantiles’ orders, which may limit applicability in certain scenarios. Could the authors elaborate on the implications of this assumption compared to Kaplan et al.'s, particularly regarding its impact on practical settings and datasets where quantiles may be closely spaced?\n\n2. Clarity of Presentation:\n   The exposition of the proposed solution could be improved for better accessibility. I suggest adding a sentence clarifying that, under suitable assumptions, $x_{(nq_i)}$ can be approximated by applying a location algorithm (e.g., a private median algorithm) to a subset of points around the quantile. The approach can be described as decoupling into two steps: (1) splitting the dataset into disjoint subsets around the quantiles using private continual counting algorithms, and (2) applying a location algorithm to each subset. This would provide a clearer high-level intuition for the SliceQuantiles algorithm.\n\n3. Potential Bias in Location Algorithm:\n   The method relies on estimating the location of data slices around quantiles. If the location algorithm used in SingleQuantile is not a median estimator (e.g., a mean estimator), this could introduce bias in the quantile estimates, unlike Kaplan et al.'s approach, which may be more robust in certain cases. Could the authors comment on whether this is a concern and how the choice of location algorithm affects bias, particularly for non-median estimators?\n\n4. Empirical CDF/Quantile Function Plots:\n   To better understand the practical improvements and potential biases, I recommend including plots of the empirical cumulative distribution functions or quantile functions for the datasets used in the experiments. These plots would help readers assess the distributions where the proposed method outperforms Kaplan et al. and whether the generic location algorithm in the $(\\varepsilon, \\delta)$-DP setting introduces bias, especially when the quantile function has high curvature.\n\n5. Literature:\n   The paper references Lalanne et al. (2023), noting that their algorithm relies on strong smoothness assumptions for randomized quantiles. I believe this refers to their histogram estimator for density, used as a plug-in for the quantile function. While Lalanne et al.'s approach requires regularity assumptions, the current paper assumes a minimum gap of 1 between data points, which is a form of regularity under which it should be possible to study histogram estimators. I suggest including a naive baseline similar to Lalanne et al.'s histogram estimator in the experiments, acknowledging the challenge of choosing the bandwidth parameter. This would provide a more comprehensive comparison and highlight the trade-offs of different assumptions.\n\n6. Notation and Clarity Issues:\n   - In line 124, the ordered statistics notation ($x_{(i)}$) is used without prior introduction, which could confuse readers unfamiliar with the convention. I recommend defining this notation explicitly.\n   - In Equation (1), the use of $d$ for a quantity in $\\{-1, 1\\}$ is potentially misleading, as $d$ is commonly used to denote dimension. Consider using a different symbol to avoid confusion.\n   - In Lemma 3.1, the equation appears to have a typo: it should likely be $\\Pr[\\mathrm{CC}(\\mathbf{r}) = \\tilde{\\mathbf{r}}] \\leq e^\\varepsilon \\Pr[\\mathrm{CC}(\\mathbf{r} + \\mathbf{e}) = \\tilde{\\mathbf{r}}]$ instead of comparing $\\mathrm{CC}(\\mathbf{r})$ to itself. Additionally, in the following line, $\\tilde{\\mathbf{r}}$ should be replaced with $\\mathrm{CC}(\\mathbf{r})$ for consistency.\n   - In lines 154–156, the bounds for applying SingleQuantile are unclear. Initially, it seems the algorithm uses the upper and lower data points as interval bounds, which could raise privacy concerns. However, it later appears that the bounds are the entire base interval $(a, b)$. Clarifying this and discussing how the domain reduction strategy compares to Kaplan et al.'s tree-based approach would improve understanding.\n   - In Lemma 4.1, the use of $F^+$ and $F^-$ for the rectifying functions is confusing, as these symbols are often associated with CDFs or quantile functions in this context. Consider alternative notation to avoid ambiguity.\n\n7. Scalability and Limitations:\n   The minimum gap requirement scaling as $\\frac{m}{n \\varepsilon}$ for $\\varepsilon$-DP makes the results less applicable in the high-quantiles regime. For $(\\varepsilon, \\delta)$-DP, the linear scaling of $\\delta$ with $m$ int the $\\delta$ part might be concerning, and the claim of utility bounds being independent of $\\delta$ suggests this dependence may be hidden in the gap assumptions. Could the authors clarify how this dependence is managed and discuss the practicality of the gap assumptions in high-quantiles scenarios?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-23"}

comments: SUMMARY:
This paper introduces SliceQuantiles, a new mechanism for privacy preserving quantile approximation. While the existing works, in $\epsilon$ $\delta$ DP are optimal in logb, the proposed method improves over the state-of-the-art in reducing the complexity of $\delta$ term (error independent of $\delta$). In pure DP setting, the proposed method saves min(logb, log^2m) factor. The improvement is achieved by combining continual counting techniques with the exponential mechanism. Using Adult data set, the authors demonstrate that the proposed SliceQuantiles outperform AQ.

STRENGTHS AND WEAKNESSES:
S1.  The use of continual counting to correlate quantile noise is clever. The main idea is sound and clearly presented.

S2. The paper provides rigorous theoretical proofs for both privacy and utility guarantees under add/remove model for approximate/pure DP. (Didn't check the proofs in detail)

S3. Using scaled Adult dataset, authors demonstrated the high accuracy of the proposed algorithm. 

W1. For clarity: please consider to i) add input dataset in algorithm 1, ii) add ticks in Figure 1, iii) compare with AppindExp (reference 12), and iv) include approximate quantile literatures in the streaming model such as `Differentially private linear sketches: Efficient implementations and applications`, `Improved utility analysis of private countsketch`, and `Bounded space differentially private quantiles`.

W2. Need more discussions or experiment on the query time complexity (perhaps appendix).

W3. What happens to the utility guarantee under substitute model?

QUESTIONS:
Please see W1 - W3

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 2
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper improved error bound in the problem of estimating multiple quantiles of a dataset under DP over prior methods. The authors introduce the SliceQuantiles mechanism, which offers both pure ε-DP and approximate (ε, δ)-DP guarantees. Their central innovation lies in leveraging continual counting techniques to inject correlated noise into the rank estimates of quantiles, enabling more efficient privacy composition.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\nThe paper presents a theoretically sound improvement over prior work, eliminating a factor in the error bound.\n\nThe use of continual counting for quantile estimation is original in this context.\n\nWeaknesses:\n\nThis paper is difficult to follow and understand in several key aspects. For example, the concept of a \"slice\" is introduced at line 75 as “m subproblems, referred to as ‘slices’.” This term is referenced multiple times before line 124, where it is only hinted that a slice contains a subset of the dataset $X$. A more concrete definition appears at line 130, where a slice is described as some subset of $X$ determined by a sufficiently large $h$; however, it is not made clear what constitutes a \"sufficiently large\" $h$. From that point onward, slices are treated as subsets of $X$ in various contexts, but it remains unclear why a slice corresponds to a subproblem in the proposed methodology.\n\nAdditionally, in line 185, the Technical Challenge section attempts to explain the difficulty of establishing the differential privacy condition by describing a failed approach. However, the exposition of this failed attempt is unclear, and it does not effectively elucidate the core technical challenge faced by the authors.\n\nWhile supported by prior literature, the problem formulation appears artificial and narrow. The paper focuses on controlling the rank error under the condition that queried quantiles are separated by a minimal gap. Although it is reasonable that rank error becomes unavoidable when quantiles are too close, the authors do not provide sufficient justification for the practical relevance of this setting, especially given that verifying the minimal quantile gap condition may be non-trivial in practice which narrow down the applications by a lot.\n\nQUESTIONS:\nCould the authors provide a clearer and earlier definition of what constitutes a \"slice\" and explain explicitly why a slice corresponds to a subproblem in their framework?\n\nThe proposed setting relies on a minimal gap between quantiles, but verifying that this condition holds may be non-trivial. Can the authors provide guidance or practical procedures for how one might check or enforce this minimal gap in realistic applications? Also, what is the motivation behind the rank error metric?\n\nThe performance deteriorates for larger $b$ and depend on minimal gap. How does $b$ and gap interacts? It appear to me that a larger minimal gap leads to a larger $b$.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a novel differentially private algorithm for the quantile estimation problem, improving upon prior utility guarantees. Specifically, the authors reduce the approximation error from the previous $O(\\log(b)\\log^2(m)/\\epsilon)$ to $O((\\log(b)+\\log^2(m))/\\epsilon)$. However, to achieve this improvement, the algorithm additionally assumes that the queried quantiles satisfy a spacing condition:\n$|q_i-q_{i+1}|\\geq\\Omega(m\\log(m)\\log(b)/n\\epsilon)$. In the case of $(\\epsilon,\\delta)$-differential privacy, this constraint is relaxed to\n$|q_i-q_{i+1}|\\geq\\Omega(\\log(m)\\log(m/\\delta)+\\log(b)/n\\epsilon)$ while maintaining the same utility bound. The paper also presents experimental results on the AdultAge and AdultHours datasets, demonstrating improved performance over previous approaches.\n\nA key technical contribution is the use of a sliced quantile approach, in which the dataset is partitioned (via noisy boundaries) into disjoint slices, and a single-quantile mechanism is applied independently within each slice. The disjointness of slices enables strong utility guarantees through statistical independence.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1.The paper is clearly structured and well written.\n\n2.The utility improvement from $O(\\log(b)\\log^2(m)/\\epsilon)$ to $O((\\log(b)+\\log^2(m))/\\epsilon)$ is substantial and well motivated.\n\n3.The theoretical analysis is sound.\n\n4.Experimental results show meaningful gains over existing methods.\n\nWeaknesses\n\n1.The core idea of slicing queries into independent subproblems is conceptually simple and may be seen as a straightforward extension.\n\n2.The spacing assumption imposed on the quantiles lacks sufficient justification or motivation, and its relationship to prior work is underexplored.\n\nQUESTIONS:\nIn the experiments section, could the authors provide a more detailed analysis of how the algorithm performs under varying values of the privacy parameters $\\epsilon$ and $\\delta$? This would help assess robustness across different privacy regimes.\n\nLIMITATIONS:\nThis is a theoretical work, and I do not see any apparent negative societal impacts.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}