Title: Time-uniform and Asymptotic Confidence Sequence of Quantile under Local Differential Privacy
Venue: NeurIPS 2025
Accepted: yes
Keywords: Confidence sequence

local differential privacy

quantile

stochastic gradient decent
Abstract: In this paper, we develop a novel algorithm for constructing time-uniform, asymptotic confidence sequences for quantiles under local differential privacy (LDP). The procedure combines dynamically chained parallel stochastic gradient descent (P-SGD) with a randomized response mechanism, thereby guaranteeing privacy protection while simultaneously estimating the target quantile and its variance. A strong Gaussian approximation for the proposed estimator yields asymptotically anytime-valid confidence sequences whose widths obey the law of the iterated logarithm (LIL). Moreover, the method is fully online, offering high computational efficiency and requiring only $\mathcal{O}(\kappa)$ memory, where $\kappa$ denotes the number of chains and is much smaller than the sample size. Rigorous mathematical proofs and extensive numerical experiments demonstrate the theoretical soundness and practical effectiveness of the algorithm.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper is the first to extend the asymptotic confidence sequences of the quantile estimation framework introduced by Waudby-Smith et al. [41] to differential privacy.  It uses a parallel SGD method that was recently applied to high-confidence interval problems by Wanrong Zhu et al. [46]. Differential privacy is ensured through a randomized response mechanism used with gradient-like updates proposed by Liu et al. [32].\n\nSTRENGTHS AND WEAKNESSES:\nThis paper is the first to bring the time uniform quantile estimation with confidence intervals problem into differential privacy. In doing so, it generalizes previous non-private works. I find the paper to be interesting, and the technical contributions are solid and non-trivial.\n\nHowever, I am not fully convinced that the asymptotic sequence framework is the most appropriate choice for practical applications. Would high-confidence intervals alone be sufficient?\n\nAdditionally, the paper would benefit from further polishing:\n\nPresentation : The fonts in all of the plots are too small and difficult to read.\n\n\nThere are issues with spacing and punctuation in citations, such as: “Privacy , see [16]”, “Privacy,see [26]”, “(e.g., 7)”\n\nQUESTIONS:\n1) The paper states that the proposed approximation rate is faster than that given by the Law of the Iterated Logarithm (LIL). However, isn't the LIL rate of order $ O\\left(\\frac{1}{\\sqrt{t \\log \\log t}}\\right)$, which appears faster than the rate you claim? \n\n2) Would high-confidence intervals alone suffice for private quantile estimation with confidence intervals? What advantages does the asymptotic sequence framework offer in practice?\n\n3) If the goal was only quantile estimation without confidence intervals, how would your procedure compare to that of Liu et al. [32]?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-21"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nAccurately estimating the quantiles under LDP is challenging because it needs asymptotic variance estimates of the estimator. For online quantile inference (LDP reports arriving in sequence), [46] proposed a parallel SGD based framework with high confidence variance estimation. However, their variance estimator does not obey consistency due to fixed length of the chains.\nThis work extends the idea of using P-SGD, and proposes a new low resource online method for estimating confidence sequences for quantiles, and an almost surely consistent estimator  of the quantile variance (derived from P-SGD iterates) under LDP.\nThey also show that Gaussian approximation for the proposed estimator provides asymptotically anytime-valid confidence sequences with lengths  following the LIL.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1)\tOnline quantile and its consistent variance estimation with asymptotic confidence sequences under LDP seems unsolved. \n2)\tThe method seems easy to implement and practical. \n\nWeakness: \n1)\tSGD based methods can be sensitive to the choice of hyperparameters and require tuning.\n\n2)\tThe continual observation model under DP (https://dl.acm.org/doi/10.1145/1806689.1806787) is well-established in the central setting. Authors do not discuss this model in the literature review.  There are some works (e.g. https://openreview.net/attachment?id=pWeQdceMHL&name=pdf) also in the LDP setting, possibly not on the same problem though.\n\nQUESTIONS:\n1)\tThe setting does not seem to assume any bound on the input? Please clarify.\n2)\tPlease explain the equation on line 128 (at least in the Appendix). It’s not immediate how G(x, \\Xi) can be viewed as a permuted stochastic gradient.   \n3)\tPlease discuss the continual observations model (or the differences from it) in the literature review, placing your solution in perspective. Some generic background on DP and LDP can be reduced in Section 1.\n4)\tMarkers and legends in Figures are barely visible. Please increase the sizes.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

comments: SUMMARY:
This work studies quantile estimation (and its variance) in the online setting under local differential privacy. The main techniques used in the paper is a parallel stochastic gradient descent (P-SGD) which partitions the data and uses multiple parallel SGD optimizations to estimate the quantile. One of the main contributions of the work is increasing the number of partitions as the number of samples increases dynamically. The algorithm is also able to measure the variance of the estimator using the sample variance from the multiple chains. To achieve privacy, the algorithm applies randomized response on the binary quantity of whether the current estimate of the quantile is smaller or larger than the data point. This signal is used to estimate the true quantile using SGD. 

The paper analyzes this algorithm theoretically to show that this allows them to derive an almost surely Gaussian approximation for the Polyak-Ruppert-type estimator of a quantile obtained by P-SGD and an almost surely consistent estimator of the quantile variance that relies only on the iterates of P-SGD.

Finally, the work also simulates some finite sample performance of the method using a Gaussian and Cauchy distributions which shows that they are able to achieve narrower intervals compared to previous works.

STRENGTHS AND WEAKNESSES:
Strengths:
- Paper is easy to read and follow. 
- The algorithm is simple, easy to implement, is fully online and requires low memory overhead.
- Work provides strong theoretical and simulation results. 

Weaknesses:
- The algorithm might require tuning many hyperparameters

QUESTIONS:
Can the authors expand on how sensitive finite sample performance is on the choice of hyperparameters?

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

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents an online algorithm for computing time-uniform, asymptotic confidence sequences for quantiles under LDP.  Specifically, the proposed solution combines dynamically chained parallel SGD (P-SGD) with a randomized response (RR) mechanism to simultaneously estimate the target quantile and its variance. A Gaussian approximation leads to asymptotically anytime-valid confidence sequences whose widths obey the law of the iterated logarithm (LIL), with optimal convergence rates. The method is also computationally efficient, requiring only O(κ) memory for κ chains.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\nS1. The paper introduces a novel approach that enables the consistent estimation of the estimator's variance without incurring any additional privacy cost.\n\nS2. The paper provides a strong, almost sure Gaussian approximation for the quantile estimator. \n\nS3. This work presents the first method for constructing time-uniform, asymptotic confidence sequences for quantiles with LDP.\n\nWeaknesses:\n\nW1. All experiments are conducted on synthetic data (mostly Normal and Cauchy). There is no demonstration on real-world datasets or application scenarios.\n\nW2: The algorithm's performance relies on several key hyperparameters, most notably the function h(t) that determines the number of parallel chains. The specific choice used in the experiments (e.g., h(t)=⌊8log₁₀(t)⌋) is presented without justification or a sensitivity analysis showing how performance changes with different functions.\n\nW3: The paper makes some empirical claims based on weak evidence. For instance, the claim that its pointwise confidence interval is \"more efficient\" than a competing method is supported only by visual inspection of a single plot (Figure 2), which is anecdotal.\n\nW4. The confidence sequences are only asymptotically valid. There are no finite-sample coverage guarantees, which limits their practical reliability in early rounds.\n\nW5. The theoretical results require a continuous and bounded derivative of the underlying density near the quantile. This excludes irregular or heavy-tailed distributions, despite claims of robustness.\n\nQUESTIONS:\nQ1: The paper claims to provide the \"first result on sequential inference for quantiles within the LDP framework\". However, prior work like Liu et al. (2023) also presents an online LDP quantile inference method. Could you clarify the distinction? \n\nQ2: Could you provide practical guidance on selecting h(t)?\n\nQ3: The experimental results show that your P-SGD-based method, in the non-private setting (r=1), produces tighter confidence sequences than the state-of-the-art non-asymptotic method from Howard and Ramdas (2022). Could you explain the reason?\n\nQ4: In theorem 2, it seems that big-O_a,s.(1) should be small-o?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies the construction of asymptotic confidence sequence for the quantile of a distribution on the line under local differential privacy; where an $\\epsilon,\\delta$-differential privacy constraint applies to each individual datum in the sequence of observations. \n\nThe method is based on parallel private SGD, where the privacy mechanism is essentially randomized response and where the estimation of the quantile and its variance is done using the SGD iterates, where the parallel chains need to be constructed in a clever way\". The authors provide both theoretical guarantees for their method as well as simulations.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n* I found the paper to be well organized and easy to read.\n* The theoretical contributions are interesting independently of the motivation and are formulated as such.\n* The method is well well explained - both in terms of its how and why.\n* The writing is clear and the authors are forward in terms of what the do and don't assume.\n\nWeakness:\n* The method seems to rely on tuning parameters which are difficult to set in practice. Furthermore, it is unclear to me whether these tuning parameters can be set in a way that does not lead to \"privacy leakage\" due to having to run the method for different hyperparameter values. \n\nA minor comment: sometimes there is some unnecessary notation overflow: On page 3, $U$ is used twice for completely different purposes. Similarly, the choice of the $\\delta_{T,i}$-notation is somewhat unfortunate given the differential privacy definition.\n\nQUESTIONS:\n* What if the user is interested in more than just one quantile, but say more general tail risk? Can the method be adapted to these types of questions?\n* What can be said about the method in terms of the efficiency theory (Steinberger 2024 - Efficiency in local differential privacy)?\n\nLIMITATIONS:\nThe main limitation I see are the hyperparameters of the method and setting these under differential privacy constraints in practice.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}