Federated Learning of Quantile Inference under Local Differential Privacy

Title: Federated Learning of Quantile Inference under Local Differential Privacy
ArXiv: 2509.21800v1
Version: v1
Authors: Leheng Cai, , Qirui Hu, and Shuyuan Wu
Sections: 19
Estimated tokens: 20.2k

## Contents
- 1 Introduction
- 2 Methodologies
  - Definition 1 (Central Differential Privacy, CDP (Dwork et al.,, 2006 ) ) .
  - Definition 2 (Local Differential Privacy, LDP (Joseph et al.,, 2019 ) ) .
  - Theorem 2.1 .
- 3 Asymptotic analysis
  - Assumption 1 .
  - Assumption 2 .
  - Assumption 3 .
  - Theorem 3.1 .
  - Theorem 3.2 .
  - Corollary 3.1 .
  - Corollary 3.2 .
- 4 Experiments
  - 4.1 Simulation setup
  - 4.2 Simulation results
  - 4.3 Real data
- 5 Concluding remark
- References

## 1 Introduction

Quantile estimation and inference are critical tools in various scientific and applied domains. In healthcare, quantile methods facilitate more informed decisions regarding the optimal distribution of scarce medical resources, thus promoting equitable and effective patient care (Yadlowsky et al.,, 2025). Similarly, quantile techniques have proven highly valuable in policy evaluation, as they reveal heterogeneous effects across different subgroups, nuances typically obscured by traditional average-based analyses (Kallus et al.,, 2024; Chernozhukov and Fernández-Val,, 2011; Chernozhukov and Hansen,, 2005). In reliability engineering, quantile-based approaches have significantly improved the assessment of system robustness, particularly under rare or extreme conditions, demonstrating their broad applicability and precision (He et al.,, 2023; Hu et al.,, 2022). Moreover, finance widely employs quantile-based metrics such as value-at-risk, essential for managing financial risks in the face of regulatory pressures and market uncertainties (Barbaglia et al.,, 2023; Chen,, 2008; Wang et al.,, 2012). In general, quantile methods excel at capturing the characteristics of skewed or extreme-valued data, delivering richer insights into complex distributions prevalent in practical scenarios (Angrist et al.,, 2006; Chen et al.,, 2023).

Traditional quantile estimation methods have been extensively studied. However, with the rapid increase in massive datasets (Jordan et al.,, 2019; Hector and Song,, 2021; Fan et al.,, 2023), traditional approaches that rely on analyzing all data on a single machine may no longer be computationally feasible. This challenge has motivated the emergence of federated learning methods (McMahan et al.,, 2017; Liu et al.,, 2020; Tian et al.,, 2023). Federated learning enables multiple distributed clients to collaboratively train a global model without exchanging raw data, effectively addressing computational efficiency and privacy concerns (Konečnỳ et al.,, 2016).
In standard federated learning, a central server coordinates iterative model updates among clients, and under suitable conditions, this process guarantees convergence (Li et al.,, 2020; Chen et al.,, 2022). To further enhance communication efficiency, local stochastic gradient descent (SGD) has been proposed, allowing clients to perform multiple local updates before synchronization. Under i.i.d. scenarios, the theoretical optimality of local SGD has been established (Stich,, 2018). However, data heterogeneity, which frequently occurs in federated learning, significantly complicates local SGD. A series of studies have investigated this issue by analyzing convergence in worst-case heterogeneous scenarios (Hu et al.,, 2024), proposing regularization techniques to ensure local models remain close to the global model (Li et al.,, 2020), and introducing momentum-based algorithms to stabilize training under non-i.i.d. conditions (Li et al.,, 2025). Moreover, inference methods have also been developed and analyzed (Li et al.,, 2022).

Federated learning aggregates individual information to enable efficient collaborative model training. These personal data power indispensable services, from facial recognition unlocking our phones to recommendation systems curating news feeds, but they also carry latent risks. Leaked genetic markers can jeopardize insurance rates, and cleverly crafted prompts can coax large language models into regurgitating fragments of their private training corpora Nasr et al., (2023). Differential privacy (DP) offers a principled defense: by bounding the statistical influence of any single participant, DP ensures that outputs remain virtually unchanged whether or not an individual opts in Dwork et al., (2006). This safeguard, however, dissolves if the data custodian is breached, coerced, or simply misconfigured access controls—scenarios illustrated by repeated healthcare leaks and high-profile cloud missteps. Local differential privacy (LDP) fortifies the pipeline by introducing randomness at the point of collection: users perturb their data locally, send only noisy summaries, and retain the key to their raw information Duchi et al., (2013). Even a fully compromised server therefore receives nothing decipherable. Industry adoption is accelerating: Google’s RAPPOR measures Chrome settings, iOS uses LDP to count emoji preferences, and Windows telemetry applies similar techniques to malware prevalence Erlingsson et al., (2014); Ding et al., (2017). Collectively, these systems prove that granular user analytics and uncompromising privacy need not be mutually exclusive; instead, LDP sets a practical, legally robust baseline for responsible data-driven innovation.

DP federated learning has attracted considerable attention recently (e.g., (Liu et al., 2023a,; Agarwal et al.,, 2018; Shi et al.,, 2022; Ma et al.,, 2022)). The additional communication layer between local clients and the global server gives rise to distinct privacy requirements. As delineated in (Lowy and Razaviyayn,, 2023), one can categorize DP at the individual record level, inter‐silo record level, shuffled‐model, and user‐level, in order of increasing trust assumptions. In particular, LDP posits that each individual does not trust any other party, including their own silo, and therefore must randomize her report before release. Extensive work has focused on this setting (e.g., (Zhao et al.,, 2020; Shen et al.,, 2023; Jiang et al.,, 2022)).

Whereas prior studies of LDP in federated learning (e.g., (Zhao et al.,, 2020; Shen et al.,, 2023; Jiang et al.,, 2022)) primarily address estimation, statistical inference, such as constructing confidence intervals and conducting hypothesis tests, poses additional challenges. Beyond deriving the limiting distribution, inference requires a consistent estimator of the asymptotic variance. For SGD‐based methods, this typically involves the Hessian matrix, which exists only for smooth loss functions (Chen et al.,, 2020). Moreover, because only privatized gradients are observed, one may need extra privacy budget or data‐splitting to estimate variance reliably. Finally, existing single‐machine LDP quantile algorithms, such as Huang et al., (2021) or Liu et al., 2023b cannot derive the inference result or do not readily extend to federated settings due to client‐heterogeneity in local loss functions.

In this paper, we propose a novel federated learning algorithm for quantile inference under LDP. Our method accommodates client‐level heterogeneity in quantile targets, privacy budgets, and data distributions, thereby enhancing the applicability of quantile inference in realistic federated environments. A key innovation is our theoretical analysis of the local SGD quantile estimator. We first design an LDP mechanism that effectively reduces the federated quantile estimation problem to an equivalent non‐private setting. Exploiting this reduction, we establish the estimator’s asymptotic normality and derive a functional central limit theorem without average‐smoothness condition on the loss function. To the best of our knowledge, this constitutes the first weak‐convergence result for local SGD when the loss does not satisfy the usual average‐smoothness condition (Li et al.,, 2022; Xie et al.,, 2024; Zhu et al.,, 2024). Building on these non‐private asymptotic results, we develop an LDP‐compliant inference procedure for federated quantile estimation. By employing a self‐normalization technique, we avoid direct estimation of the asymptotic variance, instead constructing confidence intervals that automatically eliminate the unknown variance term. To the best of our knowledge, we provide the first inference framework for federated quantile estimation, even without privacy constraints.

The remainder of the paper is organized as follows. Section [2](https://arxiv.org/html/2509.21800v1#S2) reviews background and notation. Section [3](https://arxiv.org/html/2509.21800v1#S3) presents the asymptotic analysis of the proposed estimator. Section [4](https://arxiv.org/html/2509.21800v1#S4) reports extensive numerical experiments and real data application. All technical proofs and additional simulation results are deferred to the Appendix.

## 2 Methodologies

First, we recall the definitions of central and local differential privacy. We then describe our problem setting and algorithmic details.

###### Definition 1 (Central Differential Privacy, CDP (Dwork et al.,, 2006 ) ) .

A randomized algorithm $\mathcal{A}$ operating on a dataset $S$ is *$(\epsilon,\delta)$-differentially private* if, for any pair of datasets $S$ and $S^{\prime}$ differing in a single record and for any measurable set $E$,

$$ $\Pr\bigl[\mathcal{A}(S)\in E\bigr]\leq e^{\epsilon}\,\Pr\bigl[\mathcal{A}(S^{\prime})\in E\bigr]+\delta.$ $$

When $\delta=0$, $\mathcal{A}$ is called *$\epsilon$-DP*.

###### Definition 2 (Local Differential Privacy, LDP (Joseph et al.,, 2019 ) ) .

A family of randomized mappings $R:X\to Y$ is *$(\epsilon,\delta)$-locally differentially private* if, for every pair of inputs $x,x^{\prime}\in X$ and every measurable subset $E\subseteq Y$,

$$ $\Pr\bigl[R(x)\in E\bigr]\leq e^{\epsilon}\,\Pr\bigl[R(x^{\prime})\in E\bigr]+\delta.$ $$

Under CDP, a trusted curator collects the raw data and adds noise before release; this model simplifies algorithm design and typically incurs only an $\mathcal{O}(1/n)$ loss in accuracy (Cai et al.,, 2021), where $n$ denotes the sample size. In contrast, LDP dispenses with any trust assumption: each user $i$ holds a private value $X_{i}$, applies a predetermined randomized mechanism $R_{i}$ satisfying $(\epsilon,\delta)$-DP, and submits only the perturbed output. We adopt the non‐adaptive LDP model, in which all randomizers $\{R_{i}\}$ are fixed in advance (Cheu et al.,, 2019, Definitions 2.3 and 2.6). Consequently, inference must proceed solely from locally privatized data.

In the CDP setting, the privatized estimator $\widehat{\theta}_{\mathrm{CDP}}$ typically satisfies
$\widehat{\theta}_{\mathrm{CDP}}-\theta=\mathcal{O}_{p}\bigl(n^{-1}\bigr),$
thus after $\sqrt{n}$‐scaling, it shares the same asymptotic distribution as the non‐private estimator, and one can recover its asymptotic variance with modest additional privacy cost. Under LDP, however, the error rate degrades to
$\widehat{\theta}_{\mathrm{LDP}}-\theta=\mathcal{O}_{p}\bigl(n^{-1/2}\bigr),$
which both alters the limiting law and inflates the asymptotic variance. Moreover, because only privatized data are available, consistently estimating this variance from data collected solely for point estimation is generally infeasible.

We consider a federated learning framework involving $K$ clients, each independently holding a local dataset
i.i.d. drawn from an unknown distribution $\mathcal{P}_{k}$ with cumulative distribution function (CDF) $F_{k}$ and density function $f_{k}$ (Li et al.,, 2022). The goal is to collaboratively estimate the global quantile via weighted loss, i.e., the objective is to solve the following optimization problem:

$$ $\displaystyle\mathop{\arg\min}_{Q\in{\Theta}}\mathcal{L}(Q)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\mathop{\arg\min}_{Q\in{\Theta}}\sum_{k=1}^{K}p_{k}\mathcal{L}_{\tau_{k}}(Q)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\mathop{\arg\min}_{Q\in{\Theta}}\sum_{k=1}^{K}p_{k}\mathbb{E}_{x_{k}\sim\mathcal{P}_{k}}\{\ell_{\tau_{k}}(x_{k},Q)\},$ (2.1) $$

where $p_{k}$ denotes the weight assigned to client $k$, $\tau_{k}\in(0,1)$ is local quantile level, $x_{k}$ is the sample generated from $\mathcal{P}_{k}$, and $\ell_{\tau_{k}}(x,Q)$ represents the check loss function defined as:

$$ $\displaystyle\ell_{\tau_{k}}(x,Q)=(x-Q)(\tau_{k}-\mathbb{I}(x<Q)),$ (2.2) $$

where $\mathbb{I}(\cdot)$ is the indicator function. Let $\tau:=\sum_{k=1}^{K}p_{k}\tau_{k}\in(0,1)$. For the global minimizer $Q^{\star}$ of ([2.1](https://arxiv.org/html/2509.21800v1#S2.E1)), it corresponds to the global quantile at level $\tau$ of a weighted CDF, i.e., $\sum_{k=1}^{K}p_{k}F_{k}(Q^{\star})=\tau$. In the following, we denote $F_{k}(Q^{\star})=Q_{k}$, and considers the parameter space ${\Theta}$ is bounded; see Gu and Chen, (2023).

As noted in introduction, to improve the communication efficiency, we consider a local SGD based estimator, for communication iteration sets $\mathcal{I}=\{t_{0},t_{1},\dots,t_{T}\}$, the global server will receive the local iterations and broadcast the update to $K$ clients, otherwise, the iterations are only conducted in each local clients, i.e., for $k=1,\dots K$,

$$ $\displaystyle q_{t+1}^{k}=\begin{cases}q_{t}^{k}-\eta_{t}\left\{\mathbb{I}(x_{t}^{k}<q_{t}^{k})-\tau_{k}\right\},\penalty 10000\ \penalty 10000\ &t\notin\mathcal{I},\\ \sum_{k=1}^{K}p_{k}\left[q_{t}^{k}-\eta_{t}\left\{\mathbb{I}(x_{t}^{k}<q_{t}^{k})-\tau_{k}\right\}\right],\penalty 10000\ \penalty 10000\ &t\in\mathcal{I}.\end{cases}$ $$

Here $\eta_{t}$ is the pre-determined learning rate, and $x_{t}^{k}$ represents an independent realization of $\mathcal{P}_{k}$,
The final estimator is Polyak-Ruppert type, which averages the historical iterations,

$$ $\displaystyle\widetilde{Q}_{T}=\frac{1}{t_{T}}\sum_{m=1}^{T}\sum_{k=1}^{K}p_{k}q_{t_{m}}^{k}.$ $$

The communication and statistical efficiency are determined by the interval length $E_{m}:=t_{m}-t_{m-1}$ for $m\in\mathbb{N}^{+}$. If $E_{m}=1$, the local clients must communicate with the global server at every iteration. In this scenario, the approach reduces to parallel SGD, which, as noted by Li et al., (2022), may achieve the Cramér-Rao lower bound and thus serve as an efficient estimator for certain smooth loss functions. Conversely, if $E_{m}=n$, implying only one communication at the last iteration, the estimator degenerates to a divide-and-conquer estimator. Such an estimator is consistent only when $\tau_{k}\equiv\tau$ for all $k=1,\dots,K$ and some common $\tau\in(0,1)$. In this case, minimizing the loss function ([2.1](https://arxiv.org/html/2509.21800v1#S2.E1)) becomes a distributed learning problem. However, as pointed out by Gu and Chen, (2023), the divide-and-conquer estimator may still be statistically inefficient for certain weight choices. Therefore, a careful balance must be struck between communication and statistical efficiency. For a general positive interval $E_{m}>0$, the local SGD method allows us to find an appropriate choice of $E_{m}$ that can ensure an optimal trade-off between these efficiencies.

On the other hand, the data collected from each client may be subject to privacy protection policies, particularly in surveys involving sensitive information such as income or health status. For the local quantile loss function ([2.2](https://arxiv.org/html/2509.21800v1#S2.E2)), we observe that the structure of its gradient resembles a binary response. This motivates us to incorporate an LDP mechanism based on randomized response and permutation, following the framework of Liu et al., 2023b, with a truthful response rate $r_{k}\in(0,1]$. Specifically, the mechanism allows each local client to either return a true gradient with probability $r_{k}$ or a synthetic Bernoulli random variable with probability $1-r_{k}$. This iterative mechanism ensures $\epsilon_{k}$-LDP, where the privacy parameter is given by $\epsilon_{k}=\log(1+r_{k})-\log(1-r_{k})$, as established in Liu et al., 2023b.

It is worth noting that the method of Liu et al., 2023b was originally developed in a single-machine setting. Extending it directly to federated learning raises new challenges, since federated systems inherently involve the issue of heterogeneity.
We illustrate with a simple example. Consider collaboratively estimating the national median annual income using state-level data from the United States, where each state is treated as a client. First, income distributions typically vary across states (see Figure [1](https://arxiv.org/html/2509.21800v1#S2.F1)(i)). Second, privacy preferences can differ across states due to cultural norms and development levels (Milberg et al.,, 2000; Bellman et al.,, 2004). Figure [1](https://arxiv.org/html/2509.21800v1#S2.F1)(ii) shows how the released information can vary under different privacy budgets. Due to such heterogeneity, a naive combination of local LDP estimators from Liu et al., 2023b may result in severely biased results.
To address this problem, we propose a novel federated quantile estimation algorithm under LDP, equipped with a carefully designed local SGD updating rule. This method accommodates heterogeneous data distributions, quantile targets, and privacy budgets across clients while maintaining a common global target. The complete procedure is summarized in Algorithm [1](https://arxiv.org/html/2509.21800v1#alg1), and we denote the resulting estimator as $\widehat{Q}_{T}$.

Figure: Algorithm 1 Federated quantile estimation with local SGD under LDP

In our proposed Algorithm [1](https://arxiv.org/html/2509.21800v1#alg1), each iteration integrates global information (the global quantile $\tau$) with client-specific privacy budget ($r_{k}$), thereby correcting bias arising from the aggregation of heterogeneous local LDP mechanisms and loss functions. The following theorem shows that this procedure effectively reduces the LDP inference problem to its non-private analogue.

Figure: (i) Data heterogeneity
Refer to caption: x1.png

###### Theorem 2.1 .

Denote $\widetilde{\tau}_{k}=r_{k}\tau+(1-r_{k})/2$. For a privacy budget $\epsilon_{k}=\log(1+r_{k})-\log(1-r_{k})$, there exists a dataset
consisting of i.i.d. samples drawn from some distribution $\widetilde{\mathcal{P}}_{k},1\leq k\leq K$, such that solving the federated loss ([2.1](https://arxiv.org/html/2509.21800v1#S2.E1)) with $\epsilon_{k}$-LDP using data
drawn from $\mathcal{P}_{k}$ is equivalent to solving the following non-private problem:

$$ $\displaystyle\mathop{\arg\min}_{Q}\mathcal{L}(Q)=\mathop{\arg\min}_{Q}\widetilde{\mathcal{L}}(Q):={\color[rgb]{0,0,0}\mathop{\arg\min}_{Q}\sum_{k=1}^{K}p_{k}\mathbb{E}_{x_{k}\sim\widetilde{\mathcal{P}}_{k}}\left\{r_{k}^{-1}\ell_{\widetilde{\tau}_{k}}(x_{k},Q)\right\}.}$ (2.3) $$

Therefore, by Theorem [2.1](https://arxiv.org/html/2509.21800v1#S2.Thmtheorem1), the LDP federated quantile estimation problem can be reformulated as a non-DP federated quantile estimation task under modified distributions and shifted quantile levels. The main challenge then becomes analyzing the statistical properties of the resulting non-DP estimator, particularly in the presence of the non-smooth quantile loss function.

###### Definition 1 (Central Differential Privacy, CDP (Dwork et al.,, 2006 ) ) .

###### Definition 2 (Local Differential Privacy, LDP (Joseph et al.,, 2019 ) ) .

###### Theorem 2.1 .

## 3 Asymptotic analysis

In this section, we focus on the asymptotic analysis of the proposed LDP estimator and the practical construction of confidence intervals. Before presenting the main results, we first introduce several necessary assumptions.

###### Assumption 1 .

For some constant $C_{f}>0$, $f_{k}(\cdot)$, $1\leq k\leq K$, is uniformly bounded by $C_{f}$.

###### Assumption 2 .

Define the effective step $\gamma_{m}=\eta_{m}E_{m}$, which is non-increasing in $m$ and satisfies that $\sum_{m=1}^{\infty}\gamma_{m}^{2}<\infty$, $\sum_{m=1}^{\infty}\gamma_{m}=\infty$, and ${(\gamma_{m}-\gamma_{m+1})}/{\gamma_{m}}={\scriptstyle{\mathcal{O}}}(\gamma_{m}).$

###### Assumption 3 .

The sequence $\{E_{m}\}_{m\geq 1}$ satisfies that

- (a)
$\{E_{m}\}_{m\geq 1}$ is either uniformly bounded or non-decreasing.
- (b)
There exist some $\delta>0$ and $\nu\geq 1$ such that
$\displaystyle\limsup_{T\to\infty}\frac{1}{T^{2}}\left(\sum_{m=0}^{T-1}E_{m}^{1+\delta}\right)\left(\sum_{m=0}^{T-1}E_{m}^{-1-\delta}\right)<\infty,\lim_{T\to\infty}\frac{1}{T^{2}}\left(\sum_{m=0}^{T-1}E_{m}\right)\left(\sum_{m=0}^{T-1}E_{m}^{-1}\right)=\nu.$
- (c)
Denote $t_{T}=\sum_{m=0}^{T-1}E_{m}$, satisfying
$\displaystyle\lim_{T\to\infty}\frac{\sqrt{t_{T}}}{T}\sum_{m=0}^{T}\gamma_{m}=0,\quad\lim_{T\to\infty}\frac{\sqrt{t_{T}}}{T}\frac{1}{\sqrt{\gamma_{T}}}=0$

Assumption 1 is a mild and regular condition concerning the uniform boundedness of density functions. Assumptions 2 and 3 require that the effective step sizes decay slowly and the communication intervals increase slowly; see also Li et al., (2022).

###### Theorem 3.1 .

Under Assumptions 1-3, as $T\to\infty$, the proposed LDP federated estimator enjoys

$$ $\displaystyle\sqrt{t_{T}}(\widehat{Q}_{T}-Q^{\star})\xrightarrow{d}N\left(0,{\color[rgb]{0,0,0}\nu\frac{\sum_{k=1}^{K}p_{k}^{2}\left\{r_{k}^{-2}-(2Q_{k}-1)^{2}\right\}}{4\left(\sum_{k=1}^{K}p_{k}f_{k}(Q^{\star})\right)^{2}}}\right).$ $$

Theorem [3.1](https://arxiv.org/html/2509.21800v1#S3.Thmtheorem1) establishes the asymptotic normality of the estimator $\widehat{Q}_{T}$, which theoretically allows for the theoretical construction of a confidence interval for $Q^{\star}$. However, the construction involves unknown quantities, such as the individual quantiles $Q_{k}$ and the density values $f_{k}(Q^{\star})$. Even in cases where $Q_{k}=\tau_{k}$ is known, the estimation of $f_{k}(Q^{\star})$ remains challenging. In particular, it is difficult to recover these density values using only the perturbed gradients available from Algorithm [1](https://arxiv.org/html/2509.21800v1#alg1). Moreover, in SGD-based methods, consistent variance estimation typically relies on the Hessian matrix, which is well-defined only for smooth loss functions, as previously discussed. Therefore, although Theorem [3.1](https://arxiv.org/html/2509.21800v1#S3.Thmtheorem1) provides a theoretically valid basis for confidence interval construction, it is not practically implementable due to these limitations.

Inspired by the quantile inference framework for single clients in Liu et al., 2023b, it is necessary to strengthen the pointwise result of Theorem [3.1](https://arxiv.org/html/2509.21800v1#S3.Thmtheorem1) to a functional version.

###### Theorem 3.2 .

Under Assumptions 1–3, as $T\to\infty$, we have

$$ $\mathcal{Q}_{T}({\color[rgb]{0,0,0}s}):=\frac{\sqrt{t_{T}}}{T}\sum_{m=1}^{h({\color[rgb]{0,0,0}s},T)}\left(\bar{q}_{t_{m}}-Q^{\star}\right)\xrightarrow{d}{\color[rgb]{0,0,0}\frac{\sqrt{\nu\sum_{k=1}^{K}p_{k}^{2}\left\{r_{k}^{-2}-(2Q_{k}-1)^{2}\right\}}}{2\sum_{k=1}^{K}p_{k}f_{k}(Q^{\star})}B(s)},$ $$

where $t_{T}=\sum_{m=0}^{T-1}E_{m}$, $\bar{q}_{t_{m}}=\sum_{k=1}^{K}p_{k}r_{k}q_{t_{m}}^{k}$, $B(\cdot)$ is a standard Brownian motion on $[0,1]$, and

$$ $h({\color[rgb]{0,0,0}s},T)=\max\left\{n\in\mathbb{Z}_{>0}\,\middle|\,{\color[rgb]{0,0,0}s}\sum_{m=1}^{T}\frac{1}{E_{m}}\geq\sum_{m=1}^{n}\frac{1}{E_{m}}\right\},\quad\text{for }{\color[rgb]{0,0,0}s}\in(0,1].$ $$

Theorem [3.2](https://arxiv.org/html/2509.21800v1#S3.Thmtheorem2) establishes a functional central limit theorem (FCLT) for $Q_{T}(s)$ over $s\in(0,1]$, showing that it converges weakly in the $\ell^{\infty}[0,1]$ (the space of bounded real-valued functions) to a Brownian motion, which is our another major theoretical contribution. Note that the sample quantile loss doesn’t satisfy the common $L$-average smooth conditions for weakly convergence result, such in (Li et al.,, 2022; Xie et al.,, 2024; Zhu et al.,, 2024), leading to extra challenge in deriving the almost sure and $\mathcal{L}^{2}$ convergence rates of $\bar{q}_{t_{m}}$, which are essential for handling the asymptotically negligible terms. Theorem [3.1](https://arxiv.org/html/2509.21800v1#S3.Thmtheorem1) arises as a special case of Theorem [3.2](https://arxiv.org/html/2509.21800v1#S3.Thmtheorem2) when $s=1$. Building on Theorem [3.2](https://arxiv.org/html/2509.21800v1#S3.Thmtheorem2), we proceed to construct a self-normalized test statistic and derive its asymptotic pivotal distribution via the continuous mapping theorem.

Define $r_{0}=0$ and, for $m\geq 1$,
$r_{m}=\left(\sum_{i=1}^{m}1/E_{i}\right)\left(\sum_{i=1}^{T}1/E_{i}\right)^{-1},$
which ensures that

$$ $\mathcal{Q}_{T}(r_{m})=\frac{\sqrt{t_{T}}}{T}\sum_{i=1}^{m}\left(\bar{q}_{t_{i}}-Q^{\star}\right),\,\,\,\text{and in particular, }\,\,\,\mathcal{Q}_{T}(1)=\frac{\sqrt{t_{T}}}{T}\sum_{i=1}^{T}\left(\bar{q}_{t_{i}}-Q^{\star}\right).$ $$

Following the arguments in (Shao,, 2015), once a functional central limit theorem such as Theorem [3.2](https://arxiv.org/html/2509.21800v1#S3.Thmtheorem2) is established, one can construct a self-normalized statistic that asymptotically enjoys a pivotal distribution. Specifically, define

$$ $\displaystyle\mathcal{V}_{T}=\sum_{m=1}^{T}\left(r_{m}-r_{m-1}\right)\left(\mathcal{Q}_{T}(r_{m})-\frac{m}{T}\mathcal{Q}_{T}(1)\right)^{2}.$ (3.1) $$

###### Corollary 3.1 .

Suppose Assumptions 1-3 hold and $g(r_{m})\asymp m/T$ for some continuous function $g$ on $[0,1]$. Then, as $T\to\infty$,

$$ $\frac{\mathcal{Q}_{T}(1)}{\sqrt{\mathcal{V}_{T}}}\xrightarrow{d}\frac{B(1)}{\sqrt{\int_{0}^{1}\left(B(r)-g(r)B(1)\right)^{2}dr}}.$ $$

Corollary [3.1](https://arxiv.org/html/2509.21800v1#S3.Thmcorollary1) presents the asymptotic distribution of the self-normalized statistic $Q_{T}(1)/\mathcal{V}_{T}$, which is distribution-free. As a result, there is no need to allocate additional DP budget to estimate nuisance parameters when constructing confidence intervals.

The selection of the self-normalizer is not unique, and an appropriate norm of the Gaussian process $B(r)-g(r)B(1)$ can yield similar results to those in Corollary [3.1](https://arxiv.org/html/2509.21800v1#S3.Thmcorollary1). For example, using the supremum norm and the $\mathcal{L}_{1}$ norm, one can define alternative self-normalizers as follows:

$$ $\displaystyle\mathcal{V}_{T}^{\prime}=\sup_{1\leq m\leq T}\left|\mathcal{Q}_{T}(r_{m})-\frac{m}{T}\mathcal{Q}_{T}(1)\right|,\quad\mathcal{V}_{T}^{\prime\prime}=\sum_{m=1}^{T}\left(r_{m}-r_{m-1}\right)\left|\mathcal{Q}_{T}(r_{m})-\frac{m}{T}\mathcal{Q}_{T}(1)\right|,$ $$

which are related to the processes $\sup_{0\leq r\leq 1}|B(r)-g(r)B(1)|$ and $\int_{0}^{1}|B(r)-g(r)B(1)|\,dr$, respectively. However, the self-normalizer defined in equation ([3.1](https://arxiv.org/html/2509.21800v1#S3.E1)) enjoys greater computational efficiency, as the $\mathcal{L}_{2}$ norm can be computed in an online manner, as described in Algorithm [2](https://arxiv.org/html/2509.21800v1#alg2). Let $\widehat{\mathcal{V}}_{T}$ denote the estimator of the self-normalizer in ([3.1](https://arxiv.org/html/2509.21800v1#S3.E1)), and let $v_{\alpha/2,g}$ be the $(1-\alpha/2)$ quantile of the random variable
$B(1)/\left(\int_{0}^{1}\left(B(r)-g(r)B_{1}(1)\right)^{2}dr\right)^{1/2}$.
The following corollary ensures the asymptotic validity of the constructed LDP confidence interval.

The following Corollary [3.2](https://arxiv.org/html/2509.21800v1#S3.Thmcorollary2) ensures the asymptotic validity of the constructed LDP confidence interval.

###### Corollary 3.2 .

Suppose the same conditions in Theorem [3.2](https://arxiv.org/html/2509.21800v1#S3.Thmtheorem2) hold, as $T\to\infty$, one has that

$$ $\displaystyle\mathbb{P}\left(\widehat{Q}_{T}-v_{\frac{\alpha}{2},g}\sqrt{\widehat{\mathcal{V}}_{T}}\leq Q^{\star}\leq\widehat{Q}_{T}+v_{\frac{\alpha}{2},g}\sqrt{\widehat{\mathcal{V}}_{T}}\right)\rightarrow 1-\alpha$ $$

Figure: Algorithm 2 Online Inference

###### Assumption 1 .

###### Assumption 2 .

###### Assumption 3 .

###### Theorem 3.1 .

###### Theorem 3.2 .

###### Corollary 3.1 .

###### Corollary 3.2 .

## 4 Experiments

### 4.1 Simulation setup

We first evaluate our proposed method through extensive simulation studies using synthetic data. In all experiments, we fixed $p_{k}=1/K$ for $1\leq k\leq K$, the number of clients is fixed at $K=10$. The quantile levels examined range from $0.3$ to $0.8$, and the truthful response rates vary between $0.25$ and $0.9$. We focus on the following four scenarios of heterogeneity:

- •
heterogeneous quantile levels: We investigate two distinct scenarios: (1) Case $\tau_{\operatorname{low}}$: lower quantile levels, where each client is assigned a unique quantile level $\tau_{k}$ ranging uniformly from $0.3$ to $0.5$; and (2) Case $\tau_{\operatorname{high}}$: higher quantile levels, where $\tau_{k}$ ranges uniformly from $0.5$ to $0.8$.
- •
heterogeneous response rates. Each client has a unique truthful response rate $r_{k}$, ranging uniformly from 0.25 to 0.9.
- •
heterogeneous locations (Hete L). Data for each client $k$ are independently generated from $\mathcal{N}(\mu_{k},1)$, where $\mu_{k}\sim\mathcal{N}(0,1)$.
- •
heterogeneous distribution families (Hete D). Data are generated independently across ten clients, with three drawing from $\mathcal{N}(0,1)$, three from the uniform distribution $\mathcal{U}(-1,1)$, and four from a standard Cauchy distribution $\mathcal{C}(0,1)$.

We set the step size $\gamma_{m}$ as: $\gamma_{m}=20\bar{r}/(m^{0.51}+100)$, with $\gamma_{m}=E_{m}\eta_{m}$ and $\bar{r}=K^{-1}\sum_{k=1}^{K}r_{k}$.
Following Li et al., (2022), we implement a warm-up phase, setting the communication interval $E_{m}=1$ for the first 5% of iterations. After the warm-up period, we redefine the interval sequence $\{E_{m}\}$ based on a new sequence $\{E_{m}^{\prime}\}$, specifically:
$E_{m}=E_{m-0.05\cdot T}^{\prime}$. We examine three different interval strategies for $E_{m}^{\prime}$: (1) C1: $E_{m}^{\prime}\equiv 1$ (equivalent to P-SGD),
(2) C5: $E_{m}^{\prime}\equiv 5$, and
(3) Log: $E_{m}^{\prime}=\lceil\log_{2}(m+1)\rceil$. The initial parameter estimates are set to $q_{0}^{k}=q_{0}\sim\mathcal{N}(0,1)$ for all clients $k$.
All experimental settings are replicated $R=1,000$ times. The simulations are conducted on computational resources comprising 36 Intel Xeon Gold 6271 CPUs, with a total of 128GB RAM and 500GB storage.

### 4.2 Simulation results

We first illustrate the performance of our proposed method by presenting sample iteration trajectories for estimation and inference. Specifically, we randomly select one simulation run and plot the resulting estimates and corresponding confidence intervals against $t_{T}$ (Figure [2](https://arxiv.org/html/2509.21800v1#S4.F2)). The results demonstrate that our approach accurately captures the true quantile value and provides reliable inference. Subsequently, we fix the total sample size $t_{T}$ at $10,000$ and $50,000$ and evaluate the finite sample performance under different settings. Let $\widehat{Q}^{(r)}_{T}$ denote the quantile estimator and $\operatorname{CI}^{(r)}$ represent the corresponding 95% confidence interval obtained from Algorithm [2](https://arxiv.org/html/2509.21800v1#alg2) in the $r$-th simulation. We consider two metrics: the mean absolute error (MAE), defined as $R^{-1}\sum_{r=1}^{R}|\widehat{Q}^{(r)}_{T}-Q^{*}|$, and the empirical coverage probability (ECP), defined as $R^{-1}\sum_{r=1}^{R}\mathbb{I}(Q^{*}\in\operatorname{CI}^{(r)})$. For comparison, we also consider two alternative methods:
(1) the DP-SGD method (Song et al.,, 2013), which adds noise directly to the gradients instead of introducing DP through randomized response. To align with the original paper’s setup, we focus on the case with $C=1$. In this regime, the gradient‐descent update in Algorithm [1](https://arxiv.org/html/2509.21800v1#alg1) becomes

$$ $q_{t}^{k}\;\leftarrow\;q_{t-1}^{k}+\eta_{t-1}\Big\{\tau_{k}\mathbb{I}(x_{t}^{k}>q_{t-1}^{k})-(1-\tau_{k})\mathbb{I}(x_{t}^{k}<q_{t-1}^{k})+Z_{t}^{k}\Big\},$ $$

where $Z_{t}^{k}$ is drawn from a Laplace distribution. A simple calculation shows that $Z_{t}^{k}$ has mean zero and scale parameter
$1/\log\left\{(1+r_{k})/(1-r_{k})\right\}$.
(2) the divide-and-conquer (DC) method, which corresponds to the special case $E_{m}=n$. Here we use step size $\eta_{t}=2\bar{r}/(t^{0.51}+100)$ (Goyal et al.,, 2017). The numerical results for all of the methods are reported in Tables [1](https://arxiv.org/html/2509.21800v1#S4.T1) and [2](https://arxiv.org/html/2509.21800v1#S4.T2).

From Tables [1](https://arxiv.org/html/2509.21800v1#S4.T1) and [2](https://arxiv.org/html/2509.21800v1#S4.T2), we observe that our method consistently achieves ECP close to or exceeding the nominal 95% level across all scenarios. As either the total sample size $t_{T}$ or the truthful response rate increases, the MAE decreases, which aligns with our theoretical results. Comparing the three interval strategies, we find that the C1 strategy (P-SGD) yields the smallest MAE, as it has the highest communication frequency. Comparing with the two competing methods, we find that the DC approach results in the largest errors. Notably, in certain heterogeneous cases, such as Hete L with $\tau=0.8$, the DC estimator exhibits significant bias and an ECP far below the nominal 95% level. In contrast, our proposed estimators successfully achieve approximately 95% empirical coverage in these cases. Moreover, while DP-SGD attains empirical coverage probabilities close to or even exceeding 95% in most settings, its MAE remain uniformly larger than those of our method.
To further illustrate the communication efficiency of our method, we also consider scenarios with a fixed number of communication rounds $T$. The results are summarized in Tables [3](https://arxiv.org/html/2509.21800v1#S4.T3) and [4](https://arxiv.org/html/2509.21800v1#S4.T4). We observe that our proposed method continues to provide valid inference. Additionally, under fixed communication rounds, the Log strategy generally achieves the best performance, yielding the smallest MAE.

Figure: Figure 2: Sample trajectories of the iterative estimator and corresponding confidence intervals under heterogeneous distributions (Hete L, with $r_{k}=0.9$ and $\tau=0.5$, left panel) and heterogeneous quantile levels ($\tau_{\operatorname{low}}$, with heterogeneous response rates, right panel). The horizontal dotted line indicates the true quantile value $Q^{*}$.
Refer to caption: x3.png

**Table 1: Empirical coverage probabilities (mean absolute errors) under varying quantile levels and response rates, with different $t_{T}$ and fixed $K=10$ clients and data generated from $\mathcal{N}(0,1)$. In Case $\tau_{\operatorname{low}}$, each client uses a unique quantile level $\tau_{k}$ ranging uniformly from $[0.3,0.5]$; in Case $\tau_{\operatorname{high}}$, $\tau_{k}$ is ranging from $[0.5,0.8]$. “hetero” indicates client-specific truthful response rates $r_{k}$ range from $[0.25,0.9].$**
| Quantile ($\tau$) | Rate ($r$) | C1 | C5 | Log | DP-SGD (C1) | DC |
| --- | --- | --- | --- | --- | --- | --- |
| $t_{T}=10000$ |  |  |  |  |  |  |
| 0.5 | 0.25 | 0.949(0.0133) | 0.967(0.0244) | 0.992(0.0360) | 0.949(0.0191) | 0.939(0.2503) |
| 0.5 | hetero | 0.963(0.0071) | 0.989(0.0112) | 0.997(0.0161) | 0.955(0.0100) | 1.000(0.0497) |
| 0.5 | 0.9 | 0.995(0.0023) | 1.000(0.0054) | 1.000(0.0082) | 0.980(0.0036) | 1.000(0.0158) |
| $\tau_{\operatorname{low}}$ | 0.25 | 0.947(0.0136) | 0.982(0.0253) | 0.990(0.0369) | 0.940(0.0200) | 0.969(0.2616) |
| $\tau_{\operatorname{low}}$ | hetero | 0.962(0.0072) | 0.993(0.0113) | 0.997(0.0162) | 0.949(0.0105) | 0.999(0.0530) |
| $\tau_{\operatorname{low}}$ | 0.9 | 0.999(0.0020) | 1.000(0.0055) | 1.000(0.0083) | 0.985(0.0036) | 1.000(0.0162) |
| $\tau_{\operatorname{high}}$ | 0.25 | 0.939(0.0145) | 0.987(0.0268) | 0.986(0.0399) | 0.952(0.0210) | 0.984(0.2771) |
| $\tau_{\operatorname{high}}$ | hetero | 0.968(0.0076) | 0.987(0.0126) | 0.999(0.0182) | 0.956(0.0111) | 1.000(0.0516) |
| $\tau_{\operatorname{high}}$ | 0.9 | 0.996(0.0023) | 1.000(0.0067) | 1.000(0.0102) | 0.980(0.0038) | 1.000(0.0172) |
| $t_{T}=50000$ |  |  |  |  |  |  |
| 0.5 | 0.25 | 0.956(0.0056) | 0.982(0.0081) | 0.996(0.0122) | 0.949(0.0081) | 0.988(0.0571) |
| 0.5 | hetero | 0.960(0.0032) | 0.979(0.0044) | 0.992(0.0064) | 0.950(0.0046) | 1.000(0.0115) |
| 0.5 | 0.9 | 1.000(0.0018) | 0.988(0.0027) | 0.990(0.0036) | 0.983(0.0021) | 1.000(0.0038) |
| $\tau_{\operatorname{low}}$ | 0.25 | 0.957(0.0061) | 0.981(0.0083) | 0.994(0.0125) | 0.944(0.0091) | 0.993(0.0594) |
| $\tau_{\operatorname{low}}$ | hetero | 0.953(0.0036) | 0.981(0.0046) | 0.990(0.0066) | 0.934(0.0054) | 0.999(0.0121) |
| $\tau_{\operatorname{low}}$ | 0.9 | 1.000(0.0019) | 1.000(0.0026) | 0.989(0.0038) | 0.988(0.0024) | 1.000(0.0057) |
| $\tau_{\operatorname{high}}$ | 0.25 | 0.968(0.0059) | 0.986(0.0086) | 0.997(0.0133) | 0.946(0.0089) | 0.999(0.0620) |
| $\tau_{\operatorname{high}}$ | hetero | 0.953(0.0032) | 0.990(0.0045) | 0.998(0.0065) | 0.952(0.0047) | 0.993(0.0154) |
| $\tau_{\operatorname{high}}$ | 0.9 | 0.998(0.0010) | 0.999(0.0023) | 1.000(0.0034) | 0.977(0.0016) | 0.938(0.0132) |

**Table 2: Empirical coverage probabilities (mean absolute errors) under heterogeneous distributions for different $t_{T}$. The number of clients $K$ is fixed at 10. In Hete L, data for each client $k$ are independently generated from $\mathcal{N}(\mu_{k},1)$, where $\mu_{k}\sim\mathcal{N}(0,1)$. In Hete D, data are generated from $\mathcal{N}(0,1)$, $\mathcal{U}(-1,1)$, and $\mathcal{C}(0,1)$ across different clients. “hetero” indicates client-specific truthful response rates $r_{k}$ range from $[0.25,0.9].$**
| Quantile ($\tau$) | Rate ($r$) | C1 | C5 | Log | DP-SGD (C1) | DC |
| --- | --- | --- | --- | --- | --- | --- |
| Hete L — $t_{T}=10000$ |  |  |  |  |  |  |
| 0.3 | 0.25 | 0.958(0.0184) | 0.981(0.0311) | 0.990(0.0452) | 0.942(0.0260) | 0.985(0.3066) |
| 0.3 | hetero | 0.949(0.0096) | 0.982(0.0150) | 0.993(0.0205) | 0.947(0.0142) | 0.898(0.1302) |
| 0.3 | 0.9 | 1.000(0.0029) | 1.000(0.0066) | 1.000(0.0100) | 0.981(0.0049) | 0.215(0.1273) |
| 0.5 | 0.25 | 0.950(0.0165) | 0.984(0.0315) | 0.988(0.0465) | 0.953(0.0224) | 1.000(0.2822) |
| 0.5 | hetero | 0.952(0.0085) | 0.991(0.0155) | 0.998(0.0221) | 0.955(0.0119) | 1.000(0.0525) |
| 0.5 | 0.9 | 0.996(0.0025) | 0.999(0.0078) | 1.000(0.0120) | 0.984(0.0041) | 1.000(0.0186) |
| 0.8 | 0.25 | 0.966(0.0237) | 0.995(0.0512) | 0.992(0.0791) | 0.957(0.0328) | 0.892(0.6152) |
| 0.8 | hetero | 0.962(0.0122) | 0.995(0.0227) | 0.996(0.0347) | 0.943(0.0186) | 0.709(0.2684) |
| 0.8 | 0.9 | 0.990(0.0042) | 1.000(0.0116) | 1.000(0.0185) | 0.968(0.0065) | 0.049(0.2098) |
| Hete L — $t_{T}=50000$ |  |  |  |  |  |  |
| 0.3 | 0.25 | 0.937(0.0089) | 0.981(0.0111) | 0.990(0.0165) | 0.916(0.0135) | 0.949(0.1328) |
| 0.3 | hetero | 0.911(0.0056) | 0.981(0.0056) | 0.997(0.0080) | 0.885(0.0083) | 0.093(0.1282) |
| 0.3 | 0.9 | 0.977(0.0034) | 1.000(0.0019) | 1.000(0.0030) | 0.908(0.0041) | 0.000(0.1290) |
| 0.5 | 0.25 | 0.958(0.0069) | 0.988(0.0098) | 0.995(0.0147) | 0.949(0.0099) | 1.000(0.0609) |
| 0.5 | hetero | 0.964(0.0035) | 0.994(0.0048) | 0.996(0.0069) | 0.957(0.0052) | 0.997(0.0145) |
| 0.5 | 0.9 | 1.000(0.0010) | 1.000(0.0016) | 1.000(0.0026) | 0.993(0.0018) | 0.979(0.0143) |
| 0.8 | 0.25 | 0.956(0.0102) | 0.991(0.0144) | 0.998(0.0226) | 0.931(0.0160) | 0.799(0.2829) |
| 0.8 | hetero | 0.950(0.0055) | 0.992(0.0072) | 0.997(0.0112) | 0.923(0.0092) | 0.014(0.2034) |
| 0.8 | 0.9 | 1.000(0.0013) | 1.000(0.0053) | 0.999(0.0082) | 0.985(0.0026) | 0.000(0.1929) |
| Hete D — $t_{T}=10000$ |  |  |  |  |  |  |
| 0.5 | 0.25 | 0.949(0.0132) | 0.985(0.0243) | 0.986(0.0354) | 0.953(0.0183) | 0.904(0.2496) |
| 0.5 | hetero | 0.966(0.0069) | 0.990(0.0117) | 0.989(0.0172) | 0.955(0.0098) | 0.999(0.0488) |
| 0.5 | 0.9 | 1.000(0.0023) | 1.000(0.0074) | 1.000(0.0117) | 0.991(0.0035) | 1.000(0.0163) |
| Hete D — $t_{T}=50000$ |  |  |  |  |  |  |
| 0.5 | 0.25 | 0.958(0.0057) | 0.980(0.0082) | 0.993(0.0127) | 0.943(0.0081) | 0.981(0.0589) |
| 0.5 | hetero | 0.966(0.0030) | 0.988(0.0046) | 0.998(0.0073) | 0.950(0.0041) | 1.000(0.0111) |
| 0.5 | 0.9 | 0.999(0.0008) | 1.000(0.0029) | 1.000(0.0052) | 0.990(0.0014) | 1.000(0.0037) |

**Table 3: ECP (MAE) under varying quantile levels and response rates, with different $T$ and fixed $K=10$ clients and data generated from $\mathcal{N}(0,1)$. In Case $\tau_{\operatorname{low}}$, each client uses a unique quantile level $\tau_{k}$ ranging uniformly from $[0.3,0.5]$; in Case $\tau_{\operatorname{high}}$, $\tau_{k}$ is ranging from $[0.5,0.8]$. “hetero” indicates client-specific truthful response rates $r_{k}$ range from $[0.25,0.9].$**
| Quantile ($\tau$) | Rate ($r$) | C1 | C5 | Log |
| --- | --- | --- | --- | --- |
| $T=5000$ |  |  |  |  |
| 0.5 | 0.25 | 0.954(0.0189) | 0.974(0.0129) | 0.986(0.0112) |
| 0.5 | hetero | 0.959(0.0103) | 0.976(0.0065) | 0.995(0.0052) |
| 0.5 | 0.9 | 0.999(0.0033) | 1.000(0.0040) | 1.000(0.0026) |
| $\tau_{\operatorname{low}}$ | 0.25 | 0.957(0.0200) | 0.974(0.0137) | 0.991(0.0116) |
| $\tau_{\operatorname{low}}$ | hetero | 0.957(0.0108) | 0.977(0.0067) | 0.993(0.0053) |
| $\tau_{\operatorname{low}}$ | 0.9 | 1.000(0.0033) | 1.000(0.0040) | 1.000(0.0029) |
| $\tau_{\operatorname{high}}$ | 0.25 | 0.956(0.0212) | 0.975(0.0128) | 0.993(0.0123) |
| $\tau_{\operatorname{high}}$ | hetero | 0.961(0.0112) | 0.984(0.0062) | 0.996(0.0056) |
| $\tau_{\operatorname{high}}$ | 0.9 | 0.998(0.0037) | 0.997(0.0028) | 1.000(0.0031) |
| $T=10000$ |  |  |  |  |
| 0.5 | 0.25 | 0.949(0.0133) | 0.968(0.0078) | 0.987(0.0061) |
| 0.5 | hetero | 0.963(0.0071) | 0.978(0.0037) | 0.991(0.0030) |
| 0.5 | 0.9 | 0.995(0.0023) | 0.999(0.0020) | 0.999(0.0014) |
| $\tau_{\operatorname{low}}$ | 0.25 | 0.947(0.0136) | 0.972(0.0078) | 0.984(0.0064) |
| $\tau_{\operatorname{low}}$ | hetero | 0.962(0.0072) | 0.985(0.0038) | 0.983(0.0033) |
| $\tau_{\operatorname{low}}$ | 0.9 | 0.999(0.0020) | 1.000(0.0016) | 0.967(0.0018) |
| $\tau_{\operatorname{high}}$ | 0.25 | 0.939(0.0145) | 0.974(0.0086) | 0.985(0.0066) |
| $\tau_{\operatorname{high}}$ | hetero | 0.968(0.0076) | 0.988(0.0043) | 0.985(0.0032) |
| $\tau_{\operatorname{high}}$ | 0.9 | 0.996(0.0023) | 0.999(0.0031) | 0.996(0.0014) |

**Table 4: ECP (MAE) under heterogeneous distributions for different $T$. The number of clients $K$ is fixed at 10. In Hete L, data for each client $k$ are independently generated from $\mathcal{N}(\mu_{k},1)$, where $\mu_{k}\sim\mathcal{N}(0,1)$. In Hete D, data are generated from $\mathcal{N}(0,1)$, $\mathcal{U}(-1,1)$, and $\mathcal{C}(0,1)$ across different clients. “hetero” indicates client-specific truthful response rates $r_{k}$ range from $[0.25,0.9].$**
| Quantile ($\tau$) | Rate ($r$) | C1 | C5 | Log |
| --- | --- | --- | --- | --- |
| Hete L — $T=5000$ |  |  |  |  |
| 0.3 | 0.25 | 0.942(0.0271) | 0.960(0.0168) | 0.975(0.0151) |
| 0.3 | hetero | 0.962(0.0131) | 0.966(0.0086) | 0.987(0.0067) |
| 0.3 | 0.9 | 0.998(0.0043) | 0.959(0.0063) | 1.000(0.0033) |
| 0.5 | 0.25 | 0.954(0.0254) | 0.973(0.0154) | 0.990(0.0153) |
| 0.5 | hetero | 0.963(0.0120) | 0.981(0.0072) | 0.991(0.0071) |
| 0.5 | 0.9 | 0.992(0.0042) | 0.998(0.0032) | 1.000(0.0034) |
| 0.8 | 0.25 | 0.954(0.0375) | 0.982(0.0242) | 0.998(0.0248) |
| 0.8 | hetero | 0.968(0.0181) | 0.988(0.0109) | 0.998(0.0116) |
| 0.8 | 0.9 | 0.985(0.0108) | 0.999(0.0070) | 0.982(0.0094) |
| Hete L — $T=10000$ |  |  |  |  |
| 0.3 | 0.25 | 0.958(0.0184) | 0.966(0.0102) | 0.981(0.0083) |
| 0.3 | hetero | 0.949(0.0096) | 0.965(0.0050) | 0.979(0.0040) |
| 0.3 | 0.9 | 1.000(0.0029) | 0.979(0.0022) | 0.867(0.0036) |
| 0.5 | 0.25 | 0.950(0.0165) | 0.974(0.0094) | 0.985(0.0085) |
| 0.5 | hetero | 0.952(0.0085) | 0.976(0.0045) | 0.991(0.0039) |
| 0.5 | 0.9 | 0.996(0.0025) | 0.985(0.0018) | 1.000(0.0016) |
| 0.8 | 0.25 | 0.966(0.0237) | 0.983(0.0163) | 0.990(0.0149) |
| 0.8 | hetero | 0.962(0.0122) | 0.988(0.0088) | 0.974(0.0090) |
| 0.8 | 0.9 | 0.990(0.0042) | 0.997(0.0087) | 0.645(0.0095) |
| Hete D — $T=5000$ |  |  |  |  |
| 0.5 | 0.25 | 0.954(0.0195) | 0.974(0.0129) | 0.987(0.0109) |
| 0.5 | hetero | 0.965(0.0098) | 0.974(0.0075) | 0.993(0.0049) |
| 0.5 | 0.9 | 1.000(0.0037) | 0.989(0.0060) | 1.000(0.0026) |
| Hete D — $T=10000$ |  |  |  |  |
| 0.5 | 0.25 | 0.949(0.0132) | 0.968(0.0078) | 0.982(0.0064) |
| 0.5 | hetero | 0.966(0.0069) | 0.973(0.0039) | 0.972(0.0034) |
| 0.5 | 0.9 | 1.000(0.0023) | 0.999(0.0014) | 0.966(0.0023) |

### 4.3 Real data

In this subsection, we empirically evaluate the effectiveness of our proposed method using a representative real‐world dataset widely employed in privacy research: Government Salary Dataset (Plečko et al.,, 2024). This dataset is sourced from the 2018 American Community Survey conducted by the U.S. Census Bureau and contains over 200,000 records, with annual salary (in USD) as the response variable. Since annual salary represents sensitive financial information (Gillenwater et al.,, 2021), we treat it as requiring privacy protection. To incorporate the dataset’s inherent geographic structure, we partition the sample according to the feature “economic region.” The three smallest regions are merged into a single “Others” category, yielding seven regions in total, each regarded as one client. Because region-level sample sizes vary, we apply oversampling to balance the data, resulting in $t_{T}=53,960$ observations per client. All other hyperparameters follow the settings in Section 4.1. For analysis, we apply a log transformation to the response variable and subsequently back-transform it.

We target quantile levels $\tau_{k}\equiv\tau\in\{0.3,0.5,0.8\}$ and consider response rate ranges from 0.6 to 0.9. For reference, we also compute the full-sample quantiles without LDP. The resulting estimators and confidence‐interval lengths are summarized in Table [5](https://arxiv.org/html/2509.21800v1#S4.T5). As shown, higher response rates $r$ and more communication rounds generally produce shorter confidence intervals, consistent with our simulation findings. In most cases, the empirical quantiles fall within our reported intervals, highlighting the practical utility of our method for real data.

**Table 5: Estimation results (interval lengths) on the real dataset across varying quantile levels and response rates. “Empirical” denotes the full-sample quantile estimator without LDP. “hetero” indicates client-specific truthful response rates $r_{k}$ range from $0.6$ to $0.9$.**
| Quantile ($\tau$) | Rate ($r$) | C1 | C5 | Log | Empirical |
| --- | --- | --- | --- | --- | --- |
| 0.3 | 0.6 | 33367 (1742) | 33184 (6697) | 33030 (12093) |  |
| 0.3 | hetero | 33418 (1424) | 33229 (5291) | 33140 (9788) | 34000 |
| 0.3 | 0.9 | 33547 (1548) | 33403 (4443) | 33239 (7828) |  |
| 0.5 | 0.6 | 48454 (2255) | 48212 (6315) | 47951 (11361) |  |
| 0.5 | hetero | 48462 (1435) | 48290 (4973) | 48091 (9025) | 50000 |
| 0.5 | 0.9 | 48610 (1454) | 48494 (3851) | 48311 (6863) |  |
| 0.8 | 0.6 | 78586 (2066) | 78168 (6646) | 77995 (13144) |  |
| 0.8 | hetero | 78390 (1291) | 78054 (5862) | 77722 (11101) | 80000 |
| 0.8 | 0.9 | 78657 (1138) | 78300 (4677) | 78084 (8928) |  |

## 5 Concluding remark

We propose a federated‐learning algorithm for quantile inference under LDP that flexibly accommodates client‐level heterogeneity in quantile targets, privacy budgets, and data distributions. In addition, one innovation that should be emphasized is that our developed theoretical results of local SGD quantile estimator. We first design an LDP mechanism that can transform the LDP federated quantile estimation into the non-DP case, and then derive the asymptotic normality and functional central limit theorem of the proposed estimator under non-DP cases. It is first weak‐convergence result for local SGD without the usual average‐smoothness assumption in existing literature. Building on these non‐private asymptotic results, we develop a self‐normalized inference procedure that constructs valid confidence intervals under LDP without requiring direct estimation of the asymptotic variance.

Despite these advances, our method has several limitations. First, it relies on additional regularity assumptions to handle arbitrary client‐level data heterogeneity. Second, as noted in (Shao,, 2015), self‐normalization yields heavier‐tailed limit distributions than the Gaussian, which can produce conservative confidence intervals or reduced power in hypothesis testing. Finally, our framework depends on a central server for aggregation and synchronization, which may not be available in fully decentralized environments. Addressing these challenges and extending the algorithm to decentralized settings remain important directions for future research.

## References

- Agarwal et al., (2018)
Agarwal, N., Suresh, A. T., Yu, F. X. X., Kumar, S., and McMahan, B. (2018).
cpsgd: Communication-efficient and differentially-private distributed
sgd.
Advances in Neural Information Processing Systems, 31.
- Angrist et al., (2006)
Angrist, J., Chernozhukov, V., and Fernández-Val, I. (2006).
Quantile regression under misspecification, with an application to
the us wage structure.
Econometrica, 74(2):539–563.
- Barbaglia et al., (2023)
Barbaglia, L., Consoli, S., and Manzan, S. (2023).
Forecasting with economic news.
Journal of Business & Economic Statistics, 41(3):708–719.
- Bellman et al., (2004)
Bellman, S., Johnson, E. J., Kobrin, S. J., and Lohse, G. L. (2004).
International differences in information privacy concerns: A global
survey of consumers.
The Information Society, 20(5):313–324.
- Cai et al., (2021)
Cai, T. T., Wang, Y., and Zhang, L. (2021).
The cost of privacy: Optimal rates of convergence for parameter
estimation with differential privacy.
The Annals of Statistics, 49(5):2825–2850.
- Chen et al., (2023)
Chen, L., Keilbar, G., and Wu, W. B. (2023).
Recursive quantile estimation: Non-asymptotic confidence bounds.
Journal of Machine Learning Research, 24(91):1–25.
- Chen, (2008)
Chen, S. X. (2008).
Nonparametric estimation of expected shortfall.
Journal of financial econometrics, 6(1):87–107.
- Chen et al., (2020)
Chen, X., Lee, J. D., Tong, X. T., and Zhang, Y. (2020).
Statistical inference for model parameters in stochastic gradient
descent.
The Annals of Statistics, 48(1):251–273.
- Chen et al., (2022)
Chen, X., Liu, W., and Zhang, Y. (2022).
First-order newton-type estimator for distributed estimation and
inference.
Journal of the American Statistical Association,
117(540):1858–1874.
- Chernozhukov and Fernández-Val, (2011)
Chernozhukov, V. and Fernández-Val, I. (2011).
Inference for extremal conditional quantile models, with an
application to market and birthweight risks.
The Review of Economic Studies, 78(2):559–589.
- Chernozhukov and Hansen, (2005)
Chernozhukov, V. and Hansen, C. (2005).
An iv model of quantile treatment effects.
Econometrica, 73(1):245–261.
- Cheu et al., (2019)
Cheu, A., Smith, A., Ullman, J., Zeber, D., and Zhilyaev, M. (2019).
Distributed differential privacy via shuffling.
In Advances in Cryptology–EUROCRYPT 2019: 38th Annual
International Conference on the Theory and Applications of Cryptographic
Techniques, Darmstadt, Germany, May 19–23, 2019, Proceedings, Part I 38,
pages 375–403. Springer.
- Ding et al., (2017)
Ding, B., Kulkarni, J., and Yekhanin, S. (2017).
Collecting telemetry data privately.
Advances in Neural Information Processing Systems, 30.
- Duchi et al., (2013)
Duchi, J. C., Jordan, M. I., and Wainwright, M. J. (2013).
Local privacy and statistical minimax rates.
In 2013 IEEE 54th Annual Symposium on Foundations of Computer
Science, pages 429–438. IEEE.
- Dwork et al., (2006)
Dwork, C., Kenthapadi, K., McSherry, F., Mironov, I., and Naor, M. (2006).
Our data, ourselves: Privacy via distributed noise generation.
In Annual International Conference on the Theory and
Applications of Cryptographic Techniques, pages 486–503. Springer.
- Erlingsson et al., (2014)
Erlingsson, Ú., Pihur, V., and Korolova, A. (2014).
Rappor: Randomized aggregatable privacy-preserving ordinal response.
In Proceedings of the 2014 ACM SIGSAC conference on computer and
communications security, pages 1054–1067.
- Fan et al., (2023)
Fan, J., Guo, Y., and Wang, K. (2023).
Communication-efficient accurate statistical estimation.
Journal of the American Statistical Association,
118(542):1000–1010.
- Gillenwater et al., (2021)
Gillenwater, J., Joseph, M., and Kulesza, A. (2021).
Differentially private quantiles.
In Meila, M. and Zhang, T., editors, Proceedings of the 38th
International Conference on Machine Learning, volume 139 of Proceedings
of Machine Learning Research, pages 3713–3722. PMLR.
- Goyal et al., (2017)
Goyal, P., Dollár, P., Girshick, R., Noordhuis, P., Wesolowski, L., Kyrola,
A., Tulloch, A., Jia, Y., and He, K. (2017).
Accurate, large minibatch sgd: Training imagenet in 1 hour.
arXiv preprint arXiv:1706.02677.
- Gu and Chen, (2023)
Gu, J. and Chen, S. X. (2023).
Distributed statistical inference under heterogeneity.
Journal of Machine Learning Research, 24(387):1–57.
- He et al., (2023)
He, X., Pan, X., Tan, K. M., and Zhou, W.-X. (2023).
Smoothed quantile regression with large-scale inference.
Journal of Econometrics, 232(2):367–388.
- Hector and Song, (2021)
Hector, E. C. and Song, P. X.-K. (2021).
A distributed and integrated method of moments for high-dimensional
correlated data analysis.
Journal of the American Statistical Association,
116(534):805–818.
- Hu et al., (2024)
Hu, J., Ma, Y.-T., and Eun, D.-Y. (2024).
Does worst-performing agent lead the pack? analyzing agent dynamics
in unified distributed sgd.
Advances in Neural Information Processing Systems,
37:72754–72789.
- Hu et al., (2022)
Hu, J., Peng, Y., Zhang, G., and Zhang, Q. (2022).
A stochastic approximation method for simulation-based quantile
optimization.
INFORMS Journal on Computing, 34(6):2889–2907.
- Huang et al., (2021)
Huang, Z., Liang, Y., and Yi, K. (2021).
Instance-optimal mean estimation under differential privacy.
Advances in Neural Information Processing Systems,
34:25993–26004.
- Jiang et al., (2022)
Jiang, X., Zhou, X., and Grossklags, J. (2022).
Signds-fl: Local differentially private federated learning with
sign-based dimension selection.
ACM Transactions on Intelligent Systems and Technology (TIST),
13(5):1–22.
- Jordan et al., (2019)
Jordan, M. I., Lee, J. D., and Yang, Y. (2019).
Communication-efficient distributed statistical inference.
Journal of the American Statistical Association,
114(526):668–681.
- Joseph et al., (2019)
Joseph, M., Mao, J., Neel, S., and Roth, A. (2019).
The role of interactivity in local differential privacy.
In 2019 IEEE 60th Annual Symposium on Foundations of Computer
Science (FOCS), pages 94–105.
- Kallus et al., (2024)
Kallus, N., Mao, X., and Uehara, M. (2024).
Localized debiased machine learning: Efficient inference on quantile
treatment effects and beyond.
Journal of Machine Learning Research, 25(16):1–59.
- Konečnỳ et al., (2016)
Konečnỳ, J., McMahan, H. B., Yu, F. X., Richtárik, P., Suresh,
A. T., and Bacon, D. (2016).
Federated learning: Strategies for improving communication
efficiency.
arXiv preprint arXiv:1610.05492.
- Li et al., (2025)
Li, B., Zhang, S., and Han, Q. (2025).
Federated learning with joint server-client momentum.
Scientific Reports, 15(1):15626.
- Li et al., (2020)
Li, T., Sahu, A. K., Zaheer, M., Sanjabi, M., Talwalkar, A., and Smith, V.
(2020).
Federated optimization in heterogeneous networks.
Proceedings of Machine learning and systems, 2:429–450.
- Li et al., (2022)
Li, X., Liang, J., Chang, X., and Zhang, Z. (2022).
Statistical estimation and online inference via local sgd.
In Conference on Learning Theory, pages 1613–1661. PMLR.
- (34)
Liu, C., Tian, Y., Tang, J., Dang, S., and Chen, G. (2023a).
A novel local differential privacy federated learning under
multi-privacy regimes.
Expert systems with applications, 227:120266.
- Liu et al., (2020)
Liu, W., Chen, L., Chen, Y., and Zhang, W. (2020).
Accelerating federated learning via momentum gradient descent.
IEEE Transactions on Parallel and Distributed Systems,
31(8):1754–1766.
- (36)
Liu, Y., Hu, Q., Ding, L., and Kong, L. (2023b).
Online local differential private quantile inference via
self-normalization.
In International Conference on Machine Learning, pages
21698–21714. PMLR.
- Lowy and Razaviyayn, (2023)
Lowy, A. and Razaviyayn, M. (2023).
Private federated learning without a trusted server: Optimal
algorithms for convex losses.
In The Eleventh International Conference on Learning
Representations.
- Ma et al., (2022)
Ma, X., Sun, X., Wu, Y., Liu, Z., Chen, X., and Dong, C. (2022).
Differentially private byzantine-robust federated learning.
IEEE Transactions on Parallel and Distributed Systems,
33(12):3690–3701.
- McMahan et al., (2017)
McMahan, B., Moore, E., Ramage, D., Hampson, S., and y Arcas, B. A. (2017).
Communication-efficient learning of deep networks from decentralized
data.
In Artificial intelligence and statistics, pages 1273–1282.
PMLR.
- Milberg et al., (2000)
Milberg, S. J., Smith, H. J., and Burke, S. J. (2000).
Information privacy: Corporate management and national regulation.
Organization science, 11(1):35–57.
- Nasr et al., (2023)
Nasr, M., Carlini, N., Hayase, J., Jagielski, M., Cooper, A. F., Ippolito, D.,
Choquette-Choo, C. A., Wallace, E., Tramèr, F., and Lee, K. (2023).
Scalable extraction of training data from (production) language
models.
arXiv preprint arXiv:2311.17035.
- Plečko et al., (2024)
Plečko, D., Bennett, N., and Meinshausen, N. (2024).
fairadapt: Causal reasoning for fair data preprocessing.
Journal of Statistical Software, 110:1–35.
- Shao, (2015)
Shao, X. (2015).
Self-normalization for time series: a review of recent developments.
Journal of the American Statistical Association,
110(512):1797–1817.
- Shen et al., (2023)
Shen, X., Jiang, H., Chen, Y., Wang, B., and Gao, L. (2023).
Pldp-fl: Federated learning with personalized local differential
privacy.
Entropy, 25(3):485.
- Shi et al., (2022)
Shi, S., Hu, C., Wang, D., Zhu, Y., and Han, Z. (2022).
Distributionally robust federated learning for differentially private
data.
In 2022 IEEE 42nd International Conference on Distributed
Computing Systems (ICDCS), pages 842–852. IEEE.
- Song et al., (2013)
Song, S., Chaudhuri, K., and Sarwate, A. D. (2013).
Stochastic gradient descent with differentially private updates.
In 2013 IEEE global conference on signal and information
processing, pages 245–248. IEEE.
- Stich, (2018)
Stich, S. U. (2018).
Local sgd converges fast and communicates little.
arXiv preprint arXiv:1805.09767.
- Tian et al., (2023)
Tian, Y., Weng, H., and Feng, Y. (2023).
Unsupervised federated learning: A federated gradient em algorithm
for heterogeneous mixture models with robustness against adversarial attacks.
arXiv preprint arXiv:2310.15330.
- Wang et al., (2012)
Wang, H. J., Li, D., and He, X. (2012).
Estimation of high conditional quantiles for heavy-tailed
distributions.
Journal of the American Statistical Association,
107(500):1453–1464.
- Xie et al., (2024)
Xie, C., Jin, K., Liang, J., and Zhang, Z. (2024).
Asymptotic time-uniform inference for parameters in averaged
stochastic approximation.
arXiv preprint arXiv:2410.15057.
- Yadlowsky et al., (2025)
Yadlowsky, S., Fleming, S., Shah, N., Brunskill, E., and Wager, S. (2025).
Evaluating treatment prioritization rules via rank-weighted average
treatment effects.
Journal of the American Statistical Association,
120(549):38–51.
- Zhao et al., (2020)
Zhao, Y., Zhao, J., Yang, M., Wang, T., Wang, N., Lyu, L., Niyato, D., and Lam,
K.-Y. (2020).
Local differential privacy-based federated learning for internet of
things.
IEEE Internet of Things Journal, 8(11):8836–8853.
- Zhu et al., (2024)
Zhu, W., Lou, Z., Wei, Z., and Wu, W. B. (2024).
High confidence level inference is almost free using parallel
stochastic optimization.
arXiv preprint arXiv:2401.09346.

## Peer Reviews

confidence: 3
contribution: 2
initial_rating: 4
presentation: 1
questions: See W1-W5.
soundness: 2
strengths: S1. This work designs an LDP mechanism that can transform the LDP federated quantile estimation into the non-DP case, and then derive the asymptotic normality and functional central limit theorem of the proposed estimator under non-DP cases.

S2. It is the first weak-convergence result for local SGD without the usual average-smoothness assumption in existing literature
summary: This paper considers the statistical inference of federated learning for quantile loss via local stochastic gradient decent under local differential privacy. It proposes a local-SGD algorithm with randomized response on the binary quantile “gradient” supports heterogeneous clients, and outputs a Polyak–Ruppert averaged estimator. Then, its theoretical properties were explored and experimentally verified.
weaknesses: My comments focus on presentation and experimental design issues.

W1. The introduction section reads like an extended related work. Missing the importance and/or motivation of the question. I suggest giving 1–2 concrete use cases, or moving a simplified version of Figure 1 to the intro to anchor the problem.

W2. The Methodology section lacks early notation and problem formulation. Even Q is not defined.

W3. In the Experiment section, placing synthetic results in the appendix is a bad idea. There is no direct comparison against prior work (e.g., Liu et al, 2023b). Table 1 lacks guidance on “what good looks like”, nominal 95% coverage with the shortest average CI length? Why not visually highlight when empirical quantiles fall inside/outside CIs?

W4. For the truthful response rate, I am unsure who sets $r_k$, whether it is policy-driven, user-chosen, or purely synthetic. The phrase "range from 0.6 to 0.9" in real data is missing a clear definition. I suggest adding the sensitivity of the experiment regarding the truthful response rate.

W5. In Asymptotic Analysis, are the theorems and algorithms (partially) verified in the experimental section? If not, is verification possible?

---

confidence: 3
contribution: 2
initial_rating: 4
presentation: 3
questions: 1) I cannot see any baselines in the experiments. Although the authors claim that this is the first inference framework for federated quantile estimation, related works are existing and also can be adopted as baselines; 
2) The theoretical analysis is not complete, especially for DP. The authors claim that they transform the LDP federated quantile estimation into the non-DP case. Can this transformation maintain DP guarantee? 
3) Existing results in simulation cannot evaluate the proposed methods well. Some more experimental should be added.
soundness: 2
strengths: This paper propose to use federated learning (FL) for quantile inference under local differential privacy (LDP). The core idea of this paper is to design an LDP mechanism that can transform the LDP federated quantile estimation into the non-DP case. In addition, via constructing valid confidence intervals under LDP, the authors develop a self-normalized inference procedure without requiring direct estimation of the asymptotic
variance. The extensive experiments evaluate the effectiveness of the proposed method. Overall, this paper considers a new problem in this area and well written.
summary: This paper propose to use federated learning (FL) for quantile inference under local differential privacy (LDP). The core idea of this paper is to design an LDP mechanism that can transform the LDP federated quantile estimation into the non-DP case. In addition, via constructing valid confidence intervals under LDP, the authors develop a self-normalized inference procedure without requiring direct estimation of the asymptotic
variance. The extensive experiments evaluate the effectiveness of the proposed method. Overall, this paper considers a new problem in this area. However, I have some concerns as follows: 1) I cannot see any baselines in the experiments. Although the authors claim that this is the first inference framework for federated quantile estimation, related works are existing and also can be adopted as baselines; 2) The theoretical analysis is not complete, especially for DP. The authors claim that they transform the LDP federated quantile estimation into the non-DP case. Can this transformation maintain DP guarantee? 3) Existing results in simulation cannot evaluate the proposed methods well. Some more experimental should be added.
weaknesses: However, I have some concerns as follows: 1) I cannot see any baselines in the experiments. Although the authors claim that this is the first inference framework for federated quantile estimation, related works are existing and also can be adopted as baselines; 2) The theoretical analysis is not complete, especially for DP. The authors claim that they transform the LDP federated quantile estimation into the non-DP case. Can this transformation maintain DP guarantee? 3) Existing results in simulation cannot evaluate the proposed methods well. Some more experimental should be added.

---

confidence: 2
contribution: 2
initial_rating: 4
presentation: 3
questions: 1.  Have you quantified the degree of conservatism introduced by the self-normalization approach in your experiments? Are there potential adjustments or alternative techniques to mitigate this while remaining within the LDP constraint?
2.  Can your theoretical framework be extended to more complex models, such as high-dimensional or conditional quantile regression? What would be the main challenges?
3.  How would the algorithm perform under more realistic, asynchronous client participation patterns (e.g., clients joining at random times)? Does the theory accommodate such scenarios?
soundness: 3
strengths: 1.  Important and Timely Problem: The combination of federated quantile estimation with the strong privacy guarantee of LDP is highly relevant. Quantile inference is crucial for understanding data heterogeneity (e.g., in healthcare or economics), and LDP is the gold standard for privacy in distributed settings where a central server is not trusted. Tackling inference under these constraints is a substantial contribution.
2.  Significant Theoretical Contribution: The paper's primary strength is its theoretical analysis. Proving a functional central limit theorem for a local SGD-based estimator with a non-smooth loss function is a non-trivial and valuable advancement. This provides a solid foundation for inference in a setting where standard smoothness assumptions fail.
3.  Practical and Elegant Inference Solution: The use of self-normalization to construct confidence intervals is a clever and highly practical solution. It bypasses the notoriously difficult problem of estimating the asymptotic variance under LDP, which would typically require additional privacy budget or complex techniques. The resulting method is computationally efficient and integrates seamlessly with the federated learning workflow.
4.  Comprehensive Experimental Evaluation: The paper validates its theoretical claims with extensive simulations covering a wide range of heterogeneity scenarios (in quantile levels, privacy budgets, and data distributions). The inclusion of a real-world dataset (government salaries) strengthens the practical relevance of the findings. Comparisons against appropriate baselines like DP-SGD and a divide-and-conquer method are well-executed.
summary: This paper addresses a significant and challenging problem at the intersection of federated learning, statistical inference, and privacy preservation: performing quantile inference under local differential privacy (LDP). The authors propose a novel federated learning algorithm based on local Stochastic Gradient Descent (SGD), where client gradients are perturbed using an LDP mechanism. The key innovation is a theoretical analysis that establishes the asymptotic normality and a functional central limit theorem (FCLT) for their estimator, even though the quantile loss function is non-smooth. Leveraging this FCLT, they develop a self-normalized inference procedure that constructs valid confidence intervals without needing to estimate the asymptotic variance directly, thus avoiding additional privacy costs. Extensive simulations and a real-data application on government salary data demonstrate the method's effectiveness under various heterogeneity settings.
weaknesses: 1.  Reliance on Strong Assumptions: The theoretical analysis relies on assumptions such as the boundedness of the parameter space and the uniform boundedness of the client density functions. While common in theoretical proofs, the practical implications of these assumptions could be discussed further (e.g., how to ensure parameter space boundedness in practice).
2.  Inherent Conservatism of Self-Normalization: The authors correctly note that self-normalization can lead to heavier-tailed limiting distributions than the standard normal. This often results in conservative confidence intervals (wider than necessary), which can reduce statistical power in hypothesis testing. This is a well-known trade-off of the method but is an important limitation to highlight.
3.  Centralized Server Dependency: The proposed framework requires a central server for aggregation and synchronization. The algorithm's applicability to fully decentralized or peer-to-peer federated learning environments, an active area of research, is not explored.
4.  Limited Analysis of Communication/Computation Trade-offs: While different communication strategies (C1, C5, Log) are compared, a more detailed analysis of the total communication cost and wall-clock time (computation + communication) compared to baseline methods would be beneficial for assessing real-world efficiency.

---

confidence: 4
contribution: 2
initial_rating: 4
presentation: 3
questions: • Provide a proof sketch for Theorem 2.1 explicitly showing how the randomized response
induces the r_k scaling and τ̃_k shift.
• Expand on the removal of Assumption 1 from Li et al. (2022) — clarify exactly which steps
are substituted by the new bound E|q̄−Q*|² ≲ γ_m.
• Include empirical analysis of conservative confidence intervals — e.g., actual vs nominal
coverage rates.
• Test robustness under heavy-tailed client data (e.g., Cauchy distributions) to validate
bounded-density assumptions.
• Discuss communication-computation trade-offs more quantitatively under different local
update frequencies E_m.
soundness: 3
strengths: • Novel formulation combining federated learning, quantile inference, and local differential
privacy — a previously unexplored combination.
• Theoretical depth: establishes asymptotic normality and FCLT for non-smooth quantile
loss without the average-smoothness assumption.
• Elegant reduction (Theorem 2.1) that transforms the privatised problem into an
equivalent non-private version, enabling classical analysis.
• Self-normalised inference avoids density estimation, conserving privacy budget and
improving practical applicability.
• Comprehensive experiments demonstrating good accuracy and confidence interval
coverage across various privacy levels and heterogeneity settings.
• Clear discussion of limitations and directions for future research.
summary: The paper proposes a federated algorithm for quantile estimation and inference under
Local Differential Privacy (LDP) using local stochastic gradient descent (Local-SGD). The
method perturbs local gradients via a randomised-response mechanism that ensures client-
level privacy while maintaining statistical efficiency. Key theoretical contributions include:
(i) an equivalence theorem reducing the LDP problem to a non-private one with adjusted
quantile levels, (ii) asymptotic normality and functional central limit theorem (FCLT) for the
non-smooth quantile loss, and (iii) a self-normalised confidence interval method that avoids
estimating nuisance parameters. Experiments and a real-data application validate the
theoretical results.
weaknesses: • The theoretical results depend on several strong regularity assumptions, particularly
bounded densities and step-size control, which may limit practical generality.
• The self-normalized approach yields heavier-tailed distributions, producing conservative
confidence intervals and lower test power.
• Dependence on a central aggregation server restricts use in fully decentralized federated
systems.
• Empirical validation is limited to tabular, one-dimensional quantiles — no high-
dimensional or complex FL tasks are explored.

• Certain proof steps (e.g., adaptation of Li et al., 2022 assumptions) rely on claims without
complete derivation; these could be more explicit for reproducibility.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: Why in Theorem 3.2 do you use $\ell^\infty$ instead of $D[0,1]$ (the Skorohod space)?  My understanding is that a partial sum process like this is not tight in $\ell^\infty$ so you have to work in $D[0,1]$.  Or if you want to work in $\ell^\infty$ you can linearly interpolate the partial sum process so that you have continuous paths (and then you are essentially working in C[0,1]).  But as stated, I'm not sure Thm 3.2 is actually correct since a lack of tightness implies you don't have a functional CLT (in the given space).  But let me know if I'm missing something as it's been a while since I worked on functional CLTs.

I would argue that (2.1) is not the goal, but that the goal is outlined in lines 147-149.  It just so happens that the minimizer of (2.1) gives you the desired parameter,  i.e. the goal is to estimate the quantile of some target population.

The discussion around Thm 2.1 is quite weak and should be expanded.  

The discussion/motivation for Thm 3.2 could be strengthened before being presented.  Just saying "it is necessary to strengthen" it, isn't really motivating the result.  

Line 48: "Guarantees convergence" is a bit vague.  Statistically or computationally?  

Line 61: Nasr should be in parentheses.  In general, the paper's handling of references is really all over the place.  References out of the sentence should be in parentheses, those that are part of the sentence shouldn't be, and you shouldn't have double parentheses like …(e.g.(Zhao…  I suggest the authors go through and clean these up.

Ln 124: When talking about estimation accuracy, I think the privacy parameters should be in the rate as well.  The choice of $\epsilon$ or $\delta$ play a large practical role.

Ln 130: The author should note the $\theta$ here is the one computed from the data, as opposed to the population level quantity.

Ln 151-161: The setup for the estimator is not very clear.  I suppose the authors are saying that at each step each client either updates locally or uses the current estimates from all of the other clients. But it is presented in a confusing fashion.  E.g. "$x_k^t$ is an independent realization of P_k", yes, but it is also just one of the datapoints held by a client.  "$E_m = n$" is divide an conquer?  So all clients have the same sample sizes?  I suppose $\mathcal I$ is ordered.  I would just add more details for the reader.
soundness: 3
strengths: Overall the paper tackles an interesting problem and presents extensive mathematical results.
summary: The authors consider the problem of estimating a quantile in a federated learning setting with a local differential privacy constraint.  They present a new algorithm for computing this central quantile and extensive asymptotic analysis that can be used for statistical inference.  They highlight how estimating the quantile in this LDP setting is equivalent to estimating a quantile in a non private setting, but with the distribution shifted in a known way (i.e. solving one gives a computable solution to the other).  To get around estimating nuisance parameters, the authors also present a self-normalized approach based on a functional central limit theorem.
weaknesses: The organization and writing of the paper is interesting, but feels terse or missing in details at times.  See questions.


Decision: Accept (Poster)