Differentially Private Quantile Regression

Updated 11 August 2025

Differentially private quantile regression is the estimation of conditional quantiles under privacy constraints, using techniques that balance robustness and the privacy–accuracy tradeoff.
Recent mechanisms such as counting-based approaches, the exponential mechanism, and joint estimation efficiently reduce sensitivity and control noise while providing theoretical guarantees.
Applications span risk assessment, real-time streaming, and decentralized learning, though challenges remain in optimal privacy calibration and scaling to high-dimensional settings.

Differentially private quantile regression is the theory and practice of estimating conditional quantile functions (typically in the regression context) while ensuring that the estimation process satisfies differential privacy (DP). Recent work has established a rich set of algorithmic techniques, privacy analyses, and statistical guarantees for both univariate and high-dimensional quantile regression under privacy constraints. Key advances include practical mechanisms for robustly estimating conditional quantiles, rigorous characterizations of the privacy–accuracy tradeoff, efficient distributed protocols, and valid inference methods even in high-dimensional settings. The interplay between non-smooth objective functions (such as the quantile/check loss), privacy-preserving data access patterns, and robust statistical estimation fundamentally shapes the field.

1. Differential Privacy Mechanisms for Quantile Estimation

A central challenge for differentially private quantile regression is that classical quantile queries possess very high global sensitivity, especially with non-discrete or heavy-tailed data. A principal approach to mitigate this is to reformulate the estimation process as a sequence of low-sensitivity counting queries or to smooth the objective function itself.

Counting-based Mechanisms: The bandit-based DP-SEQ algorithm (Nikolakakis et al., 2020) avoids direct noise addition to quantiles by structuring the best-arm quantile identification task as a sequence of pairwise order statistic comparisons, each of which involves a sensitivity-1 difference in counts. Noisy counts are then obtained by Laplace perturbation, and statistical inference is reconstructed from these privatized order relationships.
Exponential Mechanism: The exponential mechanism is widely used to privatize quantile estimation (Gillenwater et al., 2021, Kaplan et al., 2021, Lalanne et al., 2022, Lalanne et al., 2023, Imola et al., 19 May 2025). Instead of privatizing observed quantile values, these mechanisms select outputs with probability proportional to the exponentiated utility (typically the negative deviation in rank between a candidate and the desired quantile), yielding sharp privacy guarantees even in the presence of discretization and repeated values.
Joint/Recursive Estimation: Mechanisms such as JointExp (Gillenwater et al., 2021) and recursive partitioning (AQ, RecExp (Kaplan et al., 2021, Lalanne et al., 2023)) solve the multi-quantile estimation problem by jointly releasing several quantiles via a single DP mechanism. This design avoids the composition penalty that would arise from estimating each quantile independently, and enables linear or polylogarithmic error growth in the number of quantiles m, instead of superlinear.
Local and Distributed Mechanisms: Local DP settings employ randomized response or one-bit protocols in adaptive (noisy search) frameworks (Aamand et al., 5 Feb 2025, Liu et al., 2023), with privacy analysis tailored to sequential or shuffle-based architectures. For streaming or frugal computation, lightweight mechanisms such as DP-Frugal-1U (Cafaro et al., 27 Feb 2025) provide O(1)-space quantile tracking with post-hoc noise addition based on bounded global sensitivity.
High-dimensional and Regression Adaptations: In regression, privacy is enforced during iterative optimization or via output perturbation. For example, DP quantile regression in high-dimensions uses Noisy Hard Thresholding (NoisyHT)—combining Laplace/gaussian noise with coordinate thresholding to preserve sparsity (Shen et al., 7 Aug 2025). In distributed feature-partitioned settings, noise is added to the surrogate gradient in every optimization round (Xiao et al., 23 Apr 2025).

2. Statistical and Algorithmic Foundations

Quantile Loss and Objective Smoothing: The non-smooth nature of the quantile (check) loss function hinders direct application of classical optimization and DP mechanisms which rely on bounded gradients or sensitivity. Several works employ convolutional smoothing (e.g., kernel convolution in (Xiao et al., 23 Apr 2025)) or a Newton-type transformation to recast the non-smooth regression task as a smooth or weighted least squares problem (Shen et al., 7 Aug 2025). This enables quasi-likelihood inference, gradient-based optimization, and facilitates privacy calibration.
Optimality and Lower Bounds: Lower bounds are established for adaptive and non-adaptive protocols in the local/shuffle-DP regime (Aamand et al., 5 Feb 2025). For instance, adaptive binary search based on Bayesian screening realizes optimal sample complexity O((log B)/(ε²α²)), while non-adaptive protocols provably require additional log factors.
Complexity and Scaling: Efficiency is achieved through algorithmic innovations such as dynamic programming for the exponential mechanism (Gillenwater et al., 2021), recursive strategies yielding O(n log m) overall run time (Kaplan et al., 2021), and frugal streaming structures tracking quantiles in O(1) space (Cafaro et al., 27 Feb 2025). These advances render DP quantile regression feasible in environment with high data velocity or strict resource constraints.

3. Statistical Inference, Confidence Intervals, and Bootstrap

Valid inference under DP requires addressing both the bias induced by privatization and the often unknown variance/covariance structure of the estimators:

DP Bootstrap and Deconvolution: For quantile regression, releasing noisy estimators directly leads to inflated or flattened sampling distributions. The DP bootstrap procedure generates multiple privatized estimates via Gaussian output perturbation, then uses kernel deconvolution to infer the true underlying sampling distribution (Wang et al., 2022). Empirically, percentile-based intervals constructed from the deconvolved density attain nominal coverage.
Debiasing: In high-dimensional settings, coordinate thresholding or regularization introduces bias. Debiased estimators, constructed via sample-splitting and DP variants of CLIME precision matrix estimation, achieve Bahadur representations so that coordinatewise inference remains asymptotically normal (Shen et al., 7 Aug 2025).
Wald-type Confidence Intervals: For decentralized regression, surrogate gradients and auxiliary variables yield estimated residuals that enable consistent coefficient and variance estimation. Confidence intervals are then constructed via plug-in covariance estimates and Wald-type statistics (Xiao et al., 23 Apr 2025).
Self-Normalization and Online Inference: In local and streaming settings, self-normalization may be used to construct pivotal statistics with asymptotically valid inference even under severe privacy constraints. In quantile estimation with binary response, this allows tight confidence intervals without nuisance parameter estimation (Liu et al., 2023).

4. Practical Implementations and Applications

Differentially private quantile regression is applied in a range of settings where both robust modeling and privacy preservation are critical:

Risk and Robustness: Quantile regression and bandit identification via best-arm quantiles are prominent in risk-sensitive decision making (e.g., finance via Value-at-Risk, server latency minimization) due to their robustness to outliers and tail behavior (Nikolakakis et al., 2020).
Large-Scale and Streaming Data: Space-efficient algorithms achieve O(1)–O(log n) memory footprints, essential for IoT, network monitoring, and high-frequency event data (Alabi et al., 2022, Cafaro et al., 27 Feb 2025). Streaming algorithms are particularly attractive for real-time quantile regression with privacy guarantees.
Treatment Effects and A/B Testing: In randomized controlled trials, quantile treatment effect estimation is enabled via privacy-preserving aggregation (e.g., histogram-based quantile estimation with Laplace/Geometric noise addition) (Yao et al., 25 Jan 2024). This setting requires careful attention to bin granularity, privacy–accuracy tradeoffs, and variance inflation.
Distributed and Feature-Partitioned Learning: With data distributed across multiple devices or organizations (federated and decentralized learning), differential privacy is achieved via local perturbation of iterative updates or shuffling, often with auxiliary variables approximating global aggregates. Empirical results indicate that decentralized algorithms can match the performance of fully centralized methods while achieving robust privacy protection (Xiao et al., 23 Apr 2025).

5. Comparative Analysis and Limitations

Tradeoffs between Utility and Privacy: Across all methods, there is an inherent tradeoff governed by the privacy parameter ε: as stronger privacy is enforced (ε decreases), statistical error increases. In high-dimensional and streaming settings, appropriate scaling of ε and batch size is necessary to match non-private fits as closely as feasible.
Comparison with Classical Methods: Non-private quantile regression traditionally does not require smoothing or noise addition, and can leverage full gradient information. DP mechanisms require regularization (to control sensitivity), smoothing, or recursive composition, which can introduce bias and variance that classical estimators avoid.
Challenges with Peaked and Atomic Data: Mechanisms relying on the standard exponential mechanism or IS/JointExp can fail catastrophically in peaked or atomic distributions; in such cases, artificially jittering the data (as in HSJointExp) is necessary to avoid zero-mass blocks that otherwise dominate the privatized output (Lalanne et al., 2022).
Limits of Adaptive and Non-adaptive Protocols: For local/shuffle DP, adaptivity yields optimal accuracy, but any non-adaptive scheme must pay an exponential sample complexity penalty in domain size (Aamand et al., 5 Feb 2025). This result shapes the landscape for federated and distributed quantile regression systems.
Scaling to High Dimensions: High-dimensional regression introduces unique difficulties, especially for inference. Naïve noise addition disrupts sparsity; advances such as NoisyHT and DP–CLIME are critical to preserving both statistical validity and privacy (Shen et al., 7 Aug 2025). Bootstrap-based inference with private gradient aggregation over distributed data addresses simultaneous hypothesis testing at scale.

6. Future Directions

With core algorithmic and theoretical results established, open questions and research opportunities remain:

Extending to Richer Models: Direct application of current techniques to nonparametric, non-linear, or high-dimensional conditional quantile estimation remains an open area. Mechanisms tailored for structured data, manifold models, or generalized additive models are of increasing interest.
Automated Privacy Calibration: Implementation challenges include automatic tuning of privacy budgets and adaptive parameter selection (such as bin width, regularization strength) to fit both privacy constraints and statistical targets, potentially guided by external or historical information (Khodak et al., 2022).
Practical Deployment and Benchmarks: While the reviewed mechanisms have strong theoretical foundations and some deployments on real data, comprehensive empirical benchmarks and further validation in operational systems would solidify the utility–privacy curve in practice.
Inference and Confidence Set Construction: Development of further robust, communication-efficient, and inference-valid methods (e.g., decentralized private bootstrapping) for confidence sets across multivariate or structured output spaces, especially with non-smooth quantile loss, is an ongoing area of research (Wang et al., 2022, Shen et al., 7 Aug 2025).

Differentially private quantile regression is now a mature field with deep statistical, algorithmic, and practical roots, complemented by clear directions for continued innovation.