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Differentially Private Inference for Longitudinal Linear Regression

Getoar Sopa, Marco Avella Medina, Cynthia Rush

TL;DR

This paper develops a unified framework for estimation and inference in longitudinal linear regression under user-level differential privacy with temporal dependence. The authors introduce a private estimator that aggregates per-user regressions and a privatized, HAC-consistent covariance estimator to enable valid private inference, along with adaptive mean-estimation (DPTrimMean) that tolerates dependence without strong distributional assumptions. They extend the approach to a two-group setting for differential treatment effects and provide finite-sample and asymptotic guarantees, including privacy-preserving confidence intervals and Wald tests. Through simulations and a CAMP data example, the work demonstrates strong theoretical guarantees and competitive empirical performance, highlighting the practical viability of user-level DP for panel data. Overall, the framework offers a principled route to private longitudinal analysis with dependence, balancing privacy costs and statistical efficiency in realistic settings.

Abstract

Differential Privacy (DP) provides a rigorous framework for releasing statistics while protecting individual information present in a dataset. Although substantial progress has been made on differentially private linear regression, existing methods almost exclusively address the item-level DP setting, where each user contributes a single observation. Many scientific and economic applications instead involve longitudinal or panel data, in which each user contributes multiple dependent observations. In these settings, item-level DP offers inadequate protection, and user-level DP - shielding an individual's entire trajectory - is the appropriate privacy notion. We develop a comprehensive framework for estimation and inference in longitudinal linear regression under user-level DP. We propose a user-level private regression estimator based on aggregating local regressions, and we establish finite-sample guarantees and asymptotic normality under short-range dependence. For inference, we develop a privatized, bias-corrected covariance estimator that is automatically heteroskedasticity- and autocorrelation-consistent. These results provide the first unified framework for practical user-level DP estimation and inference in longitudinal linear regression under dependence, with strong theoretical guarantees and promising empirical performance.

Differentially Private Inference for Longitudinal Linear Regression

TL;DR

This paper develops a unified framework for estimation and inference in longitudinal linear regression under user-level differential privacy with temporal dependence. The authors introduce a private estimator that aggregates per-user regressions and a privatized, HAC-consistent covariance estimator to enable valid private inference, along with adaptive mean-estimation (DPTrimMean) that tolerates dependence without strong distributional assumptions. They extend the approach to a two-group setting for differential treatment effects and provide finite-sample and asymptotic guarantees, including privacy-preserving confidence intervals and Wald tests. Through simulations and a CAMP data example, the work demonstrates strong theoretical guarantees and competitive empirical performance, highlighting the practical viability of user-level DP for panel data. Overall, the framework offers a principled route to private longitudinal analysis with dependence, balancing privacy costs and statistical efficiency in realistic settings.

Abstract

Differential Privacy (DP) provides a rigorous framework for releasing statistics while protecting individual information present in a dataset. Although substantial progress has been made on differentially private linear regression, existing methods almost exclusively address the item-level DP setting, where each user contributes a single observation. Many scientific and economic applications instead involve longitudinal or panel data, in which each user contributes multiple dependent observations. In these settings, item-level DP offers inadequate protection, and user-level DP - shielding an individual's entire trajectory - is the appropriate privacy notion. We develop a comprehensive framework for estimation and inference in longitudinal linear regression under user-level DP. We propose a user-level private regression estimator based on aggregating local regressions, and we establish finite-sample guarantees and asymptotic normality under short-range dependence. For inference, we develop a privatized, bias-corrected covariance estimator that is automatically heteroskedasticity- and autocorrelation-consistent. These results provide the first unified framework for practical user-level DP estimation and inference in longitudinal linear regression under dependence, with strong theoretical guarantees and promising empirical performance.
Paper Structure (38 sections, 33 theorems, 227 equations, 3 figures, 5 tables, 5 algorithms)

This paper contains 38 sections, 33 theorems, 227 equations, 3 figures, 5 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gaussian Differential Privacy
  4. Probability theory
  5. Adaptive differentially private mean estimation
  6. Discussion
  7. Differentially Private longitudinal linear regression estimation
  8. Set-up and assumptions
  9. uDP Regression estimation
  10. uDP Two-group regression estimation
  11. Differentially Private HAC inference
  12. Small sample correction
  13. Inference in the two-group model
  14. Discussion
  15. Simulations
  16. ...and 23 more sections

Key Result

Proposition 2.5

Let $h$ be a statistic with user-level global sensitivity $\text{GS}({h})$. Then, the randomized statistic $\tilde{h}(X^n) =h(X^n)+ \frac{\text{GS}(h)}{\mu}Z$, with $Z \sim N(0,I_d)$, satisfies $\mu$-uGDP.

Figures (3)

  • Figure 1: An illustration of the final five iterations of Algorithm \ref{['alg:trimestTemp']}. The dashed, red circle represents the final attempted and rejected mean refinement, whereas the blue circle is the final projection ball.
  • Figure 2: The two users' Gram matrices do not concentrate uniformly. However, their local regression fits concentrate tightly around the common regression coefficients.
  • Figure 3: The unscaled RMSE for various $(n,T)$ combinations.

Theorems & Definitions (67)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5: Theorem 1 of dong2022gaussian
  • Proposition 2.6: Proposition 4 of dong2022gaussian
  • Proposition 2.7: Corollary 2 of dong2022gaussian
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 57 more