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Differentially Private Log-Location-Scale Regression Using Functional Mechanism

Jiewen Sheng, Xiaolei Fang

TL;DR

This work develops differentially private log-location-scale (DP-LLS) regression by applying the functional mechanism to SEV and logistic regressions within the LLS family. It derives sensitivity bounds, injects Laplace noise into second-order Taylor expansions of the log-likelihood, and proves ε-differential privacy for the proposed estimators. Through simulations and a NASA aircraft case study, the authors show that DP performance degrades as predictor dimension grows and privacy becomes stronger (smaller ε), but improves with larger training sizes, with DP results approaching non-DP performance when ε is moderate or large. The findings offer practical guidance on when DP-LLS regression is viable and highlight the need for dimension reduction and ample data to balance privacy and accuracy. The approach generalizes to additional LLS distributions, enabling privacy-preserving forecasting in reliability and prognostics domains.

Abstract

This article introduces differentially private log-location-scale (DP-LLS) regression models, which incorporate differential privacy into LLS regression through the functional mechanism. The proposed models are established by injecting noise into the log-likelihood function of LLS regression for perturbed parameter estimation. We will derive the sensitivities utilized to determine the magnitude of the injected noise and prove that the proposed DP-LLS models satisfy $ε$-differential privacy. In addition, we will conduct simulations and case studies to evaluate the performance of the proposed models. The findings suggest that predictor dimension, training sample size, and privacy budget are three key factors impacting the performance of the proposed DP-LLS regression models. Moreover, the results indicate that a sufficiently large training dataset is needed to simultaneously ensure decent performance of the proposed models and achieve a satisfactory level of privacy protection.

Differentially Private Log-Location-Scale Regression Using Functional Mechanism

TL;DR

This work develops differentially private log-location-scale (DP-LLS) regression by applying the functional mechanism to SEV and logistic regressions within the LLS family. It derives sensitivity bounds, injects Laplace noise into second-order Taylor expansions of the log-likelihood, and proves ε-differential privacy for the proposed estimators. Through simulations and a NASA aircraft case study, the authors show that DP performance degrades as predictor dimension grows and privacy becomes stronger (smaller ε), but improves with larger training sizes, with DP results approaching non-DP performance when ε is moderate or large. The findings offer practical guidance on when DP-LLS regression is viable and highlight the need for dimension reduction and ample data to balance privacy and accuracy. The approach generalizes to additional LLS distributions, enabling privacy-preserving forecasting in reliability and prognostics domains.

Abstract

This article introduces differentially private log-location-scale (DP-LLS) regression models, which incorporate differential privacy into LLS regression through the functional mechanism. The proposed models are established by injecting noise into the log-likelihood function of LLS regression for perturbed parameter estimation. We will derive the sensitivities utilized to determine the magnitude of the injected noise and prove that the proposed DP-LLS models satisfy -differential privacy. In addition, we will conduct simulations and case studies to evaluate the performance of the proposed models. The findings suggest that predictor dimension, training sample size, and privacy budget are three key factors impacting the performance of the proposed DP-LLS regression models. Moreover, the results indicate that a sufficiently large training dataset is needed to simultaneously ensure decent performance of the proposed models and achieve a satisfactory level of privacy protection.
Paper Structure (19 sections, 6 theorems, 28 equations, 9 figures, 2 algorithms)

This paper contains 19 sections, 6 theorems, 28 equations, 9 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. LLS Regression
  4. Differential Privacy
  5. Differentially Private LLS Regression
  6. Differentially Private SEV Regression
  7. Differentially Private Logistic Regression
  8. Simulation Study
  9. Experimental settings
  10. Results and Analysis
  11. Prediction Error vs. Predictor Dimension
  12. Prediction Error vs. Training Sample Size
  13. Prediction Error vs. Privacy Budget
  14. Case Study
  15. Dataset and Experimental Settings
  16. ...and 4 more sections

Key Result

Proposition 1

By applying Taylor expansion and truncating it at the second order, Equation eq:llk can be re-written as follows

Figures (9)

  • Figure 1: SEV Regression: Prediction Error vs. Dimensionality $d$
  • Figure 2: Logistic Regression: Prediction Error vs. Dimensionality $d$
  • Figure 3: SEV Regression: Error vs Sample Size $n$
  • Figure 4: Logistic Regression: Error vs Sample Size $n$
  • Figure 5: SEV Regression: Error vs DP privacy budget $\epsilon$
  • ...and 4 more figures

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6