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Logistic Regression Model for Differentially-Private Matrix Masked Data

Linh H Nghiem, Aidong A. Ding, Samuel Wu

TL;DR

Simulations and real data analyses demonstrate the superiority of the proposed estimators over naive logistic regression methods on privacy-preserved data sets and confirm its validity under an asymptotic framework with increasing noise magnitude to account for strict privacy requirements.

Abstract

A recently proposed scheme utilizing local noise addition and matrix masking enables data collection while protecting individual privacy from all parties, including the central data manager. Statistical analysis of such privacy-preserved data is particularly challenging for nonlinear models like logistic regression. By leveraging a relationship between logistic regression and linear regression estimators, we propose the first valid statistical analysis method for logistic regression under this setting. Theoretical analysis of the proposed estimators confirmed its validity under an asymptotic framework with increasing noise magnitude to account for strict privacy requirements. Simulations and real data analyses demonstrate the superiority of the proposed estimators over naive logistic regression methods on privacy-preserved data sets.

Logistic Regression Model for Differentially-Private Matrix Masked Data

TL;DR

Simulations and real data analyses demonstrate the superiority of the proposed estimators over naive logistic regression methods on privacy-preserved data sets and confirm its validity under an asymptotic framework with increasing noise magnitude to account for strict privacy requirements.

Abstract

A recently proposed scheme utilizing local noise addition and matrix masking enables data collection while protecting individual privacy from all parties, including the central data manager. Statistical analysis of such privacy-preserved data is particularly challenging for nonlinear models like logistic regression. By leveraging a relationship between logistic regression and linear regression estimators, we propose the first valid statistical analysis method for logistic regression under this setting. Theoretical analysis of the proposed estimators confirmed its validity under an asymptotic framework with increasing noise magnitude to account for strict privacy requirements. Simulations and real data analyses demonstrate the superiority of the proposed estimators over naive logistic regression methods on privacy-preserved data sets.

Paper Structure

This paper contains 13 sections, 3 theorems, 50 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Notations and model setup of logistic regression data collected from TM$^2$+Noise scheme
  3. Methodology
  4. Logistic model from mixture normal
  5. Conditional mixture model
  6. Estimation and inference from masked data
  7. Simulation
  8. Real data analysis
  9. Conclusion
  10. Appendix
  11. Differential privacy guarantee for the TM$^2$+Noise scheme
  12. Derivation of equation \ref{['eq:log_relationship']}
  13. Proof of Lemma \ref{['est.b1.cond.mixturemodel']}

Key Result

Lemma 1

Under the conditional mixture model conditionmixturemodel-eq:logisticZ, we have $\bm\beta_1 = \left[\textrm{Var}(\varepsilon_1)\right]^{-1} \bar{\bm{b}}_1$.

Theorems & Definitions (3)

  • Lemma 1
  • Proposition 3.1
  • Theorem 3.1