Logistic Regression Model for Differentially-Private Matrix Masked Data

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.

💡 Research Summary

This paper addresses the problem of performing logistic regression on data that have been protected by a combination of local differential privacy (LDP) noise addition and triple matrix masking (TM²), a scheme the authors refer to as TM² + Noise. In this setting, each individual adds Gaussian noise to both covariates and the binary outcome before transmitting the data, after which a random orthogonal matrix M scrambles the rows. The analyst therefore observes only real‑valued, masked covariates (X = MX^{}+U) and a continuous response (y = My^{}+v). Because the response is no longer binary and the masking preserves only first‑ and second‑order moments, standard logistic regression (maximum conditional likelihood) cannot be applied directly, and existing measurement‑error methods for generalized linear models are not suitable.

The authors’ key insight is to exploit a classic relationship between logistic regression and a Gaussian mixture model, originally noted by Haggström (1983). They show that if the raw covariates follow a class‑conditional normal distribution, \