Refined Differentially Private Linear Regression via Extension of a Free Lunch Result
For practitioners needing privacy-preserving linear regression on sensitive data, this provides a method to improve utility without additional privacy cost.
This work extends a 'free lunch' result for differentially private linear regression by designing multidimensional simplex transformations for variables bounded in [0,1], enabling refined private estimation of sufficient statistics. Analytical and numerical results demonstrate the superiority of the approach.
As data-privacy regulations tighten and statistical models are increasingly deployed on sensitive human-sourced data, privacy-preserving linear regression has become a critical necessity. For the add-remove DP model, Kulesza et al. (2024) and Fitzsimons et al. (2024) have independently shown that the size of the dataset -- an important statistic for linear regression -- can be privately estimated for "free", via a simplex transformation of bounded variables and private sum queries on the transformed variables. In this work, we extend this free lunch result via carefully crafted multidimensional simplex transformations to variables and functions that are bounded in the interval [0,1]. We show that these transformations can be applied to refine the estimates of sufficient statistics needed for private simple linear regression based on ordinary least squares. We provide both analytical and numerical results to demonstrate the superiority of our approach. Our proposed transformations have general applicability and can be readily adapted for differentially private polynomial regression.