通过改进的单纯形变换，实现更精确的隐私保护线性回归。
Refined Differentially Private Linear Regression via Extension of a Free Lunch Result
- 用多维单纯形变换处理[0,1]区间变量，提升隐私统计量估计精度。
- 在加减差分隐私下，显著降低回归参数估计误差。
- 方法可推广至多项式回归，适合高敏感数据建模场景。
随着数据隐私法规趋严，基于敏感人类数据的统计模型需具备隐私保护能力。针对加减差分隐私（add-remove DP）模型，Kulesza 等（2024）与Fitzsimons等（2024）独立证明：通过有界变量的单纯形变换及对变换后变量的私有求和查询，可“免费”估算数据集规模。本文通过精心设计的多维单纯形变换，将该“免费午餐”结果拓展至[0,1]区间内的变量与函数。我们证明，此类变换可用于精炼普通最小二乘法所需的充分统计量估计。通过解析与数值实验，验证了本方法的优越性。所提变换具有广泛适用性，可直接用于差分隐私多项式回归。
原文摘要 · Abstract (English)
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.
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