Insufficient Statistics Perturbation: Stable Estimators for Private Least Squares
This addresses privacy concerns in linear regression for datasets with bounded leverage and residuals, offering a more practical solution compared to existing approaches.
The paper tackles the problem of differentially private ordinary least squares by introducing a sample- and time-efficient algorithm with error linear in dimension and independent of the condition number, improving over prior methods that required more examples, polynomial error growth, or exponential time.
We present a sample- and time-efficient differentially private algorithm for ordinary least squares, with error that depends linearly on the dimension and is independent of the condition number of $X^\top X$, where $X$ is the design matrix. All prior private algorithms for this task require either $d^{3/2}$ examples, error growing polynomially with the condition number, or exponential time. Our near-optimal accuracy guarantee holds for any dataset with bounded statistical leverage and bounded residuals. Technically, we build on the approach of Brown et al. (2023) for private mean estimation, adding scaled noise to a carefully designed stable nonprivate estimator of the empirical regression vector.