Insufficient Statistics Perturbation: Stable Estimators for Private Least Squares
<|reference_start|>Insufficient Statistics Perturbation: Stable Estimators for Private Least Squares: 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.<|reference_end|>
@article{brown2024insufficient,
    title={Insufficient Statistics Perturbation: Stable Estimators for Private
  Least Squares},
    author={Gavin Brown, Jonathan Hayase, Samuel Hopkins, Weihao Kong, Xiyang Liu,
  Sewoong Oh, Juan C. Perdomo, Adam Smith},
    journal={arXiv preprint arXiv:2404.15409},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.15409},
    primaryClass={cs.LG cs.CR stat.ML}
}