Optimal Differentially Private PCA and Estimation for Spiked Covariance Matrices

• URL: http://arxiv.org/abs/2401.03820v2
• Date: Fri, 27 Sep 2024 14:15:12 GMT
• Title: Optimal Differentially Private PCA and Estimation for Spiked Covariance Matrices
• Authors: T. Tony Cai, Dong Xia, Mengyue Zha,
• Abstract summary: Estimating a covariance matrix and its associated principal components is a fundamental problem in contemporary statistics. We study optimal differentially private Principal Component Analysis (PCA) and covariance estimation within the spiked covariance model. We propose computationally efficient differentially private estimators and prove their minimax optimality for sub-Gaussian distributions.
• Score: 10.377683220196873
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Estimating a covariance matrix and its associated principal components is a fundamental problem in contemporary statistics. While optimal estimation procedures have been developed with well-understood properties, the increasing demand for privacy preservation introduces new complexities to this classical problem. In this paper, we study optimal differentially private Principal Component Analysis (PCA) and covariance estimation within the spiked covariance model. We precisely characterize the sensitivity of eigenvalues and eigenvectors under this model and establish the minimax rates of convergence for estimating both the principal components and covariance matrix. These rates hold up to logarithmic factors and encompass general Schatten norms, including spectral norm, Frobenius norm, and nuclear norm as special cases. We propose computationally efficient differentially private estimators and prove their minimax optimality for sub-Gaussian distributions, up to logarithmic factors. Additionally, matching minimax lower bounds are established. Notably, compared to the existing literature, our results accommodate a diverging rank, a broader range of signal strengths, and remain valid even when the sample size is much smaller than the dimension, provided the signal strength is sufficiently strong. Both simulation studies and real data experiments demonstrate the merits of our method.

Related papers

• A Geometric Unification of Distributionally Robust Covariance Estimators: Shrinking the Spectrum by Inflating the Ambiguity Set [20.166217494056916]
  We propose a principled approach to construct covariance estimators without imposing restrictive assumptions. We show that our robust estimators are efficiently computable and consistent. Numerical experiments based on synthetic and real data show that our robust estimators are competitive with state-of-the-art estimators.
  arXiv  Detail & Related papers  (2024-05-30T15:01:18Z)
• Convex Parameter Estimation of Perturbed Multivariate Generalized Gaussian Distributions [18.95928707619676]
  We propose a convex formulation with well-established properties for MGGD parameters. The proposed framework is flexible as it combines a variety of regularizations for the precision matrix, the mean and perturbations. Experiments show a more accurate precision and covariance matrix estimation with similar performance for the mean vector parameter.
  arXiv  Detail & Related papers  (2023-12-12T18:08:04Z)
• Robustness Implies Privacy in Statistical Estimation [16.061651295129302]
  We study the relationship between adversarial and differential privacy in high-dimensional statistics. We give the first blackbox reduction from privacy to robustness which can produce private estimators with optimal tradeoffs. Our algorithms are also robust to a nearly optimal fraction of adversarially-corrupted samples.
  arXiv  Detail & Related papers  (2022-12-09T18:07:30Z)
• Efficient CDF Approximations for Normalizing Flows [64.60846767084877]
  We build upon the diffeomorphic properties of normalizing flows to estimate the cumulative distribution function (CDF) over a closed region. Our experiments on popular flow architectures and UCI datasets show a marked improvement in sample efficiency as compared to traditional estimators.
  arXiv  Detail & Related papers  (2022-02-23T06:11:49Z)
• Learning to Estimate Without Bias [57.82628598276623]
  Gauss theorem states that the weighted least squares estimator is a linear minimum variance unbiased estimation (MVUE) in linear models. In this paper, we take a first step towards extending this result to non linear settings via deep learning with bias constraints. A second motivation to BCE is in applications where multiple estimates of the same unknown are averaged for improved performance.
  arXiv  Detail & Related papers  (2021-10-24T10:23:51Z)
• Near-optimal inference in adaptive linear regression [60.08422051718195]
  Even simple methods like least squares can exhibit non-normal behavior when data is collected in an adaptive manner. We propose a family of online debiasing estimators to correct these distributional anomalies in at least squares estimation. We demonstrate the usefulness of our theory via applications to multi-armed bandit, autoregressive time series estimation, and active learning with exploration.
  arXiv  Detail & Related papers  (2021-07-05T21:05:11Z)
• Benign Overfitting of Constant-Stepsize SGD for Linear Regression [122.70478935214128]
  inductive biases are central in preventing overfitting empirically. This work considers this issue in arguably the most basic setting: constant-stepsize SGD for linear regression. We reflect on a number of notable differences between the algorithmic regularization afforded by (unregularized) SGD in comparison to ordinary least squares.
  arXiv  Detail & Related papers  (2021-03-23T17:15:53Z)
• Understanding Implicit Regularization in Over-Parameterized Single Index Model [55.41685740015095]
  We design regularization-free algorithms for the high-dimensional single index model. We provide theoretical guarantees for the induced implicit regularization phenomenon.
  arXiv  Detail & Related papers  (2020-07-16T13:27:47Z)
• Fundamental Limits of Ridge-Regularized Empirical Risk Minimization in High Dimensions [41.7567932118769]
  Empirical Risk Minimization algorithms are widely used in a variety of estimation and prediction tasks. In this paper, we characterize for the first time the fundamental limits on the statistical accuracy of convex ERM for inference.
  arXiv  Detail & Related papers  (2020-06-16T04:27:38Z)
• CoinPress: Practical Private Mean and Covariance Estimation [18.6419638570742]
  We present simple differentially private estimators for the mean and covariance of multivariate sub-Gaussian data. We show that their error rates match the state-of-the-art theoretical bounds, and that they concretely outperform all previous methods.
  arXiv  Detail & Related papers  (2020-06-11T17:17:28Z)
• Instability, Computational Efficiency and Statistical Accuracy [101.32305022521024]
  We develop a framework that yields statistical accuracy based on interplay between the deterministic convergence rate of the algorithm at the population level, and its degree of (instability) when applied to an empirical object based on $n$ samples. We provide applications of our general results to several concrete classes of models, including Gaussian mixture estimation, non-linear regression models, and informative non-response models.
  arXiv  Detail & Related papers  (2020-05-22T22:30:52Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.