Covariance-Aware Private Mean Estimation Without Private Covariance Estimation

Core Concepts

Private mean estimation without needing private covariance estimation.

Abstract

The content introduces two sample-efficient differentially private mean estimators for (sub)Gaussian distributions with unknown covariance. The estimators output an approximation of the mean with a specified accuracy guarantee, overcoming the need for strong a priori bounds on the covariance matrix or a large number of samples. The first estimator utilizes Tukey depth with the exponential mechanism, while the second perturbs the empirical mean with noise calibrated to the empirical covariance. Careful preprocessing of data is essential for ensuring differential privacy. Introduction to the problem of privacy in statistical estimators and machine learning algorithms. Presentation of two novel differentially private mean estimators for Gaussian distributions. Detailed explanation of the Tukey Depth Mechanism and the Empirically Rescaled Gaussian Mechanism. Discussion on the safety of data sets and the robustness of the Tukey Depth Mechanism against corruptions. Overview of related work and lower bounds in the field of differentially private mean estimation.

Stats

Given n samples from a distribution with mean μ and covariance Σ, the estimators output an approximation μ' such that ||μ' - μ||Σ ≤ α. The sample complexity for the Tukey Depth Mechanism is approximately O(d/α^2 + d/αε). The robustness of the Tukey Depth Mechanism allows for accurate estimation even with a corruption rate of τ.

Quotes

"Our estimators output an approximation of the mean with a specified accuracy guarantee." "Careful preprocessing of data is required to satisfy differential privacy."

Key Insights Distilled From

Covariance-Aware Private Mean Estimation Without Private Covariance Estimation

by Gavin Brown,... at arxiv.org 03-27-2024

https://arxiv.org/pdf/2106.13329.pdf

Deeper Inquiries

How can the sample complexity of differentially private mean estimators be further reduced

To further reduce the sample complexity of differentially private mean estimators, several approaches can be considered. One potential method is to explore more efficient algorithms that can leverage the structure of the data to achieve accurate estimates with fewer samples. This could involve developing novel techniques that optimize the trade-off between privacy, accuracy, and sample complexity. Additionally, refining the preprocessing steps and data transformations before applying the differential privacy mechanisms could help reduce the sample size required for accurate estimation. Furthermore, exploring advanced statistical methods and machine learning algorithms tailored for differential privacy could lead to more sample-efficient estimators.

What are the implications of the robustness of the Tukey Depth Mechanism in real-world applications

The robustness of the Tukey Depth Mechanism in real-world applications has significant implications for scenarios where data may be corrupted or contain outliers. In practical settings where data integrity is crucial, having a robust estimator that can withstand adversarial corruptions ensures the reliability of the estimation process. This robustness can enhance the trustworthiness of the results obtained from the mechanism, especially in sensitive applications where data security and accuracy are paramount. By being resilient to corruptions, the Tukey Depth Mechanism can provide more reliable estimates even in the presence of noisy or tampered data, making it a valuable tool for various real-world applications.

How does the assumption of Gaussian data impact the generalizability of the estimators to other distributions

The assumption of Gaussian data can impact the generalizability of the estimators to other distributions in several ways. Firstly, the Gaussian assumption restricts the applicability of the estimators to only distributions that closely resemble Gaussian distributions. This limitation may hinder the performance of the estimators when applied to non-Gaussian data, leading to suboptimal results. Additionally, the reliance on Gaussian properties for the estimation process may introduce biases or inaccuracies when dealing with distributions that deviate significantly from Gaussian characteristics. Therefore, the generalizability of the estimators to other distributions may be limited by the underlying assumptions and properties of Gaussian data, potentially requiring modifications or adaptations to accommodate different distributional forms.