Differentially Private Least Squares Regression with Optimal Dimension Dependence

Core Concepts

This work presents a sample- and time-efficient differentially private algorithm for ordinary least squares regression, with error that depends linearly on the dimension and is independent of the condition number of the design matrix.

Abstract

The authors present a new algorithm, ISSP, for differentially private linear regression. ISSP works in two main phases: It searches for a reweighting of the dataset such that running OLS on this reweighted version is roughly stable. It computes the OLS solution on this weighted version of the data and adds appropriately shaped Gaussian noise to the solution. The key technical advances are: ISSP does not require any norm bounds on the data, only that the dataset is "good" - i.e., has bounded leverage scores and bounded residuals. This captures natural, well-studied settings where OLS is a sensible procedure. The authors prove that ISSP is differentially private and establish utility guarantees. On good datasets, the private estimator is just a slightly noisier version of the empirical OLS solution. For random-design regression with subgaussian covariates and label noise, the error of ISSP is shown to be nearly optimal, matching known lower bounds up to logarithmic factors. The algorithm can be implemented efficiently, requiring only basic linear algebraic operations.

Stats

The maximum leverage score of any observation is bounded by L. The magnitude of the residual for any observation is bounded by R.

Quotes

"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⊤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."

Key Insights Distilled From

Insufficient Statistics Perturbation: Stable Estimators for Private Least Squares

by Gavin Brown,... at arxiv.org 04-25-2024

https://arxiv.org/pdf/2404.15409.pdf

Deeper Inquiries

How can the algorithm be extended to handle heterogeneous data, where the leverage and residual bounds may vary across observations

To extend the algorithm to handle heterogeneous data with varying leverage and residual bounds across observations, we can introduce a more flexible framework that allows for individualized thresholds. This can be achieved by incorporating adaptive mechanisms that adjust the leverage and residual bounds on a per-observation basis. By dynamically updating these bounds based on the characteristics of each observation, the algorithm can effectively handle the heterogeneity present in the data. Additionally, incorporating machine learning techniques such as clustering or outlier detection algorithms can help identify patterns in the data and tailor the bounds accordingly. This adaptive approach would enhance the algorithm's ability to accommodate diverse data structures and improve its performance on heterogeneous datasets.

What are the implications of this work for private statistical inference, such as constructing confidence intervals for the regression coefficients

The implications of this work for private statistical inference, particularly in constructing confidence intervals for regression coefficients, are significant. By providing a sample- and time-efficient differentially private algorithm for linear regression, this research opens up possibilities for conducting private statistical analysis on sensitive data while preserving privacy guarantees. In the context of constructing confidence intervals for regression coefficients, the algorithm's accuracy guarantees and differential privacy properties ensure that the privacy of individual data points is maintained while still allowing for reliable statistical inference. This means that researchers and practitioners can confidently derive private confidence intervals for regression coefficients without compromising the privacy of the underlying data. The algorithm's near-optimal accuracy and privacy guarantees make it a valuable tool for private statistical inference tasks, including constructing confidence intervals in a privacy-preserving manner.

Can the techniques developed here be applied to other statistical tasks beyond linear regression, such as generalized linear models or nonparametric regression

The techniques developed in this work can be applied to a variety of statistical tasks beyond linear regression, including generalized linear models (GLMs) and nonparametric regression. For GLMs, the algorithm can be adapted to handle different types of response variables and link functions, allowing for private estimation of model parameters while maintaining differential privacy. By incorporating the principles of sufficient statistics perturbation and stable estimators, the algorithm can provide accurate and private estimates for GLMs in a sample- and time-efficient manner. Similarly, for nonparametric regression tasks, the algorithm's stability properties and differential privacy guarantees can be leveraged to ensure privacy-preserving estimation of regression functions without compromising accuracy. Overall, the techniques developed in this work have the potential to be extended to a wide range of statistical tasks, making them versatile tools for private statistical analysis across various domains.