The Cost of Privacy: Optimal Rates of Convergence for Parameter Estimation with Differential Privacy

TL;DR

This paper establishes the fundamental limits of statistical accuracy under differential privacy constraints for mean estimation and linear regression, introduces new lower bound techniques, and proposes efficient algorithms that nearly achieve these bounds.

Contribution

It develops a refined lower bound method for privacy-constrained estimation and designs algorithms that are nearly minimax optimal in high-dimensional settings.

Findings

Derived sharp minimax lower bounds for differentially private estimation.

Proposed a novel private iterative hard thresholding algorithm.

Demonstrated near-optimal performance through simulations and real data applications.

Abstract

Privacy-preserving data analysis is a rising challenge in contemporary statistics, as the privacy guarantees of statistical methods are often achieved at the expense of accuracy. In this paper, we investigate the tradeoff between statistical accuracy and privacy in mean estimation and linear regression, under both the classical low-dimensional and modern high-dimensional settings. A primary focus is to establish minimax optimality for statistical estimation with the $(\varepsilon,\delta)$-differential privacy constraint. To this end, we find that classical lower bound arguments fail to yield sharp results, and new technical tools are called for. By refining the "tracing adversary" technique for lower bounds in the theoretical computer science literature, we formulate a general lower bound argument for minimax risks with differential privacy constraints, and apply this argument to high-dimensional mean estimation and linear regression problems. We also design computationally efficient algorithms that attain the minimax lower bounds up to a logarithmic factor. In particular, for the high-dimensional linear regression, a novel private iterative hard thresholding pursuit algorithm is proposed, based on a privately truncated version of stochastic gradient descent. The numerical performance of these algorithms is demonstrated by simulation studies and applications to real data containing sensitive information, for which privacy-preserving statistical methods are necessary.