Lower Bounds for Locally Private Estimation via Communication Complexity

TL;DR

This paper establishes fundamental lower bounds for private estimation problems under local privacy constraints by linking them to communication complexity, providing sharp bounds across all privacy levels and high-dimensional settings.

Contribution

It introduces a novel equivalence between private estimation and communication complexity, enabling sharp lower bounds for all differential privacy levels and interactive mechanisms.

Findings

Minimax mean-squared error scales as d/n * d / min{ε, ε^2}

Provides sharp lower bounds for all privacy levels

Applies to arbitrary interactive privacy mechanisms

Abstract

We develop lower bounds for estimation under local privacy constraints---including differential privacy and its relaxations to approximate or R\'{e}nyi differential privacy---by showing an equivalence between private estimation and communication-restricted estimation problems. Our results apply to arbitrarily interactive privacy mechanisms, and they also give sharp lower bounds for all levels of differential privacy protections, that is, privacy mechanisms with privacy levels $\varepsilon \in [0, \infty)$. As a particular consequence of our results, we show that the minimax mean-squared error for estimating the mean of a bounded or Gaussian random vector in $d$ dimensions scales as $\frac{d}{n} \cdot \frac{d}{ \min\{\varepsilon, \varepsilon^2\}}$.