Communication Complexity in Locally Private Distribution Estimation and Heavy Hitters

TL;DR

This paper develops optimal communication-efficient local differential privacy schemes for distribution and heavy hitter estimation, revealing fundamental differences in communication requirements for these tasks.

Contribution

It introduces a sample-optimal one-bit communication scheme for distribution estimation under local differential privacy and analyzes the communication limits for heavy hitter estimation.

Findings

Hadamard Response is utility-optimal for heavy hitter estimation.

One-bit communication suffices for distribution estimation under LDP.

Heavy hitter estimation requires at least logarithmic bits per user without public randomness.

Abstract

We consider the problems of distribution estimation and heavy hitter (frequency) estimation under privacy and communication constraints. While these constraints have been studied separately, optimal schemes for one are sub-optimal for the other. We propose a sample-optimal $\varepsilon$-locally differentially private (LDP) scheme for distribution estimation, where each user communicates only one bit, and requires no public randomness. We show that Hadamard Response, a recently proposed scheme for $\varepsilon$-LDP distribution estimation is also utility-optimal for heavy hitter estimation. Finally, we show that unlike distribution estimation, without public randomness where only one bit suffices, any heavy hitter estimation algorithm that communicates $o(\min \{\log n, \log k\})$ bits from each user cannot be optimal.