# Exponential Separations in Local Differential Privacy

## TL;DR

This paper establishes exponential lower bounds on the sample complexity of locally different private protocols by connecting them to communication complexity, demonstrating significant separations based on interaction rounds.

## Contribution

It introduces a general framework linking communication and sample complexity, proving exponential separations for local differential privacy protocols using communication complexity results.

## Key findings

- Exponential sample complexity separation between sequentially and fully interactive protocols.
- Exponential separation between k-round and (k+1)-round protocols.
- Application of communication complexity bounds to privacy protocol analysis.

## Abstract

We prove a general connection between the communication complexity of two-player games and the sample complexity of their multi-player locally private analogues. We use this connection to prove sample complexity lower bounds for locally differentially private protocols as straightforward corollaries of results from communication complexity. In particular, we 1) use a communication lower bound for the hidden layers problem to prove an exponential sample complexity separation between sequentially and fully interactive locally private protocols, and 2) use a communication lower bound for the pointer chasing problem to prove an exponential sample complexity separation between $k$ round and $k+1$ round sequentially interactive locally private protocols, for every $k$.