Low Influence, Utility, and Independence in Differential Privacy: A Curious Case of $3 \choose 2$

TL;DR

This paper explores the complex relationship between low influence functions and differential privacy, revealing that while differential privacy does not imply low influence, low influence can imply approximate differential privacy, especially when mechanisms are non-independent.

Contribution

It formally analyzes the relationship between low influence and differential privacy, showing that low influence mechanisms can achieve approximate privacy only with non-independent mechanisms.

Findings

Differential privacy does not necessarily imply low influence.

Low influence implies approximate differential privacy.

Non-independent mechanisms are essential for utility and low influence.

Abstract

We study the relationship between randomized low influence functions and differentially private mechanisms. Our main aim is to formally determine whether differentially private mechanisms are low influence and whether low influence randomized functions can be differentially private. We show that differential privacy does not necessarily imply low influence in a formal sense. However, low influence implies approximate differential privacy. These results hold for both independent and non-independent randomized mechanisms, where an important instance of the former is the widely-used additive noise techniques in the differential privacy literature. Our study also reveals the interesting dynamics between utility, low influence, and independence of a differentially private mechanism. As the name of this paper suggests, we show that any two such features are simultaneously possible. However, in order to have a differentially private mechanism that has both utility and low influence, even under a very mild utility condition, one has to employ non-independent mechanisms.