Local Differential Privacy Is Equivalent to Contraction of $E_\gamma$-Divergence

TL;DR

This paper establishes an equivalence between local differential privacy guarantees and the contraction properties of the $E_eta$-divergence, enabling new analysis of privacy-utility trade-offs in statistical estimation.

Contribution

It introduces a novel characterization of local differential privacy via contraction coefficients of $E_eta$-divergences, broadening the theoretical understanding of privacy mechanisms.

Findings

LDP constraints can be expressed through contraction coefficients.

The framework applies to various $f$-divergences.

Facilitates analysis of privacy-utility trade-offs in estimation problems.

Abstract

We investigate the local differential privacy (LDP) guarantees of a randomized privacy mechanism via its contraction properties. We first show that LDP constraints can be equivalently cast in terms of the contraction coefficient of the $E_\gamma$-divergence. We then use this equivalent formula to express LDP guarantees of privacy mechanisms in terms of contraction coefficients of arbitrary $f$-divergences. When combined with standard estimation-theoretic tools (such as Le Cam's and Fano's converse methods), this result allows us to study the trade-off between privacy and utility in several testing and minimax and Bayesian estimation problems.