Factorization by extremal privacy mechanisms: new insights into efficiency

TL;DR

This paper investigates the efficiency of privacy mechanisms under local differential privacy, providing theoretical bounds, a factorization approach, and designing an extremal mechanism for uniform distribution estimation.

Contribution

It introduces a factorization lemma for privacy mechanisms, characterizes extremal mechanisms in infinite-dimensional spaces, and designs an efficient estimator for uniform distribution parameters under high privacy.

Findings

Established matching bounds on Fisher information in high privacy regime

Proved the existence of solutions for the Fisher information maximization problem for all privacy levels

Designed an extremal mechanism leading to a consistent and efficient estimator in high privacy settings

Abstract

We study the problem of efficiency under $\alpha$ local differential privacy ($\alpha$ LDP) in both discrete and continuous settings. Building on a factorization lemma, which shows that any privacy mechanism can be decomposed into an extremal mechanism followed by additional randomization, we reduce the Fisher information maximization problem to a search over extremal mechanisms. The representation of extremal mechanisms requires working in infinite dimensional spaces and invokes advanced tools from convex and functional analysis, such as Choquet's theorem. Our analysis establishes matching upper and lower bounds on the Fisher information in the high privacy regime ($\alpha \to 0$), and proves that the maximization problem always admits a solution for any $\alpha$. As a concrete application, we consider the problem of estimating the parameter of a uniform distribution on $[0, \theta]$ under $\alpha$ LDP. Guided by our theoretical findings, we design an extremal mechanism that yields a consistent and asymptotically efficient estimator in high privacy regime. Numerical experiments confirm our theoretical results.