On the role of symmetry for staircase mechanisms in local differential privacy efficiency across different privacy regimes

TL;DR

This paper explores how symmetry influences the efficiency of staircase mechanisms in local differential privacy, revealing fundamental symmetry properties and introducing asymmetric mechanisms that improve Fisher information across privacy regimes.

Contribution

It identifies a symmetry in optimal privacy mechanisms, derives Fisher information decompositions, and introduces fully asymmetric mechanisms with practical implementation.

Findings

Symmetry relates extremal values 1 and e^{α} in privacy mechanisms.

Fisher information decomposes into symmetric and asymmetric parts, scaling as α^2 and α^3.

Asymmetry becomes more beneficial as privacy constraints relax, especially in high-privacy regimes.

Abstract

We investigate the structural foundations of statistical efficiency under $\alpha$-local differential privacy, with a focus on maximizing Fisher information. Building on the role of continuous staircase mechanisms, we identify a fundamental symmetry regarding the extremal values $1$ and $e^{\alpha}$. We demonstrate that when the optimal measure satisfies this symmetry, the Fisher information admits a closed-form expression. More generally, we derive a decomposition of the Fisher information into symmetric and asymmetric components, scaling as $\alpha^{2}$ and $\alpha^{3}$, respectively, for $\alpha \to 0$. This reveals that, if in the high-privacy regime asymmetry is negligible, it is no longer the case as privacy constraints are relaxed. Motivated by this, we introduce a class of fully asymmetric privacy mechanisms constructed via pushforward mappings, proving that-unlike their symmetric counterparts-they recover the full Fisher information of the non-private model as $\alpha \to \infty$. We bridge the gap between theory and practice by providing a tractable implementation of these mechanisms, governed by a tuning parameter $c$. This parameter allows for a smooth interpolation between the symmetric regime and the fully asymmetric regime. Furthermore, we demonstrate the versatility of this framework by showing that it encompasses the binomial mechanism as a limiting case.