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

Abstract

We investigate the structural foundations of statistical efficiency under $α$-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 $e^α$. 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 $α^{2}$ and $α^{3}$, respectively, for $α\to 0$. 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 $α\to \infty$. 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.

Figures (3)

• Figure 1: $\nu(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$
• Figure 2: $\nu(x)=\frac{1}{\pi(1+x^2)}$
• Figure 3: Approximation of binomial mechanism

Theorems & Definitions (26)

• Definition 1
• Definition 2
• Definition 3
• Proposition 1
• Remark 1
• Remark 2
• Theorem 1
• Remark 3
• Proposition 2
• Remark 4