On the role of symmetry for staircase mechanisms in local differential privacy efficiency across different privacy regimes
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arXiv
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| Format: | Preprint |
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2026
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| author | Amorino, Chiara Gloter, Arnaud |
| author_facet | Amorino, Chiara Gloter, Arnaud |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_27572 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the role of symmetry for staircase mechanisms in local differential privacy efficiency across different privacy regimes Amorino, Chiara Gloter, Arnaud Statistics Theory Probability Applications 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. |
| title | On the role of symmetry for staircase mechanisms in local differential privacy efficiency across different privacy regimes |
| topic | Statistics Theory Probability Applications |
| url | https://arxiv.org/abs/2603.27572 |