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Diffusion Sensitivity Geometry and Strong Duality for Optimal Privacy-Utility Tradeoffs

Tue Do, Daniel Alabi · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.28991 · cited by 0 works (OpenAlex) · open-access PDF

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Los enfoques existentes para la privacidad diferencial en los modelos de difusión se basan en los límites de sensibilidad libres y el ruido isotrópico de Gauss, lo que conduce a errores de estimación innecesarios. Desarrollamos un marco para el comercio de privacidad-utilidad óptimo de los denoisores de difusión ideales, reemplazando los límites conservadores con características de sensibilidad exactas y la calibración de ruido probada óptima. En primer lugar, mostramos que la sensibilidad en el peor de los casos sobre conjuntos de datos de tamaño arbitrario se puede caracterizar con exactitud utilizando solo dos registros. Para un dominio de datos de tamaño de $m$, esto reduce el análisis a las limitaciones de privacidad más altas de $m(m-1)$, independientemente del tamaño del

Machine translation (Meta M2M100); the English original follows.

Abstract

Existing approaches to differential privacy in diffusion models rely on loose sensitivity bounds and isotropic Gaussian noise, leading to unnecessary estimation error. We develop a framework for the optimal privacy-utility tradeoff of ideal diffusion denoisers, replacing conservative bounds with exact sensitivity characterizations and provably optimal noise calibration. First, we show that worst-case sensitivity over datasets of arbitrary size can be characterized exactly using only two records. For a data domain of size $m$, this reduces the analysis to at most $m(m-1)$ constraints, regardless of dataset size. Our characterization can be arbitrarily tighter than conventional diameter-based sensitivity bounds, revealing potentially unbounded improvements in noise calibration. Second, we obtain exact privacy guarantees for Rényi, approximate, and Gaussian differential privacy, as well as mutual-information privacy for binary records, avoiding the looseness of previous generic privacy conversions. Finally, we show that the minimum achievable weighted mean-squared error can be computed through a semidefinite program and prove strong duality, providing certificates of global optimality.

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Topics: Sensitivity (control systems), Differential privacy, Duality (order theory), Gaussian, Mathematics, Noise (video)

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