Sparse Discrete Laplace and Gaussian Mechanisms under Local Differential Privacy

2026-05-10

AI Summary

This work investigates the design of sparse discrete perturbation mechanisms under local differential privacy, aiming to simultaneously achieve strong privacy guarantees and output sparsity. The authors propose an input-dependent sparse private channel that employs truncated discrete Laplace and Gaussian kernels, balancing privacy and sparsity by controlling the size of the output support set. They provide the first precise characterization of necessary and sufficient conditions for such mechanisms to satisfy pure and approximate local differential privacy, revealing the support size as the key parameter governing mechanism complexity. Furthermore, they derive an explicit privacy–sparsity trade-off: nontrivial approximate privacy necessitates a minimal support size, and for Gaussian mechanisms, the privacy loss grows quadratically with the support radius.

Abstract

We study sparse locally private channels of the form $M(y\mid x)\propto w(x,y) 1\{y\in S(x)\},$ where the admissible output set $S(x)$ is allowed to depend on the private input $x$ and is assumed to be small. Here, we consider the sparse discrete-Laplace family with kernel $w(x,y)=e^{-λd(x,y)}$ and the sparse Gaussian family with kernel $w(x,y)=e^{-d(x,y)^2/(2σ^2)}$. For both families we give exact characterizations of pure and approximate local differential privacy. For pure $\varepsilon$-local differential privacy, we show that input-dependent sparse supports are obtained when all supports coincide. For $(\varepsilon,δ)$-local differential privacy, we derive exact formulas for the privacy defect in terms of support leakage and excess privacy loss on the overlap region. We then specialize the analysis to radius-truncated sparse discrete-Laplace and radius-truncated sparse Gaussian mechanisms and obtain explicit privacy-sparsity tradeoffs in terms of the support size $s$. In particular, we show that nontrivial approximate local privacy requires a minimum support size, whereas larger supports reduce support leakage but increase distortion. For the Gaussian family, the overlap term exhibits an additional quadratic dependence on the support radius, which implies a sharper tradeoff between privacy and sparsity. These results identify the support cardinality as the intrinsic complexity parameter of the mechanism and yield an optimal design principle: choose the smallest support size that satisfies the target privacy constraint.

Problem

• local differential privacy
• sparse mechanisms
• discrete Laplace mechanism
• Gaussian mechanism
• privacy-sparsity tradeoff

Innovation

• sparse mechanisms
• local differential privacy
• discrete Laplace mechanism
• Gaussian mechanism
• privacy-sparsity tradeoff