Sparse Discrete Laplace and Gaussian Mechanisms under Local Differential Privacy

TL;DR

This paper characterizes the privacy-utility tradeoffs of sparse discrete Laplace and Gaussian mechanisms under local differential privacy, emphasizing support size as a key complexity parameter.

Contribution

It provides exact privacy characterizations and explicit tradeoffs for sparse mechanisms, guiding optimal support size selection under privacy constraints.

Findings

Support size determines privacy-utility tradeoff

Larger supports reduce leakage but increase distortion

Optimal support size minimizes privacy loss for given constraints

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 $w(x,y)=e^{-\lambda d(x,y)}$ and the sparse Gaussian family with kernel $w(x,y)=e^{-d(x,y)^2/(2\sigma^2)}$. 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 $(\varepsilon,\delta)$-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.