Some Constructions of Private, Efficient, and Optimal K-Norm and Elliptic Gaussian Noise
arxiv(2023)
摘要
Differentially private computation often begins with a bound on some
d-dimensional statistic's ℓ_p sensitivity. For pure differential
privacy, the K-norm mechanism can improve on this approach using a norm
tailored to the statistic's sensitivity space. Writing down a closed-form
description of this optimal norm is often straightforward. However, running the
K-norm mechanism reduces to uniformly sampling the norm's unit ball; this
ball is a d-dimensional convex body, so general sampling algorithms can be
slow. Turning to concentrated differential privacy, elliptic Gaussian noise
offers similar improvement over spherical Gaussian noise. Once the shape of
this ellipse is determined, sampling is easy; however, identifying the best
such shape may be hard.
This paper solves both problems for the simple statistics of sum, count, and
vote. For each statistic, we provide a sampler for the optimal K-norm
mechanism that runs in time Õ(d^2) and derive a closed-form expression
for the optimal shape of elliptic Gaussian noise. The resulting algorithms all
yield meaningful accuracy improvements while remaining fast and simple enough
to be practical. More broadly, we suggest that problem-specific sensitivity
space analysis may be an overlooked tool for private additive noise.
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