The sampling problem under local differential privacy has recently been studied with potential applications to generative models, but a fundamental analysis of its privacy-utility trade-off (PUT) remains incomplete. In this work, we define the fundamental PUT of private sampling in the minimax sense, using the f-divergence between original and sampling distributions as the utility measure. We characterize the exact PUT for both finite and continuous data spaces under some mild conditions on the data distributions, and propose sampling mechanisms that are universally optimal for all f-divergences. Our numerical experiments demonstrate the superiority of our mechanisms over baselines, in terms of theoretical utilities for finite data space and of empirical utilities for continuous data space.

翻译：本地差分隐私下的采样问题近期因在生成模型中的潜在应用而受到关注，但其隐私-效用权衡的基本理论分析尚不完整。本文从极小极大意义上定义了私有采样的基本隐私-效用权衡，以原始分布与采样分布之间的f-散度作为效用度量。在数据分布满足温和条件下，我们刻画了有限与连续数据空间中精确的隐私-效用权衡，并提出了适用于所有f-散度的通用最优采样机制。数值实验表明，在有限数据空间的理论效用和连续数据空间的实证效用方面，我们的机制均优于基线方法。