Exactly Minimax-Optimal Locally Differentially Private Sampling

TL;DR

This paper establishes the fundamental privacy-utility trade-off for locally differentially private sampling, providing exact minimax bounds and universally optimal mechanisms for finite and continuous data spaces.

Contribution

It introduces the first comprehensive analysis of the privacy-utility trade-off in local differential privacy sampling, with exact minimax bounds and optimal mechanisms for various data spaces.

Findings

Proposes universally optimal sampling mechanisms for all f-divergences.

Derives exact privacy-utility trade-off bounds for finite and continuous data.

Demonstrates superior performance of proposed mechanisms over baselines.

Abstract

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.