Differentially Private Sampling from Distributions via Wasserstein Projection

TL;DR

This paper introduces a new framework for differentially private sampling using Wasserstein distance, addressing geometric and support-difference limitations of previous density ratio-based methods.

Contribution

It proposes Wasserstein Projection Mechanism (WPM), a minimax optimal DP sampling method based on Wasserstein projection, with efficient algorithms and convergence guarantees.

Findings

WPM effectively captures geometric support structure.

The proposed algorithms approximate WPM with proven convergence.

WPM outperforms traditional density ratio-based methods in certain scenarios.

Abstract

In this paper, we study the problem of sampling from a distribution under the constraint of differential privacy (DP). Prior works measure the utility of DP sampling with density ratio-based measures such as KL divergence. However, such formulations suffer from two key limitations: 1) they fail to capture the geometric structure of the support, and 2) they are not applicable when the supports of the distributions differ. To deal with these issues, we develop a novel framework for DP sampling with Wasserstein distance as the utility measure. In this formulation, we propose Wasserstein Projection Mechanism (WPM), a minimax optimal mechanism based on Wasserstein projection. Furthermore, we develop efficient algorithms for computing the proposed mechanisms approximately and provide convergence guarantees.