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.

翻译：本文研究在差分隐私（DP）约束下从分布中采样的问题。现有研究采用KL散度等密度比指标衡量DP采样的效用，但此类方法存在两大局限：1）无法捕捉支撑集的几何结构；2）当分布支撑集不同时不再适用。针对这些问题，我们构建了以Wasserstein距离为效用度量的DP采样新框架。在该框架下，我们提出Wasserstein投影机制（WPM），一种基于Wasserstein投影的极小最优机制。此外，我们开发了用于近似计算该机制的高效算法，并给出了收敛性保证。