Differentially Private Sampling from Distributions via Wasserstein Projection

2026-05-11

AI Summary

Under differential privacy constraints, conventional density-ratio-based utility measures struggle to capture the geometric structure of distributional support and fail when supports are mismatched. This work introduces the Wasserstein distance as a novel utility metric, establishing a new differentially private sampling framework and proposing the Wasserstein Projection Mechanism (WPM) that achieves minimax optimality. The mechanism effectively preserves the geometric information of the underlying distribution while satisfying differential privacy guarantees. Furthermore, an efficient approximation algorithm is developed with rigorous convergence analysis, ensuring both theoretical optimality and computational tractability in practical implementations.

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.

Problem

• differential privacy
• sampling
• Wasserstein distance
• distribution
• utility measure

Innovation

• Differential Privacy
• Wasserstein Distance
• Sampling
• Wasserstein Projection Mechanism
• Minimax Optimality