Shuffling-Aware Optimization for Private Vector Mean Estimation

2026-04-30

AI Summary

This work addresses the absence of optimal mechanisms for unbiased mean estimation in high-dimensional settings under the single-message shuffle model. It introduces the notion of a shuffle index and formulates mechanism design as an explicit minimax optimization problem, thereby deriving a theoretical lower bound on the mean squared error. The study reveals that mechanisms optimal under local differential privacy may become suboptimal after shuffling. Leveraging this insight, the authors construct a novel mechanism that achieves a privacy–utility trade-off approaching that of the centralized Gaussian mechanism in high-privacy regimes, attaining asymptotically minimax-optimal performance.

Abstract

We study $d$-dimensional unbiased mean estimation in the single-message shuffle model, where each user sends a single privatized message and the analyzer only observes the shuffled multiset of reports. While minimax-optimal mechanisms are well understood in the local differential privacy setting, the corresponding notion of optimality after shuffling has remained largely unexplored. To address this gap, we introduce the recently proposed shuffle index and use it to formulate the post-shuffling mechanism design problem as an explicit optimization problem. We then establish a minimax lower bound on the achievable mean squared error in terms of the shuffle index, which implies that mechanisms that are optimal under LDP can become suboptimal once shuffling is applied. Finally, we construct an asymptotically minimax optimal mechanism in the high privacy regime, which as a consequence achieves a privacy-utility trade-off nearly identical to that of the central Gaussian mechanism.

Problem

• shuffle model
• mean estimation
• differential privacy
• minimax optimality
• privacy-utility trade-off

Innovation

• shuffle model
• shuffle index
• minimax optimality
• private mean estimation
• differential privacy