Shuffling-Aware Optimization for Private Vector Mean Estimation

TL;DR

This paper investigates optimal mean estimation in the shuffle model, introducing a new framework and mechanism that nearly match the central Gaussian mechanism's privacy-utility trade-off.

Contribution

It formulates the post-shuffling mechanism design as an optimization problem using the shuffle index and constructs an asymptotically minimax optimal mechanism.

Findings

Minimax lower bound on mean squared error in the shuffle model.

Mechanisms optimal under LDP can be suboptimal after shuffling.

Constructed mechanism achieves near-central Gaussian privacy-utility trade-off.

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.