Private Vector Mean Estimation in the Shuffle Model: Optimal Rates Require Many Messages

TL;DR

This paper introduces optimal multi-message protocols for private vector mean estimation in the shuffle model, establishing bounds on message complexity and error, and explores single-message protocols and robustness issues.

Contribution

It presents a new multi-message protocol achieving optimal error with minimal messages and proves its optimality, along with a single-message protocol and analysis of robustness against malicious users.

Findings

Multi-message protocol achieves optimal error with ilde{O}( ext{min}(n ext{ε}^2, d)) messages.

Any unbiased protocol with optimal error requires ilde{ ext{min}(n ext{ε}^2, d)} messages.

Single-message protocol attains mean squared error of O(d n^{d/(d+2)} ext{ε}^{-4/(d+2)}).

Abstract

We study the problem of private vector mean estimation in the shuffle model of privacy where $n$ users each have a unit vector $v^{(i)} \in\mathbb{R}^d$. We propose a new multi-message protocol that achieves the optimal error using $\tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right)$ messages per user. Moreover, we show that any (unbiased) protocol that achieves optimal error requires each user to send $\Omega(\min(n\varepsilon^2,d)/\log(n))$ messages, demonstrating the optimality of our message complexity up to logarithmic factors. Additionally, we study the single-message setting and design a protocol that achieves mean squared error $\mathcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)})$. Moreover, we show that any single-message protocol must incur mean squared error $\Omega(dn^{d/(d+2)})$, showing that our protocol is optimal in the standard setting where $\varepsilon = \Theta(1)$. Finally, we study robustness to malicious users and show that malicious users can incur large additive error with a single shuffler.