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.
Categories: cs.DS cs.CR cs.IT cs.LG math.IT