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

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 $O(\min(n\varepsilon^2,d))$ messages per user. Moreover, we show that any (unbiased) protocol that achieves optimal error must require 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 $O(dn^{d/(d+2)}\varepsilon^{-4/(d+2)})$. 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.