Concetti Chiave
Optimal rates for private vector mean estimation in the shuffle model require sending a large number of messages per user.
Sintesi
The paper studies the problem of private vector mean estimation in the shuffle model of privacy, where n users each have a unit vector v(i) ∈ Rd. The authors propose a new multi-message protocol that achieves the optimal error using O~(min(nε^2, d)) messages per user. They also show that any (unbiased) protocol that achieves optimal error requires each user to send Ω(min(nε^2, d)/ log(n)) messages, demonstrating the optimality of their message complexity up to logarithmic factors.
Additionally, the authors study the single-message setting and design a protocol that achieves mean squared error O(dnd/(d+2)ε^(-4/(d+2))). They also show that any single-message protocol must incur mean squared error Ω(dnd/(d+2)), showing that their protocol is optimal in the standard setting where ε = Θ(1).
Finally, the authors study robustness to malicious users and show that malicious users can incur large additive error with a single shuffler. They demonstrate that a large class of accurate protocols in the multi-message shuffle model are inherently non-robust, while the multi-shuffler model can allow for better robustness but at a significant additional cost.
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