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

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We study the problem of private vector mean estimation in the shuffle model of privacy where nnn users each have a unit vector in ddd dimensions. We propose a new multi-message protocol that achieves the optimal error using O~(min⁡(nε2,d))\tilde\mathcalO\left(\min(n\varepsilon^2,d)\right)O~(min(nε2,d)) messages per user. Moreover, we show that any (unbiased) protocol that achieves optimal error requires each user to send Ω(min⁡(nε2,d)/log⁡(n))\Omega(\min(n\varepsilon^2,d)/\log(n))Ω(min(nε2,d)/log(n)) messages, demonstrating the optimality of our message complexity up to logarithmic…

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