We study the problem of estimating the mean of private vectors in the random privacy model where north<annotation encoding="application/x-tex”>northnorth Users each have a unit vector in d<annotation encoding="application/x-tex”>dd dimensions. We propose a new multi-message protocol that achieves the optimal error using Oh~(min.(northmy2,d))<annotation encoding="application/x-tex”>\tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right)Oh~(min.(northmy2,d)) messages per user. Furthermore, we show that any (unbiased) protocol that achieves optimal error requires each user to send Oh(min.(northmy2,d)/record(north))<annotation encoding="application/x-tex”>\Omega(\min(n\varepsilon^2,d)/\log(n))Oh(min.(northmy2,d)/Itgram(north)) messages, demonstrating the optimality of our message complexity up to logarithmic factors.
Furthermore, we study the single message configuration and design a protocol that achieves mean square error. Oh(dnorthd/(d+2)my−4/(d+2))<annotation encoding="application/x-tex”>\mathcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)})Oh(dnorthd/(d+2)my−4/(d+2))Furthermore, we show that any single-message protocol must incur a mean square error. Oh(dnorthd/(d+2))<annotation encoding="application/x-tex”>\Omega(dn^{d/(d+2)})Oh(dnorthd/(d+2))which shows that our protocol is optimal in the standard environment where my=Th(1)<annotation encoding="application/x-tex”>\varepsilon = \Theta(1)my=Th(1)Finally, we study robustness to malicious users and show that malicious users can make a large additive error with a single shuffler.