Private Vector Mean Estimation in the Shuffle Model: Optimal Rates Require Many Messages
<|reference_start|>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 $\tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right)$ 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.<|reference_end|>
@article{asi2024private,
    title={Private Vector Mean Estimation in the Shuffle Model: Optimal Rates
  Require Many Messages},
    author={Hilal Asi, Vitaly Feldman, Jelani Nelson, Huy L. Nguyen, Kunal Talwar,
  Samson Zhou},
    journal={arXiv preprint arXiv:2404.10201},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.10201},
    primaryClass={cs.DS cs.CR cs.IT cs.LG math.IT}
}