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

Hilal Asi, Vitaly Feldman, Jelani Nelson, Huy Nguyen, Kunal Talwar, Samson Zhou
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:1945-1970, 2024.

Abstract

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)∈Rd. We propose a new multi-message protocol that achieves the optimal error using O(min 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-asi24a, title = {Private Vector Mean Estimation in the Shuffle Model: Optimal Rates Require Many Messages}, author = {Asi, Hilal and Feldman, Vitaly and Nelson, Jelani and Nguyen, Huy and Talwar, Kunal and Zhou, Samson}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {1945--1970}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/asi24a/asi24a.pdf}, url = {https://proceedings.mlr.press/v235/asi24a.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Private Vector Mean Estimation in the Shuffle Model: Optimal Rates Require Many Messages %A Hilal Asi %A Vitaly Feldman %A Jelani Nelson %A Huy Nguyen %A Kunal Talwar %A Samson Zhou %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-asi24a %I PMLR %P 1945--1970 %U https://proceedings.mlr.press/v235/asi24a.html %V 235 %X 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.
APA
Asi, H., Feldman, V., Nelson, J., Nguyen, H., Talwar, K. & Zhou, S.. (2024). Private Vector Mean Estimation in the Shuffle Model: Optimal Rates Require Many Messages. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:1945-1970 Available from https://proceedings.mlr.press/v235/asi24a.html.

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