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

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 $\tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right)$ messages per user. Moreover, we show that any (unbiased) protocol that achieves optimal error requires each user to send $Ω(\min(n\varepsilon^2,d)/\log(n))$ 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 $Ω(dn^{d/(d+2)})$, showing that our protocol is optimal in the standard setting where $\varepsilon = Θ(1)$. Finally, we study robustness to malicious users and show that malicious users can incur large additive error with a single shuffler.

• The paper demonstrates that achieving optimal error rates in vector mean estimation within the shuffle model requires high per-user message complexity.
• It introduces a protocol that reaches an error rate of d/ε² using approximately min(nε², d) messages per user, closely matching central model performance.
• It establishes theoretical lower bounds and discusses practical challenges, including the protocol's robustness against malicious users.

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

Introduction and Background

The paper "Private Vector Mean Estimation in the Shuffle Model: Optimal Rates Require Many Messages" (2404.10201) addresses the problem of differentially private vector mean estimation within the shuffle model of privacy. This model is particularly pertinent to federated learning scenarios, in which large-scale data originating from multiple users must be aggregated while ensuring the privacy of each individual's data. The paper's contributions are significant in the context of differential privacy (DP), especially under the local DP (LDP) model considerations, where high noise typically introduces diminished accuracy.

Differential privacy provides a mathematical framework to ensure data privacy, with the shuffle model—a specific approach where a trusted intermediary shuffles user data—offering a compromise between the highly accurate central model and the highly private but less accurate local model. The paper investigates the trade-offs between message complexity and the achievable level of privacy, focusing on whether optimal error rates can be realized with minimal communication overhead by leveraging the shuffle model.

Main Contributions

This paper's primary contribution lies in demonstrating that achieving optimal error rates in vector mean estimation within the shuffle model requires a significant communication load, quantified in terms of message complexity. Specifically, the authors establish a protocol that achieves optimal error rates of $\frac{d}{\varepsilon^2}$ using $\min(n\varepsilon^2, d)$ messages per user, largely matching the performance of the central DP model aside from logarithmic factors. This result underscores the inherent communication trade-offs in obtaining accurate estimations while guaranteeing privacy.

Further, the authors show that attaining optimal error with any unbiased protocol necessitates a per-user message complexity of $\Omega(\min(n\varepsilon^2, d)/\log(n))$. The established lower bound highlights that optimal rates under the shuffle model's constraints inherently demand significant communication overhead.

Additionally, the paper explores the single-message setting, presenting a protocol achieving mean squared error of $dn^{d/(d+2)}\varepsilon^{-4/(d+2)}$. This protocol is shown to be optimal under standard settings where $\varepsilon = \Theta(1)$. The robustness of protocols to malicious users is also addressed, elucidating the potential vulnerabilities when a single shuffler is involved.

Theoretical and Practical Implications

The paper's findings have notable implications both theoretically and practically. Theoretically, the research elaborates on the complex interactions between message complexity, privacy guarantees, and estimation accuracy in distributed systems under the shuffle model. The derivation of lower bounds effectively delineates the frontier of what is achievable given the current protocols and conceptual frameworks in DP.

Practically, understanding these trade-offs informs the design of more efficient privacy-preserving data analytics systems. Particularly in data-sensitive applications like federated learning for mobile devices, healthcare, or finance, optimizing the balance between communication costs and privacy levels will be crucial. As devices typically operate under constrained resources, minimizing communication while maintaining accuracy is essential for the widespread application of DP systems.

The robustness analysis against potential manipulations by malicious users provides a pragmatic perspective on deploying these protocols in real-world settings, where adversarial behaviors might exploit protocol vulnerabilities.

Conclusion

The paper presents a comprehensive examination of message complexity requirements for private vector mean estimation in the shuffle model, advancing both theoretical understanding and practical considerations of differential privacy applications. Future research directions may explore novel means to further reduce message complexity while retaining or even enhancing accuracy and robustness, thereby broadening the applicability of differential privacy in diverse, large-scale distributed systems.