\section{Conclusion and Limitations}

We introduce the MVU framework to jointly design scalar compression and DP mechanisms, and extend it to the vector and metric-DP settings. We show that the MVU mechanism attains a better utility-compression trade-off for both scalar and vector mean estimation compared to other approaches in the literature. Our work shows that co-designing the compression and privacy-preserving components can lead to more efficient differentially private mechanisms for federated data analysis.%Future work should further examine unbiased dithering and more scalable solutions to higher-dimensional vectors and lower values of $\epsilon$. Further work also remains to characterize the distribution of the aggregate for the MVU mechanism in order to obtain tight central DP guarantees.

\paragraph{Limitations.} Our work presents several opportunities for further improvement. 1. For vector dithering, Appendix \ref{sec:experiment_details} shows that the input vector's norm can increase by a small additive factor. Our current solution of conditional random dithering introduces a small but non-negligible bias. Future work on unbiased norm-preserving vector dithering may be able to alleviate this issue. 2. Optimizing the MVU mechanism for large values of the input/output bit width $\bin$ and $\bout$ can be prohibitively expensive, even with the alternating minimization heuristic. In order to scale the solution to higher-dimensional vectors, further effort in designing more efficient solutions for the MVU mechanism may be needed. 3. While our work focuses on local differential privacy, it may be possible to combine our approach with secure aggregation protocols to derive central differential privacy guarantees. However, since the MVU mechanism is not additive, further analysis is required to characterize the distribution of the aggregate for our mechanism, which we leave for future work.


% Limitations: optimization is challenging for small $\epsilon$. Better extension to vectors.

% Future work should look at unbiased dithering and more scalable solutions to both higher-dimensional vectors, and even higher bitrate.

% Secure aggregation

% We show that 