  Differentially private distributed mean estimation (DP-DME) is a fundamental
building block in privacy-preserving federated learning, where a central server
estimates the mean of $d$-dimensional vectors held by $n$ users while ensuring
$(\epsilon,\delta)$-DP. Local differential privacy (LDP) and distributed DP
with secure aggregation (SecAgg) are the most common notions of DP used in
DP-DME settings with an untrusted server. LDP provides strong resilience to
dropouts, colluding users, and malicious server attacks, but suffers from poor
utility. In contrast, SecAgg-based DP-DME achieves an $O(n)$ utility gain over
LDP in DME, but requires increased communication and computation overheads and
complex multi-round protocols to handle dropouts and malicious attacks. In this
work, we propose CorDP-DME, a novel DP-DME mechanism that spans the gap between
DME with LDP and distributed DP, offering a favorable balance between utility
and resilience to dropout and collusion. CorDP-DME is based on correlated
Gaussian noise, ensuring DP without the perfect conditional privacy guarantees
of SecAgg-based approaches. We provide an information-theoretic analysis of
CorDP-DME, and derive theoretical guarantees for utility under any given
privacy parameters and dropout/colluding user thresholds. Our results
demonstrate that (anti) correlated Gaussian DP mechanisms can significantly
improve utility in mean estimation tasks compared to LDP -- even in adversarial
settings -- while maintaining better resilience to dropouts and attacks
compared to distributed DP.
