  We study differentially private distributed optimization under communication
constraints. A server using SGD for optimization aggregates the client-side
local gradients for model updates using distributed mean estimation (DME). We
develop a communication-efficient private DME, using the recently developed
multi-message shuffled (MMS) privacy framework. We analyze our proposed DME
scheme to show that it achieves the order-optimal
privacy-communication-performance tradeoff resolving an open question in [1],
whether the shuffled models can improve the tradeoff obtained in Secure
Aggregation. This also resolves an open question on the optimal trade-off for
private vector sum in the MMS model. We achieve it through a novel privacy
mechanism that non-uniformly allocates privacy at different resolutions of the
local gradient vectors. These results are directly applied to give guarantees
on private distributed learning algorithms using this for private gradient
aggregation iteratively. We also numerically evaluate the private DME
algorithms.
