  We propose new techniques for reducing communication in private federated
learning without the need for setting or tuning compression rates. Our
on-the-fly methods automatically adjust the compression rate based on the error
induced during training, while maintaining provable privacy guarantees through
the use of secure aggregation and differential privacy. Our techniques are
provably instance-optimal for mean estimation, meaning that they can adapt to
the ``hardness of the problem" with minimal interactivity. We demonstrate the
effectiveness of our approach on real-world datasets by achieving favorable
compression rates without the need for tuning.
