  We consider the problem where $n$ clients transmit $d$-dimensional
real-valued vectors using $d(1+o(1))$ bits each, in a manner that allows the
receiver to approximately reconstruct their mean. Such compression problems
naturally arise in distributed and federated learning. We provide novel
mathematical results and derive computationally efficient algorithms that are
more accurate than previous compression techniques. We evaluate our methods on
a collection of distributed and federated learning tasks, using a variety of
datasets, and show a consistent improvement over the state of the art.
