  Motivated by the need for communication-efficient distributed learning, we
investigate the method for compressing a unit norm vector into the minimum
number of bits, while still allowing for some acceptable level of distortion in
recovery. This problem has been explored in the rate-distortion/covering code
literature, but our focus is exclusively on the "high-distortion" regime. We
approach this problem in a worst-case scenario, without any prior information
on the vector, but allowing for the use of randomized compression maps. Our
study considers both biased and unbiased compression methods and determines the
optimal compression rates. It turns out that simple compression schemes are
nearly optimal in this scenario. While the results are a mix of new and known,
they are compiled in this paper for completeness.
