  We consider the problem of online stochastic optimization in a distributed
setting with $M$ clients connected through a central server. We develop a
distributed online learning algorithm that achieves order-optimal cumulative
regret with low communication cost measured in the total number of bits
transmitted over the entire learning horizon. This is in contrast to existing
studies which focus on the offline measure of simple regret for learning
efficiency. The holistic measure for communication cost also departs from the
prevailing approach that \emph{separately} tackles the communication frequency
and the number of bits in each communication round.
