We study the problem of regret minimization for distributed bandits learning, in which M agents work collaboratively to minimize their total regret under the coordination of a central server. Our goal is to design communication protocols with near-optimal regret and little communication cost, which is measured by the total amount of transmitted data. For distributed multi-armed bandits, we propose a protocol with near-optimal regret and only O(M log(M K)) communication cost, where K is the number of arms. The communication cost is independent of the time horizon T , has only logarithmic dependence on the number of arms, and matches the lower bound except for a logarithmic factor. For distributed d-dimensional linear bandits, we propose a protocol that achieves near-optimal regret and has communication cost of order O ((M d + d log log d) log T ), which has only logarithmic dependence on T .
The paper considers the problem of distributed multi-arm bandit, where M players are playing in the same stochastic environment. The goal of the paper is to have small over-all regret for all the players without a significant amount of communication between the players. 
The main contribution of this paper is obtaining regret ~root(M KT) with ~M bits of communication in MAB, and regret ~d*root(MT) with ~Md bits of communication in linear bandit setting. 
The main intuition of the algorithms in this paper is to do "best arm identification" with epoching: At every epoch t, the central server sends the set of possible best arms to each player and each player pulls it for 2^t /M times, followed by a communication round. Thus, the cumulative regret is comparable to having one player doing this epoch strategy for MT iterations, where the regret follows.
The problem considered in this paper is interesting and the result is new, the technique looks simple on paper but it requires a masterful combination of known tricks in (linear) MAB to obtain the best bound.  
It seems that in the MAB setting, the lower bound could be further strengthened with a log(K) factor, since removing this factor would ultimately require "dynamic epoching" which is not possible with limited communication. This would mostly complete the picture in the distributed MAB regime.
Missing citation:
The authors are missing citations relevant to distributed MAB with collisions, see for example 
"Non-Stochastic Multi-Player Multi-Armed Bandits: Optimal Rate With Collision Information, Sublinear Without"
After Rebuttal: I have read the authors' responses and acknowledge the sensibility of the statement.