The paper proposes new algorithmic and theoretical solutions to achieving fast distributed bandit optimization over agents connected over a network, when only agents connected by a network link can exchange messages. The reviewers largely agree that the paper's exposition and results are significantly novel compared to past work on the subject, and the results improved substantially over previous ones as shown for example networks. A concern regarding the impact of acceleration of the gossip subroutine on the derived performance bounds was addressed satisfactorily by the author response. On the other hand, I must remark that a separate concern about connection to the real world -- how to capture/measure in a better and more finer way the communication cost across the entire network -- still remains unaddressed in this line of work thus far. I would urge the author(s) to take a closer look at the issue of accounting for communication at a more granular level other than just allowing real-valued messages across links (perhaps by drawing upon ideas from the communication networks/information theory communities), and expose a more concrete dependence of performance on the information bottlenecks in messaging between adjacent nodes.