  In cooperative bandits, a framework that captures essential features of
collective sequential decision making, agents can minimize group regret, and
thereby improve performance, by leveraging shared information. However, sharing
information can be costly, which motivates developing policies that minimize
group regret while also reducing the number of messages communicated by agents.
Existing cooperative bandit algorithms obtain optimal performance when agents
share information with their neighbors at \textit{every time step}, i.e., full
communication. This requires $\Theta(T)$ number of messages, where $T$ is the
time horizon of the decision making process. We propose \textit{ComEx}, a novel
cost-effective communication protocol in which the group achieves the same
order of performance as full communication while communicating only $O(\log T)$
number of messages. Our key step is developing a method to identify and only
communicate the information crucial to achieving optimal performance. Further
we propose novel algorithms for several benchmark cooperative bandit frameworks
and show that our algorithms obtain \textit{state-of-the-art} performance while
consistently incurring a significantly smaller communication cost than existing
algorithms.
