  We consider the following problem in this paper: given a set of $n$
distributions, find the top-$m$ ones with the largest means. This problem is
also called {\em top-$m$ arm identifications} in the literature of
reinforcement learning, and has numerous applications. We study the problem in
the collaborative learning model where we have multiple agents who can draw
samples from the $n$ distributions in parallel. Our goal is to characterize the
tradeoffs between the running time of learning process and the number of rounds
of interaction between agents, which is very expensive in various scenarios. We
give optimal time-round tradeoffs, as well as demonstrate complexity
separations between top-$1$ arm identification and top-$m$ arm identifications
for general $m$ and between fixed-time and fixed-confidence variants. As a
byproduct, we also give an algorithm for selecting the distribution with the
$m$-th largest mean in the collaborative learning model.
