  We study best arm identification in a federated multi-armed bandit setting
with a central server and multiple clients, when each client has access to a
{\em subset} of arms and each arm yields independent Gaussian observations. The
goal is to identify the best arm of each client subject to an upper bound on
the error probability; here, the best arm is one that has the largest {\em
average} value of the means averaged across all clients having access to the
arm. Our interest is in the asymptotics as the error probability vanishes. We
provide an asymptotic lower bound on the growth rate of the expected stopping
time of any algorithm. Furthermore, we show that for any algorithm whose upper
bound on the expected stopping time matches with the lower bound up to a
multiplicative constant ({\em almost-optimal} algorithm), the ratio of any two
consecutive communication time instants must be {\em bounded}, a result that is
of independent interest. We thereby infer that an algorithm can communicate no
more sparsely than at exponential time instants in order to be almost-optimal.
For the class of almost-optimal algorithms, we present the first-of-its-kind
asymptotic lower bound on the expected number of {\em communication rounds}
until stoppage. We propose a novel algorithm that communicates at exponential
time instants, and demonstrate that it is asymptotically almost-optimal.
