Almost Cost-Free Communication in Federated Best Arm Identification

We study the problem of best arm identification in a federated learning multi-armed bandit setup with a central server and multiple clients. Each client is associated with a multi-armed bandit in which each arm yields {\em i.i.d.}\ rewards following a Gaussian distribution with an unknown mean and known variance. The set of arms is assumed to be the same at all the clients. We define two notions of best arm -- local and global. The local best arm at a client is the arm with the largest mean among the arms local to the client, whereas the global best arm is the arm with the largest average mean across all the clients. We assume that each client can only observe the rewards from its local arms and thereby estimate its local best arm. The clients communicate with a central server on uplinks that entail a cost of $C\ge0$ units per usage per uplink. The global best arm is estimated at the server. The goal is to identify the local best arms and the global best arm with minimal total cost, defined as the sum of the total number of arm selections at all the clients and the total communication cost, subject to an upper bound on the error probability. We propose a novel algorithm {\sc FedElim} that is based on successive elimination and communicates only in exponential time steps and obtain a high probability instance-dependent upper bound on its total cost. The key takeaway from our paper is that for any $C\geq 0$ and error probabilities sufficiently small, the total number of arm selections (resp.\ the total cost) under {\sc FedElim} is at most~$2$ (resp.~$3$) times the maximum total number of arm selections under its variant that communicates in every time step. Additionally, we show that the latter is optimal in expectation up to a constant factor, thereby demonstrating that communication is almost cost-free in {\sc FedElim}. We numerically validate the efficacy of {\sc FedElim}.