  This paper introduces a general multi-agent bandit model in which each agent
is facing a finite set of arms and may communicate with other agents through a
central controller in order to identify, in pure exploration, or play, in
regret minimization, its optimal arm. The twist is that the optimal arm for
each agent is the arm with largest expected mixed reward, where the mixed
reward of an arm is a weighted sum of the rewards of this arm for all agents.
This makes communication between agents often necessary. This general setting
allows to recover and extend several recent models for collaborative bandit
learning, including the recently proposed federated learning with
personalization (Shi et al., 2021). In this paper, we provide new lower bounds
on the sample complexity of pure exploration and on the regret. We then propose
a near-optimal algorithm for pure exploration. This algorithm is based on
phased elimination with two novel ingredients: a data-dependent sampling scheme
within each phase, aimed at matching a relaxation of the lower bound.
