  We study a decentralized cooperative multi-agent multi-armed bandit problem
with $K$ arms and $N$ agents connected over a network. In our model, each arm's
reward distribution is same for all agents, and rewards are drawn independently
across agents and over time steps. In each round, agents choose an arm to play
and subsequently send a message to their neighbors. The goal is to minimize
cumulative regret averaged over the entire network. We propose a decentralized
Bayesian multi-armed bandit framework that extends single-agent Bayesian bandit
algorithms to the decentralized setting. Specifically, we study an information
assimilation algorithm that can be combined with existing Bayesian algorithms,
and using this, we propose a decentralized Thompson Sampling algorithm and
decentralized Bayes-UCB algorithm. We analyze the decentralized Thompson
Sampling algorithm under Bernoulli rewards and establish a problem-dependent
upper bound on the cumulative regret. We show that regret incurred scales
logarithmically over the time horizon with constants that match those of an
optimal centralized agent with access to all observations across the network.
Our analysis also characterizes the cumulative regret in terms of the network
structure. Through extensive numerical studies, we show that our extensions of
Thompson Sampling and Bayes-UCB incur lesser cumulative regret than the
state-of-art algorithms inspired by the Upper Confidence Bound algorithm. We
implement our proposed decentralized Thompson Sampling under gossip protocol,
and over time-varying networks, where each communication link has a fixed
probability of failure.
