  We study decentralized stochastic linear bandits, where a network of $N$
agents acts cooperatively to efficiently solve a linear bandit-optimization
problem over a $d$-dimensional space. For this problem, we propose DLUCB: a
fully decentralized algorithm that minimizes the cumulative regret over the
entire network. At each round of the algorithm each agent chooses its actions
following an upper confidence bound (UCB) strategy and agents share information
with their immediate neighbors through a carefully designed consensus procedure
that repeats over cycles. Our analysis adjusts the duration of these
communication cycles ensuring near-optimal regret performance
$\mathcal{O}(d\log{NT}\sqrt{NT})$ at a communication rate of
$\mathcal{O}(dN^2)$ per round. The structure of the network affects the regret
performance via a small additive term - coined the regret of delay - that
depends on the spectral gap of the underlying graph. Notably, our results apply
to arbitrary network topologies without a requirement for a dedicated agent
acting as a server. In consideration of situations with high communication
cost, we propose RC-DLUCB: a modification of DLUCB with rare communication
among agents. The new algorithm trades off regret performance for a
significantly reduced total communication cost of $\mathcal{O}(d^3N^{2.5})$
over all $T$ rounds. Finally, we show that our ideas extend naturally to the
emerging, albeit more challenging, setting of safe bandits. For the recently
studied problem of linear bandits with unknown linear safety constraints, we
propose the first safe decentralized algorithm. Our study contributes towards
applying bandit techniques in safety-critical distributed systems that
repeatedly deal with unknown stochastic environments. We present numerical
simulations for various network topologies that corroborate our theoretical
findings.
