  Standard Multi-Armed Bandit (MAB) problems assume that the arms are
independent. However, in many application scenarios, the information obtained
by playing an arm provides information about the remainder of the arms. Hence,
in such applications, this informativeness can and should be exploited to
enable faster convergence to the optimal solution. In this paper, we introduce
and formalize the Global MAB (GMAB), in which arms are globally informative
through a global parameter, i.e., choosing an arm reveals information about all
the arms. We propose a greedy policy for the GMAB which always selects the arm
with the highest estimated expected reward, and prove that it achieves bounded
parameter-dependent regret. Hence, this policy selects suboptimal arms only
finitely many times, and after a finite number of initial time steps, the
optimal arm is selected in all of the remaining time steps with probability
one. In addition, we also study how the informativeness of the arms about each
other's rewards affects the speed of learning. Specifically, we prove that the
parameter-free (worst-case) regret is sublinear in time, and decreases with the
informativeness of the arms. We also prove a sublinear in time Bayesian risk
bound for the GMAB which reduces to the well-known Bayesian risk bound for
linearly parameterized bandits when the arms are fully informative. GMABs have
applications ranging from drug and treatment discovery to dynamic pricing.
