  We study the bandit problem where the underlying expected reward is a Bounded
Mean Oscillation (BMO) function. BMO functions are allowed to be discontinuous
and unbounded, and are useful in modeling signals with infinities in the
do-main. We develop a toolset for BMO bandits, and provide an algorithm that
can achieve poly-log $\delta$-regret -- a regret measured against an arm that
is optimal after removing a $\delta$-sized portion of the arm space.
