  We consider a combinatorial multi-armed bandit problem for maximum value
reward function under maximum value and index feedback. This is a new feedback
structure that lies in between commonly studied semi-bandit and full-bandit
feedback structures. We propose an algorithm and provide a regret bound for
problem instances with stochastic arm outcomes according to arbitrary
distributions with finite supports. The regret analysis rests on considering an
extended set of arms, associated with values and probabilities of arm outcomes,
and applying a smoothness condition. Our algorithm achieves a
$O((k/\Delta)\log(T))$ distribution-dependent and a $\tilde{O}(\sqrt{T})$
distribution-independent regret where $k$ is the number of arms selected in
each round, $\Delta$ is a distribution-dependent reward gap and $T$ is the
horizon time. Perhaps surprisingly, the regret bound is comparable to
previously-known bound under more informative semi-bandit feedback. We
demonstrate the effectiveness of our algorithm through experimental results.
