  We consider the bandit problem of selecting $K$ out of $N$ arms at each time
step. The reward can be a non-linear function of the rewards of the selected
individual arms. The direct use of a multi-armed bandit algorithm requires
choosing among $\binom{N}{K}$ options, making the action space large. To
simplify the problem, existing works on combinatorial bandits {typically}
assume feedback as a linear function of individual rewards. In this paper, we
prove the lower bound for top-$K$ subset selection with bandit feedback with
possibly correlated rewards. We present a novel algorithm for the combinatorial
setting without using individual arm feedback or requiring linearity of the
reward function. Additionally, our algorithm works on correlated rewards of
individual arms. Our algorithm, aDaptive Accept RejecT (DART), sequentially
finds good arms and eliminates bad arms based on confidence bounds. DART is
computationally efficient and uses storage linear in $N$. Further, DART
achieves a regret bound of $\tilde{\mathcal{O}}(K\sqrt{KNT})$ for a time
horizon $T$, which matches the lower bound in bandit feedback up to a factor of
$\sqrt{\log{2NT}}$. When applied to the problem of cross-selling optimization
and maximizing the mean of individual rewards, the performance of the proposed
algorithm surpasses that of state-of-the-art algorithms. We also show that DART
significantly outperforms existing methods for both linear and non-linear joint
reward environments.
