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We consider the bandit problem of selecting $K$ out of $N$ arms at each timestep. The reward can be a non-linear function of the rewards of the selectedindividual arms. The direct use of a multi-armed bandit algorithm requireschoosing among $\binom{N}{K}$ options, making the action space large. Tosimplify the problem, existing works on combinatorial bandits {typically}assume feedback as a linear function of individual rewards. In this paper, weprove the lower bound for top-$K$ subset selection with bandit feedback withpossibly correlated rewards. We present a novel algorithm for the combinatorialsetting without using individual arm feedback or requiring linearity of thereward function. Additionally, our algorithm works on correlated rewards ofindividual arms. Our algorithm, aDaptive Accept RejecT (DART), sequentiallyfinds good arms and eliminates bad arms based on confidence bounds. DART iscomputationally efficient and uses storage linear in $N$. Further, DARTachieves a regret bound of $\tilde{\mathcal{O}}(K\sqrt{KNT})$ for a timehorizon $T$, which matches the lower bound in bandit feedback up to a factor of$\sqrt{\log{2NT}}$. When applied to the problem of cross-selling optimizationand maximizing the mean of individual rewards, the performance of the proposedalgorithm surpasses that of state-of-the-art algorithms. We also show that DARTsignificantly outperforms existing methods for both linear and non-linear jointreward environments.

DART: aDaptive Accept RejecT for non-linear top-K subset identification