  We consider the combinatorial bandits problem, where at each time step, the
online learner selects a size-$k$ subset $s$ from the arms set $\mathcal{A}$,
where $\left|\mathcal{A}\right| = n$, and observes a stochastic reward of each
arm in the selected set $s$. The goal of the online learner is to minimize the
regret, induced by not selecting $s^*$ which maximizes the expected total
reward. Specifically, we focus on a challenging setting where 1) the reward
distribution of an arm depends on the set $s$ it is part of, and crucially 2)
there is \textit{no total order} for the arms in $\mathcal{A}$.
  In this paper, we formally present a reward model that captures set-dependent
reward distribution and assumes no total order for arms. Correspondingly, we
propose an Upper Confidence Bound (UCB) algorithm that maintains UCB for each
individual arm and selects the arms with top-$k$ UCB. We develop a novel regret
analysis and show an $O\left(\frac{k^2 n \log T}{\epsilon}\right)$
gap-dependent regret bound as well as an $O\left(k^2\sqrt{n T \log T}\right)$
gap-independent regret bound. We also provide a lower bound for the proposed
reward model, which shows our proposed algorithm is near-optimal for any
constant $k$. Empirical results on various reward models demonstrate the broad
applicability of our algorithm.
