We consider the combinatorial multi-armed bandit (CMAB) problem, where the reward function is nonlinear. In this setting, the agent chooses a batch of arms on each round and receives feedback from each arm of the batch. The reward that the agent aims to maximize is a function of the selected arms and their expectations. In many applications, the reward function is highly nonlinear, and the performance of existing algorithms relies on a global Lipschitz constant to encapsulate the function's nonlinearity. This may lead to loose regret bounds, since by itself, a large gradient does not necessarily cause a large regret, but only in regions where the uncertainty in the reward's parameters is high. To overcome this problem, we introduce a new smoothness criterion, which we term \textit{Gini-weighted smoothness}, that takes into account both the nonlinearity of the reward and concentration properties of the arms. We show that a linear dependence of the regret in the batch size in existing algorithms can be replaced by this smoothness parameter. This, in turn, leads to much tighter regret bounds when the smoothness parameter is batch-size independent. For example, in the probabilistic maximum coverage (PMC) problem, that has many applications, including influence maximization, diverse recommendations and more, we achieve dramatic improvements in the upper bounds. We also prove matching lower bounds for the PMC problem and show that our algorithm is tight, up to a logarithmic factor in the problem's parameters.

针对组合多臂赌博机问题中的非线性奖励函数和批处理选择，引入一种新的平滑度标准——基尼加权平滑度，证明了现有算法中的线性后悔量与批大小的关系可以被该平滑度参数取代，并在概率最大覆盖问题中获得了明显的精度提高。

组合多臂老虎机问题的批量独立遗憾界