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# Combinatorial Multi-Armed Bandit: General Framework and Applications

*Proceedings of the 30th International Conference on Machine Learning*, PMLR 28(1):151-159, 2013.

#### Abstract

We define a general framework for a large class of combinatorial multi-armed bandit (CMAB) problems, where simple arms with unknown istributions form \em super arms. In each round, a super arm is played and the outcomes of its related simple arms are observed, which helps the selection of super arms in future rounds. The reward of the super arm depends on the outcomes of played arms, and it only needs to satisfy two mild assumptions, which allow a large class of nonlinear reward instances. We assume the availability of an (α,β)-approximation oracle that takes the means of the distributions of arms and outputs a super arm that with probability βgenerates an αfraction of the optimal expected reward. The objective of a CMAB algorithm is to minimize \em (α,β)-approximation regret, which is the difference in total expected reward between the αβfraction of expected reward when always playing the optimal super arm, and the expected reward of playing super arms according to the algorithm. We provide CUCB algorithm that achieves O(\log n) regret, where n is the number of rounds played, and we further provide distribution-independent bounds for a large class of reward functions. Our regret analysis is tight in that it matches the bound for classical MAB problem up to a constant factor, and it significantly improves the regret bound in a recent paper on combinatorial bandits with linear rewards. We apply our CMAB framework to two new applications, probabilistic maximum coverage (PMC) for online advertising and social influence maximization for viral marketing, both having nonlinear reward structures.