text: We study a multiplayer stochastic multi-armed bandit problem in which players cannot communicate, and if two or more players pull the same arm, a collision occurs and the involved players receive zero reward. We consider the challenging heterogeneous setting, in which different arms may have different means for different players, and propose a new, efficient algorithm that combines the idea of leveraging forced collisions for implicit communication and that of performing matching eliminations. We give a finite-time analysis of our algorithm, bounding its regret by O((log T)^ 1+ ) for any fixed >0. If the optimal assignment of players to arms is unique, we further show that it attains the optimal O(log(T)) regret, solving an open question raised at NeurIPS 2018.
  @cite_14: We formulate the following combinatorial multi-armed bandit (MAB) problem: There are N random variables with unknown mean that are each instantiated in an i.i.d. fashion over time. At each time multiple random variables can be selected, subject to an arbitrary constraint on weights associated with the selected variables. All of the selected individual random variables are observed at that time, and a linearly weighted combination of these selected variables is yielded as the reward. The goal is to find a policy that minimizes regret, defined as the difference between the reward obtained by a genie that knows the mean of each random variable, and that obtained by the given policy. This formulation is broadly applicable and useful for stochastic online versions of many interesting tasks in networks that can be formulated as tractable combinatorial optimization problems with linear objective functions, such as maximum weighted matching, shortest path, and minimum spanning tree computations. Prior work on multi-armed bandits with multiple plays cannot be applied to this formulation because of the general nature of the constraint. On the other hand, the mapping of all feasible combinations to arms allows for the use of prior work on MAB with single-play, but results in regret, storage, and computation growing exponentially in the number of unknown variables. We present new efficient policies for this problem that are shown to achieve regret that grows logarithmically with time, and polynomially in the number of unknown variables. Furthermore, these policies only require storage that grows linearly in the number of unknown parameters. For problems where the underlying deterministic problem is tractable, these policies further require only polynomial computation. For computationally intractable problems, we also present results on a different notion of regret that is suitable when a polynomial-time approximation algorithm is used.
summary: In the centralized setting of our problem, a central controller selects at each time step a matching that maps the @math players to different arms in @math . In other words, introducing @math with means @math for @math and @math , the central controller selects a subset of @math elementary arms whose indices form a matching from @math to @math . Then, the utility of each elementary arm is observed and the reward is the sum of those. This is a particular combinatorial bandit problem with semi-bandit feedback, first studied by @cite_14 .