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.
Our problem can be viewed as a challenging extension of two complex multi-armed bandit problems. First, relaxing the need for decentralization, i.e. when a central controller is requested to jointly select @math , the problem coincides with a combinatorial bandit problem with semi-bandit feedback, as explained in , where we review the achievable regret in the centralized setting. Second, the particular case of a common utility for all players, i.e. @math for all @math , has been studied extensively, and we review in the achievable regret in this multi-player bandit problem, presenting in particular some inspiring ideas to design algorithms for our more general problem. Finally in we discuss the Game-of-Thrones algorithm of @cite_16 , which is our only competitor in the fully distributed setting considered in this paper.
@cite_16
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We consider the problem of learning in single-player and multiplayer multiarmed bandit models. Bandit problems are classes of online learning problems that capture exploration versus exploitation tradeoffs. In a multiarmed bandit model, players can pick among many arms, and each play of an arm generates an i.i.d. reward from an unknown distribution. The objective is to design a policy that maximizes the expected reward over a time horizon for a single player setting and the sum of expected rewards for the multiplayer setting. In the multiplayer setting, arms may give different rewards to different players. There is no separate channel for coordination among the players. Any attempt at communication is costly and adds to regret. We propose two decentralizable policies, @math ( @math - @math ) and @math - @math , that can be used in both single player and multiplayer settings. These policies are shown to yield expected regret that grows at most as O( @math ). It is well known that @math is the lower bound on the rate of growth of regret even in a centralized case. The proposed algorithms improve on prior work where regret grew at O( @math ). More fundamentally, these policies address the question of additional cost incurred in decentralized online learning, suggesting that there is at most an @math -factor cost in terms of order of regret. This solves a problem of relevance in many domains and had been open for a while. We consider the problem of learning in single-player and multiplayer multiarmed bandit models. Bandit problems are classes of online learning problems that capture exploration versus exploitation tradeoffs. In a multiarmed bandit model, players can pick among many arms, and each play of an arm generates an i.i.d. reward from an unknown distribution. The objective is to design a policy that maximizes the expected reward over a time horizon for a single-player setting and the sum of expected rewards for the multiplayer setting. In the multiplayer setting, arms may give different rewards to different players. There is no separate channel for coordination among the players. Any attempt at communication is costly and adds to regret. We propose two decentralizable policies, @math ( @math - @math ) and @math - @math , that can be used in both single-player and multiplayer settings. These policies are shown to yield expected regret that grows, at most, as @math (and @math under some assumption). It is well known that @math is the lower bound on the rate of growth of regret even in a centralized case. The proposed algorithms improve on prior work where regret grew at @math . More fundamentally, these policies address the question of additional cost incurred in decentralized online learning, suggesting that there is, at most, an @math -factor cost in terms of the order of regret. This solves a problem of relevance in many domains and had been open for a while. In this paper, we study the online learning problem involving rested and restless bandits, in both a centralized and a decentralized setting. In a centralized setting, the system consists of a single player user and a set of K finite-state discrete-time Markov chains (arms) with unknown state spaces (rewards) and statistics. The objective of the player is to decide in each step which M of the K arms to play over a sequence of trials so as to maximize its long-term reward. In a decentralized setting, multiple uncoordinated players each makes its own decision on which arm to play in a step, and if two or more players select the same arm simultaneously, a collision results and none of the players selecting that arm gets a reward. The objective of each player again is to maximize its long-term reward. We first show that logarithmic regret algorithms exist both for the centralized rested and restless bandit problems. For the decentralized setting, we propose an algorithm with logarithmic regret with respect to the optimal centralized arm allocation. Numerical results and extensive discussion are also provided to highlight insights obtained from this study. The fundamental problem of multiple secondary users contending for opportunistic spectrum access over multiple channels in cognitive radio networks has been formulated recently as a decentralized multi-armed bandit (D-MAB) problem. In a D-MAB problem there are @math users and @math arms (channels) that each offer i.i.d. stochastic rewards with unknown means so long as they are accessed without collision. The goal is to design a decentralized online learning policy that incurs minimal regret, defined as the difference between the total expected rewards accumulated by a model-aware genie, and that obtained by all users applying the policy. We make two contributions in this paper. First, we consider the setting where the users have a prioritized ranking, such that it is desired for the @math -th-ranked user to learn to access the arm offering the @math -th highest mean reward. For this problem, we present the first distributed policy that yields regret that is uniformly logarithmic over time without requiring any prior assumption about the mean rewards. Second, we consider the case when a fair access policy is required, i.e., it is desired for all users to experience the same mean reward. For this problem, we present a distributed policy that yields order-optimal regret scaling with respect to the number of users and arms, better than previously proposed policies in the literature. Both of our distributed policies make use of an innovative modification of the well known UCB1 policy for the classic multi-armed bandit problem that allows a single user to learn how to play the arm that yields the @math -th largest mean reward.