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.
The only previous paper on the setting we study here is by @cite_16 , who use a very different approach from ours. The Game-of-Thrones algorithm proceeds in epochs, and each epoch is divided into three phases. The first is an exploration phase in which each player select arms at random in order to estimate the mean of each arm. In the second phase (called the GOT phase) the players jointly run a Markov chain (which is called the games of thrones dynamics) whose unique stochastically stable state corresponds to a maximum matching of the empirical means, on which the Markov chain will spend most of its time. The players then enter an exploitation phase in which they keep playing the arm they visited most often during the GOT phase.
@cite_16
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We consider the restless multi-armed bandit (RMAB) problem with unknown dynamics. At each time, a player chooses K out of N (N > K) arms to play. The state of each arm determines the reward when the arm is played and transits according to Markovian rules no matter the arm is engaged or passive. The Markovian dynamics of the arms are unknown to the player. The objective is to maximize the long-term expected reward by designing an optimal arm selection policy. The performance of a policy is measured by regret, defined as the reward loss with respect to the case where the player knows which K arms are the most rewarding and always plays these K best arms. We construct a policy, referred to as Restless Upper Confidence Bound (RUCB), that achieves a regret with logarithmic order of time when an arbitrary nontrivial bound on certain system parameters is known. When no knowledge about the system is available, we extend the RUCB policy to achieve a regret arbitrarily close to the logarithmic order. In both cases, the system achieves the maximum average reward offered by the K best arms. Potential applications of these results include cognitive radio networks, opportunistic communications in unknown fading environments, and financial investment. We consider the problem of distributed online learning with multiple players in multiarmed bandit (MAB) models. Each player can pick among multiple arms. When a player picks an arm, it gets a reward. We consider both independent identically distributed (i.i.d.) reward model and Markovian reward model. In the i.i.d. model, each arm is modeled as an i.i.d. process with an unknown distribution with an unknown mean. In the Markovian model, each arm is modeled as a finite, irreducible, aperiodic and reversible Markov chain with an unknown probability transition matrix and stationary distribution. The arms give different rewards to different players. If two players pick the same arm, there is a collision, and neither of them get any reward. There is no dedicated control channel for coordination or communication among the players. Any other communication between the users is costly and will add to the regret. We propose an online index-based distributed learning policy called dUCB4 algorithm that trades off exploration versus exploitation in the right way, and achieves expected regret that grows at most as near- O(log2T). The motivation comes from opportunistic spectrum access by multiple secondary users in cognitive radio networks wherein they must pick among various wireless channels that look different to different users. This is the first distributed learning algorithm for multiplayer MABs with heterogeneous players (that have player-dependent rewards) to the best of our knowledge. We consider the restless multiarmed bandit problem with unknown dynamics in which a player chooses one out of N arms to play at each time. The reward state of each arm transits according to an unknown Markovian rule when it is played and evolves according to an arbitrary unknown random process when it is passive. The performance of an arm selection policy is measured by regret, defined as the reward loss with respect to the case where the player knows which arm is the most rewarding and always plays the best arm. We construct a policy with an interleaving exploration and exploitation epoch structure that achieves a regret with logarithmic order. We further extend the problem to a decentralized setting where multiple distributed players share the arms without information exchange. Under both an exogenous restless model and an endogenous restless model, we show that a decentralized extension of the proposed policy preserves the logarithmic regret order as in the centralized setting. The results apply to adaptive learning in various dynamic systems and communication networks, as well as financial investment. 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.