The problem of opportunistic spectrum access in cognitive radio networks has been recently formulated as a non-Bayesian restless multi-armed bandit problem. In this problem, there are N arms (corresponding to channels) and one player (corresponding to a secondary user). The state of each arm evolves as a finite-state Markov chain with unknown parameters. At each time slot, the player can select K < N arms to play and receives state-dependent rewards (corresponding to the throughput obtained given the activity of primary users). The objective is to maximize the expected total rewards (i.e., total throughput) obtained over multiple plays. The performance of an algorithm for such a multi-armed bandit problem is measured in terms of regret, defined as the difference in expected reward compared to a model-aware genie who always plays the best K arms. In this paper, we propose a new continuous exploration and exploitation (CEE) algorithm for this problem. When no information is available about the dynamics of the arms, CEE is the first algorithm to guarantee near-logarithmic regret uniformly over time. When some bounds corresponding to the stationary state distributions and the state-dependent rewards are known, we show that CEE can be easily modified to achieve logarithmic regret over time. In contrast, prior algorithms require additional information concerning bounds on the second eigenvalues of the transition matrices in order to guarantee logarithmic regret. Finally, we show through numerical simulations that CEE is more efficient than prior algorithms.
There have been a few recent attempts to solve the restless multi-arm bandit problem under unknown models. @cite_2 , Tekin and Liu use a weaker definition of regret and propose a policy (RCA) that achieves logarithmic regret when certain knowledge about the system is known. However, the algorithm only exploits part of observing data and leaves space to improve performances. @cite_0 , Haoyang Liu proposed a policy, referred to as RUCB, achieving a logarithmic regret over time when certain system parameters are known. The regret they adopt is the same as in @cite_2 . They also extend the RUCB policy to achieve a near-logarithmic regret over time when no knowledge about the system is available. Conclusions on multi-arm selections are given in @cite_11 . However, they only give the upper bound of regret at the end of a certain time point referred as . When no information about the system is known, their analysis of regret gives the upper bound over time only asymptotically, not uniformly.
@cite_0 @cite_11 @cite_2
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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 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 the classic Bayesian restless multi-armed bandit (RMAB) problem, there are N arms, with rewards on all arms evolving at each time as Markov chains with known parameters. A player seeks to activate K ≥ 1 arms at each time in order to maximize the expected total reward obtained over multiple plays. RMAB is a challenging problem that is known to be PSPACE-hard in general. We consider in this work the even harder non-Bayesian RMAB, in which the parameters of the Markov chain are assumed to be unknown a priori. We develop an original approach to this problem that is applicable when the corresponding Bayesian problem has the structure that, depending on the known parameter values, the optimal solution is one of a prescribed finite set of policies. In such settings, we propose to learn the optimal policy for the non-Bayesian RMAB by employing a suitable meta-policy which treats each policy from this finite set as an arm in a different non-Bayesian multi-armed bandit problem for which a single-arm selection policy is optimal. We demonstrate this approach by developing a novel sensing policy for opportunistic spectrum access over unknown dynamic channels. We prove that our policy achieves near-logarithmic regret (the difference in expected reward compared to a model-aware genie), which leads to the same average reward that can be achieved by the optimal policy under a known model. This is the first such result in the literature for a non-Bayesian RMAB. We consider a class of restless multi-armed bandit (RMAB) problems with unknown arm dynamics. At each time, a player chooses an arm out of @math arms to play, referred to as an active arm, and receives a random reward from a finite set of reward states. The reward state of the active arm transits according to an unknown Markovian dynamic. The reward state of passive arms (which are not chosen to play at time t) evolves according to an arbitrary unknown random process. The objective is an arm-selection policy that minimizes the regret, defined as the reward loss with respect to a player that always plays the most rewarding arm. This class of RMAB problems has been studied recently in the context of communication networks and financial investment applications. We develop a strategy that selects arms to be played in a consecutive manner in which the selection sequencing rules are adaptively updated controlled by the current sample reward means, referred to as Adaptive Sequencing Rules (ASR) algorithm. By designing judiciously the adaptive sequencing rules of the chosen arms, we show that ASR algorithm achieves a logarithmic regret order with time and a finite-sample bound on the regret is established. Although existing methods have shown a logarithmic regret order with time in this RMAB setting, the theoretical analysis presents significant improvement in the regret scaling with respect to the system parameters under ASR. Extensive simulation results support the theoretical study and demonstrate strong performance of the algorithm as compared to existing methods.