We present a novel extension of Thompson Sampling for stochastic sequential decision problems with graph feedback, even when the graph structure itself is unknown and or changing. We provide theoretical guarantees on the Bayesian regret of the algorithm, linking its performance to the underlying properties of the graph. Thompson Sampling has the advantage of being applicable without the need to construct complicated upper confidence bounds for different problems. We illustrate its performance through extensive experimental results on real and simulated networks with graph feedback. More specifically, we tested our algorithms on power law, planted partitions and Erdo's-Renyi graphs, as well as on graphs derived from Facebook and Flixster data. These all show that our algorithms clearly outperform related methods that employ upper confidence bounds, even if the latter use more information about the graph.
Optimal policies for the stochastic multi-armed bandit problem were first characterised by @cite_14 , while index-based optimal policies for general non-parametric problems were given by @cite_24 . Later @cite_8 proved finite-time regret bounds for a number of UCB (Upper Confidence Bound) index policies, while @cite_13 proved finite-time bounds for index policies similar to those of @cite_24 , with problem-dependent bounds @math . Recently, a number of policies based on sampling from the posterior distribution (i.e. Thompson sampling @cite_4 ) were analysed in both the frequentist @cite_9 and Bayesian setting @cite_11 and shown to obtain the same order of regret bound for the stochastic case. For the bandit problem the bounds are of order @math . The analysis for the full information case generally results in @math bounds on the regret @cite_0 , i.e. with a much lower dependence on the number of arms.
@cite_14 @cite_4 @cite_8 @cite_9 @cite_24 @cite_0 @cite_13 @cite_11
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We consider a non-Bayesian infinite horizon version of the multi-armed bandit problem with the objective of designing simple policies whose regret increases sldwly with time. In their seminal work on this problem, Lai and Robbins had obtained a O(logn) lower bound on the regret with a constant that depends on the KullbackLeibler number. They also constructed policies for some specific families of probability distributions (including exponential families) that achieved the lower bound. In this paper we construct index policies that depend on the rewards from each arm only through their sample mean. These policies are computationally much simpler and are also applicable much more generally. They achieve a O(logn) regret with a constant that is also based on the Kullback-Leibler number. This constant turns out to be optimal for one-parameter exponential families; however, in general it is derived from the optimal one via a 'contraction' principle. Our results rely entirely on a few key lemmas from the theory of large deviations. This paper studies the deviations of the regret in a stochastic multi-armed bandit problem.When the total number of plays n is known beforehand by the agent, (2009) exhibit a policy such that with probability at least 1-1 n, the regret of the policy is of order log n. They have also shown that such a property is not shared by the popular ucb1 policy of (2002). This work first answers an open question: it extends this negative result to any anytime policy. The second contribution of this paper is to design anytime robust policies for specific multi-armed bandit problems in which some restrictions are put on the set of possible distributions of the different arms. We study the stochastic multi-armed bandit (MAB) problem in the presence of side-observations across actions. In our model, choosing an action provides additional side observations for a subset of the remaining actions. One example of this model occurs in the problem of targeting users in online social networks where users respond to their friends's activity, thus providing information about each other's preferences. Our contributions are as follows: 1) We derive an asymptotic (with respect to time) lower bound (as a function of the network structure) on the regret (loss) of any uniformly good policy that achieves the maximum long term average reward. 2) We propose two policies - a randomized policy and a policy based on the well-known upper confidence bound (UCB) policies, both of which explore each action at a rate that is a function of its network position. We show that these policies achieve the asymptotic lower bound on the regret up to a multiplicative factor independent of network structure. The upper bound guarantees on the regret of these policies are better than those of existing policies. Finally, we use numerical examples on a real-world social network to demonstrate the significant benefits obtained by our policies against other existing policies. 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.