We consider stochastic multi-armed bandit problems with graph feedback, where the decision maker is allowed to observe the neighboring actions of the chosen action. We allow the graph structure to vary with time and consider both deterministic and Erd o s-R 'enyi random graph models. For such a graph feedback model, we first present a novel analysis of Thompson sampling that leads to tighter performance bound than existing work. Next, we propose new Information Directed Sampling based policies that are graph-aware in their decision making. Under the deterministic graph case, we establish a Bayesian regret bound for the proposed policies that scales with the clique cover number of the graph instead of the number of actions. Under the random graph case, we provide a Bayesian regret bound for the proposed policies that scales with the ratio of the number of actions over the expected number of observations per iteration. To the best of our knowledge, this is the first analytical result for stochastic bandits with random graph feedback. Finally, using numerical evaluations, we demonstrate that our proposed IDS policies outperform existing approaches, including adaptions of upper confidence bound, @math -greedy and Exp3 algorithms.
The Thompson Sampling algorithm as analyzed in @cite_4 does not explicitly use the graph structure in each step for its operation (similar to UCB-N algorithm proposed in @cite_14 ). This is attractive when such graphical information is difficult to obtain, a case also studied in @cite_22 . However, in many cases such as the problem of promotions in online social networks and sensor networks, the graph structure is revealed or can be learned a priori. When such knowledge is available, @cite_0 show that using graphical information to make choices in each time helps in obtaining the optimal regret both in terms of network structure and time asymptotically. This work motivated us to investigate extensions of IDS policies that exploit the knowledge of network structure in our work. Using numerical evaluations, we find that IDS policies outperform asymptotically optimal policies presented in @cite_0 .
@cite_0 @cite_14 @cite_4 @cite_22
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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. 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. This paper considers the use of a simple posterior sampling algorithm to balance between exploration and exploitation when learning to optimize actions such as in multi-armed bandit problems. The algorithm, also known as Thompson Sampling, offers significant advantages over the popular upper confidence bound (UCB) approach, and can be applied to problems with finite or infinite action spaces and complicated relationships among action rewards. We make two theoretical contributions. The first establishes a connection between posterior sampling and UCB algorithms. This result lets us convert regret bounds developed for UCB algorithms into Bayesian regret bounds for posterior sampling. Our second theoretical contribution is a Bayesian regret bound for posterior sampling that applies broadly and can be specialized to many model classes. This bound depends on a new notion we refer to as the eluder dimension, which measures the degree of dependence among action rewards. Compared to UCB algorithm Bayesian regret bounds for specific model classes, our general bound matches the best available for linear models and is stronger than the best available for generalized linear models. Further, our analysis provides insight into performance advantages of posterior sampling, which are highlighted through simulation results that demonstrate performance surpassing recently proposed UCB algorithms. Thompson Sampling has recently been shown to achieve the lower bound on regret in the Bernoulli Multi-Armed Bandit setting. This bandit problem assumes stationary distributions for the rewards. It is often unrealistic to model the real world as a stationary distribution. In this paper we derive and evaluate algorithms using Thompson Sampling for a Switching Multi-Armed Bandit Problem. We propose a Thompson Sampling strategy equipped with a Bayesian change point mechanism to tackle this problem. We develop algorithms for a variety of cases with constant switching rate: when switching occurs all arms change (Global Switching), switching occurs independently for each arm (PerArm Switching), when the switching rate is known and when it must be inferred from data. This leads to a family of algorithms we collectively term Change-Point Thompson Sampling (CTS). We show empirical results in 4 articial environments, and 2 derived from real world data: news click-through[Yahoo!, 2011] and foreign exchange data[Dukascopy, 2012], comparing them to some other bandit algorithms. In real world data CTS is the most eective.