We consider a variant of the multi-armed bandit model, which we call multi-armed bandit problem with known trend, where the gambler knows the shape of the reward function of each arm but not its distribution. This new problem is motivated by different online problems like active learning, music and interface recommendation applications, where when an arm is sampled by the model the received reward change according to a known trend. By adapting the standard multi-armed bandit algorithm UCB1 to take advantage of this setting, we propose the new algorithm named A-UCB that assumes a stochastic model. We provide upper bounds of the regret which compare favourably with the ones of UCB1. We also confirm that experimentally with different simulations
In @cite_11 author study specific classes of drifting restless bandits selected for their relevance to modelling an online website optimization process. The contribution was a feasible weighted least squares technique capable of utilizing contextual arm parameters while considering the parameter space drifting non-stationary within reasonable bounds.
@cite_11
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Restless bandits model the exploration vs. exploitation trade-off in a changing (non-stationary) world. Restless bandits have been studied in both the context of continuously-changing (drifting) and change-point (sudden) restlessness. In this work, we study specific classes of drifting restless bandits selected for their relevance to modelling an online website optimization process. The contribution in this work is a simple, feasible weighted least squares technique capable of utilizing contextual arm parameters while considering the parameter space drifting non-stationary within reasonable bounds. We produce a reference implementation, then evaluate and compare its performance in several different true world states, finding experimentally that performance is robust to time drifting factors similar to those seen in many real world cases. The restless bandit problem is one of the most well-studied generalizations of the celebrated stochastic multi-armed bandit (MAB) problem in decision theory. In its ultimate generality, the restless bandit problem is known to be PSPACE-Hard to approximate to any nontrivial factor, and little progress has been made on this problem despite its significance in modeling activity allocation under uncertainty. In this article, we consider the Feedback MAB problem, where the reward obtained by playing each of n independent arms varies according to an underlying on off Markov process whose exact state is only revealed when the arm is played. The goal is to design a policy for playing the arms in order to maximize the infinite horizon time average expected reward. This problem is also an instance of a Partially Observable Markov Decision Process (POMDP), and is widely studied in wireless scheduling and unmanned aerial vehicle (UAV) routing. Unlike the stochastic MAB problem, the Feedback MAB problem does not admit to greedy index-based optimal policies. We develop a novel duality-based algorithmic technique that yields a surprisingly simple and intuitive (2+e)-approximate greedy policy to this problem. We show that both in terms of approximation factor and computational efficiency, our policy is closely related to the Whittle index, which is widely used for its simplicity and efficiency of computation. Subsequently we define a multi-state generalization, that we term Monotone bandits, which remains subclass of the restless bandit problem. We show that our policy remains a 2-approximation in this setting, and further, our technique is robust enough to incorporate various side-constraints such as blocking plays, switching costs, and even models where determining the state of an arm is a separate operation from playing it. Our technique is also of independent interest for other restless bandit problems, and we provide an example in nonpreemptive machine replenishment. Interestingly, in this case, our policy provides a constant factor guarantee, whereas the Whittle index is provably polynomially worse. By presenting the first O(1) approximations for nontrivial instances of restless bandits as well as of POMDPs, our work initiates the study of approximation algorithms in both these contexts. The class of restless bandits as proposed by Whittle [21] have long been known to be intractable. This paper presents an optimality result which extends that of Weber and Weiss [19] for restless bandits to a more general setting in which individual bandits have multiple levels of activation but are subject to an overall resource constraint. The contribution is motivated by recent works of [10, 11] who discuss the performance of index heuristics for resource allocation in such systems. Hitherto, index heuristics have been shown, under a condition of full indexability, to be optimal for a natural Lagrangian relaxation of such problems in which resource is purchased rather than constrained. We find that under key assumptions about the nature of solutions to a deterministic differential equation that the index heuristics above are asymptotically optimal in a sense described by Whittle. We then demonstrate that these assumptions always hold for three-state bandits. 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. The problem of rested and restless multi-armed bandits with constrained availability (RMAB-CA) of arms is considered. The states of arms evolve in Markovian manner and the exact states are hidden from the decision maker. First, some structural results on value functions are claimed. Following these results, the optimal policy turns out to be a threshold policy . Furthermore, indexability is established for both rested and restless RMAB-CAs. An index formula is derived for the rested model, while an algorithm is provided for restless case. Contextual bandits are widely used in Internet services from news recommendation to advertising, and to Web search. Generalized linear models (logistical regression in particular) have demonstrated stronger performance than linear models in many applications where rewards are binary. However, most theoretical analyses on contextual bandits so far are on linear bandits. In this work, we propose an upper confidence bound based algorithm for generalized linear contextual bandits, which achieves an @math regret over @math rounds with @math dimensional feature vectors. This regret matches the minimax lower bound, up to logarithmic terms, and improves on the best previous result by a @math factor, assuming the number of arms is fixed. A key component in our analysis is to establish a new, sharp finite-sample confidence bound for maximum-likelihood estimates in generalized linear models, which may be of independent interest. We also analyze a simpler upper confidence bound algorithm, which is useful in practice, and prove it to have optimal regret for certain cases. 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 consider a class of restless multiarmed bandit processes (RMABs) that arises in dynamic multichannel access, user server scheduling, and optimal activation in multiagent systems. For this class of RMABs, we establish the indexability and obtain Whittle index in closed form for both discounted and average reward criteria. These results lead to a direct implementation of Whittle index policy with remarkably low complexity. When arms are stochastically identical, we show that Whittle index policy is optimal under certain conditions. Furthermore, it has a semiuniversal structure that obviates the need to know the Markov transition probabilities. The optimality and the semiuniversal structure result from the equivalence between Whittle index policy and the myopic policy established in this work. For nonidentical arms, we develop efficient algorithms for computing a performance upper bound given by Lagrangian relaxation. The tightness of the upper bound and the near-optimal performance of Whittle index policy are illustrated with simulation examples. We consider the multi armed bandit problem in non-stationary environments. Based on the Bayesian method, we propose a variant of Thompson Sampling which can be used in both rested and restless bandit scenarios. Applying discounting to the parameters of prior distribution, we describe a way to systematically reduce the effect of past observations. Further, we derive the exact expression for the probability of picking sub-optimal arms. By increasing the exploitative value of Bayes' samples, we also provide an optimistic version of the algorithm. Extensive empirical analysis is conducted under various scenarios to validate the utility of proposed algorithms. A comparison study with various state-of-the-arm algorithms is also included. We show that if performance measures in a stochastic scheduling problem satisfy a set of so-called partial conservation laws (PCL), which extend previously studied generalized conservation laws (GCL), then the problem is solved optimally by a priority-index policy for an appropriate range of linear performance objectives, where the optimal indices are computed by a one-pass adaptive-greedy algorithm, based on Klimov's. We further apply this framework to investigate the indexability property of restless bandits introduced by Whittle, obtaining the following results: (1) we identify a class of restless bandits (PCL-indexable) which are indexable; membership in this class is tested through a single run of the adaptive-greedy algorithm, which also computes the Whittle indices when the test is positive; this provides a tractable sufficient condition for indexability; (2) we further indentify the class of GCL-indexable bandits, which includes classical bandits, having the property that they are indexable under any linear reward objective. The analysis is based on the so-called achievable region method, as the results follow from new linear programming formulations for the problems investigated. We study exploration in Multi-Armed Bandits in a setting where @math players collaborate in order to identify an @math -optimal arm. Our motivation comes from recent employment of bandit algorithms in computationally intensive, large-scale applications. Our results demonstrate a non-trivial tradeoff between the number of arm pulls required by each of the players, and the amount of communication between them. In particular, our main result shows that by allowing the @math players to communicate only once, they are able to learn @math times faster than a single player. That is, distributing learning to @math players gives rise to a factor @math parallel speed-up. We complement this result with a lower bound showing this is in general the best possible. On the other extreme, we present an algorithm that achieves the ideal factor @math speed-up in learning performance, with communication only logarithmic in @math . Motivated by online recommendation and advertising systems, we consider a causal model for stochastic contextual bandits with a latent low-dimensional confounder. In our model, there are @math observed contexts and @math arms of the bandit. The observed context influences the reward obtained through a latent confounder variable with cardinality @math ( @math ). The arm choice and the latent confounder causally determines the reward while the observed context is correlated with the confounder. Under this model, the @math mean reward matrix @math (for each context in @math and each arm in @math ) factorizes into non-negative factors @math ( @math ) and @math ( @math ). This insight enables us to propose an @math -greedy NMF-Bandit algorithm that designs a sequence of interventions (selecting specific arms), that achieves a balance between learning this low-dimensional structure and selecting the best arm to minimize regret. Our algorithm achieves a regret of @math at time @math , as compared to @math for conventional contextual bandits, assuming a constant gap between the best arm and the rest for each context. These guarantees are obtained under mild sufficiency conditions on the factors that are weaker versions of the well-known Statistical RIP condition. We further propose a class of generative models that satisfy our sufficient conditions, and derive a lower bound of @math . These are the first regret guarantees for online matrix completion with bandit feedback, when the rank is greater than one. We further compare the performance of our algorithm with the state of the art, on synthetic and real world data-sets. In a multi-armed bandit (MAB) problem, an online algorithm makes a sequence of choices. In each round it chooses from a time-invariant set of alternatives and receives the payoff associated with this alternative. While the case of small strategy sets is by now well-understood, a lot of recent work has focused on MAB problems with exponentially or infinitely large strategy sets, where one needs to assume extra structure in order to make the problem tractable. In particular, recent literature considered information on similarity between arms. We consider similarity information in the setting of contextual bandits, a natural extension of the basic MAB problem where before each round an algorithm is given the context--a hint about the payoffs in this round. Contextual bandits are directly motivated by placing advertisements on web pages, one of the crucial problems in sponsored search. A particularly simple way to represent similarity information in the contextual bandit setting is via a similarity distance between the context-arm pairs which bounds from above the difference between the respective expected payoffs. Prior work on contextual bandits with similarity uses "uniform" partitions of the similarity space, so that each context-arm pair is approximated by the closest pair in the partition. Algorithms based on "uniform" partitions disregard the structure of the payoffs and the context arrivals, which is potentially wasteful. We present algorithms that are based on adaptive partitions, and take advantage of "benign" payoffs and context arrivals without sacrificing the worst-case performance. The central idea is to maintain a finer partition in high-payoff regions of the similarity space and in popular regions of the context space. Our results apply to several other settings, e.g., MAB with constrained temporal change (Slivkins and Upfal, 2008) and sleeping bandits (, 2008a). 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. This paper considers stochastic bandits with side observations, a model that accounts for both the exploration exploitation dilemma and relationships between arms. In this setting, after pulling an arm i, the decision maker also observes the rewards for some other actions related to i. We will see that this model is suited to content recommendation in social networks, where users' reactions may be endorsed or not by their friends. We provide efficient algorithms based on upper confidence bounds (UCBs) to leverage this additional information and derive new bounds improving on standard regret guarantees. We also evaluate these policies in the context of movie recommendation in social networks: experiments on real datasets show substantial learning rate speedups ranging from 2.2x to 14x on dense networks. This paper studies a generalized class of restless multi-armed bandits with hidden states and allow cumulative feedback, as opposed to the conventional instantaneous feedback. We call them lazy restless bandits (LRBs) as the events of decision making are sparser than the events of state transition. Hence, feedback after each decision event is the cumulative effect of the following state transition events. The states of arms are hidden from the decision maker and rewards for actions are state dependent. The decision maker needs to choose one arm in each decision interval, such that the long-term cumulative reward is maximized. As the states are hidden, the decision maker maintains and updates its belief about them. It is shown that LRBs admit an optimal policy which has threshold structure in belief space. The Whittle-index policy for solving the LRB problem is analyzed; indexability of LRBs is shown. Further, the closed-form index expressions are provided for two sets of special cases; for more general cases, an algorithm for index computation is provided. An extensive simulation study is presented; Whittle-index, modified Whittle-index, and myopic policies are compared. The Lagrangian relaxation of the problem provides an upper bound on the optimal value function; it is used to assess the degree of sub-optimality various 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. We propose novel multi-armed bandit (explore exploit) schemes to maximize total clicks on a content module published regularly on Yahoo! Intuitively, one can explore'' each candidate item by displaying it to a small fraction of user visits to estimate the item's click-through rate (CTR), and then exploit'' high CTR items in order to maximize clicks. While bandit methods that seek to find the optimal trade-off between explore and exploit have been studied for decades, existing solutions are not satisfactory for web content publishing applications where dynamic set of items with short lifetimes, delayed feedback and non-stationary reward (CTR) distributions are typical. In this paper, we develop a Bayesian solution and extend several existing schemes to our setting. Through extensive evaluation with nine bandit schemes, we show that our Bayesian solution is uniformly better in several scenarios. We also study the empirical characteristics of our schemes and provide useful insights on the strengths and weaknesses of each. Finally, we validate our results with a side-by-side'' comparison of schemes through live experiments conducted on a random sample of real user visits to Yahoo! We introduce a novel algorithmic approach to content recommendation based on adaptive clustering of exploration-exploitation "bandit") strategies. We provide a sharp regret analysis of this algorithm in a standard stochastic noise setting, demonstrate its scalability properties, and prove its effectiveness on a number of artificial and real-world datasets. Our experiments show a significant increase in prediction performance over state-of-the-art methods for bandit problems. Generalized Linear Bandits (GLBs), a natural extension of the stochastic linear bandits, has been popular and successful in recent years. However, existing GLBs scale poorly with the number of rounds and the number of arms, limiting their utility in practice. This paper proposes new, scalable solutions to the GLB problem in two respects. First, unlike existing GLBs, whose per-time-step space and time complexity grow at least linearly with time @math , we propose a new algorithm that performs online computations to enjoy a constant space and time complexity. At its heart is a novel Generalized Linear extension of the Online-to-confidence-set Conversion (GLOC method) that takes online learning algorithm and turns it into a GLB algorithm. As a special case, we apply GLOC to the online Newton step algorithm, which results in a low-regret GLB algorithm with much lower time and memory complexity than prior work. Second, for the case where the number @math of arms is very large, we propose new algorithms in which each next arm is selected via an inner product search. Such methods can be implemented via hashing algorithms (i.e., "hash-amenable") and result in a time complexity sublinear in @math . While a Thompson sampling extension of GLOC is hash-amenable, its regret bound for @math -dimensional arm sets scales with @math , whereas GLOC's regret bound scales with @math . Towards closing this gap, we propose a new hash-amenable algorithm whose regret bound scales with @math . Finally, we propose a fast approximate hash-key computation (inner product) with a better accuracy than the state-of-the-art, which can be of independent interest. We conclude the paper with preliminary experimental results confirming the merits of our methods.