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.
In the centralized setting of our problem, a central controller selects at each time step a matching that maps the @math players to different arms in @math . In other words, introducing @math with means @math for @math and @math , the central controller selects a subset of @math elementary arms whose indices form a matching from @math to @math . Then, the utility of each elementary arm is observed and the reward is the sum of those. This is a particular combinatorial bandit problem with semi-bandit feedback, first studied by @cite_14 .
@cite_14
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We consider the problem of learning in single-player and multiplayer multiarmed bandit models. Bandit problems are classes of online learning problems that capture exploration versus exploitation tradeoffs. In a multiarmed bandit model, players can pick among many arms, and each play of an arm generates an i.i.d. reward from an unknown distribution. The objective is to design a policy that maximizes the expected reward over a time horizon for a single-player setting and the sum of expected rewards for the multiplayer setting. In the multiplayer setting, arms may give different rewards to different players. There is no separate channel for coordination among the players. Any attempt at communication is costly and adds to regret. We propose two decentralizable policies, @math ( @math - @math ) and @math - @math , that can be used in both single-player and multiplayer settings. These policies are shown to yield expected regret that grows, at most, as @math (and @math under some assumption). It is well known that @math is the lower bound on the rate of growth of regret even in a centralized case. The proposed algorithms improve on prior work where regret grew at @math . More fundamentally, these policies address the question of additional cost incurred in decentralized online learning, suggesting that there is, at most, an @math -factor cost in terms of the order of regret. This solves a problem of relevance in many domains and had been open for a while. We consider the problem of learning in single-player and multiplayer multiarmed bandit models. Bandit problems are classes of online learning problems that capture exploration versus exploitation tradeoffs. In a multiarmed bandit model, players can pick among many arms, and each play of an arm generates an i.i.d. reward from an unknown distribution. The objective is to design a policy that maximizes the expected reward over a time horizon for a single player setting and the sum of expected rewards for the multiplayer setting. In the multiplayer setting, arms may give different rewards to different players. There is no separate channel for coordination among the players. Any attempt at communication is costly and adds to regret. We propose two decentralizable policies, @math ( @math - @math ) and @math - @math , that can be used in both single player and multiplayer settings. These policies are shown to yield expected regret that grows at most as O( @math ). It is well known that @math is the lower bound on the rate of growth of regret even in a centralized case. The proposed algorithms improve on prior work where regret grew at O( @math ). More fundamentally, these policies address the question of additional cost incurred in decentralized online learning, suggesting that there is at most an @math -factor cost in terms of order of regret. This solves a problem of relevance in many domains and had been open for a while. 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. Many real-world problems face the dilemma of choosing best @math out of @math options at a given time instant. This setup can be modelled as combinatorial bandit which chooses @math out of @math arms at each time, with an aim to achieve an efficient tradeoff between exploration and exploitation. This is the first work for combinatorial bandit where the reward received can be a non-linear function of the chosen @math arms. The direct use of multi-armed bandit requires choosing among @math -choose- @math options making the state space large. In this paper, we present a novel algorithm which is computationally efficient and the storage is linear in @math . The proposed algorithm is a divide-and-conquer based strategy, that we call CMAB-SM. Further, the proposed algorithm achieves a regret bound of @math for a time horizon @math , which is sub-linear in all parameters @math , @math , and @math . The evaluation results on different reward functions and arm distribution functions show significantly improved performance as compared to standard multi-armed bandit approach with @math choices. We consider the stochastic bandit problem in the sublinear space setting, where one cannot record the win-loss record for all @math arms. We give an algorithm using @math words of space with regret [ i=1 ^ K 1 T ] where @math is the gap between the best arm and arm @math and @math is the gap between the best and the second-best arms. If the rewards are bounded away from @math and @math , this is within an @math factor of the optimum regret possible without space constraints. We present algorithms for reducing the Dueling Bandits problem to the conventional (stochastic) Multi-Armed Bandits problem. The Dueling Bandits problem is an online model of learning with ordinal feedback of the form "A is preferred to B" (as opposed to cardinal feedback like "A has value 2.5"), giving it wide applicability in learning from implicit user feedback and revealed and stated preferences. In contrast to existing algorithms for the Dueling Bandits problem, our reductions -- named @math , @math and @math -- provide a generic schema for translating the extensive body of known results about conventional Multi-Armed Bandit algorithms to the Dueling Bandits setting. For @math and @math we prove regret upper bounds in both finite and infinite settings, and conjecture about the performance of @math which empirically outperforms the other two as well as previous algorithms in our experiments. In addition, we provide the first almost optimal regret bound in terms of second order terms, such as the differences between the values of the arms. The multi-armed bandit problem is a popular model for studying exploration exploitation trade-off in sequential decision problems. Many algorithms are now available for this well-studied problem. One of the earliest algorithms, given by W. R. Thompson, dates back to 1933. This algorithm, referred to as Thompson Sampling, is a natural Bayesian algorithm. The basic idea is to choose an arm to play according to its probability of being the best arm. Thompson Sampling algorithm has experimentally been shown to be close to optimal. In addition, it is efficient to implement and exhibits several desirable properties such as small regret for delayed feedback. However, theoretical understanding of this algorithm was quite limited. In this paper, for the first time, we show that Thompson Sampling algorithm achieves logarithmic expected regret for the multi-armed bandit problem. More precisely, for the two-armed bandit problem, the expected regret in time @math is @math . And, for the @math -armed bandit problem, the expected regret in time @math is @math . Our bounds are optimal but for the dependence on @math and the constant factors in big-Oh. Multi-armed bandit (MAB) is a class of online learning problems where a learning agent aims to maximize its expected cumulative reward while repeatedly selecting to pull arms with unknown reward distributions. In this paper, we consider a scenario in which the arms' reward distributions may change in a piecewise-stationary fashion at unknown time steps. By connecting change-detection techniques with classic UCB algorithms, we motivate and propose a learning algorithm called M-UCB, which can detect and adapt to changes, for the considered scenario. We also establish an @math regret bound for M-UCB, where @math is the number of time steps, @math is the number of arms, and @math is the number of stationary segments. and @math is the gap between the expected rewards of the optimal and best suboptimal arms. Comparison with the best available lower bound shows that M-UCB is nearly optimal in @math up to a logarithmic factor. We also compare M-UCB with state-of-the-art algorithms in a numerical experiment based on a public Yahoo! dataset. In this experiment, M-UCB achieves about @math regret reduction with respect to the best performing state-of-the-art algorithm. 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 . We study a variant of the stochastic @math -armed bandit problem, which we call "bandits with delayed, aggregated anonymous feedback". In this problem, when the player pulls an arm, a reward is generated, however it is not immediately observed. Instead, at the end of each round the player observes only the sum of a number of previously generated rewards which happen to arrive in the given round. The rewards are stochastically delayed and due to the aggregated nature of the observations, the information of which arm led to a particular reward is lost. The question is what is the cost of the information loss due to this delayed, aggregated anonymous feedback? Previous works have studied bandits with stochastic, non-anonymous delays and found that the regret increases only by an additive factor relating to the expected delay. In this paper, we show that this additive regret increase can be maintained in the harder delayed, aggregated anonymous feedback setting when the expected delay (or a bound on it) is known. We provide an algorithm that matches the worst case regret of the non-anonymous problem exactly when the delays are bounded, and up to logarithmic factors or an additive variance term for unbounded delays. The fundamental problem of multiple secondary users contending for opportunistic spectrum access over multiple channels in cognitive radio networks has been formulated recently as a decentralized multi-armed bandit (D-MAB) problem. In a D-MAB problem there are @math users and @math arms (channels) that each offer i.i.d. stochastic rewards with unknown means so long as they are accessed without collision. The goal is to design a decentralized online learning policy that incurs minimal regret, defined as the difference between the total expected rewards accumulated by a model-aware genie, and that obtained by all users applying the policy. We make two contributions in this paper. First, we consider the setting where the users have a prioritized ranking, such that it is desired for the @math -th-ranked user to learn to access the arm offering the @math -th highest mean reward. For this problem, we present the first distributed policy that yields regret that is uniformly logarithmic over time without requiring any prior assumption about the mean rewards. Second, we consider the case when a fair access policy is required, i.e., it is desired for all users to experience the same mean reward. For this problem, we present a distributed policy that yields order-optimal regret scaling with respect to the number of users and arms, better than previously proposed policies in the literature. Both of our distributed policies make use of an innovative modification of the well known UCB1 policy for the classic multi-armed bandit problem that allows a single user to learn how to play the arm that yields the @math -th largest mean reward. In this paper, we study the stochastic combinatorial multi-armed bandit (CMAB) framework that allows a general nonlinear reward function, whose expected value may not depend only on the means of the input random variables but possibly on the entire distributions of these variables. Our framework enables a much larger class of reward functions such as the @math function and nonlinear utility functions. Existing techniques relying on accurate estimations of the means of random variables, such as the upper confidence bound (UCB) technique, do not work directly on these functions. We propose a new algorithm called stochastically dominant confidence bound (SDCB), which estimates the distributions of underlying random variables and their stochastically dominant confidence bounds. We prove that SDCB can achieve @math distribution-dependent regret and @math distribution-independent regret, where @math is the time horizon. We apply our results to the @math -MAX problem and expected utility maximization problems. In particular, for @math -MAX, we provide the first polynomial-time approximation scheme (PTAS) for its offline problem, and give the first @math bound on the @math -approximation regret of its online problem, for any @math . We study a general class of online learning problems where the feedback is specified by a graph. This class includes online prediction with expert advice and the multi-armed bandit problem, but also several learning problems where the online player does not necessarily observe his own loss. We analyze how the structure of the feedback graph controls the inherent difficulty of the induced @math -round learning problem. Specifically, we show that any feedback graph belongs to one of three classes: strongly observable graphs, weakly observable graphs, and unobservable graphs. We prove that the first class induces learning problems with @math minimax regret, where @math is the independence number of the underlying graph; the second class induces problems with @math minimax regret, where @math is the domination number of a certain portion of the graph; and the third class induces problems with linear minimax regret. Our results subsume much of the previous work on learning with feedback graphs and reveal new connections to partial monitoring games. We also show how the regret is affected if the graphs are allowed to vary with time. A stochastic combinatorial semi-bandit is an online learning problem where at each step a learning agent chooses a subset of ground items subject to constraints, and then observes stochastic weights of these items and receives their sum as a payoff. In this paper, we close the problem of computationally and sample efficient learning in stochastic combinatorial semi-bandits. In particular, we analyze a UCB-like algorithm for solving the problem, which is known to be computationally efficient; and prove @math and @math upper bounds on its @math -step regret, where @math is the number of ground items, @math is the maximum number of chosen items, and @math is the gap between the expected returns of the optimal and best suboptimal solutions. The gap-dependent bound is tight up to a constant factor and the gap-free bound is tight up to a polylogarithmic factor. We consider a stochastic bandit problem with infinitely many arms. In this setting, the learner has no chance of trying all the arms even once and has to dedicate its limited number of samples only to a certain number of arms. All previous algorithms for this setting were designed for minimizing the cumulative regret of the learner. In this paper, we propose an algorithm aiming at minimizing the simple regret. As in the cumulative regret setting of infinitely many armed bandits, the rate of the simple regret will depend on a parameter @math characterizing the distribution of the near-optimal arms. We prove that depending on @math , our algorithm is minimax optimal either up to a multiplicative constant or up to a @math factor. We also provide extensions to several important cases: when @math is unknown, in a natural setting where the near-optimal arms have a small variance, and in the case of unknown time horizon. 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. 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. We consider a linear stochastic bandit problem where the dimension @math of the unknown parameter @math is larger than the sampling budget @math . In such cases, it is in general impossible to derive sub-linear regret bounds since usual linear bandit algorithms have a regret in @math . In this paper we assume that @math is @math sparse, i.e. has at most @math non-zero components, and that the space of arms is the unit ball for the @math norm. We combine ideas from Compressed Sensing and Bandit Theory and derive algorithms with regret bounds in @math . We present Epoch-Greedy, an algorithm for multi-armed bandits with observable side information. Epoch-Greedy has the following properties: No knowledge of a time horizon @math is necessary. The regret incurred by Epoch-Greedy is controlled by a sample complexity bound for a hypothesis class. The regret scales as @math or better (sometimes, much better). Here @math is the complexity term in a sample complexity bound for standard supervised learning. In this paper we consider the multiarmed bandit problem where the arms are chosen from a subset of the real line and the mean rewards are assumed to be a continuous function of the arms. The problem with an infinite number of arms is much more difficult than the usual one with a finite number of arms because the built-in learning task is now infinite dimensional. We devise a kernel estimator-based learning scheme for the mean reward as a function of the arms. Using this learning scheme, we construct a class of certainty equivalence control with forcing schemes and derive asymptotic upper bounds on their learning loss. To the best of our knowledge, these bounds are the strongest rates yet available. Moreover, they are stronger than the @math required for optimality with respect to the average-cost-per-unit-time criterion.