Mobile Context-Aware Recommender Systems can be naturally modelled as an exploration exploitation trade-off (exr exp) problem, where the system has to choose between maximizing its expected rewards dealing with its current knowledge (exploitation) and learning more about the unknown user's preferences to improve its knowledge (exploration). This problem has been addressed by the reinforcement learning community but they do not consider the risk level of the current user's situation, where it may be dangerous to recommend items the user may not desire in her current situation if the risk level is high. We introduce in this paper an algorithm named R-UCB that considers the risk level of the user's situation to adaptively balance between exr and exp. The detailed analysis of the experimental results reveals several important discoveries in the exr exp behaviour.
Very frequently used in reinforcement learning to study exr exp, the multi-armed bandit problem was originally described by @cite_10 . The @math -greedy is one of the most used strategies to solve this problem. It chooses a random document with @math -frequency, and chooses otherwise the document with the highest estimated mean, the estimation being based on the observed rewards. The @math is chosen by the user in the open interval ]0, 1[.
@cite_10
1998498767
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. 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. The probability that a user will click a search result depends both on its relevance and its position on the results page. The position based model explains this behavior by ascribing to every item an attraction probability, and to every position an examination probability. To be clicked, a result must be both attractive and examined. The probabilities of an item-position pair being clicked thus form the entries of a rank- @math matrix. We propose the learning problem of a Bernoulli rank- @math bandit where at each step, the learning agent chooses a pair of row and column arms, and receives the product of their Bernoulli-distributed values as a reward. This is a special case of the stochastic rank- @math bandit problem considered in recent work that proposed an elimination based algorithm Rank1Elim, and showed that Rank1Elim's regret scales linearly with the number of rows and columns on "benign" instances. These are the instances where the minimum of the average row and column rewards @math is bounded away from zero. The issue with Rank1Elim is that it fails to be competitive with straightforward bandit strategies as @math . In this paper we propose Rank1ElimKL which simply replaces the (crude) confidence intervals of Rank1Elim with confidence intervals based on Kullback-Leibler (KL) divergences, and with the help of a novel result concerning the scaling of KL divergences we prove that with this change, our algorithm will be competitive no matter the value of @math . Experiments with synthetic data confirm that on benign instances the performance of Rank1ElimKL is significantly better than that of even Rank1Elim, while experiments with models derived from real data confirm that the improvements are significant across the board, regardless of whether the data is benign or not. 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. 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. 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. In a multi-armed bandit problem, an online algorithm chooses from a set of strategies in a sequence of @math trials so as to maximize the total payoff of the chosen strategies. While the performance of bandit algorithms with a small finite strategy set is quite well understood, bandit problems with large strategy sets are still a topic of very active investigation, motivated by practical applications such as online auctions and web advertisement. The goal of such research is to identify broad and natural classes of strategy sets and payoff functions which enable the design of efficient solutions. In this work we study a very general setting for the multi-armed bandit problem in which the strategies form a metric space, and the payoff function satisfies a Lipschitz condition with respect to the metric. We refer to this problem as the "Lipschitz MAB problem". We present a complete solution for the multi-armed problem in this setting. That is, for every metric space (L,X) we define an isometry invariant Max Min COV(X) which bounds from below the performance of Lipschitz MAB algorithms for @math , and we present an algorithm which comes arbitrarily close to meeting this bound. Furthermore, our technique gives even better results for benign payoff functions. 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 . 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. 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. 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. Space-economical estimation of the pth frequency moments, defined as , for p> 0, are of interest in estimating all-pairs distances in a large data matrix [14], machine learning, and in data stream computation. Random sketches formed by the inner product of the frequency vector f 1 , ..., f n with a suitably chosen random vector were pioneered by Alon, Matias and Szegedy [1], and have since played a central role in estimating F p and for data stream computations in general. The concept of p-stable sketches formed by the inner product of the frequency vector with a random vector whose components are drawn from a p-stable distribution, was proposed by Indyk for estimating F p , for 0 < p< 2, and has been further studied in Li [13]. In this paper, we consider the problem of estimating F p , for 0 p , for 0 sketch [7] and the structure [5]. Our algorithms require space @math to estimate F p to within 1 ±i¾?factors and requires expected time @math to process each update. Thus, our technique trades an @math factor in space for much more efficient processing of stream updates. We also present a stand-alone iterative estimator for F 1 . 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. 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. Thompson Sampling is one of the oldest heuristics for multi-armed bandit problems. It is a randomized algorithm based on Bayesian ideas, and has recently generated significant interest after several studies demonstrated it to have better empirical performance compared to the state-of-the-art methods. However, many questions regarding its theoretical performance remained open. In this paper, we design and analyze a generalization of Thompson Sampling algorithm for the stochastic contextual multi-armed bandit problem with linear payoff functions, when the contexts are provided by an adaptive adversary. This is among the most important and widely studied versions of the contextual bandits problem. We provide the first theoretical guarantees for the contextual version of Thompson Sampling. We prove a high probability regret bound of @math (or @math ), which is the best regret bound achieved by any computationally efficient algorithm available for this problem in the current literature, and is within a factor of @math (or @math ) of the information-theoretic lower bound for this problem. The multi-armed bandit problem for a gambler is to decide which arm of a K-slot machine to pull to maximize his total reward in a series of trials. Many real-world learning and optimization problems can be modeled in this way. Several strategies or algorithms have been proposed as a solution to this problem in the last two decades, but, to our knowledge, there has been no common evaluation of these algorithms. This paper provides a preliminary empirical evaluation of several multi-armed bandit algorithms. It also describes and analyzes a new algorithm, Poker (Price Of Knowledge and Estimated Reward) whose performance compares favorably to that of other existing algorithms in several experiments. One remarkable outcome of our experiments is that the most naive approach, the e-greedy strategy, proves to be often hard to beat. We present a new algorithm for the contextual bandit learning problem, where the learner repeatedly takes one of @math actions in response to the observed context, and observes the reward only for that chosen action. Our method assumes access to an oracle for solving fully supervised cost-sensitive classification problems and achieves the statistically optimal regret guarantee with only @math oracle calls across all @math rounds, where @math is the number of policies in the policy class we compete against. By doing so, we obtain the most practical contextual bandit learning algorithm amongst approaches that work for general policy classes. We further conduct a proof-of-concept experiment which demonstrates the excellent computational and prediction performance of (an online variant of) our algorithm relative to several baselines. 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. 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.