Domain name registrars and URL shortener service providers place advertisements on the parked domains (Internet domain names which are not in service) in order to generate profits. As the web contents have been removed, it is critical to make sure the displayed ads are directly related to the intents of the visitors who have been directed to the parked domains. Because of the missing contents in these domains, it is non-trivial to generate the keywords to describe the previous contents and therefore the users intents. In this paper we discuss the adaptive keywords extraction problem and introduce an algorithm based on the BM25F term weighting and linear multi-armed bandits. We built a prototype over a production domain registration system and evaluated it using crowdsourcing in multiple iterations. The prototype is compared with other popular methods and is shown to be more effective.
In @cite_25 the authors propose a combination algorithm of using upper confidence bound (UCB) and @math -greedy to solve the exploration-exploitation dilemma. Their target is to select high profit ads which would be fed in contextual advertising platforms. The feature vectors of ads are not used in their system as side information, instead they use standard bandits considering the reward following an unknown stochastic process. Our research is partially inspired by their work.
@cite_25
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We study contextual bandits with budget and time constraints, referred to as constrained contextual bandits.The time and budget constraints significantly complicate the exploration and exploitation tradeoff because they introduce complex coupling among contexts over time.Such coupling effects make it difficult to obtain oracle solutions that assume known statistics of bandits. To gain insight, we first study unit-cost systems with known context distribution. When the expected rewards are known, we develop an approximation of the oracle, referred to Adaptive-Linear-Programming (ALP), which achieves near-optimality and only requires the ordering of expected rewards. With these highly desirable features, we then combine ALP with the upper-confidence-bound (UCB) method in the general case where the expected rewards are unknown a priori . We show that the proposed UCB-ALP algorithm achieves logarithmic regret except for certain boundary cases. Further, we design algorithms and obtain similar regret analysis results for more general systems with unknown context distribution and heterogeneous costs. To the best of our knowledge, this is the first work that shows how to achieve logarithmic regret in constrained contextual bandits. Moreover, this work also sheds light on the study of computationally efficient algorithms for general constrained contextual bandits. 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. Multi-armed bandit problems are considered as a paradigm of the trade-off between exploring the environment to find profitable actions and exploiting what is already known. In the stationary case, the distributions of the rewards do not change in time, Upper-Confidence Bound (UCB) policies have been shown to be rate optimal. A challenging variant of the MABP is the non-stationary bandit problem where the gambler must decide which arm to play while facing the possibility of a changing environment. In this paper, we consider the situation where the distributions of rewards remain constant over epochs and change at unknown time instants. We analyze two algorithms: the discounted UCB and the sliding-window UCB. We establish for these two algorithms an upper-bound for the expected regret by upper-bounding the expectation of the number of times a suboptimal arm is played. For that purpose, we derive a Hoeffding type inequality for self normalized deviations with a random number of summands. We establish a lower-bound for the regret in presence of abrupt changes in the arms reward distributions. We show that the discounted UCB and the sliding-window UCB both match the lower-bound up to a logarithmic factor. 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. Contextual bandit algorithms provide principled online learning solutions to find optimal trade-offs between exploration and exploitation with companion side-information. Most contextual bandit algorithms simply assume the learner would have access to the entire set of features, which govern the generation of payoffs from a user to an item. However, in practice it is challenging to exhaust all relevant features ahead of time, and oftentimes due to privacy or sampling constraints many factors are unobservable to the algorithm. Failing to model such hidden factors leads a system to make constantly suboptimal predictions. In this paper, we propose to learn the hidden features for contextual bandit algorithms. Hidden features are explicitly introduced in our reward generation assumption, in addition to the observable contextual features. A scalable bandit algorithm is achieved via coordinate descent, in which closed form solutions exist at each iteration for both hidden features and bandit parameters. Most importantly, we rigorously prove that the developed contextual bandit algorithm achieves a sublinear upper regret bound with high probability, and a linear regret is inevitable if one fails to model such hidden features. Extensive experimentation on both simulations and large-scale real-world datasets verified the advantages of the proposed algorithm compared with several state-of-the-art contextual bandit algorithms and existing ad-hoc combinations between bandit algorithms and matrix factorization methods. 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 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 . Many problems, such as cognitive radio, parameter control of a scanning tunnelling microscope or internet advertisement, can be modelled as non-stationary bandit problems where the distributions of rewards changes abruptly at unknown time instants. In this paper, we analyze two algorithms designed for solving this issue: discounted UCB (D-UCB) and sliding-window UCB (SW-UCB). We establish an upperbound for the expected regret by upper-bounding the expectation of the number of times suboptimal arms are played. The proof relies on an interesting Hoeffding type inequality for self normalized deviations with a random number of summands. We establish a lower-bound for the regret in presence of abrupt changes in the arms reward distributions. We show that the discounted UCB and the sliding-window UCB both match the lower-bound up to a logarithmic factor. Numerical simulations show that D-UCB and SW-UCB perform significantly better than existing soft-max methods like EXP3.S. 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 this paper, we consider the challenge of maximizing an unknown function f for which evaluations are noisy and are acquired with high cost. An iterative procedure uses the previous measures to actively select the next estimation of f which is predicted to be the most useful. We focus on the case where the function can be evaluated in parallel with batches of fixed size and analyze the benefit compared to the purely sequential procedure in terms of cumulative regret. We introduce the Gaussian Process Upper Confidence Bound and Pure Exploration algorithm (GP-UCB-PE) which combines the UCB strategy and Pure Exploration in the same batch of evaluations along the parallel iterations. We prove theoretical upper bounds on the regret with batches of size K for this procedure which show the improvement of the order of @math for fixed iteration cost over purely sequential versions. Moreover, the multiplicative constants involved have the property of being dimension-free. We also confirm empirically the efficiency of GP-UCB-PE on real and synthetic problems compared to state-of-the-art competitors. 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 show how an ensemble of @math -functions can be leveraged for more effective exploration in deep reinforcement learning. We build on well established algorithms from the bandit setting, and adapt them to the @math -learning setting. We propose an exploration strategy based on upper-confidence bounds (UCB). Our experiments show significant gains on the Atari benchmark. Contextual bandit algorithms provide principled online learning solutions to find optimal trade-offs between exploration and exploitation with companion side-information. They have been extensively used in many important practical scenarios, such as display advertising and content recommendation. A common practice estimates the unknown bandit parameters pertaining to each user independently. This unfortunately ignores dependency among users and thus leads to suboptimal solutions, especially for the applications that have strong social components. In this paper, we develop a collaborative contextual bandit algorithm, in which the adjacency graph among users is leveraged to share context and payoffs among neighboring users while online updating. We rigorously prove an improved upper regret bound of the proposed collaborative bandit algorithm comparing to conventional independent bandit algorithms. Extensive experiments on both synthetic and three large-scale real-world datasets verified the improvement of our proposed algorithm against several state-of-the-art contextual bandit algorithms. In the stochastic multi-armed bandit problem we consider a modification of the UCB algorithm of [4]. For this modified algorithm we give an improved bound on the regret with respect to the optimal reward. While for the original UCB algorithm the regret in K-armed bandits after T trials is bounded by const · ( K (T) ), where Δ measures the distance between a suboptimal arm and the optimal arm, for the modified UCB algorithm we show an upper bound on the regret of const · ( K (T ^2 ) ). Reinforcement learning policies face the exploration versus exploitation dilemma, i.e. the search for a balance between exploring the environment to find profitable actions while taking the empirically best action as often as possible. A popular measure of a policy's success in addressing this dilemma is the regret, that is the loss due to the fact that the globally optimal policy is not followed all the times. One of the simplest examples of the exploration exploitation dilemma is the multi-armed bandit problem. Lai and Robbins were the first ones to show that the regret for this problem has to grow at least logarithmically in the number of plays. Since then, policies which asymptotically achieve this regret have been devised by Lai and Robbins and many others. In this work we show that the optimal logarithmic regret is also achievable uniformly over time, with simple and efficient policies, and for all reward distributions with bounded support. The multi-armed bandit is a model of exploration and exploitation, where one must select, within a finite set of arms, the one which maximizes the cumulative reward up to the time horizon T. For the adversarial multi-armed bandit problem, where the sequence of rewards is chosen by an oblivious adversary, the notion of best arm during the time horizon is too restrictive for applications such as ad-serving, where the best ad could change during time range. In this paper, we consider a variant of the adversarial multi-armed bandit problem, where the time horizon is divided into unknown time periods within which rewards are drawn from stochastic distributions. During each time period, there is an optimal arm which may be different from the optimal arm at the previous time period. We present an algorithm taking advantage of the constant exploration of EXP3 to detect when the best arm changes. Its analysis shows that on a run divided into N periods where the best arm changes, the proposed algorithms achieves a regret in O(N √T log T). We propose the contextual combinatorial cascading bandits, a combinatorial online learning game, where at each time step a learning agent is given a set of contextual information, then selects a list of items, and observes stochastic outcomes of a prefix in the selected items by some stopping criterion. In online recommendation, the stopping criterion might be the first item a user selects; in network routing, the stopping criterion might be the first edge blocked in a path. We consider position discounts in the list order, so that the agent's reward is discounted depending on the position where the stopping criterion is met. We design a UCB-type algorithm, C3-UCB, for this problem, prove an n-step regret bound O(√n) in the general setting, and give finer analysis for two special cases. Our work generalizes existing studies in several directions, including contextual information, position discounts, and a more general cascading bandit model. Experiments on synthetic and real datasets demonstrate the advantage of involving contextual information and position discounts. 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. We consider stochastic multi-armed bandits where the expected reward is a unimodal function over partially ordered arms. This important class of problems has been recently investigated in (Cope 2009, Yu 2011). The set of arms is either discrete, in which case arms correspond to the vertices of a finite graph whose structure represents similarity in rewards, or continuous, in which case arms belong to a bounded interval. For discrete unimodal bandits, we derive asymptotic lower bounds for the regret achieved under any algorithm, and propose OSUB, an algorithm whose regret matches this lower bound. Our algorithm optimally exploits the unimodal structure of the problem, and surprisingly, its asymptotic regret does not depend on the number of arms. We also provide a regret upper bound for OSUB in non-stationary environments where the expected rewards smoothly evolve over time. The analytical results are supported by numerical experiments showing that OSUB performs significantly better than the state-of-the-art algorithms. For continuous sets of arms, we provide a brief discussion. We show that combining an appropriate discretization of the set of arms with the UCB algorithm yields an order-optimal regret, and in practice, outperforms recently proposed algorithms designed to exploit the unimodal structure. We consider the sequential Bayesian optimization problem with bandit feedback, adopting a formulation that allows for the reward function to vary with time. We model the reward function using a Gaussian process whose evolution obeys a simple Markov model. We introduce two natural extensions of the classical Gaussian process upper confidence bound (GP-UCB) algorithm. The first, R-GP-UCB, resets GP-UCB at regular intervals. The second, TV-GP-UCB, instead forgets about old data in a smooth fashion. Our main contribution comprises of novel regret bounds for these algorithms, providing an explicit characterization of the trade-off between the time horizon and the rate at which the function varies. We illustrate the performance of the algorithms on both synthetic and real data, and we find the gradual forgetting of TV-GP-UCB to perform favorably compared to the sharp resetting of R-GP-UCB. Moreover, both algorithms significantly outperform classical GP-UCB, since it treats stale and fresh data equally.