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24
MultiArmed Bandits in Metric Spaces
 STOC'08
, 2008
"... In a multiarmed bandit problem, an online algorithm chooses from a set of strategies in a sequence of n 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 larg ..."
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Cited by 47 (8 self)
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In a multiarmed bandit problem, an online algorithm chooses from a set of strategies in a sequence of n 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 multiarmed 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 multiarmed problem in this setting. That is, for every metric space (L, X) we define an isometry invariant MaxMinCOV(X) which bounds from below the performance of Lipschitz MAB algorithms for X, and we present an algorithm which comes arbitrarily close to meeting this bound. Furthermore, our technique gives even better results for benign payoff functions.
An Adaptive Algorithm for Selecting Profitable Keywords for SearchBased Advertising Services
 In EC ’06: Proceedings of the 7th ACM conference on Electronic commerce
, 2006
"... Increases in online searches have spurred the growth of searchbased advertising services offered by search engines, enabling companies to promote their products to consumers based on search queries. With millions of available keywords whose clickthru rates and profits are highly uncertain, identify ..."
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Cited by 45 (0 self)
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Increases in online searches have spurred the growth of searchbased advertising services offered by search engines, enabling companies to promote their products to consumers based on search queries. With millions of available keywords whose clickthru rates and profits are highly uncertain, identifying the most profitable set of keywords becomes challenging. We formulate a stylized model of keyword selection in searchbased advertising services. Assuming known profits and unknown clickthru rates, we develop an approximate adaptive algorithm that prioritizes keywords based on a prefix ordering – sorting of keywords in a descending order of expectedprofittocost ratio (or “bangperbuck”). We show that the average expected profit generated by our algorithm converges to nearoptimal profits, with the convergence rate that is independent of the number of keywords and scales gracefully with the problem’s parameters. By leveraging the special structure of our problem, our algorithm trades off bias with faster convergence rate, converging very quickly but with only nearoptimal profit in the limit. Extensive numerical simulations show that when the number of keywords is large, our algorithm outperforms existing methods, increasing profits by about 20 % in as little as 40 periods. We also extend our algorithm to the setting when both the clickthru rates and the expected profits are unknown. 1
Algorithms for Infinitely ManyArmed Bandits
"... We consider multiarmed bandit problems where the number of arms is larger than the possible number of experiments. We make a stochastic assumption on the meanreward of a new selected arm which characterizes its probability of being a nearoptimal arm. Our assumption is weaker than in previous work ..."
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Cited by 29 (4 self)
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We consider multiarmed bandit problems where the number of arms is larger than the possible number of experiments. We make a stochastic assumption on the meanreward of a new selected arm which characterizes its probability of being a nearoptimal arm. Our assumption is weaker than in previous works. We describe algorithms based on upperconfidencebounds applied to a restricted set of randomly selected arms and provide upperbounds on the resulting expected regret. We also derive a lowerbound which matches (up to a logarithmic factor) the upperbound in some cases. 1
Contextual Bandits with Similarity Information
 24TH ANNUAL CONFERENCE ON LEARNING THEORY
, 2011
"... In a multiarmed bandit (MAB) problem, an online algorithm makes a sequence of choices. In each round it chooses from a timeinvariant set of alternatives and receives the payoff associated with this alternative. While the case of small strategy sets is by now wellunderstood, a lot of recent work ha ..."
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Cited by 19 (4 self)
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In a multiarmed bandit (MAB) problem, an online algorithm makes a sequence of choices. In each round it chooses from a timeinvariant set of alternatives and receives the payoff associated with this alternative. While the case of small strategy sets is by now wellunderstood, 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 webpages, 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 contextarm pairs which bounds from above the difference between the respective expected payoffs. Prior work
HighProbability Regret Bounds for Bandit Online Linear Optimization
"... We present a modification of the algorithm of Dani et al. [8] for the online linear optimization problem in the bandit setting, which with high probability has regret at most O ∗ ( √ T) against an adaptive adversary. This improves on the previous algorithm [8] whose regret is bounded in expectatio ..."
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Cited by 16 (0 self)
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We present a modification of the algorithm of Dani et al. [8] for the online linear optimization problem in the bandit setting, which with high probability has regret at most O ∗ ( √ T) against an adaptive adversary. This improves on the previous algorithm [8] whose regret is bounded in expectation against an oblivious adversary. We obtain the same dependence on the dimension (n 3/2) as that exhibited by Dani et al. The results of this paper rest firmly on those of [8] and the remarkable technique of Auer et al. [2] for obtaining highprobability bounds via optimistic estimates. This paper answers an open question: it eliminates the gap between the highprobability bounds obtained in the fullinformation vs bandit settings. 1
Dynamic Assortment Optimization with a Multinomial Logit Choice Model and Capacity Constraint
, 2008
"... The paper considers a stylized model of a dynamic assortment optimization problem, where given a limited capacity constraint, we must decide the assortment of products to offer to customers to maximize the profit. Our model is motivated by the problem faced by retailers of stocking products on a she ..."
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Cited by 12 (1 self)
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The paper considers a stylized model of a dynamic assortment optimization problem, where given a limited capacity constraint, we must decide the assortment of products to offer to customers to maximize the profit. Our model is motivated by the problem faced by retailers of stocking products on a shelf with limited capacities and by the problem of placing a limited number of ads on a web page. We assume that each customer chooses to purchase the product (or to click on the ad) that maximizes her utility. We use the multinomial logit choice model to represent demand. However, we do not know the demand for each product. We can learn the demand distribution by offering different product assortments, observing resulting selections, and inferring the demand distribution from past selections and assortment decisions. We present an adaptive policy for joint parameter estimation and assortment optimization. To evaluate our proposed policy, we define a benchmark profit as the maximum expected profit that we can earn if we know the underlying demand distribution in advance. We show that the running average expected profit generated by our policy converges to the benchmark profit and establish its convergence rate. Numerical experiments based on sales data from an online retailer indicate that our policy performs well, generating over 90 % of the optimal profit after less than two days of sales. 1.
Adapting to a Changing Environment: the Brownian Restless Bandits
"... In the multiarmed bandit (MAB) problem there are k distributions associated with the rewards of playing each of k strategies (slot machine arms). The reward distributions are initially unknown to the player. The player iteratively plays one strategy per round, observes the associated reward, and de ..."
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Cited by 12 (3 self)
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In the multiarmed bandit (MAB) problem there are k distributions associated with the rewards of playing each of k strategies (slot machine arms). The reward distributions are initially unknown to the player. The player iteratively plays one strategy per round, observes the associated reward, and decides on the strategy for the next iteration. The goal is to maximize the reward by balancing exploitation: the use of acquired information, with exploration: learning new information. We introduce and study a dynamic MAB problem in which the reward functions stochastically and gradually change in time. Specifically, the expected reward of each arm follows a Brownian motion, a discrete random walk, or similar processes. In this setting a player has to continuously keep exploring in order to adapt to the changing environment. Our formulation is (roughly) a special case of the notoriously intractable restless MAB problem. Our goal here is to characterize the cost of learning and adapting to the changing environment, in terms of the stochastic rate of the change. We consider an infinite time horizon, and strive to minimize the average cost per step which we define with respect to a hypothetical algorithm that at every step plays the arm with the maximum expected reward at this step. A related line of work on the adversarial MAB problem used a significantly weaker benchmark, the best timeinvariant policy. The dynamic MAB problem models a variety of practical online, gameagainst nature type optimization settings. While building on prior work, algorithms and steadystate analysis for the dynamic setting require a novel approach based on different stochastic tools.
Dynamic CostPerAction Mechanisms and Applications to Online Advertising
"... We study the CostPerAction or CostPerAcquisition (CPA) charging scheme in online advertising. In this scheme, instead of paying per click, the advertisers pay only when a user takes a specific action (e.g. fills out a form) or completes a transaction on their websites. We focus on designing effi ..."
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Cited by 9 (0 self)
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We study the CostPerAction or CostPerAcquisition (CPA) charging scheme in online advertising. In this scheme, instead of paying per click, the advertisers pay only when a user takes a specific action (e.g. fills out a form) or completes a transaction on their websites. We focus on designing efficient and incentive compatible mechanisms that use this charging scheme. We describe a mechanism based on a samplingbased learning algorithm that under suitable assumptions is asymptotically individually rational, asymptotically Bayesian incentive compatible and asymptotically exante efficient. In particular, we demonstrate our mechanism for the case where the utility functions of the advertisers are independent and identicallydistributed random variables as well as the case where they evolve like independent reflected Brownian motions.
Mortal MultiArmed Bandits
"... We formulate and study a new variant of the karmed bandit problem, motivated by ecommerce applications. In our model, arms have (stochastic) lifetime after which they expire. In this setting an algorithm needs to continuously explore new arms, in contrast to the standard karmed bandit model in wh ..."
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Cited by 8 (0 self)
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We formulate and study a new variant of the karmed bandit problem, motivated by ecommerce applications. In our model, arms have (stochastic) lifetime after which they expire. In this setting an algorithm needs to continuously explore new arms, in contrast to the standard karmed bandit model in which arms are available indefinitely and exploration is reduced once an optimal arm is identified with nearcertainty. The main motivation for our setting is onlineadvertising, where ads have limited lifetime due to, for example, the nature of their content and their campaign budgets. An algorithm needs to choose among a large collection of ads, more than can be fully explored within the typical ad lifetime. We present an optimal algorithm for the stateaware (deterministic reward function) case, and build on this technique to obtain an algorithm for the stateoblivious (stochastic reward function) case. Empirical studies on various reward distributions, including one derived from a realworld ad serving application, show that the proposed algorithms significantly outperform the standard multiarmed bandit approaches applied to these settings. 1
The KnowledgeGradient Algorithm for Sequencing Experiments in Drug Discovery
 INFORMS J. on Computing
, 2010
"... We present a new technique for adaptively choosing the sequence of molecular compounds to test in drug discovery. Beginning with a base compound, we consider the problem of searching for a chemical derivative of the molecule that best treats a given disease. The problem of choosing molecules to test ..."
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Cited by 7 (3 self)
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We present a new technique for adaptively choosing the sequence of molecular compounds to test in drug discovery. Beginning with a base compound, we consider the problem of searching for a chemical derivative of the molecule that best treats a given disease. The problem of choosing molecules to test to maximize the expected quality of the best compound discovered may be formulated mathematically as a rankingandselection problem in which each molecule is an alternative. We apply a recently developed algorithm, known as the knowledgegradient algorithm, that uses correlations in our Bayesian prior distribution between the performance of different alternatives (molecules) to dramatically reduce the number of molecular tests required, but it has heavy computational requirements that limit the number of possible alternatives to a few thousand. We develop computational improvements that allow the knowledgegradient method to consider much larger sets of alternatives, and we demonstrate the method on a problem with 87,120 alternatives.