We consider an adversarial online learning setting where a decision maker can choose an action in every stage of the game. In addition to observing the reward of the chosen action, the decision maker gets side observations on the reward he would have obtained had he chosen some of the other actions. The observation structure is encoded as a graph, where node i is linked to node j if sampling i provides information on the reward of j. This setting naturally interpolates between the well-known "experts" setting, where the decision maker can view all rewards, and the multi-armed bandits setting, where the decision maker can only view the reward of the chosen action. We develop practical algorithms with provable regret guarantees, which depend on non-trivial graph-theoretic properties of the information feedback structure. We also provide partially-matching lower bounds.
Our work is also somewhat related to the contextual bandit problem (e.g., @cite_4 @cite_3 ), where the standard multi-armed bandits setting is augmented with some side-information provided in each round, which can be used to determine which action to pick. While we also consider additional side-information, it is in a more specific sense. Moreover, our goal is still to compete against the best single action, rather than some set of policies which use this side-information.
@cite_4 @cite_3
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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). Multi-armed bandit problems are the predominant theoretical model of exploration-exploitation tradeoffs in learning, and they have countless applications ranging from medical trials, to communication networks, to Web search and advertising. In many of these application domains the learner may be constrained by one or more supply (or budget) limits, in addition to the customary limitation on the time horizon. The literature lacks a general model encompassing these sorts of problems. We introduce such a model, called "bandits with knapsacks", that combines aspects of stochastic integer programming with online learning. A distinctive feature of our problem, in comparison to the existing regret-minimization literature, is that the optimal policy for a given latent distribution may significantly outperform the policy that plays the optimal fixed arm. Consequently, achieving sub linear regret in the bandits-with-knapsacks problem is significantly more challenging than in conventional bandit problems. We present two algorithms whose reward is close to the information-theoretic optimum: one is based on a novel "balanced exploration" paradigm, while the other is a primal-dual algorithm that uses multiplicative updates. Further, we prove that the regret achieved by both algorithms is optimal up to polylogarithmic factors. We illustrate the generality of the problem by presenting applications in a number of different domains including electronic commerce, routing, and scheduling. As one example of a concrete application, we consider the problem of dynamic posted pricing with limited supply and obtain the first algorithm whose regret, with respect to the optimal dynamic policy, is sub linear in the supply. A bandit problem with side observations is an extension of the traditional two-armed bandit problem, in which the decision maker has access to side information before deciding which arm to pull. In this paper, essential properties of the side observations that allow achievability results with respect to optimal regret are extracted and formalized. The sufficient conditions for good side information obtained here admit various types of random processes as special cases, including i.i.d. sequences, Markov chains, deterministic periodic sequences, etc. A simple necessary condition for optimal regret is given, providing further insight into the nature of bandit problems with side observations. A game-theoretic approach simplifies the analysis and justifies the viewpoint that the side observation serves as an index specifying different sub-bandit machines. In this paper we study the contextual bandit problem (also known as the multi-armed bandit problem with expert advice) for linear payoff functions. For T rounds, K actions, and d dimensional feature vectors, we prove an O (√ Td ln(KT ln(T ) δ) ) regret bound that holds with probability 1− δ for the simplest known (both conceptually and computationally) efficient upper confidence bound algorithm for this problem. We also prove a lower bound of Ω( √ Td) for this setting, matching the upper bound up to logarithmic factors.