Selecting the right web links for a website is important because appropriate links not only can provide high attractiveness but can also increase the website's revenue. In this work, we first show that web links have an intrinsic . For example, consider a @math -level feedback web link: the @math st level feedback provides the Click-Through Rate (CTR) and the @math nd level feedback provides the potential revenue, which collectively produce the compound @math -level revenue. We consider the context-free links selection problem of selecting links for a homepage so as to maximize the total compound @math -level revenue while keeping the total @math st level feedback above a preset threshold. We further generalize the problem to links with @math -level feedback structure. The key challenge is that the links' multi-level feedback structures are unobservable unless the links are selected on the homepage. To our best knowledge, we are the first to model the links selection problem as a constrained multi-armed bandit problem and design an effective links selection algorithm by learning the links' multi-level structure with provable regret and violation bounds. We uncover the multi-level feedback structures of web links in two real-world datasets. We also conduct extensive experiments on the datasets to compare our proposed algorithm with two state-of-the-art context-free bandit algorithms and show that algorithm is the most effective in links selection while satisfying the constraint.
@cite_14: In the multiarmed bandit problem, a gambler must decide which arm of K nonidentical slot machines to play in a sequence of trials so as to maximize his reward. This classical problem has received much attention because of the simple model it provides of the trade-off between exploration (trying out each arm to find the best one) and exploitation (playing the arm believed to give the best payoff). Past solutions for the bandit problem have almost always relied on assumptions about the statistics of the slot machines. In this work, we make no statistical assumptions whatsoever about the nature of the process generating the payoffs of the slot machines. We give a solution to the bandit problem in which an adversary, rather than a well-behaved stochastic process, has complete control over the payoffs. In a sequence of T plays, we prove that the per-round payoff of our algorithm approaches that of the best arm at the rate O(T-1 2). We show by a matching lower bound that this is the best possible. We also prove that our algorithm approaches the per-round payoff of any set of strategies at a similar rate: if the best strategy is chosen from a pool of N strategies, then our algorithm approaches the per-round payoff of the strategy at the rate O((log N1 2 T-1 2). Finally, we apply our results to the problem of playing an unknown repeated matrix game. We show that our algorithm approaches the minimax payoff of the unknown game at the rate O(T-1 2).
@cite_9: We define a general framework for a large class of combinatorial multi-armed bandit (CMAB) problems, where simple arms with unknown distributions form super arms. In each round, a super arm is played and the outcomes of its related simple arms are observed, which helps the selection of super arms in future rounds. The reward of the super arm depends on the outcomes of played arms, and it only needs to satisfy two mild assumptions, which allow a large class of nonlinear reward instances. We assume the availability of an (α, β)-approximation oracle that takes the means of the distributions of arms and outputs a super arm that with probability β generates an β fraction of the optimal expected reward. The objective of a CMAB algorithm is to minimize (α, β)- approximation regret, which is the difference in total expected reward between the αbeta; fraction of expected reward when always playing the optimal super arm, and the expected reward of playing super arms according to the algorithm. We provide CUCB algorithm that achieves O(log n) regret, where n is the number of rounds played, and we further provide distribution-independent bounds for a large class of reward functions. Our regret analysis is tight in that it matches the bound for classical MAB problem up to a constant factor, and it significantly improves the regret bound in a recent paper on combinatorial bandits with linear rewards. We apply our CMAB framework to two new applications, probabilistic maximum coverage (PMC) for online advertising and social influence maximization for viral marketing, both having nonlinear reward structures.
@cite_19: Given a repeatedly issued query and a document with a not-yet-confirmed potential to satisfy the users' needs, a search system should place this document on a high position in order to gather user feedback and obtain a more confident estimate of the document utility. On the other hand, the main objective of the search system is to maximize expected user satisfaction over a rather long period, what requires showing more relevant documents on average. The state-of-the-art approaches to solving this exploration-exploitation dilemma rely on strongly simplified settings making these approaches infeasible in practice. We improve the most flexible and pragmatic of them to handle some actual practical issues. The first one is utilizing prior information about queries and documents, the second is combining bandit-based learning approaches with a default production ranking algorithm. We show experimentally that our framework enables to significantly improve the ranking of a leading commercial search engine.
@cite_15: We discuss a multiple-play multi-armed bandit (MAB) problem in which several arms are selected at each round. Recently, Thompson sampling (TS), a randomized algorithm with a Bayesian spirit, has attracted much attention for its empirically excellent performance, and it is revealed to have an optimal regret bound in the standard single-play MAB problem. In this paper, we propose the multiple-play Thompson sampling (MP-TS) algorithm, an extension of TS to the multiple-play MAB problem, and discuss its regret analysis. We prove that MP-TS for binary rewards has the optimal regret upper bound that matches the regret lower bound provided by (1987). Therefore, MP-TS is the first computationally efficient algorithm with optimal regret. A set of computer simulations was also conducted, which compared MP-TS with state-of-the-art algorithms. We also propose a modification of MP-TS, which is shown to have better empirical performance.
@cite_20: Adversarial bandit problems studied by [4] are multi-armed bandit problems in which no stochastic assumption is made on the nature of the process generating the rewards for actions. In this paper, we extend their theories to the case where k(≥1) distinct actions are selected at each time step. As algorithms to solve our problem, we analyze an extension of Exp3 [4] and an application of a bandit online linear optimization algorithm [1] in addition to two existing algorithms (Exp3, ComBand [5]) in terms of time and space efficiency and the regret for the best fixed action set. The extension of Exp3, called Exp3.M, performs best with respect to all the measures: it runs in O(K(log k+1)) time and O(K) space, and suffers at most O(√kTK log(K k)) regret, where K is the number of possible actions and T is the number of iterations. The upper bound of the regret we proved for Exp3.M is an extension of that proved by for Exp3.
Our bandit formulation is related to the bandit models with multiple plays, where multiple arms are selected in each round. @cite_20 presented the bandit algorithm that extends the single-played algorithm @cite_14 to multiple-played cases using exponential weights. @cite_9 proposed an algorithm that selects multiple arms with the highest upper confidence bound (UCB) indices. @cite_15 presented the multiple-play Thompson Sampling algorithm (MP-TS) for arms with binary rewards. @cite_19 proposed a bandit-based ranking algorithm for ranking search queries. Our bandit model differs from these bandit models as we further consider the constraint on the total @math st level rewards in selecting the multiple arms.